Geometry from Dynamics, Classical and Quantum

José F. CariñenaAlberto IbortGiuseppe MarmoGiuseppe Morandi

Geometry from Dynamics, Classical and Quantum

José F. Cariñena • Alberto IbortGiuseppe Marmo • Giuseppe Morandi

Geometry from Dynamics,Classical and Quantum

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José F. CariñenaDepartamento de Física TeóricaUniversidad de ZaragozaZaragozaSpain

Alberto IbortDepartamento de MatemáticasUniversidad Carlos III de MadridMadridSpain

Giuseppe MarmoDipartimento di FisicheUniversita di Napoli “Federico II”NapoliItaly

Giuseppe MorandiINFN Sezione di BolognaUniversitá di BolognaBolognaItaly

ISBN 978-94-017-9219-6 ISBN 978-94-017-9220-2 (eBook)DOI 10.1007/978-94-017-9220-2

Library of Congress Control Number: 2014948056

Springer Dordrecht Heidelberg New York London

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Foreword

The Birth and the Long Gestation of a Project

Starting a book is always a difficult task. Starting a book with the characteristics ofthis one is, as we hope will become clear at the end of this introduction, evenharder. It is difficult because the project underlying this book began almost 20 yearsago and, necessarily, during such a long period of time, has experienced ups anddowns, turning points where the project changed dramatically and moments wherethe success of the endeavor seemed dubious.

However the authors are all very grateful that things have turned out as they did.The road followed during the elaboration of this book, the innumerable discussionsand arguments we had during preparation of the different sections, the puzzlinguncertainties we suffered when facing some of the questions raised by the problemstreated, has been a major part of our own scientific evolution and have madeconcrete contributions toward the shaping of our own thinking on the role ofgeometry in the description of dynamics. In this sense we may say with the poet:

Caminante, son tus huellas1

el camino y nada más;Caminante, no hay camino,se hace camino al andar.Al andar se hace el camino,y al volver la vista atrasse ve la senda que nuncase ha de volver a pisar.Caminante no hay caminosino estelas en la mar.

Antonio Machado, Proverbios y Cantares.

1 Wanderer, your footsteps are// the road, and no more;// wanderer, there is no road,// the road ismade when we walk.// By walking the path is done,// and upon glancing back// one sees the path//that never will be trod again.// Wanderer, there is no road// only foam upon the sea.

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Thus, contrary to what happens with other projects that represent the culminationof previous work, in this case the road that we have traveled was not there beforethis enterprise was started. We can see from where we are now that this work has tobe pursued further to try to uncover the unknowns surrounding some of thebeautiful ideas that we have tried to put together. Thus the purpose of this book is toshare with the reader some of the ideas that have emerged during the process ofreflection on the geometrical foundations of mechanics that we have come up withduring the preparation of the book itself. In this sense it would be convenient toexplain to the reader some of the major conceptual problems that were seeding themilestones marking the evolution of this intellectual adventure.

The original idea of this book, back in the early 1990s, was to offer in anaccessible way to young Ph.D. students some completely worked significantexamples of physical systems where geometrical and topological ideas play afundamental role. The consolidation of geometrical and topological ideas andtechniques in Yang-Mills theories and other branches of Physics, not only theo-retical, such as in Condensed Matter Physics with the emergence of new collectivephenomena or the fractional quantum Hall effect or High Tc superconductivity,were making it important to have a rapid but well-founded access to geometry andtopology at a graduate level; this was rather difficult for the young student or theresearcher needing a fast briefing on the subject. The timeliness of this idea wasconfirmed by the fact that a number of books describing the basics of geometry andtopology delved into the modern theories of fields and other physical models thathad appeared during these years. Attractive as this idea was, it was immediatelyclear to us that offering a comprehensive approach to the question of why somegeometrical structures played such an important role in describing a variety ofsignificant physical examples such as the electron-monopole system, relativisticspinning particles, or particles moving in a non-abelian Yang-Mills field, requiredus to present a set of common guiding principles and not just an enumeration ofresults, no matter how fashionable they were.

Besides, the reader must be warned that because of the particular idiosyncrasies ofthe authors, we were prone to take such a road. So we joyously jumped into theoceanic deepness of the foundations of the Science of Mechanics, trying to discussthe role that geometry plays in it, probably believing that the work that we hadalready done on the foundations of Lagrangian and Hamiltonian mechanics qualifiedus to offer our own presentation of the subject. Most probably it is unnecessary torecall here that, in the more than 20 years that had passed since publication of thebooks on the mathematical foundations of mechanics by V.I. Arnold [Ar76],R. Abraham and J. Marsden [Ab78] and J.M. Souriau [So70], the use of geometry, orbetter, the geometrical approach to Mechanics, had gained a widespread acceptationamong many practitioners and the time was ripe for a second wave of the literatureon the subject. Again our attitude was timely, as a number of books deepening andexploring complementary avenues in the realm of mechanics had started to appear.

When trying to put together a good recollection of the ideas embracingGeometry and Mechanics, including our own contributions to the subject, a feelingof uneasiness started to come over us as we realized that we were not completely

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satisfied with the various ways that geometrical structures were currently introducedinto the description of a given dynamical system. They run from the “axiomatic”way as in Abraham and Marsden’s book Foundations of Mechanics to the “con-structive” way as in Souriau’s book Structure des Systèmes Dynamiques where ageometrical structure, the Lagrange form, was introduced in the space of “move-ments” of the system, passing through the “indirect” justification by means ofHamilton’s principle, leading to a Lagrangian description in Arnold’s Méthodesmathématiques de la mécanique classique. All these approaches to the geometry ofMechanics were solidly built upon ideas deeply rooted in the previous work ofLagrange, Hamilton, Jacobi, etc. and the geometric structures that were brought tothe front row on them had been laboriously uncovered by some of the most brilliantthinkers of all times. Thus, in this sense, there was very little to object in the variouspresentations of the subject commented above. However, it was also beginning tobe clear at that time, that some of the geometrical structures that played such aprominent role in the description of the dynamical behavior of a physical systemwere not univocally determined. For instance there are many alternative Lagrangiandescriptions for such a simple and fundamental system as the harmonic oscillator.Thus, which one is the preferred one, if there is one, and why? Moreover, thecurrent quantum descriptions of many physical systems are based on either aLagrangian or Hamiltonian description of a certain classical one. Thus, if theLagrangian and/or the Hamiltonian description of a given classical system is notunique, which quantum description prevails? Even such a fundamental notion aslinearity was compromised at this level of analysis as it is easy to show the exis-tence of nonequivalent linear structures compatible with a given “linear” dynamics,for instance that of the harmonic oscillator again.

It took some time, but soon it became obvious that from an operational point ofview, the geometrical structures introduced to describe a given dynamics were not apriori entities, but they accompanied the given dynamics in a natural way. Thus,starting from raw observational data, a physical system will provide us with afamily of trajectories on some “configuration space” Q, like the trajectories pho-tographed on a fog chamber displayed below (see Fig. 1) or the motion of celestialbodies during a given interval of time. From these data we would like to build adifferential equation whose solution will include the family of the observed tra-jectories. However we must point out here that a differential equation is not, ingeneral, univocally determined by experimental data. The ingenuity of the theo-retician regarding experimental data will provide a handful of choices to startbuilding up the theory. At this point we stand with A. Einstein’s famous quote:

Physical concepts are free creations of the human mind, and are not, however it may seem,uniquely determined by the external world. In our endeavor to understand reality we aresomewhat like a man trying to understand the mechanism of a closed watch. He sees the faceand the moving hands, even hears its ticking, but he has no way of opening the case. If he isingenious he may form some picture of a mechanism which could be responsible for all thethings he observes, but he may never be quite sure his picture is the only one which couldexplain his observations. He will never be able to compare his picture with the real mech-anism and he cannot even imagine the possibility or the meaning of such a comparison. But

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he certainly believes that, as his knowledge increases, his picture of reality will becomesimpler and simpler and will explain a wider and wider range of his sensuous impressions.He may also believe in the existence of the ideal limit of knowledge and that it is approachedby the human mind. He may call this ideal limit the objective truth.

A. Einstein, The Evolution of Physics (1938) (co-written with Leopold Infeld).

For instance, the order of the differential equation will be postulated followingan educated guess of the theoretician. Very often from differential equations weprefer to go to vector fields on some (possibly) larger carrier space, so that evolutionis described in terms of one parameter groups (or semigroups). Thus a first initialgeometrization of the theory is performed.

At this point we decided to stop assuming additional structures for a givendescription of the dynamics and, again, following Einstein, we assumed that allgeometrical structures should be considered equally placed with respect to theproblem of describing the given physical system, provided that they were com-patible with the given dynamics, id est2 with the data gathered from it. Thus thisnotion of operational compatibility became the Occam’s razor in our analysis ofdynamical evolution, as geometrical structures should not be postulated butaccepted only on the basis of their consistency with the observed data. The way totranslate such criteria into mathematical conditions will be discussed at lengththroughout the text; however, we should stress here that such emphasis on thesubsidiary character of geometrical structures with respect to a given set of data isalready present, albeit in a different form, in Einstein’s General Relativity, wherethe geometry of space–time is dynamically determined by the distribution of massand energy in the universe. All solutions of Einstein’s equations for a given energy–

Fig. 1 Trajectories of particles on a fog chamber

2 i.e., ‘which is to say’ or ‘in other words’.

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momentum tensor are acceptable geometrical descriptions of the universe. Only ifthere exists a Cauchy surface (i.e., only if we are considering a globally hyperbolicspace–time) we may, after fixing some initial data, determine (locally) the particularsolution of equations compatible with a given energy–momentum tensor Fig. 2.

From this point on, we embarked on the systematic investigation of geometricalstructures compatible with a given dynamical system. We have found that such atask has provided in return a novel view on some of the most conspicuous geo-metrical structures already filling the closet of mathematical tools used in the theoryof mechanical and also dynamical systems in general, such as linear structures,symmetries, Poisson and symplectic structures, Lagrangian structures, etc. It isapparent that looking for structures compatible with a given dynamical systemconstitutes an “Inverse Problem” a description in terms of some additional struc-tures. The inverse problem of the calculus of variations is a paradigmatic example ofthis. The book that we present to your attention offers at the same time a reflection onthe geometrical structures that could be naturally attached to a given dynamicalsystem and the variety of them that could exist, creating in this way a hierarchy onthe family of physical systems according with their degree of compatibility withnatural geometrical structures, a system being more and more “geometrizable” asmore structures are compatible with it. Integrable systems have played a key role inthe development of Mechanics as they have constituted the main building blocks forthe theory, both because of their simple appearance, centrality in the development ofthe theories, and their ubiquity in the description of the physical world. The avenuewe follow here leads to such a class of systems in a natural way as the epitome ofextremely geometrizable systems in the previous sense.

We may conclude this exposition of motives by saying that if any work has amotto, probably the one encapsulating the spirit of this book could be:

All geometrical structures used in the description of the dynamics of a given physicalsystem should be dynamically determined.

Fig. 2 The picture shows the movements of several planets over the course of several years. Themotion of the planets relative to the stars (represented as unmoving points) produces continuousstreaks on the sky. (Courtesy of the Museum of Science, Boston)

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What you will Find and What you will not in This Book

This is a book that pursues an analysis of the geometrical structures compatible witha given dynamical system, thus you will not find in it a discussion on such crucialissues such as determination of the physical magnitudes relevant for description ofmechanical systems, be they classical or quantum, or an interpretation of theexperiments performed to gain information on it, that is on any theoreticaldescription of the measurement process. Neither will we extend our enquiries to thedomain of Field Theory (Fig. 3) (even though we included in the preparation of thisproject such key points but we had to discard them to keep the present volume at areasonable size) where new structures with respect to the ones described here areinvolved. It is a work that focuses on a mathematical understanding of some fun-damental issues in the Theory of Dynamics, thus in this sense both the style and thescope will be heavily determined by these facts.

Chapter 1 of the book will be devoted to a discussion of some elementaryexamples in finite and infinite dimensions where some of the standard ideas indealing with mechanical systems like constants of motion, symmetries, Lagrangian,and Hamiltonian formalisms, etc., are recalled. In this way, we pretend to help thereader to have a strong foothold on what is probably known to him/her with respectto the language and notions that are going to be developed in the main part of thetext. The examples chosen are standard: The harmonic oscillator, an electronmoving on a constant magnetic field, the free particle on the finite-dimensional side,and the Klein–Gordon equation, Maxwell equations, and the Schrödinger equationas prototypes of systems in infinite dimensions. We have said that field theory willnot be addressed in this work, that is actually so because the examples in infinitedimensions are treated as evolution systems, i.e., time is a privileged variable and

Fig. 3 Counter rotating vortex generated at the tip of a wing. (American Physical Society’s 2009Gallery of Fluid Motion)

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no covariant treatment of them are pursued. Dealing with infinite-dimensionalsystems, already at the level of basic examples, shows that many of the geometricalideas that are going to appear are not restricted by the number of degrees offreedom. Even though a rigorous mathematical treatment of them in the case ofinfinite dimensions will be out of the scope of this book, the geometrical argumentsapply perfectly well to them as we will try to show throughout the book.

Another interesting characteristic of the examples chosen in the first part ofChap. 1 is that they are all linear systems. Linear systems are going to play aninstrumental role in the development of our discourse because they provide aparticularly nice bridge between elementary algebraic ideas and geometricalthinking. Thus we will show how a great deal of differential geometry can beconstructed from linear systems. Finally, the third and last part of the first chapterwill be devoted to a discussion of a number of nonlinear systems that have managedto gain their own relevant place in the gallery of dynamics, like the Calogero-Mosersystem, and that all share the common feature of being obtained from simpler affinesystems. The general method of obtaining these systems out of simpler ones iscalled “reduction” and we will offer to the reader an account of such procedures byexample working out explicitly a number of interesting ones. These systems willprovide also a source of interesting situations where the geometrical analysis isparamount because their configuration/phase spaces fail to be open domains on anEuclidean space. The general theory of reduction together with the problem ofintegrability will be discussed again at the end of the book in Chap. 7.

Geometry plays a fundamental role in this book. Geometry is so pervasive that ittends very quickly to occupy a central role in any theory where geometricalarguments become relevant. Geometrical thinking is synthetic so it is natural toattach to it an a priori or relatively higher position among the ideas used to constructany theory. This attitude spreads in many occasions to include also geometricalstructures relevant for analysis of a given problem. We have deliberately subvertedthis approach here considering geometrical structures as subsidiaries to the givendynamics; however, geometrical thinking will be used always as a guide, almost asa metalanguage, in analyses of the problems. In Chap. 2 we will present the basicgeometrical ideas needed to continue the discussion started here. It would be almostimpossible to present all details of the foundations of geometry, in particular dif-ferential geometry, which would be necessary to make the book self-consistent.This would make the book hard to use. However, we are well aware that manystudents who could be interested in the contents of this book do not possess thenecessary geometrical background to read it without introducing (with some care)some of the fundamental geometrical notions that are necessarily used in anydiscussion where differential geometrical ideas become relevant; just to name a few:manifolds, bundles, vector fields, Lie groups, etc. We have decided to take apragmatic approach and try to offer a personal view of some of these fundamentalnotions in parallel with the development of the main stream of the book. However,we will refer to standard textbooks for more detailed descriptions of some of theideas sketched here.

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Linearity plays a fundamental role in the presentation of the ideas of this book.Because of that some care is devoted to the description of linearity from a geo-metrical perspective. Some of the discourse in Chap. 3 is oriented toward this goaland a detailed description of the geometrical description of linear structures bymeans of Euler or dilation vector fields is presented. We will show how a smallgeneralization of this presentation leads naturally to the description of vectorbundles and to their characterization too. Some care is also devoted to describe thefundamental concepts in a dual way, i.e., from the set-theoretical point of view andfrom the point of view of the algebras of functions on the corresponding carrierspaces. The second approach is instrumental in any physical conceptualization ofthe mathematical structures appearing throughout the book; they are not usuallytreated from this point of view in standard textbooks.

After the preparation offered by the first two chapters we are ready to startexploring geometrical structures compatible with a given dynamics. Chapter 4 willbe devoted to it. Again we will use as paradigmatic dynamics the linear ones and wewill start by exploring systematically all geometrical structures compatible withthem: zero order, i.e., constants of motion, first order, that is symmetries, andimmediately after, second-order invariant structures. The analysis of constants ofmotion and infinitesimal symmetries will lead us immediately to pose questionsrelated with the “integrability” of our dynamics, questions that will be answeredpartially there and that will be recast in full in Chap. 8. The most significantcontribution of Chap. 4 consists in showing how, just studying the compatibilitycondition for geometric structures of order two in the case of linear dynamics, wearrive immediately to the notion of Jacobi, Poisson, and Hamiltonian dynamics.Thus, in this sense, standard geometrical descriptions of classical mechanical sys-tems are determined from given dynamics and are obtained by solving the corre-sponding inverse problems. All of them are analyzed with care, putting specialemphasis on Poisson dynamics as it embraces both the deep geometrical structurescoming from group theory and the fundamental notions of Hamiltonian dynamics.The elementary theory of Poisson manifolds is reviewed from this perspective andthe emerging structure of symplectic manifolds is discussed. A number of examplesderived from group theory and harmonic analysis are discussed as well as appli-cations to some interesting physical systems like massless relativistic systems.

The Lagrangian description of dynamical systems arises as a further step in theprocess of requiring additional properties to the system. In this sense, the lastsection of Chap. 5 can be considered as an extended exposition of the classicalFeynman’s problem together with the inverse problem of the calculus of variationsfor second-order differential equations. The geometry of tangent bundles, which isreviewed with care, shows its usefulness as it allows us to greatly simplify expo-sition of the main results: necessary and sufficient conditions will be given for theexistence of a Lagrangian function that will describe a given dynamics and thepossible forms that such a Lagrangian function can take under simple physicalassumptions (Fig. 4).

Once the classical geometrical pictures of dynamical systems have been obtainedas compatibility conditions for ð2; 0Þ and ð0; 2Þ tensors on the corresponding carrier

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space, it remains to explore a natural situation where there is also a complexstructure compatible with the given dynamics. The fundamental instance of thissituation happens when there is an Hermitean structure admissible for ourdynamics. Apart from the inherent interest of such a question, we should stress thatthis is exactly the situation for the dynamical evolution of quantum systems. Let uspoint out that the approach developed here does not preclude their being an a priorigiven Hermitean structure. But under what conditions there will exist an Hermiteanstructure compatible with the observed dynamics. Chapter 6 will be devoted tosolving such a problem and connecting it with various fundamental ideas inQuantum Mechanics. We must emphasize here that we do not pretend to offer aself-contained presentation of Quantum Mechanics but rather insist that evolutionof quantum systems can be dealt within the same geometrical spirit as otherdynamics, albeit the geometrical structures that emerge from such activity are ofdiverse nature. Therefore no attempt has been made to provide an analysis of thevarious geometrical ideas that are described in this chapter regarding the physics ofquantum systems, even though a number of remarks and observations pertinent tothat are made and the interested reader will be referred to the appropriate literature.

At this point we consider that our exploration of geometrical structures obtainedfrom dynamics has exhausted the most notorious ones. However, not all geomet-rical structures that have been relevant in the discussion of dynamical systems arecovered here. Notice that we have not analyzed, for instance, contact structures thatplay an important role in treatment of the Hamilton–Jacobi theory or Jacobistructures. Neither have we considered relevant geometrical structures arising infield theories or the theory of integrable systems (or hierarchies to be precise) likeYang–Baxter equations, Hopf algebras, Chern-Simons structures, Frobenius man-ifolds, etc. There is a double reason for that. On one side it will take us far beyondthe purpose of this book and, more important, some of these structures are char-acteristic of a very restricted, although extremely significant, class of dynamics.

Fig. 4 Quantum stroboscopebased on a sequence of iden-tical attosecond pulses that areused to release electrons into astrong infrared (IR) laser fieldexactly once per laser cycle

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However we have decided not to finish this book without entering, once we arein possession of a rich baggage of ideas, some domains in the vast land of the studyof dynamics, where geometrical structures have had a significant role. In particularwe have chosen the analysis of symmetries by means of the so-called reductiontheory and the problem of the integrability of a given system. These issues will becovered in Chap. 7 were the reduction theory of systems will be analyzed for themain geometrical structures described before.

Once one of the authors was asked by E. Witten, “how does it come that somesystems are integrable and others not?” The question was rather puzzling takinginto account the large amount of literature devoted to the subject of integrability andthe attitude shared by most people that integrability is a “non-generic” property,thus only possessed by a few systems. However, without trying to interpret Witten,it is clear that the emergence of systems in many different contexts (by that timeWitten had realized the appearance of Ramanujan’s τ-function in quantum 2Dgravity) was giving him a certain uneasiness on the true nature of “integrability” asa supposedly well-established notion. Without oscillating too much towardV. Arnold’s answer to a similar question raised by one of the authors: “An inte-grable system is a system that can be integrated”, we may try to analyze theproblem of the integrability of systems following the spirit of these notes: given adynamics, what are the fundamental structures determined by the structural char-acteristics of the flow that are instrumental in the “integrability” problem?

Chapter 8 will be devoted to a general perspective regarding the problem ofintegrability of dynamical systems. Again we do not pretend to offer an inclusiveapproach to this problem, i.e., we are not trying to describe and much less to unify,the many theories and results on integrability that are available in the literature. Thatwould be an ill-posed problem. However, we will try to exhibit from an elementaryanalysis some properties shared by an important family of systems lying within theclass of integrable systems and that can be analyzed easily with the notionsdeveloped previously in this book. We will close our excursion on the geometriesdetermined by dynamics by considering in detail a special class of them that exhibitmany of the properties described before, the so-called Lie–Scheffers systems whichprovide an excellent laboratory to pursue the search on this field.

Finally, we have to point out that the book is hardly uniform both in style andcontent. There are wide differences among its different parts. As we have tried toexplain before a substantial part of it is in a form designed to make it accessible to alarge audience, hence it can be read by assuming only a basic knowledge of linearalgebra and calculus. However there are sections that try to bring the understandingof the subject further and introduce more advanced material. These sections aremarked with an asterisk and their style is less self-contained. We have collected inthe form of appendices some background mathematical material that could behelpful for the reader.

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References

Abraham, R, Marsden, J.E.: Foundations of mechanics, (2nd ed.). Benjamin, Massachussetts(1978)

Arnol’d, V.I.: Méthodes mathématiques de la mécanique classique (Edition Mir), 1976.Mathematical Methods of Classical Mechanics. Springer, New York (1989)

Souriau, J.-M.: Structure des systemes dynamiques. Dunod, Paris (1970)

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Acknowledgments

As mentioned in the introduction, we have been working on this project for over20 years. First we would like to thank our families for their infinite patience andsupport. Thanks Gloria, Conchi, Patrizia and Maria Rosa.

During this long period we discussed various aspects of the book with a lot ofpeople in different contexts and situations. We should mention some particular oneswho have been regular through the years.

All of us have been participating regularly in the “International Workshop onDifferential Geometric Methods in Theoretical Mechanics”; other regular partici-pants with whom we have interacted the most have been Frans Cantrjin, MikeCrampin, Janusz Grabowski, Franco Magri, Eduardo Martinez, Enrico Pagani,Willy Sarlet and Pawel Urbanski.

A long association with the Erwin Schrödinger Institute has seen many of usmeeting there on several occasions and we have benefited greatly from the col-laboration with Peter Michor and other regular visitors.

In Naples we held our group seminar each Tuesday and there we presented manyof the topics that are included in the book. Senior participants of this seminar werePaolo Aniello, Giuseppe Bimonte, Giampiero Esposito, Fedele Lizzi and PatriziaVitale and of course, for even longer time, Alberto Simoni, Wlodedk Tulczyjew,Franco Ventriglia, Gaetano Vilasi and Franco Zaccaria.

Our long association with A.P. Balachandran, N. Mukunda and G. Sudarshanhas influenced many of us and contributed to most of our thoughts.

In the last part of this long term project we were given the opportunity to meet inMadrid and Zaragoza quite often, in particular in Madrid, under the auspices of a“Banco de Santander/UCIIIM Excellence Chair”, so that during the last 2 yearsmost of us have been able to visit there for an extended period.

We have also had the befit of ongoing discussions with Manolo Asorey, ElisaErcolessi, Paolo Facchi, Volodya Man’ko and Saverio Pascazio of particular issuesconnected with quantum theory.

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During the fall workshop on Geometry and Physics, another activity that hasbeen holding us together for all these years, we have benefited from discussionswith Manuel de León, Miguel Muñoz-Lecanda, Narciso Román-Roy and XavierGracia.

xviii Acknowledgments

Contents

1 Some Examples of Linear and Nonlinear Physical Systemsand Their Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Equations of Motion for Evolution Systems . . . . . . . . . . . . . . . 2

1.2.1 Histories, Evolution and Differential Equations . . . . . . . . 21.2.2 The Isotropic Harmonic Oscillator . . . . . . . . . . . . . . . . . 41.2.3 Inhomogeneous or Affine Equations . . . . . . . . . . . . . . . 51.2.4 A Free Falling Body in a Constant Force Field . . . . . . . . 71.2.5 Charged Particles in Uniform and Stationary Electric

and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.6 Symmetries and Constants of Motion. . . . . . . . . . . . . . . 121.2.7 The Non-isotropic Harmonic Oscillator . . . . . . . . . . . . . 161.2.8 Lagrangian and Hamiltonian Descriptions

of Evolution Equations. . . . . . . . . . . . . . . . . . . . . . . . . 211.2.9 The Lagrangian Descriptions of the Harmonic

Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.10 Constructing Nonlinear Systems Out of Linear Ones . . . . 281.2.11 The Reparametrized Harmonic Oscillator . . . . . . . . . . . . 291.2.12 Reduction of Linear Systems . . . . . . . . . . . . . . . . . . . . 34

1.3 Linear Systems with Infinite Degrees of Freedom . . . . . . . . . . . 411.3.1 The Klein-Gordon Equation and the Wave Equation . . . . 411.3.2 The Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . 441.3.3 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . 501.3.4 Symmetries and Infinite-Dimensional Systems . . . . . . . . 531.3.5 Constants of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2 The Language of Geometry and Dynamical Systems:The Linearity Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 Linear Dynamical Systems: The Algebraic Viewpoint . . . . . . . . 64

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2.2.1 Linear Systems and Linear Spaces. . . . . . . . . . . . . . . . . 642.2.2 Integrating Linear Systems: Linear Flows . . . . . . . . . . . . 662.2.3 Linear Systems and Complex Vector Spaces. . . . . . . . . . 732.2.4 Integrating Time-Dependent Linear Systems:

Dyson’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.2.5 From a Vector Space to Its Dual: Induced Evolution

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.3 From Linear Dynamical Systems to Vector Fields . . . . . . . . . . . 84

2.3.1 Flows in the Algebra of Smooth Functions . . . . . . . . . . . 842.3.2 Transformations and Flows. . . . . . . . . . . . . . . . . . . . . . 862.3.3 The Dual Point of View of Dynamical Evolution . . . . . . 872.3.4 Differentials and Vector Fields: Locality . . . . . . . . . . . . 892.3.5 Vector Fields and Derivations on the Algebra

of Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 912.3.6 The ‘Heisenberg’ Representation of Evolution. . . . . . . . . 932.3.7 The Integration Problem for Vector Fields . . . . . . . . . . . 95

2.4 Exterior Differential Calculus on Linear Spaces . . . . . . . . . . . . . 1002.4.1 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.4.2 Exterior Differential Calculus: Cartan Calculus . . . . . . . . 1022.4.3 The ‘Easy’ Tensorialization Principle . . . . . . . . . . . . . . . 1082.4.4 Closed and Exact Forms . . . . . . . . . . . . . . . . . . . . . . . 111

2.5 The General ‘Integration’ Problem for Vector Fields . . . . . . . . . 1132.5.1 The Integration Problem for Vector Fields:

Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.5.2 Foliations and Distributions . . . . . . . . . . . . . . . . . . . . . 115

2.6 The Integration Problem for Lie Algebras . . . . . . . . . . . . . . . . . 1182.6.1 Introduction to the Theory of Lie Groups:

Matrix Lie Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.6.2 The Integration Problem for Lie Algebras* . . . . . . . . . . . 130

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3 The Geometrization of Dynamical Systems . . . . . . . . . . . . . . . . . . 1353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.2 Differentiable Spaces* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.2.1 Ideals and Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.2.2 Algebras of Functions and Differentiable Algebras . . . . . 1413.2.3 Generating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.2.4 Infinitesimal Symmetries and Constants of Motion . . . . . 1453.2.5 Actions of Lie Groups and Cohomology . . . . . . . . . . . . 147

3.3 The Tensorial Characterization of Linear Structuresand Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.3.1 A Tensorial Characterization of Linear Structures . . . . . . 1533.3.2 Partial Linear Structures . . . . . . . . . . . . . . . . . . . . . . . . 1573.3.3 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

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3.4 The Holonomic Tensorialization Principle* . . . . . . . . . . . . . . . . 1633.4.1 The Natural Tensorialization of Algebraic Structures . . . . 1633.4.2 The Holonomic Tensorialization Principle . . . . . . . . . . . 1653.4.3 Geometric Structures Associated to Algebras . . . . . . . . . 169

3.5 Vector Fields and Linear Structures . . . . . . . . . . . . . . . . . . . . . 1713.5.1 Linearity and Evolution . . . . . . . . . . . . . . . . . . . . . . . . 1713.5.2 Linearizable Vector Fields . . . . . . . . . . . . . . . . . . . . . . 1723.5.3 Alternative Linear Structures: Some Examples . . . . . . . . 175

3.6 Normal Forms and Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 1803.6.1 The Conjugacy Problem. . . . . . . . . . . . . . . . . . . . . . . . 1803.6.2 Separation of Vector Fields . . . . . . . . . . . . . . . . . . . . . 1843.6.3 Symmetries for Linear Vector Fields . . . . . . . . . . . . . . . 1863.6.4 Constants of Motion for Linear Dynamical Systems. . . . . 188

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4 Invariant Structures for Dynamical Systems: Poisson Dynamics . . . 1934.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1934.2 The Factorization Problem for Vector Fields . . . . . . . . . . . . . . . 194

4.2.1 The Geometry of Noether’s Theorem. . . . . . . . . . . . . . . 1944.2.2 Invariant 2-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.2.3 Factorizing Linear Dynamics: Linear Poisson

Factorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.3 Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

4.3.1 Poisson Algebras and Poisson Tensors . . . . . . . . . . . . . . 2104.3.2 The Canonical ‘Distribution’ of a Poisson Structure. . . . . 2144.3.3 Poisson Structures and Lie Algebras . . . . . . . . . . . . . . . 2154.3.4 The Coadjoint Action and Coadjoint Orbits . . . . . . . . . . 2194.3.5 The Heisenberg–Weyl, Rotation and Euclidean

Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214.4 Hamiltonian Systems and Poisson Structures . . . . . . . . . . . . . . . 227

4.4.1 Poisson Tensors Invariant Under Linear Dynamics . . . . . 2274.4.2 Poisson Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2314.4.3 Symmetries and Constants of Motion. . . . . . . . . . . . . . . 233

4.5 The Inverse Problem for Poisson Structures:Feynman’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2434.5.1 Alternative Poisson Descriptions . . . . . . . . . . . . . . . . . . 2444.5.2 Feynman’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.5.3 Poisson Description of Internal Dynamics. . . . . . . . . . . . 2494.5.4 Poisson Structures for Internal and External Dynamics . . . 253

4.6 The Poincaré Group and Massless Systems . . . . . . . . . . . . . . . . 2604.6.1 The Poincaré Group. . . . . . . . . . . . . . . . . . . . . . . . . . . 2604.6.2 A Classical Description for Free Massless Particles . . . . . 267

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

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5 The Classical Formulations of Dynamics of Hamiltonand Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.2 Linear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 272

5.2.1 Symplectic Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . 2735.2.2 The Geometry of Symplectic Linear Spaces . . . . . . . . . . 2765.2.3 Generic Subspaces of Symplectic Linear Spaces . . . . . . . 2815.2.4 Transformations on a Symplectic Linear Space . . . . . . . . 2825.2.5 On the Structure of the Group SpðωÞ . . . . . . . . . . . . . . . 2865.2.6 Invariant Symplectic Structures . . . . . . . . . . . . . . . . . . . 2885.2.7 Normal Forms for Hamiltonian Linear Systems . . . . . . . . 292

5.3 Symplectic Manifolds and Hamiltonian Systems . . . . . . . . . . . . 2955.3.1 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 2955.3.2 Symplectic Potentials and Vector Bundles . . . . . . . . . . . 3005.3.3 Hamiltonian Systems of Mechanical Type . . . . . . . . . . . 303

5.4 Symmetries and Constants of Motionfor Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3055.4.1 Symmetries and Constants of Motion:

The Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3055.4.2 Symplectic Realizations of Poisson Structures . . . . . . . . . 3065.4.3 Dual Pairs and the Cotangent Group . . . . . . . . . . . . . . . 3085.4.4 An Illustrative Example: The Harmonic Oscillator . . . . . . 3115.4.5 The 2-Dimensional Harmonic Oscillator . . . . . . . . . . . . . 312

5.5 Lagrangian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3205.5.1 Second-Order Vector Fields . . . . . . . . . . . . . . . . . . . . . 3215.5.2 The Geometry of the Tangent Bundle . . . . . . . . . . . . . . 3265.5.3 Lagrangian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 3415.5.4 Symmetries, Constants of Motion

and the Noether Theorem . . . . . . . . . . . . . . . . . . . . . . . 3515.5.5 A Relativistic Description for Massless Particles . . . . . . . 358

5.6 Feynman’s Problem and the Inverse Problemfor Lagrangian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3605.6.1 Feynman’s Problem Revisited . . . . . . . . . . . . . . . . . . . . 3605.6.2 Poisson Dynamics on Bundles and the Inclusion

of Internal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 3665.6.3 The Inverse Problem for Lagrangian Dynamics . . . . . . . . 3745.6.4 Feynman’s Problem and Lie Groups . . . . . . . . . . . . . . . 383

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

6 The Geometry of Hermitean Spaces: Quantum Evolution. . . . . . . . 4076.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4076.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

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6.3 Invariant Hermitean Structures. . . . . . . . . . . . . . . . . . . . . . . . . 4096.3.1 Positive-Factorizable Dynamics . . . . . . . . . . . . . . . . . . . 4096.3.2 Invariant Hermitean Metrics . . . . . . . . . . . . . . . . . . . . . 4176.3.3 Hermitean Dynamics and Its Stability Properties . . . . . . . 4206.3.4 Bihamiltonian Descriptions . . . . . . . . . . . . . . . . . . . . . . 4216.3.5 The Structure of Compatible Hermitean Forms . . . . . . . . 424

6.4 Complex Structures and Complex Exterior Calculus . . . . . . . . . . 4306.4.1 The Ring of Functions of a Complex Space . . . . . . . . . . 4306.4.2 Complex Linear Systems . . . . . . . . . . . . . . . . . . . . . . . 4336.4.3 Complex Differential Calculus and Kähler Manifolds. . . . 4356.4.4 Algebras Associated with Hermitean Structures . . . . . . . . 437

6.5 The Geometry of Quantum Dynamical Evolution. . . . . . . . . . . . 4396.5.1 On the Meaning of Quantum Dynamical Evolution . . . . . 4396.5.2 The Basic Geometry of the Space of Quantum States . . . 4446.5.3 The Hermitean Structure on the Space of Rays . . . . . . . . 4486.5.4 Canonical Tensors on a Hilbert Space . . . . . . . . . . . . . . 4496.5.5 The Kähler Geometry of the Space of Pure

Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4536.5.6 The Momentum Map and the Jordan–Scwhinger Map . . . 4566.5.7 A Simple Example: The Geometry of a Two-Level

System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4596.6 The Geometry of Quantum Mechanics and the GNS

Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4626.6.1 The Space of Density States . . . . . . . . . . . . . . . . . . . . . 4636.6.2 The GNS Construction. . . . . . . . . . . . . . . . . . . . . . . . . 467

6.7 Alternative Hermitean Structures for Quantum Systems . . . . . . . 4716.7.1 Equations of Motion on Density States

and Hermitean Operators . . . . . . . . . . . . . . . . . . . . . . . 4716.7.2 The Inverse Problem in Various Formalisms. . . . . . . . . . 4716.7.3 Alternative Hermitean Structures for Quantum Systems:

The Infinite-Dimensional Case . . . . . . . . . . . . . . . . . . . 481References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

7 Folding and Unfolding Classical and Quantum Systems . . . . . . . . . 4897.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4897.2 Relationships Between Linear and Nonlinear Dynamics . . . . . . . 489

7.2.1 Separable Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 4907.2.2 The Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.2.3 Burgers Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4937.2.4 Reducing the Free System Again. . . . . . . . . . . . . . . . . . 4957.2.5 Reduction and Solutions of the Hamilton-Jacobi

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

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7.3 The Geometrical Description of Reduction . . . . . . . . . . . . . . . . 5007.3.1 A Charged Non-relativistic Particle in a Magnetic

Monopole Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5037.4 The Algebraic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

7.4.1 Additional Structures: Poisson Reduction . . . . . . . . . . . . 5067.4.2 Reparametrization of Linear Systems . . . . . . . . . . . . . . . 5087.4.3 Regularization and Linearization of the Kepler

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5147.5 Reduction in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 520

7.5.1 The Reduction of Free Motion in the Quantum Case . . . . 5207.5.2 Reduction in Terms of Differential Operators . . . . . . . . . 5227.5.3 The Kustaanheimo–Stiefel Fibration. . . . . . . . . . . . . . . . 5247.5.4 Reduction in the Heisenberg Picture . . . . . . . . . . . . . . . 5277.5.5 Reduction in the Ehrenfest Formalism . . . . . . . . . . . . . . 532

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

8 Integrable and Superintegrable Systems . . . . . . . . . . . . . . . . . . . . 5398.1 Introduction: What Is Integrability? . . . . . . . . . . . . . . . . . . . . . 5398.2 A First Approach to the Notion of Integrability: Systems

with Bounded Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.2.1 Systems with Bounded Trajectories . . . . . . . . . . . . . . . . 542

8.3 The Geometrization of the Notion of Integrability . . . . . . . . . . . 5468.3.1 The Geometrical Notion of Integrability

and the Erlangen Programme . . . . . . . . . . . . . . . . . . . . 5488.4 A Normal Form for an Integrable System . . . . . . . . . . . . . . . . . 550

8.4.1 Integrability and Alternative Hamiltonian Descriptions . . . 5508.4.2 Integrability and Normal Forms. . . . . . . . . . . . . . . . . . . 5528.4.3 The Group of Diffeomorphisms of an Integrable

System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5558.4.4 Oscillators and Nonlinear Oscillators . . . . . . . . . . . . . . . 5568.4.5 Obstructions to the Equivalence of Integrable Systems . . . 557

8.5 Lax Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5588.5.1 The Toda Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

8.6 The Calogero System: Inverse Scattering . . . . . . . . . . . . . . . . . 5638.6.1 The Integrability of the Calogero-Moser System . . . . . . . 5638.6.2 Inverse Scattering: A Simple Example . . . . . . . . . . . . . . 5648.6.3 Scattering States for the Calogero System. . . . . . . . . . . . 565

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

9 Lie–Scheffers Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5699.1 The Inhomogeneous Linear Equation Revisited . . . . . . . . . . . . . 5699.2 Inhomogeneous Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 5719.3 Non-linear Superposition Rule . . . . . . . . . . . . . . . . . . . . . . . . . 5789.4 Related Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

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9.5 Lie–Scheffers Systems on Lie Groups and HomogeneousSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

9.6 Some Examples of Lie–Scheffers Systems . . . . . . . . . . . . . . . . 5899.6.1 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5899.6.2 Euler Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5959.6.3 SODE Lie–Scheffers Systems . . . . . . . . . . . . . . . . . . . . 5979.6.4 Schrödinger–Pauli Equation . . . . . . . . . . . . . . . . . . . . . 5989.6.5 Smorodinsky–Winternitz Oscillator . . . . . . . . . . . . . . . . 599

9.7 Hamiltonian Systems of Lie–Scheffers Type . . . . . . . . . . . . . . . 6009.8 A Generalization of Lie–Scheffers Systems . . . . . . . . . . . . . . . . 605References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

10 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712

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Chapter 1Some Examples of Linear and NonlinearPhysical Systems and Their DynamicalEquations

An instinctive, irreflective knowledge of the processes of naturewill doubtless always precede the scientific, consciousapprehension, or investigation, of phenomena. The former is theoutcome of the relation in which the processes of nature stand tothe satisfaction of our wants.

Ernst Mach, The Science of Mechanics (1883).

1.1 Introduction

This chapter is devoted to the discussion of a few simple examples of dynamicsby using elementary means. The purpose of that is twofold, on one side after thediscussion of these examples we will have a catalogue of systems to test the ideas wewould be introducing later on; on the other hand this collection of simple systemswill help to illustrate how geometrical ideas actually are born from dynamics.

The chosen examples are at the same time simple, however they are ubiquitousin many branches of Physics, not just theoretical, and they constitute part of a physi-cist’s wardrobe. Most of them are linear systems, even though we will show howto construct non-trivial nonlinear systems out of them, and they are both finite andinfinite-dimensional.

We have chosen to present this collection of examples by using just elementarynotions from calculus and the elementary theory of differential equations. Moreadvanced notions will arise throughout that will be given a preliminary treatment;however proper references to the place in the book where the appropriate discussionis presented will be given.

Throughout the book we will refer back to these examples, even though new oneswill be introduced. We will leave most of the more advanced discussions on theirstructure for later chapters, thus we must consider this presentation as a warmup andalso as an opportunity to think back on basic ideas.

© Springer Science+Business Media Dordrecht 2015J.F. Cariñena et al., Geometry from Dynamics, Classical and Quantum,DOI 10.1007/978-94-017-9220-2_1

1

2 1 Some Examples of Linear and Nonlinear Physical . . .

1.2 Equations of Motion for Evolution Systems

1.2.1 Histories, Evolution and Differential Equations

A physical system is primarily characterized by histories, histories told by observers:trajectories in a bubble chamber of an elementary particle, trajectories in the sky forcelestial bodies, or changes in the polarization of light. The events composing thesehistories must be localized in some carrier space, for instance the events composingthe trajectories in a bubble chamber can be localized in space and time as well asthe motion of celestial bodies, but the histories of massless particles can be localizedonly in momentum space.

In the Newtonian approach to the time evolution of a classical physical system,a configuration space Q is associated with the system, that at this moment will beassumed to be identified with a subset of R

N , and space-time is replaced by Q × R

that will be the carrier space where trajectories can be localized. Usually, temporalevolution is determined by solving a system of ordinary differential equations on Q×R which because of experimental reasons combined with the theoretician ingenuityfreedom, are chosen to be a system of second-order differential equations:

d2qi

dt2= Fi

(q1, . . . , q N ,

dq1

dt, . . . ,

dq N

dt

), i = 1, . . . , N . (1.1)

How differential equations are arrived at from families of ‘experimental trajectories’is discussed in [Mm85]. Assuming evolution is described by a second-order differ-ential equation was the point of view adopted by Joseph-Louis Lagrange and it ledhim to find for the first time a symplectic structure on the space of motions [La16].

The evolution of the system will be described by solving the system of Eq. (1.1)for each one of the initial data posed by the experimentalist, i.e., at a given time t0,both the positions and velocities q0 and v0 of the system must be determined. Thesolution q(t), that will be assumed to exist, of the initial value problem posed byEq. (1.1) and q(t0) = q0, q(t0) = v0, will be the trajectory described by the systemon the carrier space Q. The role of the theoretician should be quite clear now. Westarted from a necessarily finite number of ‘known’ trajectories and we have founda way to make previsions for an infinite number of them, for each initial condition.

If we are able to solve the evolution Eq. (1.1) for all possible initial data q0, v0,then we may alternatively think of the family of solutions obtained in this way asdefining a transformation ϕt mapping each pair (q0, v0) to (q(t), q(t)) (for each tsuch that the solution q(t) exists). The one-parameter family of transformations ϕt

will be called the flow of the system and knowing it we can determine the state ofthe system at each time t provided that we know the state of the system (describedin this case by a pair of points (q, v)) at a time t0.

To turn the description of evolution into a one-parameter family of transformationswe prefer to work with an associated system of first-order differential equations. Inthis way there will be a one-to-one correspondence between solutions and initial

1.2 Equations of Motion for Evolution Systems 3

data. A canonical way to do that is to replace our Eq. (1.1) by the system of 2Nequations

dqi

dt= vi ,

dvi

dt= Fi (q, v) , i = 1, . . . , N , (1.2)

where additional variables vi , the velocities of the system, have been introduced.If we start with a system in the form given by Eq. (1.1) we can consider as

equivalent any other description that gives us the possibility to recover the trajectoriesof our starting system. This extension however has some ambiguities. The one we aredescribing is a ‘natural one’. However other possibilities exist as has been pointed outin [Mm85]. In particular we can think of, for instance, a coordinate transformationthatwould transformour starting system into a linear onewhere such a transformationwould exist. We will consider this problem in depth in relation with the existence ofnormal forms for dynamical systems at various places throughout the text. A largefamily of examples fitting in this scheme is provided by the theory of completelyintegrable systems.

A completely integrable system is characterized by the existence of variables,called action-angle variables denoted by (Ji , φ

i ), such that when written in this newset of variables our evolution Eq. (1.2) look like:

dφi

dt= νi (J ) ,

d J i

dt= 0, i = 1, ..., N . (1.3)

The general solution of such a system is given by:

φi = φi0 + νi

0 t, Ji = Ji (t0).

where νi0 = νi (J (t0)) and Ji (t0) is the value taken by each one of the variables Ji at

a given initial time t0.If det(∂νi /∂ J j ) �= 0, this system can be given an equivalent description as follows:

d

dt

(φi

νi

)=

(0 I

0 0

)(φi

νi

), (1.4)

and the 2N × 2N matrix in Eq. (1.4) is nilpotent of index 2. We will discuss theclassical theory of completely integrable systems and we will offer a general viewon integrability in Chap.8.

The family of completely integrable systems is the first of a large class of sys-tems that can be written in the form of a homogeneous first-order linear differentialequations:

dx

dt= A · x , (1.5)

where A is an n × n real matrix and x ∈ Rn . Here and hereafter use is made of

Einstein’s convention of summation over repeated indices. The Eq. (1.5) is then thesame as:

http://dx.doi.org/10.1007/978-94-017-9220-2_8


dxi

dt= Ai

j x j . (1.6)

Then the solution of Eq. (1.5) for a given Cauchy datum x(0) = x0 is given by:

x (t) = exp (t A) x0 , (1.7)

where the exponential function is defined as the power series:

exp t A =∞∑

k=0

t k

k! Ak,

(see Sect. 2.2.4 for a detailed discussion on the definition and properties of exp A).

1.2.2 The Isotropic Harmonic Oscillator

As a first example let us consider an m-dimensional isotropic harmonic oscillatorof unit mass and proper frequency ω. Harmonic oscillators are ubiquitous in thedescription of physical systems. For instance the electromagnetic field in a cavitymaybe described by an infinite number of oscillators. An LC oscillator circuit consistingof an inductor (L) and a capacitor (C) connected together is described by the harmonicoscillator equation. In classical mechanics, any system described by kinetic energyplus potential energy, say V (q), assumed to have a minimum at point q0, may beapproximated by an oscillator in the following manner. We Taylor expand V (q)

around the equilibrium point q0, provided that V is regular enough, and on takingonly the first two non-vanishing terms in the expansion we have: V (q) ≈ V (q0) +12V ′′(q0)(q − q0)2. By using

ω =√

V ′′(q0)m

,

m being the mass, and removing the constant value V (x0) one finds and approximat-ing potential U (q) = 1

2mω2q2 for our system. It is this circumstance that places theharmonic oscillator in a pole position in the description of physical systems.

Then, extending the previous considerations to systems with m degrees of free-dom,wecouldwrite the equations ofmotion for anm-dimensional isotropic harmonicoscillator as the system of second-order differential equations on R

m :

d2qi

dt2= −ω2qi , i = 1, . . . , m.

We may write this system as a first-order linear system:

dx/dt = A · x,

http://dx.doi.org/10.1007/978-94-017-9220-2_2


by introducing the vectors q, v ∈ Rm , x = (q, v)T ∈ R

2m and the 2m × 2m matrix:

A =(

0 Im

−ω2Im 0

). (1.8)

Then the equations of motion read in the previous coordinates:

dqi

dt= vi ,

dvi

dt= −ω2qi , i = 1, . . . , m . (1.9)

A direct computation shows:

A2 j = (−1) jω2 jIN , A2 j+1 = (−1) jω2 j A, j ≥ 0 , (1.10)

and we find at once:

exp(t A) = cos(ωt) IN + 1

ωsin(ωt) A , (1.11)

as well as the standard solution for (1.9):

x(t) = et Ax0.

given explicitly by:

q(t) = q0 cos(ωt) + v0

ωsin(ωt), v(t) = v0 cos(ωt) − ωq0 sin(ωt) . (1.12)

1.2.3 Inhomogeneous or Affine Equations

Because of external interactions very often systems exhibit inhomogeneous terms inthe evolution equations. Let us show now how we can deal with them in the samesetting as the homogeneous linear ones. Thus we will consider an inhomogeneousfirst-order differential equation:

dx

dt= A · x + b . (1.13)

First of all we can split off b in terms of its components in the range of A, and asupplementary space, i.e., we can write:

b = b1 + b2 , (1.14)

where b2 = A · c form some c ∈ Rn . Then Eq. (1.13) becomes


dx

dt= A · x + b1 , (1.15)

with x = x + c. Note that the splitting of b is not unique but depends on the choiceof a supplementary space to the range of A. If b1 = 0, we are back to the previoushomogeneous case. If not, we can define a related dynamical system on R

n+1 bysetting:

dξ

dt= A · ξ (1.16)

with ξ = (x, a)T , a ∈ R, and,

A =(

A b10 0

). (1.17)

Explicitly, this leads to the equations of motion

dx

dt= A · x + b1,

da

dt= 0 , (1.18)

and the solutions of Eq. (1.15) will correspond to those of Eq. (1.16) with a(0) = 1(and vice versa). The latter will be obtained again by exponentiating A. Note that wecan write

A = A1 + A2, A1 =(

A 00 0

), A2 =

(0 b10 0

), (1.19)

and that the commutator of A1 and A2 is given by:

[ A1, A2] =(

0 A · b10 0

). (1.20)

Hence an important case happens when the kernel of A is a supplementary spacefor the image of A, because had we chosen such space as the supplementary spacegiving raise to Eq. (1.14), in such case A · b1 = 0, and then,

exp(t A) = exp(t A1) exp(t A2) (1.21)

where, explicitly

exp(t A1) = exp t

(A 00 0

)=

(et A 00 In

);


and

exp(t A2) = exp t

(0 b10 0

)=

(In tb10 In

).

Hence the solution of Eq. (1.15) would be written as:

x (t) = et A (x0 + tb1) , (1.22)

or,x (t) = et A (x0 + c + tb1) − c , (1.23)

with initial value x(0) = x0, and only the explicit exponentiation of A is required.Very often this particular situation is referred to as the ‘composition of independentmotions’. Note that the fact that b can be decomposed into a part that is in the rangeand a part that is in the kernel of A is only guaranteed when ker A ⊕ ran A = R

n .Let us consider now two examples that illustrate this situation.

1.2.4 A Free Falling Body in a Constant Force Field

We start by considering the case of the motion of a particle in constant force field,a simple example being free fall in a constant gravitational field. As for any choiceof the initial conditions the motion takes place in a plane, we can consider directlyQ = R

2 with the acceleration g pointing along the negative y-axis direction.Then, again we take x = (q, v)T ∈ R

4, q, v ∈ R2 and Newton’s equations of

motion are:dq1

dt= v1,

dq2

dt= v2,

dv1

dt= 0,

dv2

dt= −g . (1.24)

If the initial velocity is not parallel to g, i.e., v10 �= 0, then the solutions of Eq. (1.24)will be the family of parabolas:

q1(t) = q10 + v10 t, q2 = q2

0 +(

v20

v10

)(q1 − q1

0 ) −(

g

2(v10)2

)(q1 − q1

0 )2,

where q0 = (q10 , q2

0 ) and v0 = (v10, v20) are the initial data. Equation (1.24) can be

recasted in the form of Eq. (1.13) by setting, in terms of 2 × 2 blocks

A =(

0 I20 0

), b = −g

⎛⎜⎜⎝0001

⎞⎟⎟⎠ . (1.25)


A free particle would have been described by the previous equation with g = 0. Insuch a case the solutions would have been a family of straight lines.

Now, ker A = ran A and it consists of vectors of the form (x, 0)T . Hence b isneither in ker A nor in ran A, and in order to obtain the solution the simple methoddiscussed in the previous section cannot be used.Wewould use the general procedureoutlined above and would be forced to exponentiate a 5 × 5 matrix. In this specificcase, using the decomposition Eq. (1.14), one might observe that, due to the fact thatA is nilpotent of order 2 (A2 = 0), both A1 and A2 commute with [ A1, A2], andthis simplifies greatly the procedure of exponentiating A. However, that is specificto this case, that can be solved, as we did, by direct and elementary means, so wewill not insist on this point.

1.2.5 Charged Particles in Uniform and Stationary Electricand Magnetic Fields

Let us consider now the motion of a charged particle in an electromagnetic field inR3. Denoting the electric and magnetic fields respectively by E and B, and by q,

and v the position and velocity of the particle (all vectors in R3), we have that the

equations of motion of the particle are given by Lorentz equations of motion:

dqdt

= v,dvdt

= e

mE + e

mv × B , (1.26)

where m and e are the mass and charge of the particle respectively, and we havechosen physical units such that c = 1, with c the speed of light. Of course, we canconsider the second equation in (1.26) as an autonomous inhomogeneous equationon R

3. Let us work then on the latter.We begin by defining a matrix B by setting B · u = u × B for any u ∈ R

3, i.e.,Bi j = εi jk Bk . The matrix B is a 3×3 skew–symmetric matrix, hence degenerate. Itskernel, ker B, is spanned by B (we are assuming that B is not identically zero), andranB is spanned by the vectors that are orthogonal to B. Hence, R3 = ker B⊕ ranBand we are under the circumstances described after Eq. (1.20). We can decompose Ealong ker B and ranB as follows:

E = 1

B2 [(E · B) B + B× (E × B)] , B2 = B · B . (1.27)

In the notation of Eq. (1.13), we have:

x = v, A = e

mB, b1 = e

m B2 (E · B) B,

and,

b2 = A · c = e

m B2 B× (E × B) with c = 1

B2 B × E.


The equations of motion (1.15) become:

d

dt

(v − E × B

B2

)= − e

mB ×

(v − E × B

B2

)+ e

m B2 (E · B) B,

and we find from Eq. (1.23)

v − E × BB2 = eetB/m

(v0 − E × B

B2 + et

m B2 (E · B) B)

,

or, noticing that exp (etB/m) · B = B,

v − E × BB2 = eetB/m

(v0 − E × B

B2

)+ et

m B2 (E · B) B.

If S is the matrix sending B into its normal form, i.e., S is the matrix defining achange of basis in which the new basis is made up by an orthonormal set with twoorthogonal vectors to B such that e1, e2, B/‖B‖ is an oriented orthonormal set,

SBS−1 =⎛⎝ 0 1 0

−1 0 00 0 0

⎞⎠ ,

then we have

S

(v − E × B

B2

)= e

(et SBS−1)

S

(v0 − E × B

B2 + et

m B2 (E · B) B)

,

and we find that eetB/m is a rotation around the axis defined by B with angularvelocity given by the cyclotron (or Larmor) frequency = eB/m (recasting thelight velocity c in its proper place we would find = eB/mc).

Proceeding further, the first of Eq. (1.26) can be rewritten as

d

dt

(q − E × B

B2 t − et2

2m B2 (E · B) B)

= eetB/m(

v0 − E × BB2

).

We can decompose v0 as well along ker B and ranB, i.e.,

v0 = 1

B2 [(v0 · B)B + B × (v0 × B)] . (1.28)

Hence,

v0− E × BB2 = (v0 ·B)

BB2 +

(B × v0

B2 − EB2

)×B = v0 · B

B2 B+ 1

B2B (B × v0 − E) .


By applying eetB/m to this decomposition we get

eetB/m(

v0 − E × BB2

)= v0 · B

B2 B + 1

B2 eetB/m · B (B × v0 − E) (1.29)

which can be written as

eetB/m(

v0 − E × BB2

)= d

dt

[tv0 · B

B2 B + m

eB2 eetB/m · B (B × v0 − E)

],

(1.30)and we find from Eq. (1.29) and (1.30) that the solution is:

q (t) −[

E × BB2 + v0 · B

B2 B]

t − et2

2m B2 (E · B) B − m

eB2 eetB/m (B × v0 − E) = const.

The constant can be determined in terms of the initial position q0, and we find even-tually for the general motion of a charged particle in some external electromagneticfields (E, B):

q (t) = q0 +[

E × BB2 + v0 · B

B2 B]

t + et2

2m B2 (E · B) B

+ m

eB2 (exp (etB/m) − 1) (B × v0 − E) . (1.31)

We can examine now various limiting cases:

1. When E × B = 0, E is in ker B and (exp (etB/m) − 1)E = 0. So,

q (t) = q0 +[

tv0 · B

B2 + et2

2m B2 (E · B)

]B + m

eB2 (exp (etB/m) − 1)B × v0,

and themotion consists of a rotation around the B axis composedwith a uniformlyaccelerated motion along the direction of B itself.

2. As a subcase, ifE = 0, we have a rotation aroundB plus uniformmotion alongB:

q (t) = q0 + tv0 · B

B2 B + m

eB2 (exp (etB/m) − 1)B × v0.

3. B ≈ 0. In this case we can expand

exp(etB/m) − 1 ∼= etBm

+ 1

2

(etBm

)2

+ O(B3).

Now,B(B × v0 − E) = B2v0 − ((v0 · B)B + E × B)


while

B2(B × v0 − E) = −(E × B) × B + O(B3) = B2E − (E · B)B + O(B3).

We see that there are quite a few cancellations and that we are left with

q(t) = q0 + v0t + e E t2/2m,

as it should be if we solve directly Eq. (1.26) with B = 0.4. E · B = 0 .

This is perhaps the most interesting case because this condition is Lorentz-invariant and this geometry of fields is precisely that giving raise to the Hall effect.

First of all, we notice that the motion along B will be a uniform motion. We willdecouple it by assuming v0 · B = 0 (or by shifting to a reference frame that movesalong B with uniform velocity (v0 · B)/B2). Then, we may write Eq. (1.31) as:

q (t) = vDt + Q0 + exp (etB/m) R,

where

vD = E × B/B2, Q0 = q0 − m

eB2 (B × v0 − E), R = m

eB2 (B × v0 − E).

The first term represents a uniform motion, at right angles with both E and B, withthe drift velocity vD . The second term represents a circular motion around B withcenter at Q0, radius R = ‖R‖ and angular frequency . Correspondingly, we have

v (t) = vD + eetB/m (v0 − vD) . (1.32)

Remark 1.1 Here we can set E = 0 too. Then vD = 0 and we recover case 2. Theradius of the (circular) orbit is the Larmor radius

RL = ‖R‖ = v0/ . (1.33)

More generally, the standard formulae for transformation of the fields under Lorentzboosts [Ja62] show that, if (in units c = 1): ‖vD‖ = ‖E‖/‖B‖ < 1, a Lorentz boostwith velocity vD leads to

E′ = 0, B′ = B + O((E/B)2

). (1.34)

So, if E · B = 0 and ‖E‖ < ‖B‖ (actually under normal experimental conditions‖E‖ � ‖B‖) there is a frame in which the electric field can be boosted away.


1.2.5.1 Classical Hall Effect

If we have a sample with n charged particles per unit volume, the total electric currentwill be j = nev(t), with v(t) given by Eq. (1.32). Averaging over times of order−1,the second term in Eq. (1.32) will average to zero, and the average current J = j(t)will be given by

J = nevD . (1.35)

Defining a conductivity tensor σi j as

Ji = σi j E j , i, j = 1, 2, (1.36)

and taking the magnetic field in the z–direction, we find the (classical) Hall conduc-tivity,

σi j = ne

Bεi j , ε12 = −ε21 = 1, εi i = 0 (1.37)

or

σ = σH

(0 1

−1 0

), σH = ne

B. (1.38)

1.2.6 Symmetries and Constants of Motion

A symmetry of the inhomogeneous equation (1.13) will be, for the moment, anysmooth and invertible transformation: x �→ x ′ = F(x) that takes solutions intosolutions. Limiting ourselves to affine transformations, it can be easily shown thatthe transformation x ′ = M · x + d, with M and d constant, will satisfy

dx ′

dt= A · x ′ + b

iff [M, A] = 0 and A · d = (M − I ) · b.Using Eq. (1.6) we can compute, for any smooth function f in R

n (the set of suchfunctions will be denoted henceforth as C∞ (Rn), F (Rn), or simply as F if there isno risk of confusion):

d f

dt= ∂ f

∂x j

dx j

dt= ∂ f

∂x jA j

i xi . (1.39)

Then a constant ofmotionwill be any (at leastC1) function f (x) such thatd f/dt = 0.Limiting ourselves to functions that are at most quadratic in x , i.e., having the form:

fN (x) = xt N x + at x = Ni j xi x j + ai xi (1.40)

where N is a constant symmetric matrix, we find that


d fN (x)

dt= xt (At N + N A

)x + (

at A)

x + at b + 2(bt N

)x .

Hence, fN will be a constant of motion if and only if:

At N + N A = 0, at b = 0, at A + 2bt N = 0.

If we want to apply these considerations to the motion of a charged particle forinstance, we have to rewrite Eq. (1.26) in the compact form of Eq. (1.13). To thiseffect, let us introduce the six-dimensional vectors

x =(

qv

), b =

(0b

), b = e E

m, (1.41)

and the 6 × 6 matrix (written in terms of 3 × 3 blocks)

A =(

0 I

0 C

), C · u = e

mu × B . (1.42)

Writing M and d in a similar form, i.e.,

M =(

α β

γ δ

), d =

(dq

dv

),

(with α, . . . , δ being 3×3matrices), we find that the condition [A, M] = 0 becomes:

γ = 0; δ = α + βC, [δ, C] = 0,

while the condition Ad = (M − 1)b, yields:

dv = βb, Cdv = (δ − I )b.

There are no conditions on dq , and this corresponds to the fact that arbitrary trans-lations of q alone are trivial symmetries of Eq. (1.26). Let us consider in particularthe case in which M is a rotation matrix. Then the condition MT M = I leads to theadditional constraints:

αT α = δT δ + βT β = I, αT β = 0 . (1.43)

But then, as α is an orthogonal matrix, β = 0, and we are left with

γ = α, αt = α−1, dv = 0, αt b = b,[δ, C

] = 0 . (1.44)

As C itself is proportional to the infinitesimal generator of rotations about the direc-tion of B, this implies that α must represent a rotation about the same axis, and that b


must be an eigenvector of α with eigenvalue one. As b is parallel to E, this implies,of course, that E×B = 0. It is again pretty obvious that, if E and B are parallel, thenrotations about their common direction are symmetries.

More generally, a simple counting shows that because of Eq. (1.43) thematrices α,β, δ, generate in general a six–parameter family of symmetries. Special relationshipsbetween E and B (or vanishing of some of them) may enlarge the family.

The transformations determined by Eq. (1.44) (whether it is a symmetry or not)is an example of a point transformation, i.e., a transformation q �→ q′ = q′(q), ofthe coordinates together with the transformation: v �→ v′ = dq′/dt , that preservethe relation between the position and the velocity. Such transformations are calledalso Newtonian.

For a given system of second-order differential equations one can permit alsotransformations (in particular, symmetries) that preserve the relationship betweenq and v without being point transformations (also-called ‘Newtonoid’ transforma-tions).

Let us consider an example of the latter in the case of the motion of a chargedparticle. Consider, at the infinitesimal level, the transformation

q �→ q′ = q + δq, δq = λe

m(v × B + E) = λ (C · v + b) , (1.45)

(with λ a small parameter) and the (infinitesimal) transformation:

v �→ v′ = v + δv, δv = d

dtδq.

This transformation is clearly Newtonoid and is also a symmetry,

d

dtq′ = v′, d

dtv′ = C · v′ + b , (1.46)

because

δv = λd

dt[Cv + b] = λC[Cv + b] , (1.47)

and thend

dtδv = λC2 [Cv + b] = Cδv . (1.48)

1.2.6.1 Non-point Transformation Symmetries for Linear Systems

We describe now briefly a way of obtaining symmetries that are non-point transfor-mations starting from a system of differential equations written in the form Eq. (1.5).

Let us first remark that if we start from a homogeneous linear system


dξ

dt= A · ξ , (1.49)

where A is constant, we can consider for any natural number k, the infinitesimaltransformation

ξ �→ ξ ′ = ξ + δξ, δξ = λAkξ , (1.50)

then, it is easy to prove thatdξ ′

dt= A · ξ ′ , (1.51)

i.e., Eq. (1.50) is a symmetry. As long as taking powers generates independent matri-ces, this procedure will generate new symmetries at each step. Notice howeverthat only a finite number of them, will be independent, because of the celebratedHamilton–Cayley theorem [Ga59], according to which any matrix satisfies its char-acteristic equation.

In the particular case we are considering here,

A =⎡⎣0 I 00 C b0 0 0

⎤⎦ , (1.52)

and the infinitesimal symmetry Eq. (1.45) is precisely that generated by A2, because

A2 =⎡⎣0 C b0 C2 Cb0 0 0

⎤⎦ (1.53)

leads to δq = λ(Cv + b) and δv = λC(Cv + b).

Let us discuss now briefly the constants of motion. A general symmetric matrixN can be written, in terms of 3 × 3 blocks, as

N =[

α β

β t γ

], αt = α, γ t = γ , (1.54)

Writing the vector a of Eq. (1.40) as a =[

aqav

]and using Eqs. (1.41) and (1.42), we

can spell out explicitly the condition for fN to be a constant of motion as

1. α + βC = 0.2. (β + γ C) + (β + γ C)t = 0.3. β · b = 0.4. av · b = 0.5. aq = C · av − 2γ · b .


Let us examine the case in which

α = β = 0, γ = 1

2m I . (1.55)

Then, as C + Ct = 0, Eq. (1.2.6) are automatically satisfied, and we are left with

av · b = 0 (1.56)

which implies av = E × ξ for an arbitrary vector ξ , and

aq = e

m[(E · B) ξ − (ξ · B)E] − eE . (1.57)

Then,

1. ξ = 0 leads to

fN = Etot = 1

2m [v]2 − eE · q , (1.58)

where Etot is the total energy.2. For ξ �= 0 we find then that

fξ = e

m

[(E · B) (ξ · q) − (ξ · B) (E · q)

] − (E × v) · ξ (1.59)

is another constant of motion.

We have discussed at length this example to show that the subsequent physicalinterpretation of the solutions of a given system contains much more than the generalsolution provided by Eq. (1.7), which from the mathematical point of view is ratherexhaustive.

1.2.7 The Non-isotropic Harmonic Oscillator

Let us consider now inmoredetail them-dimensional anysotropic harmonic oscillatorIf the oscillator is not isotropic, the system of equations (1.8) generalizes to:

A =(

0 Im

− 0

), =

⎛⎜⎝

ω21

. . .

ω2m

⎞⎟⎠ (1.60)

with different frequencies ωk , and corresponds to the equations of motion:

dqk

dt= vk,

dvk

dt= −ω2

k qk, k = 1, . . . , m , (1.61)

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whose general solution is the obvious generalization of Eq. (1.12).It is more convenient to reorder the variables and denote by ξ ∈ R

2m the collectivevariable ξ = (q1, p1, q2, p2, . . . , qm , pm), where pi = vi/ωi , i = 1, . . . , m. Then,the equations of motion still have the form (1.8) but now, in the new coordinates, thematrix A takes the form:

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 ω1 0 0 . . . 0−ω1 0 0 0 . . . 00 0 0 ω2 0 . . 00 0 −ω2 0 0 . . 0. . . . . . . .

. . . . . . . .

. . . . . . 0 ωm

0 0 0 0 . . −ωm 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(1.62)

i.e., A is block diagonal. The diagonal blocks are the 2 × 2 matrices

(0 ωk

−ωk 0

)= ωk J, J =

(0 1

−1 0

).

Thus the matrix A is a direct sum of 2 × 2 commuting matrices.We can introduce complex variables

zk = qk + i pk, k = 1, . . . , m.

In terms of the latter the equations of motion (1.61) become simply:

dzk

dt= −iωk zk , (1.63)

or, in matrix form:dz

dt= B · z , (1.64)

where z = (z1, . . . , zm)T ∈ Cn , and B is a diagonal matrix

B = −i

⎛⎜⎜⎜⎜⎜⎜⎝

ω1 0 . . . 00 ω2 . . . 0. . . . . .

. . . . . .

. . . . . .

0 0 . . . ωm

⎞⎟⎟⎟⎟⎟⎟⎠

. (1.65)

After an essentially trivial reshuffling of variables, it can be shown that the matrix Ais precisely the realified counterpart of the complex matrix B (see Sect. 2.2.3).

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We can now adapt to the complex formalism the discussion of symmetries andconstants of motion for the harmonic oscillator. A complex affine transformation,

z �→ z′ = Mz + b , (1.66)

will be a symmetry iff [M, B] = 0 and B · b = 0, i.e., b ∈ ker B. Of course, unlesssome of the frequencies vanish, ker B = 0. Leaving aside this case, complex linearsymmetries will be homogeneous transformations with

[M, B] = 0 . (1.67)

It is clear that symmetries will include those generated by all the powers of B, i.e.,M = Bk will generate a symmetry for all integer k’s, and this will be true forany linear system. In the generic case (see below) the powers of B will exhaustall possibilities, i.e., any matrix M satisfying the condition (1.67) will be a linearcombination thereof.

We can consider now two extreme cases, namely:

1. The generic case in which the frequencies are all pairwise different: ωi �= ω j forall i �= j . Then M has to be diagonal, and the only symmetries are the dilations

zk �→ λkzk, λk ∈ C∗,

which are obviously symmetries of the equation (1.63).2. In the isotropic case, ω1 = · · · = ωm = ω �= 0, instead, B = −iω Im×m and if

we require the transformation to be invertible, then M ∈ GL(m, C). The linearsymmetry group of the m-dimensional isotropic oscillator is then the full generalcomplex linear group GL(m, C).

Intermediate cases can be worked out in a similar way.Let us turn now to consider real quadratic constants of motion. In complex nota-

tion,fa,N (z) = z†N z + a†z + z†a, N † = N .

where z† = (z∗1, . . . , z∗

m), and N†i j = N ∗

j i .Again, d fa,N /dt = 0 leads to

B†N + N B = 0, B†a = 0.

But B† = −B and, unless some of the frequencies vanish, this implies: a = 0.Quadratic constants of motion are therefore of the form

fN (z) = z†N z, N † = N , [N , B] = 0 . (1.68)

We are here in a situation that is quite similar to the one we have just discussedin connection with symmetries, except that N is required now to be a Hermitean


matrix. In the generic case, the requirement that N should commute with B forcesN to be diagonal, and we have only the m independent constants of motion

Ei = |zi |2, i = 1, . . . , m,

and any other quadratic constant of motion will be a linear combination with realcoefficients of the functions Ei . The corresponding matrices Ni will of course com-mute with each other: (Ni ) jk = δi jδ jk . Explicitly,

Ei = q2i + v2i /ω2

i ,

and will be proportional to the energy associated with the i-th coordinate of the oscil-lator, the proportionality constant being mω2

i /2. In the isotropic case, any Hermiteanmatrix will generate a constant of the motion, and we will have a total of (as a basisof) m2 independent constants of motion, namely,

R(i j) = zi z∗j + z j z∗

i , (1.69)

I (i j) = −i(zi z∗j − z j z

∗i ), for i �= j (1.70)

(notice that there are m(m +1)/2 R(i j)’s and m(m −1)/2 I (i j)’s), and again: R(i i) =2Ei , I (i i) = 0. The constants of motion can be written as quadratic forms

R(i j) =m∑

k,l=1

z∗k M(i j)

kl zl , I (i j) =m∑

k,l=1

z∗k N (i j)

kl zl ,

and the associated Hermitean matrices M (i j) and N (i j) will be given by:

M(i j)kl = δi

kδjl + δi

l δjk , k �= l, M(i j)

kk = δikδi

k , k = l, N (i j)kl = i(δi

kδjl − δi

l δjk , ) .

(1.71)

In particular the matrix associated with the total energy E = z†z will be M = I. Theremaining constants of motion will be associated with traceless Hermitean matrices.A basis of (m2 − 1) such matrices will provide us with a basis in the Lie algebraof the special unitary group SU (m), that has a natural action on C

m . Recall that amatrix U is unitary if U†U = I and the set of all unitary matrices is a subgroup ofGL(m, C) called the unitary group. The special unitary group SU (m) is the subgroupof unitary matrices with determinant 1 (see Sect. 2.6.1 for a detailed discussion ofthese notions).

Hence, a basis of (quadratic) constants of motion will be provided by sesquilinearexpressions of the form: fN = z†N z, where N is either the identity or a generator ofthe Lie algebra of SU (m). Let us remark that there is an obvious action of U (m) onC

m by matrix multiplication. We recall thatU (m) ≈ (U (1)× SU (m))/Zm , with Zm

being the cyclic group of order m, the correspondence given by (eiθ , U ) �→ eiθU .

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Thus the U (1) subgroup acts by multiplication through a phase, i.e.

U (1) : z �→ exp(iθ)z

and this action will be generated by N = I. As the dynamics is described by

z (t) = exp(−iωt)z0 , (1.72)

it is clear that there is a one-to-one correspondence between constants of the motionand elements in a basis for the Lie algebra of U (m), with the total energy beingassociated with the subgroup U (1).

In view of this U (1) action, it is clear that any function invariant under U (1) (i.e.,a constant of motion for H = z†z) must be at least a Hermitean quadratic invariantfunction. That is why there are no linear constants of motion.

We will discuss in Sect. 5.4.3 the topological features of the orbit space of a har-monic oscillator. It will define the Hopf fibration, that will be of great importance indiscussing several examples throughout this book. Because of its interest, it deservesa special attention and careful analysis (see the cover picture where the trajectoriesof a two-dimensional non-isotropic harmonic oscillator with irrational ratio ω1/ω2,are depicted on a fixed energy surface).

There are other symmetries for our oscillator dynamics which are not linear. Wefirst describe some of them and then make some general considerations. We haveseen that the isotropic harmonic oscillator has constants of motion given by

fN = N i j z∗i z j .

Now we can perform any coordinate transformation of the kind

ξ j = B ji zi , ξ∗

j = B ji z∗

i ,

with B = (B ji ) being now a matrix whose entries are real constants of the motion

such that det B �= 0. The equations of motion in the new coordinates ξk are given by:

ξk = −iωkξk; ξ∗k = iωkξ

∗k .

These equations are still linear but they are connected to the previous ones,Eq. (1.61), by a nonlinear coordinate transformation. It is clear however that sym-metries that are now linear in the coordinates (ξ, ξ∗) will not come from linearsymmetries in the coordinates (z, z∗). Thus, even though we are dealing with vectorspaces, it is natural to allow also for nonlinear transformations. We have here aninstance of a nonlinear transformation that connects to alternative linear descriptionsof the same system.

If our interest is primarily on dynamical systems, we will be interested in alltransformations that help visualizing the dynamics (think for instance of polar

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coordinates). As a matter of fact, a physical process will take place irrespectiveof which coordinate frame the observer may choose to observe it. For this reasonit would be very useful to have an identification of our dynamical system with amathematical entity that transforms naturally with respect to any transformation ofcoordinates. Of course these entities are known to be tensor fields on our space. Wedigress further in Chap. 2, Sect. 2.4.4. on how to associate matrices with tensor fields.

1.2.8 Lagrangian and Hamiltonian Descriptions of EvolutionEquations

We will say that a given system whose evolution equations are of the form (1.1) or(1.2) admit of Lagrangian description if there exists a function L = L(q, v) suchthat they may be written as

dqi

dt= vi ,

d

dt

∂L

∂vi= ∂L

∂qi, i = 1, . . . , n.

The function L = L(q, v)will be called theLagrangian of the system. If this happens,then the second equation in (1.2.8) implies that

∂2L

∂vi∂v jF j (q, v) = ∂L

∂qi− ∂2L

∂vi∂q jv j , i = 1, . . . , n , (1.73)

We will discuss later on some of the foundational ideas behind the Lagrangian for-malism and their geometrical form (see Chap. 5, Sect. 5.5) but now we will be onlyconcentrated on discussing some consequences of the existence of such formalismfor the systems presented before.

Whether or not Eq. (1.73) can be solved for L constitutes what is known as theInverse Problem of Lagrangian Dynamics. It may admit no solutions at all or morethan one solution. Notice that if L is a solution of (1.73), then λL is also a solutionfor any λ ∈ R. Moreover, if L is a Lagrangian for (1.2), then L ′ = L + qi∂α/∂qi

with α(q) an arbitrary function, is also a solution of (1.73) in the sense that:

d

dt

∂α

∂vi= ∂α

∂qi,

with α = dα/dt = vi∂α/∂qi the total time derivative of the functionα, is an identity.The latter case, when the solutions are not (trivially) proportional to each other ordo not differ merely by a total time derivative, leads to the possibility of alternativeLagrangian descriptions for the same dynamical system. We will give examples ofthis later on.

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We shall limit ourselves, although more general cases can be envisaged, toLagrangians that are quadratic in the velocities. Also, we shall consider here onlycases in which the Lagrangian is regular, i.e., such that

det(Wi j ) �= 0, with Wi j = ∂2L

∂vi∂v j.

The matrix (Wi j ) is called the Hessian matrix of the Lagrangian. If that is the case,and if W −1 = (W i j ) is the inverse of the Hessian matrix W = (Wi j ), then Eq. (1.73)can be solved for the forces once the Lagrangian is given, i.e.,

Fi = W i j(

∂L

∂q j− ∂2L

∂v j∂qkvk)

, i = 1, . . . , n.

Defining now the canonical momenta pi as

pi = ∂L

∂vi, i = 1, . . . , n , (1.74)

and using the implicit function theorem, one can invert this relation between the vi ’sand the pk’s, vi = φi (q, p), and pass over to the Hamiltonian description of thesystem (1.2), where the equations of motion are transformed into the well-knownHamilton equations:

dqi

dt= ∂ H

∂pi,

dpi

dt= −∂ H

∂qi, (1.75)

whereH = piφ

i (q, p) − L(q, φ(q, p)) = H (q, p)

is the Hamiltonian function of the system, and the velocities have been expressedas functions of the qi ’s and pk’s by inverting Eq. (1.74). The space whose pointsare labelled by coordinates qi ’s and pi ’s is called the phase space of the dynamicalsystem above and it will be denoted1 as T ∗ Q if Q ⊂ R

n denotes the configurationspace of the original system (1.1), hence T ∗Q ⊂ R

2n .It is a well known fact, to be explained later on, that if Q is the configuration

space of a mechanical system, there is a natural Poisson bracket structure on thephase space. Denoting collectively by ξ i , i = 1, . . . , 2n, the coordinates on T ∗ Q,(ξ i = qi , ξ i+n = pi , i = 1, . . . , n), the fundamental Poisson brackets are defined as

{ξ i , ξ j

}= J i j (1.76)

1 For the time being that is just a convenient notation. The geometrical aspects of T ∗Q and itsgeometrical definition will be considered later (see Sect. 5.3.2).

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where J = (J i j ) is the 2n × 2n symplectic matrix:

J =(

J i j)

=(

0 I

−I 0

).

More explicitly, {qi , q j

}= {

pi , p j} = 0,

{qi , p j

}= δi

j .

The Poisson bracket of any two functions f, g on T ∗ Q is defined as:

{ f, g} =∑

i j

J i j ∂ f

∂ξ i

∂g

∂ξ j. (1.77)

It is easy to show then that:

1. ThePoissonbracket definedbyEq. (1.77)with J i j givenby (1.76), is anR-bilinearskew-symmetric map { f, g} = − {g, f }, that satisfies the Jacobi identity,

{{ f, g} , h} + {{h, f } , g} + {{g, h} , f } = 0,

for every triple of functions f, g, h, and that:2. The Poisson bracket { f, gh} can be evaluated using the Leibnitz rule, i.e.,

{ f, gh} = { f, g} h + g { f, h} .

3. If { f, g} = 0 for any function g, then f is a constant function. We will say thenthat the Poisson bracket we have defined is nondegenerate.

4. In arbitrary coordinates, say {ηα}, the Poisson bracket above can be written

{ f, g} =2n∑

α,β=1

∂ f

∂ηα

{ηα, ηβ

} ∂g

∂ηβ

because

{ f, g} =2n∑

α,β,γ,λ=1

∂ f

∂ηγ

∂ηγ

∂ξαJαβ ∂ηλ

∂ξβ

∂g

∂ηλ=

2n∑γ,λ=1

∂ f

∂ηγ

{ηγ , ηλ

} ∂g

∂ηλ.

Note that, in particular, under an arbitrary change of coordinates in Q: qi �→qi = qi (q), the velocities will change as vk �→ vk = vi

(∂qk/∂qi

), and then,

∂vk/∂vi = ∂qk/∂qi . Consequently, the change in the coordinates pi is given by

pi = ∂L

∂vi= ∂vk

∂vi

∂L

∂vk= ∂qk

∂qipk .


Then, it can be shown easily that, under such a change of coordinates, the fundamentalPoisson brackets become

{qi , qk

}= {

pi , pk} = 0,

{qi , pk

}= δi

k , (1.78)

and this proves that the Poisson bracket defined by Eq. (1.76) does not depend onthe choice of coordinates on Q , i.e., it is an intrinsic object.

One of the first applications of the Lagrangian (and/or Hamiltonian) formalism isthat they allow for procedures that associate constants of motion with infinitesimalsymmetries, and vice versa. Let us start with the association of constants of motionwith symmetries. We recall that the energy function

EL = vi ∂L

∂vi − L

associated with the Lagrangian L , is a constant of motion. Indeed

d EL

dt= vi

(d

dt

∂L

∂vi− ∂L

∂qi

)= 0.

Working at the infinitesimal level, let:

qi �→ qi + δqi , vi �→ vi + δvi (1.79)

be an infinitesimal symmetry. At this stage, we will not require the transformation(1.79) to be a point symmetry, i.e., one such that δqi = δqi (q).

The condition for being a symmetry are

d

dtδqi = δvi = δ

dqi

dt,

d

dtδvi = δFi = δ

dvi

dt

i.e., that the operations of taking variations and of taking time derivatives commute.This property being true for the variations of the fundamental functions qiand vi , itwill be true for the variations of any function. If we consider then

εK = δEL = ∂ EL

∂qiδqi + ∂ EL

∂viδvi ,

where ε is the infinitesimal parameter, it turns out that

εd K

dt= d

dtδEL = δ

d EL

dt= 0,

and hence K (if it is not trivially a constant) is a constant of motion.As already said, it may happen that this procedure leads only to trivial results.

Consider for example, the two-dimensional isotropic oscillator with the Lagrangian


L = 1

2v2 − 1

2ω2q2,

and then

EL = 1

2(v2 + ω2q2).

Let q = (q1, q2), v = (v1, v2), and denote by k a unit vector in the directionorthogonal to the plane, k = (0, 0, 1) . Then,

δq = ε k × v, δv = d

dtδq = −ε ω2k × q,

is an infinitesimal symmetry, as indeed,

d

dtδv = −εω2k × v = −ω2δq,

and we find

ε K = 1

2δ(v2 + ω2q2) = 2ε ω2(q2v1 − q1v2) = −2ε ω2k · (q × v)

i.e., K is proportional to the angular momentum of the oscillator. If we take instead

δq1 = ε v2, δq2 = ε v1, δv = d

dtδq,

(δv1 = −ε ω2q2, δv2 = −ε ω2q1), we find K = 0, i.e., the above procedure turnsout to be empty in this case.

The search for symmetries for systems admitting of a Lagrangian description iseasier in the case of point symmetries. These symmetries correspond to symmetriesfor the Lagrangian. The standard way of associating constants of the motion withsymmetries goes through Noether’s theorem, that can be stated in a local languageas follows.

Let Eq. (1.79) be a point symmetry, i.e., one such that: δqi = δqi (q), for thesystem described by Eq. (1.2.8). If under Eq. (1.79),

L �→ L + δL , δL = εdg

dt, g = g(q),

then K given by

ε K = ∂L

∂viδqi − ε g

is a constant of motion.


As an example, let us consider again the two-dimensional isotropic oscillator. Theequations of motion are clearly invariant under rotations in the plane, and therefore,

δq = εk × q, δv = εk × v , (1.80)

is a point symmetry. Under Eq. (1.80) δL = 0 and Noether’s constant of motion is

ε K = ∂L

∂v· δq = ε v · k × q = ε (q × v) · k,

which gives us again the angular momentum. We see here that different procedurescan associate the same constant of motion with different symmetries.

We can also try to work backwards, and see if and how one can associate symme-tries with constants of motion. If the system happens to have a constant of motion Kthat is linear in the momenta, i.e., if

K = piαi − g, pi = ∂L

∂vi, αi = αi (q), g = g(q),

is a constant of motion, i.e., using the equations of motion

0 = d K

dt= ∂L

∂qiαi + ∂L

∂vi

dαi

dt− dg

dt,

then, defining

δqi = εαi , δvi = d

dtδqi = ε

dαi

dt(1.81)

it is easily seen that

δL = εdg

dt.

But, as the Lagrangian gets altered by a total time derivative, the equations of motionare unchanged and Eq. (1.81) is a (point and Noether) symmetry.

More generally, we can take advantage of the fact that (for regular Lagrangians) aPoisson bracket can be defined both at the Lagrangian and at the Hamiltonian level,and use it in the followingmanner. Let F be any function, and define the infinitesimaltransformation

g �→ g + δg, δg = ε {g, F} ,

(in particular, g = qi or pi (on T ∗ Q) or g = qi or vi ). Considering now the timeevolution we find, using the Jacobi identity,

d

dtδg = ε {{g, F} , H} = ε {{g, H} , F} + ε {g, {F, H}} .


But when F is a constant of motion the last term vanishes, as being a constant ofmotion implies a vanishing Poisson bracket with the Hamiltonian, and we are leftwith

d

dtδg = δ

d

dtg,

which implies precisely that ‘δ’ commutes with the time evolution, i.e., that it is asymmetry.

1.2.9 The Lagrangian Descriptions of the Harmonic Oscillator

We have seen that different procedures for associating constants of motion withsymmetries may lead to different kinds of associations. Even if we stick to pointsymmetries and to Noether’s theorem, the association still depends in a crucial wayon the Lagrangian, and it may be not be unique if the latter is not unique. To illustratethe ambiguities that may arise in such a case, let us consider again for simplicitythe isotropic harmonic oscillator in m dimensions. Then, it is easy to prove that theequations of motion Eq. (1.61) (with ω1 = · · · = ωm = ω) can be derived from anyLagrangian of the form

L B = 1

2Bi j (v

iv j − ω2qi q j ),

where (Bi j ) is any constant nonsingular real symmetric matrix.Let us consider, for example, the following two Lagrangians, both appropriate for

the description of the two-dimensional isotropic harmonic oscillator, namely

L1 = 1

2[(v1)2 + (v2)2 − ω2((q1)2 + (q2)2)],

andL2 = v1v2 − ω2q1q2 . (1.82)

The Lagrangian function L1 is invariant under rotations and Noether’s theorem asso-ciates the angularmomentum, l = q1v2−q2v1, with this rotational symmetry.On thecontrary, L2 is not rotationally invariant. It is however invariant under the ‘squeeze’transformations

(q1, q2) �→ (λq1, λ−1q2), (v1, v2) �→ (λv1, λ−1v2), λ ∈ R+.

Setting λ = eτ , we find, at the infinitesimal level:

δq1 = ε q1, δq2 = −ε q2 ,


and similarly for δv1 and δv2. The associated Noether constants of motion will be

∂L

∂viδqi = (v1δq2 + v2δq1) = ε (q1v2 − q2v1),

i.e., it is again the angular momentum. However, it is associated via the Lagrangian(1.82) with a symmetry that has nothing to do with rotations.

Similarly, for m = 3 we may consider the standard Lagrangian:

L1 = 1

2(v2 − ω2q2)

for the three-dimensional harmonic oscillator, together with, exempli gratia2

L2 = 1

2

[(v1)2 +

(v2)2 −

(v3)2 − ω2

(q1)2 − ω2

(q2)2 + ω2

(q3)2]

.

While L1 is invariant under the orthogonal group O(3), it is quite clear that L2 willonly be invariant under the pseudorthogonal group O(2, 1). Infinitesimal transfor-mations of O(2, 1) corresponding to three generators of its Lie algebra are of theform:

δq1 = εq3, δq2 = 0, δq3 = εq1

δq1 = 0, δq2 = εq3, δq3 = εq2

δq1 = εq2, δq2 = −εq1, δq3 = 0

together with δv = d(δq)/dt .Notice that the two first infinitesimal symmetries correspond to ‘boosts’ in the

q1 and q2 directions respectively, while the third one is a rotation in the (q1, q2)

plane. It can be easily shown that the associated Noether constants are again thethree components l1, l2 and l3 of the angular momentum: l = q × v. While l3 is stillassociated to a rotation, l1 and l2 are now associated with boosts.

1.2.10 Constructing Nonlinear Systems Out of Linear Ones

Up to now we have been discussing a variety of examples of linear systems. Ourmain motivation has been based, as in the rest of this book, on the idea that most, ifnot all, of the geometric structures associated with dynamical systems can be fullyunderstood already at the (apparently) elementary level of linear systems.

However, a possible objection to this approach could be that there are quite a fewdynamical systems (the Riccati equation, Calogero-Moser systems, Toda molecules,

2 e.g.,‘for example’.


to quote only a few of them) that are definitely nonlinear but are nonetheless (orperhaps just because of that) of a great interest.

All of these systems share with the linear ones the feature of being explicitlyintegrable. We will try to argue in this Section (conducted again on examples, andwithout pretension to full generality) that many nonlinear systems can be ‘related’(in manners that will be specified immediately below) to linear ones, in such a waythat (obvious) integrability of the latter will entail integrability of the former (whichmight look instead not so obvious). Many interesting geometrical structures can beunveiled in the study of these systems. In the spirit of this introductory chapter, wekeep the exposition of the examples at the most possible elementary level, deferringa more detailed study of the same examples to the chapter on Integrability.

We will sketch here three different ways of constructing nonlinear systems whichare associated with a linear one and that in addition are integrable in the same senseas the original dynamical system is, namely:

1. Reparametrization.2. Restriction to invariant surfaces.3. Reduction and quotienting by equivalence relations.

The first idea lies in the fact that changing the parametrization of a linear systemsometimes preserves its explicit integrability. In this way the new system will notbe linear anymore but still it will be integrable. The second idea consists simply inextracting a subsystem out of a linear system. Subsystems do not have to be linear,however they are obviously explicitly integrable if the total system is. Finally, thethird procedure listed above relies in restricting the observable space of a system to asubset, to be more precise to a subalgebra, i.e., one associated with a quotient space,which need not to be linear.

A further way of constructing nonlinear systems out of linear ones by usingnonlinear transformations will be discussed in Sect. 3.5.2, when discussing at lengththe possibility of constructing alternative (i.e. nonlinearly related) linear structures.This procedure will exploit the fact that a system can sometimes be brought to linearform using a nonlinear transformation, or in other words, that some systems are linearsystems in disguise.

We illustrate next the use of the methods listed above with some elementaryexamples and we will leave a more detailed account of this subject to Chap. 7.

1.2.11 The Reparametrized Harmonic Oscillator

Wewill start by considering a very elementary example, namely the one-dimensionalharmonic oscillator with unit mass q = −ω2q. If u denotes the quotient u = q/ω

the equations of motion are:

dq

dt= ω u,

du

dt= −ω q . (1.83)

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Fig. 1.1 Orbits of the 1Dreparametrized harmonicoscillator. The angularvelocity decays with thedistance

In polar coordinates: q = r cos θ , u = r sin θ , we obtain

dr

dt= 0,

dθ

dt= −ω (1.84)

and the integral curves are obviously circles r = constant, whose points are movinguniformly with angular velocity ω, i.e.,

θ(t) = θ0 − ωt.

Therefore the general solution of (1.83) is (r0 cos(θ0 − ωt), r0 sin(θ0 − ωt)).If we look for instance to a spiral galaxy from a zenithal position, or to a viscous

fluid rotating in a circular basin, the motion of its particles (or ‘stars’) is describedto a first approximation by equations of motion similar to the previous ones, but theangular velocity of its integral curves is a function on the radius, where we assumethe angular frequency ω to be a smooth function of r2 at r = 0: ω = ω(r2), and thusthe dynamics will be given by an equation like (Fig. 1.1):

dθ

dτ= −ω

(r2)

. (1.85)

The integral curves are again circles r = constant, but now the particles on eachcircle ‘move’ with an angular velocity ω(r2). Notice that r2, and not r , is a smoothconstant of motion, and any constant of the motion is a function of it. In the originalcoordinates (q, u) we obtain

dq

dτ= ω(q2 + u2) u,

du

dτ= −ω(q2 + u2) q . (1.86)


We say that the new dynamics is obtained from the previous one by time-reparametri-zation, and the ‘time-parameter’ t of the former is related to the ‘time-parameter’ τ

of the latter by:dt

dτ= ω(r2)

ω,

or explicitly in integral form:

τ =t∫

0

ω

ω(r2)dt,

where the integral is computed along the integral curves of the initial dynamics.The new dynamics is not linear anymore, even though we can integrate it explic-

itly. This phenomenon is general, as we will show in what follows. Before doingthat, let us remark that the previous discussion can be carried out in the originallinear coordinates and the ideas above do not depend on finding a ‘clever’ system ofcoordinates. Because ω(q2 + u2) is a constant of motion for the system , when wefix Cauchy data, say q0, u0, our equation can be evaluated on the curve

� ={ω(q2 + u2) = ω(q2

0 + u20)}

. (1.87)

This restriction of a dynamics to a surface can be done every time the latter is aninvariant surface under the associated flow.When restricted to the (one-dimensional)surface �, selected by the choice of the Cauchy data, the dynamics becomes

dx

dt= ω(x20 + y20) y = ω0 y ,

dy

dt= −ω(x20 + y20) x = ω0 x , (1.88)

and we find now a linear system whose solution is

q(t) = a cos(ω0t + α), u(t) = −a sin(ω0t + α) (1.89)

with a2 = q20 + u2

0, tan α = −u0/q0.

1.2.11.1 The Free System in R3 and Invariant Surfaces

The second method consists in restricting a linear system to a nonlinear subspace.This procedure is conceptually very simple. We start with a linear system on a vectorspace E and restrict our Cauchy data to an invariant surface � which is not a linearsubspace on E . Then it is obvious that if x0 and x1 are Cauchy data on �, it will notbe true that x0 + x1 ∈ �. Therefore, the superposition rule does not hold and therestriction of � to � is not linear. In particular, if we adopt coordinates for � thedifferential equation in these coordinates will be a nonlinear differential equation.


A very simple but interesting example of this method is provided by the Newtonequations of the free motion in R

3 of a particle of unit mass:

r = 0 , (1.90)

which can be rewritten in the form of a system

{ ·r = vv = 0

(1.91)

and is therefore associated to the second-order vector field � in T R3,

� = v∂

∂r(1.92)

and has as constants of motion the velocity and the angular momentum:

d

dtv = 0 ,

d

dt(r × v) = 0.

Invariance under rotations suggests us introducing spherical polar coordinates inR3

r = r n n · n = 1, r > 0,

where n = r/‖r‖ = r/r is the unit vector in the direction of r, and taking derivatives

r = r n + r n , r = r n + 2 r n + r n .

Moreover, we have the identities

n · n = 1 , n · n = 0, n2 = −n · n ,

and we see that r·n = ·r , and r · r = r r .

The equations r = 0 are written as

r = −r n · n = r n2 ,

and we have for the angular momentum the expression

r × v = r2n × n.

Then r cannot be expressed in terms of only the variable r and its derivativebecause of the term n2.

However, by making use of constants of motion, we can choose a family ofinvariant surfaces � for � such that restricted to them we can associate with this


equation an equation of motion involving only r, r and some “coupling constants”related to the values of the constants of motion determining the surface.

As a first instance we can consider the constant level sets of the energy function,which is a constant of motion. The leaves �E are invariant under � and we take therestrictions of � on such subsets. Then

2E = r · r = r2 + r2n · n =⇒ n2 = 1

r2(2E − r2).

Therefore the equation of motion turns out to be

r = 2E

r− r2

r.

We obtain in this way an equation of motion involving only r , r and some ‘couplingconstants’ related to the values of the constants of motion.

We can proceed similarly with other constants of motion, for instance with theangular momentum, which is also rotationally invariant. We restrict ourselves toinitial conditions with a fixed value of the angular momentum, and the correspondingsurface �� say, for instance,

�2 = r4 n2,

in order to get

r = �2

r3.

More generally, by selecting an invariant submanifold ofR3 bymeans of a convex

combination of energy and angular momentum, i.e. α(r × r)2 + (1−α) r2 = k, with0 ≤ α ≤ 1, we would find

r = α�2 + (1 − α)(2E − r2)r2

r3(1.93)

which represents a completely integrable system interpolating between the two abovementioned completely integrable systems obtained for α = 0 and α = 1.

We can use other constants of motion to eliminate n2. For instance we can usethe invariant surfaces defined by

S = E �2 = r4

2

(r2n2 + r2n4

), (1.94)

and then we find

r = − r2

2r±√

r2 r2 + 8 S . (1.95)

If we use a time-dependent surface �t , or a time-dependent constant of the motion


r2 + t2v2 − 2r · v t = k2 , (1.96)

we find

n2 = 1

r2[(k2 + 2r · v t − r2)t−2 − r2] , (1.97)

and replacing it in the equations of motion we find the time-dependent equation ofmotion,

r = k2

rt−2 + 2r t−1 − t−2

r− r2

r. (1.98)

Now it should be clear that to solve these equations it is enough to solve them on thevelocity phase space T R

3.

1.2.12 Reduction of Linear Systems

The previous discussions have shown that starting from linear systems we can obtainby elementary means nonlinear systems which are still explicitly integrable. In par-ticular we have discussed reparametrized harmonic oscillator and generalized repara-metrized systems. By restricting to surfaces we have found several nonlinear systemsobtained from free motion in R

3. In fact, a further identification by an equivalencerelation is already implicit on it. Such final process of quotienting by an equivalencerelation has been discussed in the context of the Riccati equation in the previoussection.

Of course reparametrization and reduction via invariant surfaces and invariantequivalence relations can be used together to get nonlinear systems out of linearones that will be called reduced dynamical systems. To be more precise, given adynamical system, that will be supposed to be linear, the restriction to an invariantsurface and the quotienting by a compatible equivalence leads to a ‘reduced’ system.It is obvious from what we said that the reduced dynamical system will be integrableif the initial system is.

Now we will restrict ourselves to show how it works using a couple of examples.In additionwewill show how to obtain a family of the so called completely integrablesystems, Calogero–Moser, Toda, etc., by reduction from simple systems.

1.2.12.1 Reducing the Free System on R3: The Calogero–Moser System

We consider again the free system on R3, namely the equations of motion

r = 0. (1.99)

This equation has a very large symmetry group and a large number of constants ofmotion. In fact, the Lie symmetry group of linear transformations is of dimension 11,


the semidirect productR3�GL(3, R) (see later Sect. 4.6.1 for definitions). Therefore

we have a large choice of invariant surfaces and symmetry groups.We can use for instance an equivalence relation defined by the vector field

Y = x2∂

∂x1−x1

∂

∂x2+x3

∂

∂x2−x2

∂

∂x3+v2

∂

∂v1−v1

∂

∂v2+v3

∂

∂v2−v2

∂

∂v3. (1.100)

By using rectifying coordinates (q1, q2, ϕ) and the corresponding velocities (ormomenta) in such a way that Y = ∂/∂ϕ, say

x1 = q1 cos2 ϕ + q2 sin2 ϕ

x2 = sin 2ϕ√2

(q2 − q1)

x3 = q1 sin2 ϕ + q2 cos2 ϕ

(1.101)

and ‘momenta’

p1 = v1 cos2 ϕ + v3 sin2 ϕ − v2sin 2ϕ√

2p2 = v1 cos2 ϕ + v3 sin2 ϕ + v2

sin 2ϕ√2

pϕ = (q2 − q1)(v1 − v3) sin 2ϕ + √2(q2 − q1) cos 2ϕ

(1.102)

we get1

2p2 = 1

2(p21 + p22) + p2ϕ

4(q2 − q1)2. (1.103)

Therefore, on the invariant surface pϕ = 2k associated with ∂/∂ϕ we find a quo-tient manifold diffeomorphic to T ∗

R2 with dynamics described by the Hamiltonian

function

H = 1

2(p21 + p22) + k2

(q2 − q1)2, (1.104)

i.e., we have obtained the so-called Calogero–Moser potential, and k behaves like acoupling constant.

Now we shall continue the discussion of the example in Eq. (7.47) out of whichwe may find various interesting interactions.

1.2.12.2 Interactions from a Simple Cubic System on R3

Consider again the cubic system defined by Eq. (7.47). We shall consider the caseswhen it is a sphere and the case when it is an hyperboloid. For that we can considernow the invariant surface

� ={(q, p) ∈ T ∗

R3 | 〈q, p〉 = 0, 〈q, q〉 = 1

}, (1.105)

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where 〈·, ·〉 can denote either the pseudometric onR3 of signature−++or theEuclid-

ian metric. The second constraint defines either the two-sphere S2 or an hyperboloid.By virtue of the first one, instead, at each point the momenta p span in both cases theEuclidean plane R

2. All in all, the phase space of the system turns out to be in theEuclidean case T S2 which locally looks like S2×R

2, and a similar four-dimensionalsurface in the pseudo-Euclidean case. On this surface the equations ofmotion acquirethe form

d

dt

(qp

)=

(0 1

−〈p, p〉 0)(

qp

)(1.106)

Let us examine then separately the two cases.

Euclidean Metric: The Sphere

In this case we have〈q, q〉 = q2

0 + q21 + q2

2 = 1 , (1.107)

and then � is given by

� = {(q, p) | 〈q, p〉 = q0 p0 + q1 p1 + q2 p2 = 0 . (1.108)

The equations of motion on � give the geodesical equations of motion on phasespace. We have

·q = p ,

·p = −p2q (1.109)

with the constraints on q and p being understood.Now we introduce polar spherical coordinates on the unit sphere. We have

q0 = cos θ , q1 = sin θ cosϕ , q2 = sin θ sin ϕ . (1.110)

In these coordinates we obtain

pθ = p1 cos θ cosϕ + p2 cos θ sin ϕ − p0 sin θ, pϕ = p2 sin θ cosϕ − p1 sin θ sin ϕ

(1.111)and the Hamiltonian becomes

H = 1

2

(p2θ + p2ϕ

sin2 θ

). (1.112)

Therefore on the surface � ={(q, p) |pϕ = √

2g}, by using the equivalence

relation associated with ∂/∂ϕ, i.e., by the curves ϕ = 1, we find


H = 1

2p2x + g2

sin2 x(1.113)

where we have denoted by x the angle 0 < θ < π . The equations of motion on thesphere are easily integrated to

q(t) = a cos kt + b sin kt (1.114)

with (a, a) = 1, (a, b) = 0, (b, b) = 1. Hence we find:

x(t) = cos−1(cos q0 cos kt) (1.115)

where k = pϕ/ sin q0.

Hyperboloid Metric: The Pseudo–Sphere

Wenotice that� can be identifiedwith the phase space of the pseudo–sphere (q, q) =1.

When we are on a hyperboloid of one sheet, say

(q, q) = q20 − q2

1 − q22 = −1 , (1.116)

we introduce pseudo–spherical polar coordinates

q0 = sinh x , q1 = cosh x cosϕ , q2 = cosh x sin ϕ , (1.117)

and proceed in a similar way to find the following reduced system

H = 1

2p2x − g2

cosh2 x. (1.118)

However now there are three different types of geodetical orbits.Wewill thereforehave three types of orbits for x(t) depending on the initial conditions.

(a) Initial conditions such that 〈a, a〉 = −1, 〈a, b〉 = 0, 〈b, b〉 = 1. Then

q (t) = a cosh kt + b sinh kt (1.119)

andx(t) = sinh−1(α sinh kt) , k = √

2g/√

α2 − 1 . (1.120)

(b) Initial conditions such that 〈a, a〉 = −1, 〈a, b〉 = 0, 〈b, b〉 = −1. We obtain

q(t) = a cos kt + b sin kt (1.121)


x(t) = sinh−1(α cos kt) , k = √2g/

√α2 + 1 . (1.122)

(c) Initial conditions such that 〈a, a〉 = −1, 〈a, b〉 = 0, 〈b, b〉 = 0.

q(t) = a + bt (1.123)

x(t) = sinh−1(αt) , k = √2g/α . (1.124)

1.2.12.3 Exponential Interaction

As indicated before, we can obtain different interactions, i.e., nonlinear mechani-cal systems with a non-trivial potential term, out of the geodesical motion on thehyperboloid, by using a different surface � and a different equivalence relation. Weconsider the above defined pseudoscalar product. We recall that the equations ofmotion are

..q = −2(p × q)2q . (1.125)

Introducing new coordinates

q0 − q1 = ex , q0 + q1 = ey , e2z = q22 , (1.126)

it is not difficult to findx = −g2e−2x (1.127)

where g2 = (x2 + (p × q)2)e2x evaluated on � is a constant of motion and definesan invariant surface �1. The Hamiltonian function has the expression

H = 1

2p2x + g2e−2x (1.128)

which is equivalent to the Toda interaction of two particles.For the equations of motion one has Eq. (1.119). If for simplicity we use the

following initial conditions a = (a0, a1, 0), b = (0, 0, 1), we get

x(t) = log(α cosh kt) ,

where kα = √2g.

These various systems can be extended to higher dimensions. The main idea is touse spaces of matrices and use the group SO(p, q) to introduce pseudo–sphericalpolar coordinates and SU (n) and the Hermitian matrices. We refer to the literaturefor further details [Pe90]. Here we limit ourselves to derive the Calogero–Moserpotential from this different viewpoint.


1.2.12.4 Free Motion on Matrix Spaces: The Calogero–Moser System

We consider a parametrization of R3 in terms of symmetric 2 × 2 matrices,

X = 1√2(x1 I + x2 σ1 + x3 σ3) = 1√

2

(x1 + x3 x2

x2 x1 − x3

). (1.129)

The free motion equations become in this notation

X = 0 (1.130)

and then we see that:

M = [X, X ] = −(x2 x3 − x2 x3) σ ,

where σ denotes

σ = i σ2 =[

0 1−1 0

]

for which σ 2 = −I , is a matrix constant of motion whose nonzero elements areproportional to the third component �3 of the angular momentum. In fact,

M = [X , X ] + [X, X ] = 0 .

We can introduce some new coordinates by using the rotation group: any sym-metric matrix X can be diagonalized by means of an orthogonal transformation G,thus, X can be written as

X = G Q G−1 , (1.131)

with

Q =[

q1 00 q2

], G =

[cosϕ sin ϕ

− sin ϕ cosϕ

]

and therefore, as

G Q G−1 =[

q1 cos2 ϕ + q2 sin2 ϕ (q2 − q1) sin ϕ cosϕ

(q2 − q1) sin ϕ cosϕ q1 sin2 ϕ + q2 cos2 ϕ

]

we get the relation

x1 = 1√2(q1 + q2) , x2 = 1√

2(q2 − q1) sin 2ϕ , x3 = 1√

2(q1 − q2) cos 2ϕ .

We also note that G σ = σ G.


Then, using:d

dtG−1 = −G−1

·GG−1,

we see that

·X = ·

G Q G−1 − G Q G−1·

GG−1 + G Q G−1 = G

([G−1

·G, Q] + Q

)G−1,

with

G−1 G = G G−1 = ϕ

[0 1

−1 0

]= ϕ σ ,

i.e.X = G(Q + ϕ [σ, Q]) G−1.

Notice that [σ, Q] = (q2 − q1) σ and [Q, Q] = 0. Consequently,

M = [X, X ] = G [Q, ϕ [σ, Q] + Q] G−1 = ϕ (q2 − q1)G [Q, σ ] G−1 = −ϕ (q2 − q1)2 σ ,

and then �3 is given by�3 = ϕ (q2 − q1)

2.

The equations of motion in the new coordinates are given by:

d

dtTr Mσ = 0 ,

d

dtQ = ϕ2[σ, [σ, Q]] . (1.132)

By setting:

− 1

2Tr Mσ = ϕ(q2 − q1)

2 = g (1.133)

we find the equations of motion:

q1 = − 2g2

(q2 − q1)3, q2 = 2g2

(q2 − q1)3, (1.134)

which are the (nonlinear) Euler–Lagrange equations associated with the Lagrangianfunction

L = 1

2

(q21 + q2

2

)− g2

(q2 − q1)2. (1.135)

and called the Calogero-Moser equations.

1.3 Linear Systems with Infinite Degrees of Freedom 41

1.3 Linear Systems with Infinite Degrees of Freedom

Although themain bodyof this bookdealswith the simpler case of dynamical systemsmodelled on finite-dimensional linear spaces, there are many instances where wehave to face the description of dynamical systems whose state spaces are infinite-dimensional. That is the case, for instance, of continuum mechanics, field theory,etc., and many other cases where the framework for the description of the system isthat of systems of (linear) partial differential equation.

Examples of this situation are the wave equation, the Schrödinger equation, theKlein–Gordon equation, the Pauli-Dirac equation and the Proca and Maxwell equa-tions. In all these cases the state space of the system consists of a set of functionsφ, called the fields of the theory, defined on an Euclidean space,3 and therefore, it isan infinite-dimensional linear space that in addition supports a representation of thecorresponding symmetry group of the theory.

From an historical viewpoint these equations arose first on phenomenologicalgrounds and later on, their Lagrangian and Hamiltonian descriptions were investi-gated,mainly because of the quantizationproblemand the descriptionof conservationlaws.

These equations can be given the form of an ordinary (linear) differential equationin some infinite-dimensional linear spaceF = {φ} once a time-coordinate t has beenfixed and a splitting of the space–time of the theory has been chosen. Thus the initialdata of the equation is fixed by choosing a Cauchy surface {t = 0}, and the valuesof the fields on it:

d

dtφ = Aφ , φ(t = 0) = φ0 . (1.136)

Very often the linear operator A is a differential operator (see Chap. 10), thesimplest cases beingmultiplication by a function, V (x), the derivative operator d/dx(the gradient in the three-dimensional case), or functions of them.

1.3.1 The Klein-Gordon Equation and the Wave Equation

The Klein-Gordon equation (K G) is usually written in the form

utt (x, t) − �u(x, t) + m2u(x, t) = 0; x ∈ Rn,

with initial conditions (Cauchy data): u(x, 0) = f (x), ut(x, 0) = g(x).This equation corresponds to the relationship between linear momentum and

energy:E2 = p2 + m2,

3 More generally, on manifolds.

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(in natural units � = c = 1), with the formal substitution of E by i ∂/∂t and p by−i ∇, as operators acting on wave functions.

In order to treat this equation in the general setting of first-order systems we mayuse different approaches. The first one is very much similar to our procedure toassociate first-order equations with second-order Newtonian ones. We set: v(x, t) =ut (x, t) and write the K G equation as the system

ut − v = 0

vt − �u + m2u = 0 (1.137)

with the initial conditions: u(x, 0) = f (x), v(x, 0) = g(x).We may introduce now a column vector:

φ(x, t) =(

u(x, t)v(x, t)

),

and rewrite the K G equation as a first-order system:

d

dtφ =

(0 I

� − m2 0

)φ, φ(x, 0) =

(f (x)

g(x)

).

Remark 1.2 The operator: H = −� + m2 is a positive self-adjoint operator onL2(Rn), the space of square integrable functions on R

n , whose domain are the func-tions f ∈ L2(Rn) such that (k2 + m2) f (k) ∈ L2(Rn), where f (·) denotes theFourier transform of f .

Let us denote now by B the positive, self-adjoint square root of H = −� + m2,B = H1/2 ≥ 0, defined as it is usual in functional analysis. Since B is strictlypositive and closed, its domain D(B), is a Hilbert space under the inner product(Bu, Bv)L2 . Then,we may introduce a Hilbert space: HB = D(B) ⊕ L2(Rn) withthe inner product

〈(

uv

),

(u′v′)

〉 = (Bu, Bu′)L2 + (v, v′)L2 .

In analogy with what we did in the case of the harmonic oscillator (Sect. 1.2.7),we set

A = i

(0 I

−B2 0

),

and check that A is a self-adjoint operator on HB with domain: D(A) = D(B2) ⊕D(B). Moreover, A is closed since B and B2 are closed.

We may now ‘integrate’ the equation:


d

dtφ = −iAφ, φ(0) = φ0

by considering, for each t ,

W(t) =(

cos(t B) B−1 sin(t B)

−B sin(t B) cos(t B)

)

where each entry is defined by the usual procedure with respect to B on L2(Rn).One may check directly that {W(t)}t∈R is a strongly continuous one-parameter

group, that is,W(t) ◦ W(s) = W(t + s), ∀t, s ∈ R,

and the map t �→ W(t) f is continuous for any f ∈ L2(Rn). Moreover for anyψ ∈ D(A) the strong derivative of W(t)ψ exists and is equal to −iAψ . Therefore,by Stone’s theorem [Re80] the operator A is the infinitesimal generator of a one-parameter unitary group, and the equations ofmotionwill integrate by exponentiationas

φ(t) = exp (−iAt) φ0 = W(t)φ0 . (1.138)

Alternatively we may write the KG equation as a first-order differential equationby considering the change of variables:

ξ = Buη = v = ut

and we find that the K G equation can be recasted in the form

d

dt

(ξ

η

)=

(0 B

−B 0

)(ξ

η

).

This form is very similar to the way one can write the Maxwell equations (seebelow) and the harmonic oscillator Sect. 1.2.7. The unitary time-evolution operatoris given now by

W(t) = exp(−iAt

),

where

A = i

(0 B

−B 0

),

and (ξ(t)η(t)

)= exp

(−iAt) ( ξ(0)

η(0)

).

Amore elementary approach to integrate the linear system above consists in usingthe elementary system of ‘solutions’ (the plane waves):


uk,ω(x, t) = exp {i (kx − ωt)} ,

where k and ω have to satisfy the dispersion relation:

− ω2 + k2 + m2 = 0 . (1.139)

Then, the uk,ω’s will be locally integrable solutions and the linear superposition prin-ciple can be used to build up wave packets satisfying the assigned initial conditions.

Setting m = 0 we get the scalar wave equation from the K G equation. Forexample, in spherical polar coordinates (r, θ, ψ) the wave equation becomes

∂2u

∂r2+ 2

r

∂u

∂r+ 1

r2 sin θ

∂

∂θ

(sin θ

∂u

∂θ

)+ 1

r2 sin2 θ

∂2u

∂ψ2 = 1

c2∂2u

∂t2.

If we are interested in solutions possessing spherical symmetry we have to solvethe simpler equation

∂2u

∂r2+ 2

r

∂u

∂r= 1

c2∂2u

∂t2,

that can be rewritten as∂2

∂r2(ru) = 1

c2∂2

∂t2(ru),

showing that the solutions acquire the form

u = 1

rf (r − ct) + 1

rg(r + ct) . (1.140)

If we look instead for solutions that, apart from time, depend only on one coor-dinate, x say, then they are of the form u = f (x − ct) + g(x + ct). By using thegenerators of rotations (x∂/∂y − y∂/∂x , etc.), or the Lorentz boosts it is possible togenerate new solutions out of the given ones. This gives an example of the use ofsymmetries to generate new solutions, but more on this later on.

1.3.2 The Maxwell Equations

The well-knownMaxwell equations describing the electric and magnetic fields, sup-posed to be measurable at any point of space-time ( x, t) ∈ R

4, are:

1

c

∂B∂t

= − ∇ × E = −rot E

1

c

∂D∂t

+ 4π

cj = ∇ × H = rot H


∇ · B = divB = 0

∇ · D = divD = ρ

where E and H are the electric and magnetic fields, respectively, D and B the electricand magnetic inductions and ρ and j the macroscopic charge and current densities,respectively. The fields D and H are not independent of E and B, but are related bythe constitutive equations.

In the absence of dielectrics and diamagnetics we may avoid dealing with all fourfieldsD,E,B andH, and consider only the electric fieldE and themagnetic inductionB. The Maxwell equations, in Gaussian units, are then:

∇ · E = divE = 4πρ , Coulomb′s law (1.141)

∇ · B = divB = 0 , Absence of monopoles (1.142)

1

c

∂B∂t

= −∇ × E = −rot E , Faraday′s law (1.143)

1

c

∂E∂t

+ 4π

cj = ∇ × B = rot B , Ampere′s law (1.144)

The first three equations are the expression in differential form of experimentalfacts. The first equation is the Coulomb law for electric charges, while the secondone is the Coulomb law for magnetic poles, supplemented by the fact that no freemagnetic poles do seem to exist in nature. The third equation is Faraday’s inductionlaw while the fourth is the generalization of Ampère’s law

∇ × B = rot B = 4π

cj (1.145)

by the addition of the term 1c ∂E/∂t , which is called the Maxwell displacement term.

Maxwell added this term because otherwise the equations lead to ∇ · j = 0, and thiswould contradict the experience for a non-steady flow of current. When this term isincluded we get instead

∇ · j = −1

c

∂ρ

∂t(1.146)

which by putting

j = 1

cρv

gives raise to

∇ · (ρv) + ∂ρ

∂t= 0. (1.147)

Maxwell’s additional term is also necessary for E and B to satisfy the waveequation in free space. If the term is not included we would get:

∇2E = 0 ∇2B = 0,


which do not admit of plane wave solutions and again would contradict theexperience.

These equations have to be supplemented with the Lorentz force law:

f = ρ

[E + 1

cv × B

]= ρ E + 1

cj × B , (1.148)

when considering also interactions of fields with charged particles.

In this section we shall concentrate on Maxwell equations in free space. Puttingρ = 0 and j = 0 we find

∇ · E = 0,1

c

∂B∂t

= −∇ × E ,

∇ · B = 0,1

c

∂E∂t

= ∇ × B .

These equations can be written in the form

1

c

∂

∂t

(EB

)=

(0 rot

−rot 0

)(EB

)=

(0 1

−1 0

)(rot Erot B

)(1.149)

∇ ·(

EB

)= 0. (1.150)

The first equation appears as a first-order (in time) evolution equation. It can beformally integrated to yield (as in the KG equation):

(E(t)B(t)

)= W(t)

(E(0)B(0)

)(1.151)

with

W(t) =(

cos(ct (rot )) sin(ct (rot ))− sin(ct (rot )) cos(ct (rot ))

)= exp (ict (rot )σ2) (1.152)

where the differential operators (see Appendix G) in the entries are defined through

their series expansions, and σ2 is the Pauli sigma matrix

(0 −ii 0

).

The second equation (1.150) appears instead as a constraint. It is immediate toshow that, in view of the known fact that ∇ · rot ≡ 0, it will be satisfied at all timesif it is at the initial time, i.e., evolution is compatible with constraints.

It will be shown (Sect. 4.2.3) that the appearance of the skew-symmetric matrix�,

� =(

0 rot−rot 0

),

means that the Maxwell equations can be written in Hamiltonian form.

http://dx.doi.org/10.1007/978-94-017-9220-2_4


To close this section, let us recall how the Maxwell equations can be rewritten ina manifestly covariant form. We shall assume as known that if we ‘lump’ togetherthe electric and magnetic fields E and B to form the Faraday tensor:

F = (Fμν) =

⎛⎜⎜⎝

0 Ex Ey Ez

−Ex 0 −Bz By−Ey Bz 0 −Bx

−Ez −By Bx 0

⎞⎟⎟⎠ , (1.153)

the latter transforms as a rank-two (skew-symmetric) covariant tensor.Using the standard Minkowski metric gμν on R

4 whose diagonal entries are:diag(1,−1,−1,−1), the contravariant components of the Faraday tensor will begiven by

∗F = (∗Fμν) = gμηgξν Fηξ .

Explicitly,

(Fμν) =

⎛⎜⎜⎝

0 −Ex −Ey −EzEx 0 −Bz By

Ey Bz 0 −BxEz −By Bx 0

⎞⎟⎟⎠ .

Exercise 1.1 Show that under rotations and/or inversions �00 = 1,�0

μ = 0, μ =1, 2, 3 the F0k ’s and the Fi j ’s i, j, �= 0 transform separately as the components ofa vector and of a pseudo-vector respectively. Show also that under a Lorentz boostwith velocity v (and setting c = 1) the fields transform as:

E′ = (E · n) n + ( n×E) × n + v×B√1 − v2

and,

B′ = (B · n) + ( n×B)× n − v×E√1 − v2

,

where: n = v/|v| .Introducing now the totally antisymmetric tensor εμνρη,4 we can define the

dualtensor

Fμν = 1

2εμνρη Fρη

or, explicitly

4 Defined as: εμνρη = 1 if (μ, ν, ρ, η) is an even permutation of (0, 1, 2, 3), εμνρη = −1 if(μ, ν, ρ, η) is an odd permutation, and 0 otherwise. It is left as an exercise to show that εμνρη isinvariant under the transformations of the Poincaré group and that: εμνρη = −εμνρη.


(Fμν) =

⎛⎜⎜⎝

0 Bx By Bz

−Bx 0 Ez −Ey−By −Ez 0 Ex

−Bz Ey −Ex 0

⎞⎟⎟⎠ ,

the Maxwell equations can be rewritten (again setting c = 1) as

∂μ Fμν = 0 ∂μFμν = 4π jν (1.154)

where: ( jν) = (ρ, j) is the four-current. Antisymmetry of Fμν ensures at once thatjμ is a conserved current, i.e., that it satisfies the continuity equation

∂μ jμ = 0 (1.155)

Remark 1.3 The way it is written, the continuity equation is manifestly Poincaré-invariant if jμ does indeed transform as a four-vector. It is actually the requirementthat the total charge, being a measurable quantity, be an invariant that leads to thisrequirement, and this entails in turn that Fμν should transform as a rank-two tensor.

1.3.2.1 The Electromagnetic Potentials

Expressing the dual tensor in terms of the Faraday tensor the homogeneous Maxwellequations can be rewritten as:

∂μFνη + ∂η Fμν + ∂ν Fημ = 0.

Let us show in an elementaryway that this implies (and is implied by) the existenceof a four-vector Aμ , the four-potential, such that

Fμν = ∂μ Aν − ∂ν Aμ . (1.156)

That if Aμ exists, then the derived Faraday tensor satisfies the homogeneousMaxwell equations is obvious.Viceversa, given Fμν , let us define in the neighborhoodof anypoint, thatwewill take as the origin (x = 0),where the homogeneous equationshold:

Aμ(x) = −1∫

0

dλ λFμν(λx)xν |.

Then, using again the homogeneous equations, one can prove with some algebraiceffort that:

∂μ Aν(x) − ∂ν Aμ(x) =1∫

0

dλd

dλ

(λ2Fμν(λx)

)= Fμν(x) , (1.157)


and this completes the proof.Spelling out explicitly the dependence of the fields from the potentials we find

the more familiar relations

B = ∇ × A, E = −∂A∂t

− ∇φ

where: (φ, A) = (Aμ) and Aμ = gμν Aν .In terms of the four-potential, the inhomogeneous Maxwell equations become

∂μ∂μ Aν − ∂ν(∂μ Aμ) = 4π jν . (1.158)

It is known that gauge transformations of the form:

Aμ �→ A′μ = Aμ + ∂μχ (1.159)

with χ an arbitrary (C2) function will leave the Faraday tensor invariant. This ‘gaugefreedom’ can be used to choose, e.g., the Lorentz gauge, that is defined by thecovariant condition

∂μ Aμ = 0 (1.160)

whereby the equation(s) for the potentials (1.158) acquire the form

∂μ∂μ Aν = 4π jν , (1.161)

i.e., in the Lorentz gauge, the potentials satisfy the inhomogeneous wave equationwith the four-current jμ as a source term.

Remark 1.4 Fixing the Lorentz gauge leaves a residual gauge freedom in which the‘gauge function’χ is obliged to satisfy the homogeneouswave equation: ∂μ∂μχ = 0.

Remark 1.5 The explicitly covariant form of the equation for the potentials can be,to some extent, misleading. Indeed, spelling out explicitly the equations, that for thescalar potential φ reads:

�φ + ∂

∂t(∇ · A) = −4πρ , (1.162)

and choosing then the (non covariant) Coulomb gauge:

∇ · A = 0 (1.163)

(with the residual gauge freedom dictated by: �χ = 0) makes it explicit that thescalar potential obeys the Poisson equation

�φ = −4πρ , (1.164)


which does not contain time derivatives. Therefore, it is not an evolution, but rathera constraint equation (see also the previous discussion).

1.3.3 The Schrödinger Equation

The Schrödinger equation arising in one-dimensional problems inQuantumMechan-ics for a quantum mechanical particle moving under the action of a potential V (x),is

i �dψ

dt= − �

2

2m

d2ψ

dx2+ V (x)ψ . (1.165)

It corresponds to the ‘Newtonian’ relationship between linearmomentumand energy:

E = p2

2m+ V (x) , (1.166)

with the formal substitution of E by i � ∂/∂t and p by−i � ∂/∂x , as operators actingon wave functions ψ . A general discussion on the phenomenological origins of thisequation can be found for instance in [Es04].

In this case the linear space of the theory is the set of square-integrable functionsL2(R) and the linear operator is (in natural units � = 2m = 1):

A = − d2

dx2+ V (x) .

The corresponding time-dependent Schrödinger equation describing the dynam-ical evolution of a quantum system in a three-dimensional space will be:

idψ

dt= −�ψ + V ( x)ψ , (1.167)

where� denotes the Laplace operator onR3. Equation (1.167) is a complex equation

that can be rewritten in terms of the real and imaginary parts of the wave function ψ

[Ma96]:q( x, t) = Reψ , p( x, t) = Imψ , (1.168)

as follows:

d

dt

(qp

)=

(0 −d2/dx2 + V (x)

d2/dx2 − V (x) 0

)(qp

)(1.169)

i.e.,d

dt

(qp

)=

(0 1

−1 0

)(−d2/dx2 + V (x)

)( qp

)= �

(qp

)(1.170)


which will be shown to be of a Hamiltonian form because of the skew symmetry ofthe operator �.

Remark 1.6 The space of square-integrable complex wave functions is endowedwith the standard scalar product that we will denote as 〈·, ·〉. Then, if ψ = q + i pand ψ ′ = q ′ + i p′:

〈ψ,ψ ′〉 = 〈q, q ′〉 + 〈p, p′〉 + i(〈q, p′〉 − 〈p, q ′〉).

On the other hand, the real linear space of the two-dimensional real vectors ζ =(

qp

)

has the natural scalar product:

〈ζ, ζ ′〉 = 〈q, q ′〉 + 〈p, p′〉,

but this product captures only the real part of the complex scalar product. Noticehowever that the norms of the vectors will coincide in the two cases.

Let us consider now the case of a free particle (i.e., V (x) ≡ 0) in an arbitrarynumber of space dimensions. Then the realified Schrödinger equation can be writtenas

∂

∂t

(q(x, t)p(x, t)

)= −i K

(q(x, t)p(x, t)

)

where

K = i��

2m

(0 −11 0

)= ��

2mσ2 .

Then the evolution of initial Cauchy data is provided by:

(q(x, t)p(x, t)

)= W(t)

(q(x, 0)p(x, 0)

)

where

W(t) = exp[−i K t] =(cos(�t�/2m) − sin(�t�/2m)

sin(�t�/2m) cos(�t�/2m)

).

The action of W(t) on the (realified) wave functions is best seen by going toFourier space. If we define the Fourier transforms of q and p as

q(k, t) = 1

(2π)r/2

∫dr xei〈k,x〉q(x, t)

(with r being the space dimension) and similarly for p, thenWwill act on the Fouriertransform as the matrix (a numerical matrix for every k)


W(t) =(

cos(�tk2/2m) sin(�tk2/2m)

− sin(�tk2/2m) cos(�tk2/2m)

).

Each entry of W(t) will act on square-integrable functions f (k) ( f = q orp) and the domain of W(t) will be that of the square-integrable f ’s such that f (k)

cos(�tk2/2m) ( f (k) sin(�tk2/2m)) is also square-integrable.5 The same will be truethen for W(t) and this, together with W W

† = W†W = I, shows in an elementary

way that W is actually a unitary operator.In the more general case of a nonvanishing potential, W(t) will have the same

expression at the price of the replacement

� �→ � − 2m

�2V (x) (1.171)

and, by Stone’s theorem, W(t) will be unitary to the extent that the potential is suchthat � − 2mV (x)/�

2 is self-adjoint.

1.3.3.1 Transformation Properties of the Schrödinger Equation

Consider the Galilei transformation:

x ′ = x − x0 − vt, t ′ = t , (1.172)

which implies that x ′ = x − v.Let us consider the effect of the transformation (1.172) on the differential operators

(see Appendix G) occurring in the Schrödinger equation. The elementary rules ofdifferentiation of composite functions yield

∂

∂t ′= ∂

∂t+ v

∂

∂xand

∂

∂x= ∂

∂x ′

which implies

i�∂

∂t ′+ �

2

2m

∂2

∂x ′2 = i�

(∂

∂t+ v

∂

∂x

)+ �

2

2m

∂2

∂x2.

Ifψ ′(x ′, t ′) = e

i�

( 12mv2t−mv·x) ψ(x, t) (1.173)

5 But, as: | f (k) cos(�tk2/2m)|2 (| f (k) sin(�tk2/2m)|2) ≤ | f (k)|2, the domain will be the wholespace of square-integrable functions.


in the frame (t ′, x ′), one finds that ψ ′(x ′, t ′) solves the equation:(

i�∂

∂t ′+ �

2

2m

∂2

∂x ′2

)ψ ′ = 0 (1.174)

if ψ solves the equation

(i�

∂

∂t+ �

2

2m

∂2

∂x2

)ψ = 0, (1.175)

in the frame (t, x). This happens because the left-hand sides of Eqs. (1.174) and(1.175) differ by the function:

ei f/�

[− ∂ f

∂tψ − v · ∂ f

∂xψ + i�v · ∂ψ

∂x+ �

2

2m

3∑l=1

(2i

�

∂ f

∂xl

∂ψ

∂xl+ i

�

∂2 f

∂x2lψ − 1

�2

(∂ f

∂xl

)2

ψ

)]

which is found to vanish if f = 12mv2t − mv · x .

Remark 1.7 Note that the phase of a plane wave: i�(p ·x− Et) is not invariant under

Galilei transformations, because:

p′ · x′ − E ′t ′ = (p · x − Et) + 1

2mv2t − mv · x. (1.176)

From the relation p·dx−Hdt = Ldt, it follows that the phase changes exactly bythe quantity occurring in the variation of the Lagrangian. The transformation (1.173)is called a gauge transformation on the wave function, and the physical quantitieswhich remain invariant under such transformations are called gauge-invariant. Theanalysis performed here shows therefore that thewave function of QuantumMechan-ics does not change as a (scalar) function under transformations of reference frames.Its deeper meaning can only become clear after a thorough investigation of the geo-metric formulation of modern physical theories, but this goes somehow beyond thescope of this introductory chapter.

1.3.4 Symmetries and Infinite-Dimensional Systems

All field equations that we have considered so far can be written in the general form:

Dφ = 0

where φ is the field (scalar, real or complex, vector and so on) and D a suitabledifferential operator, see Appendix G, (e.g., D = ∂t t −�+m2 for the K G equation,D = i�∂t − (

�2/2m

)� + V for the Schrödinger equation, etc.).


A symmetry for the field equations will be, just as in the finite-dimensional caseof Sect. 1.2.6, any transformation of the space-time coordinates and (if it is the case)of the fields that ‘sends solutions into solutions’, i.e., such that the transformed fieldssatisfy the same wave equation. In general one considers transformations belongingto a group. These will be either discrete transformations (like, e.g., parity and time-reversal) or transformations belonging to the connected component of the identityof the group. There can be also transformations that act on the fields alone, like thegauge transformations that we will consider later on. For example, under translations

xμ �→ x ′μ = xμ + aμ, then φ(x) �→ φ′(x ′) = φ(x + a) (1.177)

and that is true for all kinds of fields, i.e., the fields do not transform (except for theirdependence on the coordinates) under space-time translations. As a second example,under a homogeneous Lorentz transformation

xμ �→ x ′μ = �μν xν (1.178)

we will haveφ(x) �→ φ′(x ′) = φ(�x) (1.179)

for a scalar field while, e.g.

Aμ(x) �→ A′μ(x ′) = �μν Aν(�x) (1.180)

for a vector field, and so on. Actually, these transformation laws define for us whatwe mean for a field being a ‘scalar’, a ‘vector’ and so on.

Let then g be a transformation belonging to a group G. Requiring that g be asymmetry amounts to

Dφ = 0 =⇒ D(g(φ)) = 0, (1.181)

where, in a rather succinct notation, we denote by g(·) the action of g both on thecoordinates and on the form of the field.

Notice that all we are requiring here is that g(φ) be a solution whenever φ is (andvice versa, since g is invertible by assumption). That is a less stringent requirementthan the requirement that g should commute with D, i.e., that: gD = Dg, this lastrequirement defining the invariance group of the differential operator itself, indeeda transformation which changes D into eλ D would still satisfy Eq. (1.181). Thisinvariance group will of course be a group of symmetries, but it may well be thatthere are additional symmetries that do not necessarily leave D invariant. Of courseit is the former that can be often identified by simple inspection, while finding thelatter can be a considerably more complicated task.

For example, the full Poincaré group will be an invariance group for the K Goperator and hence a symmetry group for the K G equation (and, as a special case, forthe scalar wave equation) as well as ofMaxwell’s equations once it is recognized thatthe electric field E and the magnetic induction B can be lumped together to form the


contravariant Faraday rank-two tensor Fμν . Translations in time will be symmetriesfor the Schrödinger equation for time-independent potentials. Space translationswill be also symmetries in the absence of a potential, or some of them will again besymmetries if the potential is invariant under translations in some space directions.Asthe Schrödinger equation is manifestly nonrelativistic, the relevant group of space-time transformationswill be theGalilei group. In the absence of a potential, we expectthe Galilei group to be a symmetry group. Rotations and (spacetime) translations willbe obvious symmetries in this case. We will digress now briefly on the covarianceproperties of the Schrödinger equation under the remaining part of the group, namelyunder the Galilei boosts.

1.3.5 Constants of Motion

A constant of motion will be a function or a functional of the fields that remainsconstant in time when the solutions of the field equation are substituted for the fields,i.e., a function (or a functional) that is ‘constant along the trajectories’ of the fieldequation. Normally, in Classical Field Theory, there appear conserved currents, i.e.four-vectors jμ (but they could be higher-rank tensors as well) that are built up usingthe fields and such that they satisfy a continuity equation of the form

∂μ jμ = 0. (1.182)

Remark 1.8 The formally ‘covariant’ notation that we are using here has to beintended only as a convenient notation. We will employ the same notation in non-relativistic cases as well, like when it will be the case of the Schrödinger equation.So, the previous equation stands simply for (we will set c = 1 everywhere in whatfollows whenever the velocity of light enters the equations):

∂ρ

∂t+ ∇ · j = 0 (1.183)

where ρ = j0 and j = ( j1, j2, j3).

If a continuity equation holds, integrating it on a constant-time surface (or, forthat matter,on any space-like surface) and defining

Q =∫

R3

ρ(x, t) d3x (1.184)

we findd Q

dt=

∫

R3

∂ρ(x, t)

∂td3x = −

∫

R3

∇ · j(x, t) d3x (1.185)


and, if the fields are ‘well behaved’, i.e., they vanish fast enough at space infinity,Gauss’s theorem shows that we can drop the resulting surface term and end up with

d Q

dt= 0. (1.186)

i.e., Q is the required constant of motion.We give below a few examples of conserved currents and of the associated con-

stants of motion, warning that is far from being an exhaustive list.

1. The real K G field.

Consider the four-vector (h, jH ), where

h(x, t) = 1

2

{(∂0φ)2 + (∇φ)2 + m2φ2

}(1.187)

andjH = −(∂0φ)∇φ . (1.188)

It is not hard to see that, by virtue of the equation of motion obeyed by the K Gfield, it satisfies the continuity equation

∂h

∂t+ ∇ · jh = 0 . (1.189)

The quantity

E =∫

R3

h(x, t) d3x (1.190)

is therefore a constant ofmotion and, indeed, aswill be seen in the next Subsection,it can be identified with the total Hamiltonian, and hence with the total energy,for the K G field.

Remark 1.9 The conditions of ‘good behavior at infinity’ that allows us to deducethe conservation law from the continuity equation are just the same that ensure thatthe total energy of the field is finite.

2. The complex K G field.

It is left as an exercise to show that if φ is complex-valued the definitions of hand jH modify into

h(x, t) = 1

2

{|∂0φ|2 + ∇φ · ∇φ + m2|φ|2

}(1.191)


and

jH (x, t) = −1

2

{(∂0φ)∇φ + ∂0φ∇φ

}(1.192)

and the same equation of continuity holds.

Again using the equations of motion it can be proved immediately that the four-vector

jμ = − i

2φ←→∂ μφ (1.193)

whereφ←→∂ μφ = φ∂μφ − φ∂μφ (1.194)

satisfies a continuity equation. The associated constant of motion will be

Q = − i

2

∫

R3

d3x φ←→∂ 0φ (1.195)

It will be useful to express the conservation laws that we have just found for theK G equation in terms of the Fourier transform of the field.

For reasons that will become clear shortly, the Fourier expansion of the field isdone customarily with a slightly different normalization, namely as

φ(x) =√2

(2π)3/2

∫d4k exp(−i〈k, x〉)φ(k) (1.196)

where: 〈k, x〉 = kμxμ. The K G equation becomes then simply

(kμkμ − m2)φ(k) = 0 (1.197)

The solutions will be therefore distributions concentrated on the ‘mass shell’:kμkμ − m2 = 0. Explicitly, with: (kμ) = (k0, k), (kμ) = (k0,−k), the massshell will be defined by the equation: k20 = k2 + m2. It will be then the union ofthe two disjoint hyperboloids M+ and M− defined by

M± = {(k0, k) ∈ R4| k0 = ±E(k), E(k) =

√k2 + m2 . (1.198)

The solution(s) of the K G equation in Fourier space will be of the form:

φ(k) ∝ δ(kμkμ − m2) ≡ δ(k20 − k2 − m2) (1.199)

Using some standard results in the manipulation of δ functions


δ(k20 − k2 − m2) = 1

2E(k){δ(k0 − E(k)) + δ(k0 + E(k))} . (1.200)

As the mass shell is the union of two disjoint components, φ will be fixed by theindependent values φ+(k) and φ−(k) it assumes on M+ and M− respectively,and the most general solution of the K G equation will be of the form

φ(k) = 1

2E(k)

{φ+(k)δ(k0 − E(k)) + φ−(k)δ(k0 + E(k))

}. (1.201)

Eventually, the most general solution of the K G equation in space-time can bewritten as

φ(x) = φ+(x) + φ−(x) (1.202)

with

φ±(x) =∫

d3k

(2π)3/2√2E(k)

exp[i(k · x ∓ E(k)x0)]φ±(k) . (1.203)

It will be convenient also to introduce the amplitudes

a(k) = [E(k)]−1/2φ+(k) (1.204)

andb(k) = [E(k)]−1/2φ−(−k) . (1.205)

Then

φ(x) =∫

d3k

(2π)3/2E(k)

[a(k) exp(−i〈k, x〉 + b(k) exp(i〈k, x〉

](1.206)

where: 〈k, x〉 = E(k)x0 − k · x.

In terms of the new amplitudes that we have just introduced it is not hard to checkthat the two constants of motion that we have just found take the form

E = 1

2

∫d3k E(k)

{[a(k)]2 + [b(k)]2

}(1.207)

and

Q = 1

2

∫d3k

{[a(k)]2 − [b(k)]2

}(1.208)

and are both additive in Fourier space. This leads to interpreting (1/2) |a(k)|2((1/2) |b(k)|2) as a density of positive (negative) charges with energy E(k) asso-ciated with the plane wave of momentum k.


Remark 1.10 Of course the same construction applies to a real K G field, in whichcase reality of the field implies: a(k) = b(k), and hence Q = 0 for any solution ofthe K G equation. With the interpretation outlined above, a real K G field is a neutralfield, while a complex K G field is a charged field.

3. The electromagnetic field. Let us reconsider now the Maxwell equations invacuum, namely the inhomogeneous equations

∇ · E = 4πρ,1

c

∂E∂t

+ 4π

cj = ∇ × B (1.209)

and the homogeneous ones

∇ · B = 0,1

c

∂B∂t

= −∇ × E . (1.210)

As a first example of a locally conserved current associated with the Maxwellequations, it is easy to deduce from the first pair of equations that the charge andcurrent densities obey the continuity equation

∂ρ

∂t+ ∇ · j = 0 (1.211)

that is a local conservation law which implies that, under rather mild assumptionson the distribution of charges and currents (e.g., if they are localized) the totalcharge

Q =∫

d3xρ(x, t) (1.212)

will be a conserved quantityd Q

dt= 0 . (1.213)

In Electrostatics the energy density associated with the electric field is (inGaussian units) E2/8π . Quite similarly, the energy density of the field is B2/8πin Magnetostatics. It makes therefore sense to consider

h(x, t) = E2 + B2

8π(1.214)

as the energy density associated with a general electromagnetic field. Taking thentime derivatives and using again Maxwell’s equations one finds easily that

∂h

∂t= − c

4π[B · ∇ × E − E · ∇ × B] − E · j (1.215)


ButB · ∇ × E − E · ∇ × B ≡ ∇ · (E × B) (1.216)

and therefore, defining the Poynting vector

S = c

4πE × B (1.217)

we obtain∂h

∂t+ ∇ · S = −E · j (1.218)

In the absence of charges and currents this becomes the local conservation law

∂h

∂t+ ∇ · S = 0 (1.219)

and this implies that, under suitable assumptions on the distribution of the fields

H =∫

d3xE(x, t)2 + B(x, t)2

8π(1.220)

will be a conserved quantity, and will represent the total energy stored in the elec-tromagnetic field. Whenever j �= 0, the term on the right-hand side of Eq. (1.218)can be interpreted as the rate at which energy is being exchanged with the systemof sources, and a more general conservation law can be established in that contextas well.

4. The Schrödinger equation.

Now we consider the Schrödinger equation and its complex conjugate equationwhich is automatically satisfied:

− i�∂ψ∗

∂t=

(− �

2

2m� +V

)ψ∗(x, t) (1.221)

andwemultiply the Schrödinger equation byψ∗( x, t), and its complex conjugateby ψ( x, t). On subtracting one equation from the other one then finds

− �2

2m(ψ∗ � ψ − ψ � ψ∗) = i�

(ψ∗ ∂ψ

∂t+ ψ

∂ψ∗

∂t

). (1.222)

Notice that this equation does not depend on the potential V as long as the poten-tial is real. Thus the conclusions of this paragraph are valid for all Schrödingeroperators. Interestingly, on defining


j ≡ �

2mi(ψ∗�(ψ) − ψ�(ψ∗)) (1.223)

and ρ ≡ ψ∗ψ , Eq. (1.222) takes the form of a continuity equation for the current

∂ρ

∂t+ divj = 0 (1.224)

that is a local conservation law. If one integrates the continuity equation on avolume V , the divergence theorem yields

∫V

∂ψ∗∂t

ψd3x +∫V

ψ∂ψ∗∂t

d3x = d

dt

∫V

ψ∗ψd3x = − �

2mi

∫�

(ψ∗ ∂ψ

∂n− ψ

∂ψ∗∂n

)dσ

(1.225)where� = ∂V is the boundary surface of V and ∂

∂n denotes differentiation alongthe direction normal to �. Thus, if ψ vanishes in a sufficiently rapid way as‖x‖ → ∞, and if its first derivatives remain bounded in that limit, the integralon the right-hand side of (1.225) vanishes when the surface � is pushed off toinfinity. The volume V extends then to the whole of R3, and one finds the globalconservation property

d

dt

∫

R3

ψ∗ψ d3x = 0 (1.226)

which shows that the integral of |ψ |2 over the whole space is independent of t .We have thus found a conservation law for the dynamics defined by Eq. (1.221).By virtue of the linearity of the Schrödinger equation, if ψ is square-integrable,one can then rescale the wave function so that the integral is set to 1

∫

R3

ψ∗ψ d3x = 1 (1.227)

The interpretation of |ψ |2 as a probability density will be discussed later on (seeSect. 6.4.1).

References

[Mm85] Marmo, G., Saletan, E.J., Simoni, A., Vitale, B.: Dynamical Systems: A DifferentialGeometric Approach to Symmetry and Reduction. John Wiley, Chichester (1985)

[La16] Lagrange, J.L.: Mécanique Analytique, 1816. A. Blanchard, Paris (1965)[Ja62] Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1962)[Ga59] Gantmacher, F.R.: The Theory of Matrices, vol. II. Chelsea, New York (1959)[Pe90] Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras.

Birkhauser, Boston (1990)

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[Re80] Reed, M., Simon, B.: Functional Analysis, vol. I. Academic Press, New York (1980)[Es04] Esposito, G., Marmo, G., Sudarshan, G.: From Classical to Quantum Mechanics: An

Introduction to the Formalism. Cambridge University Press, Cambridge (2004)[Ma96] Marmo, G., Vilasi, G.: Symplectic structures and quantummechanics.Modern Phys. Lett.

B 10, 545–553 (1996)

Chapter 2The Language of Geometry and DynamicalSystems: The Linearity Paradigm

La filosofia è scritta in questo grandissimo libro checontinuamente ci sta aperto innanzi agli occhi (io dicol’universo), ma non si può intendere, se prima non s’impara aintender la lingua, e conoscer i caratteri ne’quali è scritto. Egliè scritto in lingua matematica, e i caratteri son triangoli, cerchied altre figure geometriche, senza i quali mezzi è impossibile aintenderne umanamente parola; senza questi è un aggirarsivanamente per un oscuro laberinto.Philosophy is written in this grand book, the universe, whichstands continually open to our gaze. But the book cannot beunderstood unless one first learns to comprehend the languageand read the letters in which it is composed. It is written in thelanguage of mathematics, and its characters are triangles,circles, and other geometric figures without which it is humanlyimpossible to understand a single word of it; without these, onewanders about in a dark labyrinth.

Galileo Galilei Il Saggiatore

2.1 Introduction

We can infer from the examples given in Chap.1 that linear dynamical systems areinteresting on their own. Moreover they can be explicitly integrated providing there-fore a laboratory to explore new ideas and methods. We will use them systematicallyto illustrate all new notions and ideas to be introduced in this book.

Webegin by elaboratingmore systematically the elementary, i.e., algebraic theory,for finite-dimensional linear dynamical systems, whose discussion was only initiatedin the previous chapter. Later on, we will see how these algebraic ideas can bebetter expressed using a combination of geometry and analysis, that is, differentialgeometry.

Wewill use our experiencewith linear systems to build the foundations of differen-tial geometry on vector spaces and from there to move tomore general carrier spaces.This simple relation between linear algebra and elementary differential geometry ishighlighted at the end of the chapter under the name of the ‘easy’ tensorializationprinciple, a simple idea that will prove to be very useful throughout the book.


63

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64 2 The Language of Geometry and Dynamical Systems . . .

Thus in building a geometrical language for the theory of dynamical evolution wewill start by discussing the notion of vector fields versus dynamical systems and theaccompanying notions of tangent space, flows, forms, exterior algebra and the moreabstract notion of derivations of the algebra of smooth functions on a linear space.This will be the content of Sects. 2.3 and 2.4.

Finally we will address in Sect. 2.5 the integration problem not just for a singlevector field, but for a family of them, stating a simple form of the Frobenius theoremthat provides necessary and sufficient conditions for the simultaneous integrationof a family of vector fields. Then we will use a variation of this idea to solve theintegration problem for Lie algebras, offering in this way a proof of Lie’s thirdtheorem and introducing in a more systematic way the theory of Lie groups.

2.2 Linear Dynamical Systems: The Algebraic Viewpoint

2.2.1 Linear Systems and Linear Spaces

A mathematical setting that embraces most examples in Chap. 1 is provided by alinear space E (also sometimes called a vector space) and a linear map A : E → Ethat helps us in defining the dynamics we are interested in. Later on we will discussthe extent to which these assumptions are reasonable so that these considerationswill become an argument throughout all the book.

Thus we will consider a real linear space E , i.e. there are two binary operationsdefined on it: addition, denoted by u + v for any u, v ∈ E , and multiplication byreal numbers (scalars), denoted by λu for any λ ∈ R. The linear map A satisfiesA(u + v) = Au + Av and A(λu) = λAu.

The interpretation of these objects regarding a given dynamics is that the vectorsof the linear space E characterize our knowledge of the system at a given time t ,thus we may think of vectors of E as describing partially the ‘state’ of the systemwe are studying (if the knowledge we can obtain from the system under scrutiny ismaximal, i.e., no further information on the system can be obtained besides the oneprovided by the vectors u, then wewould say that description provided by the vectorsu is complete and they actually characterize the ‘states’ of the system). This physicalinterpretation is the reason that leads in many occasions to consider that there is agiven linear structure in E , even though that is at this moment just an assumption weare introducing. In this setting, the trajectories u(t) of our system, the data we canobserve, are functions t �→ u(t), i.e., curves on E . The parameter t used to describethe evolution of the system is associated to the ‘observer’ of the system, that is, tothe people who are actually performing the experiments that allow description of thechange on the vectors u, thus the parameter t has the meaning of a ‘time’ providedby an observer’s clock.

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2.2 Linear Dynamical Systems: The Algebraic Viewpoint 65

We will assume first that the linear space is finite-dimensional with finitedimension n, thus the choice of any linear basis B = {ei | i = 1, . . . , n} allowsus to identify it with R

n by means of u �→ (ui ), u = ui ei , thus all notions ofdifferential calculus of several variables can be transported to E by means of theprevious identification. Hence if the trajectories are differentiable functions, they arecharacterized by the tangent vectors du/dt .

The expression for du/dt has to be determined by the experimental data, i.e., theobservation of the actual trajectories of the system. If exhaustive experiments couldbe performed that will give us the value F(u, t) of du/dt for all values of u at allpossible times t , then we will have an expression of the form

du

dt= F(u, t),

that will be the mathematical law describing the dynamics of our system. Howeverthat is not the case, because performing such exhaustive measurements is impossible.Thus the determination of du/dt must be done combining experimental data and theingenuity of the theoretician (at this point we should recall again Einstein’s quotementioned in the introduction (p. 12) of this book). On how to construct a vectorfield out of experimental data see [MS85].

The simplest situation we may envisage happens when the system has the form:

du

dt= A · u. (2.1)

(Let us recall that in Sect. 1.2.3 it was shown how it is possible by a simple manip-ulation to transform a system possessing an inhomogeneous term into a linear one,thus we will omit in what follows possible inhomogeneous terms in Eq. (2.1)). Inother words, we assume that the tangent vector to the curve u(t) depends just on thevector u(t) at any time and does it linearly. Any description of a dynamics in theform given by Eq. (2.1) will be called a linear dynamical system or a linear systemfor short. Notice that the full description of a linear system involves the prescriptionof a space E with its linear structure +, · and a linear map A on it.

It is interesting to observe that linearity appears at two different levels here. Onone side, we are assuming that the vectors describing states of the system can becomposed and that the composition law we use for them satisfies the axioms of alinear space. On the other hand, we are considering that the infinitesimal variationsto the trajectories of the states of the system, which are vectors by themselves, aretied to the linear structure of the states themselves by means of a linear map. Thuswe are identifying a natural mathematical linear structure, that possessed by tangentvectors to curves, with a linear structure on the space of states which depends on theexperimental setting we are preparing to describe the system under study. The natureof the exact relation between both structures will be the substance ofmuch discussionin what follows. It will be enough to point out here that it cannot be assumed thatthe linear structure on the space of states will be uniquely determined in general

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(what can be thought to be a trivial statement) and that among all possible linearstructures on such space, there could be some of them compatible with the dynamicswe are trying to describe, in other words, a single dynamics can have various differentdescriptions as a linear system.

We must also notice here that most of the finite-dimensional examples discussedin the previous chapter were not linear systems. Actually none of the systems forwhich we have had a direct experience, like the free falling system (Sect. 1.2.4),the motion of celestial bodies, the Kepler system (Sect. 7.4.3), or other systems likethe motion of charged particles in constant magnetic fields like the ones exhibitedin picture 0.1 (Sect. 1.2.5) are linear systems (notice that we describe them usingvery specific parameters). Among these simple systems only the harmonic oscillator(Sect. 1.2.7) is a linear system. However all of them can be related to linear systemsin different ways, either by direct manipulation or by the general theory of reductionas it will be done exhaustively in Chap.7.

Contrary to systems in finite dimensions, all infinite-dimensional systems we dis-cussed before, the Klein-Gordon field (Sect. 1.3.1), Maxwell equations (Sect. 1.3.2),and on top of all them, the Schrödinger equation (Sect. 1.3.3), were linear systems.The paramount role played by linearity in the analysis of infinite-dimensional systemshas overcast its dynamical character and has taken over its physical interpretation.Without trying to elaborate a full physical analysis of them from a different perspec-tive where linearity will not play a primitive role, we will emphasize later on someof the geometrical structures attached to them not depending on linearity.

2.2.2 Integrating Linear Systems: Linear Flows

Linear systems can be easily integrated. A linear system (2.1) defines a linear homo-geneous (and autonomous) differential equation on E . Thus the problem we wantto solve is to find differentiable curves u(t) with t , defined on some interval I in R,t0 ∈ I with values on E , such that

d

dtu(t) = A · u(t), ∀t ∈ I.

Such a curve will be called an integral curve or a (parametrized) solution of the linearsystem (2.1). We may think of this equation as an initial value problem if we selecta vector u0 ∈ E and we look for solutions u(t) that at time t0 satisfies u(t0) = u0.The pair t0,u0 are called the Cauchy data of the initial value problem. The generalexistence and uniqueness theorem on the theory of differential equations guaranteesthat if E is a finite-dimensional linear space there exists a unique smooth curve u(t),t ∈ R solving such a problem. However we will not need to use such a theorem aswe will provide a direct constructive proof of this fact in Sect. 2.2.4. For the momentit will be enough for us to assume that such solutions exist and are unique.

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Without loss of generality, see later, we can set t0 = 0 in the present case. Anatural way of thinking about solving this equation is to find a family of linear maps

φt : E → E, t ∈ R, (2.2)

differentiable1 on the variable t such that for any initial condition u0 ∈ E the curveu(t) = φt (u0) is a solution of Eq. (2.1) with the given Cauchy data, namely,

d

dtφt (u0) = A(φt(u0)), φ0(u0) = u0. (2.3)

The family of maps {φt | t ∈ R} will be called the flow of the linear system (2.1).Characteristic properties of a flow are:

φt ◦ φs = φt+s, φ0 = I. (2.4)

(with I the identity map on E). As an example, it is easy to check that the flow of thedifferential equation associated to the identity map A = I is given by φt (x) = et x .From the additive properties of the family of maps ϕt it is immediate to get that allof them are invertible and

ϕ−1t = ϕ−t .

Bothproperties in (2.4) are immediate consequences of the uniqueness of solutionsfor the differential equation (2.1) (see for instance [Ar73] and [HS74] for a generaldiscussion on the subject).

Exercise 2.1 Prove properties (a), (b) of Eq. (2.4).

Definition 2.1 Given a vector space E , a one-parameter flow (or just a flow) is afamily of linear maps ϕt , t ∈ R depending smoothly on t and such that they satisfyproperties (2.4) above.

In the case of E being finite-dimensional, it is easy to show from (2.4) that thesmooth dependence on t of the family ϕt of linear maps defining a flow is equivalentto the much weaker property of the curve t �→ ϕt (u) from R into E is continuousfor all u ∈ E . In the later case we will say that the family ϕt is a strongly continuousone-parameter family of linear maps. In infinite dimensions the situation is muchmore subtle leading eventually to Stone’s theorem that will be discussed in Chap.6.

It is also evident from the linearity of Eq. (2.1) that the space of its solutions isa linear space. Thus if u(t) = φt · u0 is the solution with initial condition u0 andv(t) = φt · v0 is the solution with initial data v0, then, for any real number λ, λu(t)is the solution with initial data λu0 and u(t) + v(t) is the solution with initial data

1 ‘Differentiable’ here could be understood simply as the statement that the maps t �→ φt ei , with{ei } a basis in E , are differentiable.

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u0 + v0. It follows from Eq. (2.4) that φ−t = φ−1t , and it is also easy to show that

the transformation sending t �→ −t sends solutions of the linear equation given byA into solutions of the linear equation given by −A, because

d

dt(φ−t · u0) = − d

ds(φs · x0)|s=−t = −A · (φsu0)|s=−t = −A · (φ−t · u0).

We have just proved that

Proposition 2.2 Let E be a finite-dimensional space and A a linear map on it.Then the space of solutions S of the linear system du/dt = A · u is a linear spaceisomorphic to E. Such isomorphism is not canonical and a linear basis of the spaceof solutions is called a system of fundamental solutions.

Proof A way to establish an isomorphism between S and E is by mapping u(t) intou(0). Clearly this map is invertible with inverse u0 �→ u(t) = φt (u0). ��

For any initial condition u0 ∈ E , Eq. (2.3) can be written as

(dφt

dt

)(u0) = (A ◦ φt ) (u0) ,

and therefore we can also set2

dφt

dt= A ◦ φt , (2.5)

or, equivalently,(

dφt

dt

)· φ−1

t =(

dφt

dt

)· φ−t = A. (2.6)

If we think of t �→ φt as a curve on the space of linear maps on E , fancily writtenas End(E), then Eq. (2.5) can be thought as an equation on End(E) whose solutionsare the flow of the linear system we are looking for. In order to put such a problemas a genuine initial value problem we should provide it with the initial data φ0,which we can take without losing generality as being the identity matrix I . Noticethat because of the existence and uniqueness theorem for solutions of differentialequations, the solution of this initial value problem exists and must coincide withthe flow of the linear system du/dt = A(u) described in (2.4). This means that theevolution described by Eq. (2.5) takes place actually in the subset GL(E) ⊂ End(E)

of invertible linearmaps on E . It is clear that the setGL(E) has the algebraic structure

2 The derivative of φt could be easily understood as thinking of φt as a curve of matrices, once wehave selected any basis on E , then the space of n × n matrices can be identified with R

n2 .


of a group, that is it carries its binary associative composition law given by thestandard composition of linear maps, the identity linear map is the unit element and,finally, any element has an inverse (see later on Sect. 2.6.1 for a detailed descriptionof the class of groups that will be of interest in this book). We call the initial valueproblem

dφt/dt = A ◦ φt , φ0 = I,

the group theoretical picture of the linear system (2.1). It is remarkable that the spaceof linear maps of E denoted before by End(E) carries a natural Lie algebra structuregiven by the standard commutator [·, ·] of linear maps, that is:

[A, B] = A ◦ B − B ◦ A. (2.7)

(Let us recall that a Lie algebra structure on a linear space L is a skew symmetricbilinear composition law [·, ·] : L × L → L such that it satisfies Jacobi identity, i.e.,[a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0, ∀a, b, c ∈ L , see 10 for more details on thebasic algebraic notions concerning Lie algebras and groups). In what follows we willstart exploring the relation between the Lie algebra structure in the space End(E)

and the group structure in the group of automorphisms of E , GL(E).From the form of our equation on Aut (E) it is clear that φt can be found easily

by exponentiation, namely,3

φt = exp(t A) = et A =∞∑

k=0

tk

k! Ak. (2.8)

Indeed, since

d

dt

( ∞∑n=0

tn

n! An

)=

∞∑n=1

tn−1

(n − 1)! An = A∞∑

m=0

tm

m! Am = A exp(t A) , (2.9)

we see that(

d

dtet A)

· e−t A = A. (2.10)

The operator A is called the infinitesimal generator of the one-parameter groupφt = exp (t A), t ∈ R. We recall some useful properties of the exponential map expdefined above, Eq. (2.8):

3 Notice that the operator-valued power series∑∞

k=0tkk! Ak is convergent because it can be bounded

by the numerical series∑∞

k=0tkk! ||A||k with || · || any norm in the finite-dimensional linear space

of linear maps on E .


1. If B is an isomorphismof E and A is in End (E), we have B−1An B = (B−1AB)n ,for any integer number n, and therefore,

eB−1 AB = B−1eA B. (2.11)

2. If A, B ∈ End (E), and A ◦ B = B ◦ A, then, eA+B = eAeB .3. Every exponential is a square, for eA = (eA/2)2. If E is a real, finite-dimensional

vector space, then det eA > 0. It follows that the map exp : End(E) → GL(E)

is not surjective.4. For a finite-dimensional vector space E ,

det eA = eTr A,

where det stays for the determinant and Tr for the trace.

The flow φt is a symmetry of the system for any value of t according with thenotion of symmetry presented in Sect. 1.2.6. If φt (u0) is a solution of Eq. (2.1) withinitial condition u0, then (φs ◦ φt )(u0) is again a solution starting at φs(u0). On theother hand is obvious that:

[φt , A] = 0 . (2.12)

Having written u(t) = et A ·u(0) as a solution of our equation u = A ·u, we mightbelieve that the problem of analyzing the linear system is solved. However that isnot so. Indeed the series Eq. (2.8) defining et A may be unsuitable for computationsor we could have questions on the dynamics that are not easy to answer from theexpression before. For instance we may be interested in knowing if the solutionet Au(0) is periodic or, if et Au(0) is bounded when t → ±∞, and so on.

There are however some situations where most of the questions we may raiseabout the properties of the solutions have an easy answer:

1. A is diagonalizable. In this generic case E has a basis{e j}of eigenvectors of

A with corresponding eigenvalues λ j , i.e., Ae j = λ j e j . It follows that Ane j =(λ j )

ne j and therefore the set of curves

ek(t) = et A (ek) = etλk (ek) (2.13)

forms a fundamental system of solutions. Hence any solution u(t) will have theform u(t) = ckek(t) (sum over k) and ck determined by the initial conditions.However we must point it out that even if we know in advance that the operator Ais diagonalizable, for instance if it is symmetric, solving the eigenvalue problemis often a difficult problem.

2. The endomorphism A is nilpotent. We recall that A ∈ End (E) is nilpotent withindex n if n is the smallest positive integer for which An = 0. In this case et A

reduces to a polynomial and the general solution will have the form:

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u(t) =(1 + t A + · · · + (t A)n−1

(n − 1)!)

u(0). (2.14)

The general solution exp(t A)u(0) will be a vector with components which arepolynomials in t of degree less than n.As a particular instance, if we consider E to be the set of real polynomials in x ofdegree less than n, then E is a vector space of dimension n on which the operatord/dx is nilpotent of index n. On the basis {1, x, x2, . . . , xn−1}, the linear operatord/dx is represented by the matrix:

(d

dx

)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0

0 0 2. . . 0 0

0 0 0. . . 0 0

......

.... . .

......

0 0 0 n − 2 00 0 0 · · · 0 n − 10 0 0 · · · 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

and

exp(td/dx) = 1 + td

dx+ · · · + tn−1

(n − 1)!d(n−1)

dx (n−1).

If p ∈ E we find that

[exp(td/dx)p

](x) = p(x) + tp′ (x) + · · · + tn−1

(n − 1)! p(n−1) (x) = p(x + t).

Indeed the operator exp(td/dx) is the translation operator by −t . It is clear thatthe series expansion above would still make sense for any real analytic functionon the real line, while the translation operator makes sense for any function, evenif it is not differentiable.

3. The case of a general endomorphism A. A general endomorphism A can bedecomposed in an essentially unique way as A = D + N with D diagonalizableand N nilpotent (see later on the discussion of Jordan canonical forms) such that[D, N ] = 0. Therefore the general situation will reduce to the preceding cases,but again a similar warning as in the diagonalizable case should be raised: findingD and N is often unpracticable.

2.2.2.1 Linear Changes of Coordinates

If we perform a linear change of coordinates x �→ x ′ = Px in the vector spaceE where the linear system (2.1) is defined, clearly we obtain a new equation ofmotion with: A �→ A′ = P AP−1. That is clear because: x ′ = Px = P A · x =


P AP−1Px = P AP−1 · x ′. Of course all the elements of the family of equationsof motion obtained in this way are equivalent under all respects and the corre-sponding flows are then related by φ′

t = Pφt P−1. In fact: d(Pφt P−1

)/dt =

P (dφt/dt) P−1 = P Aφt P−1 = A′ Pφt P−1.

From the active viewpoint, if � : E → E is a linear isomorphism and the curvex(t) is a solution of (2.1), then using the chain rule it is easy to see that the curve(� ◦ x)(t) is a solution of,

d

dt(� · x) = (� ◦ A ◦ �−1)(� ◦ x) (2.15)

A linear isomorphism� will be a linear symmetry for our linear dynamical systemif it maps solutions of the linear system into solutions of the same system. That meansthat if x(t) = exp(t A) ·x(0) is a solution, then also (� ◦x)(t) = (� ◦exp(t A)) ·x(0)is a solution of x = A(x). This clearly implies that � ◦ A ◦�−1 = A. Conversely, if� ◦ A ◦ �−1 = A, then the linear map � sends solutions into solutions. The set ofsymmetries obtained in this way is thus a group characterized as the subgroup G A

of GL(E) of those isomorphisms � such that [A, �] = 0.

2.2.2.2 Symmetries

Given the evolution equation: dx/dt = A · x , it is clear that we can construct newevolution equations as

dx

ds(k)

= Ak · x, (2.16)

for any positive integer k (we emphasize here that the evolution parameters are alltaken to be independent. Of course: s(1) ≡ t). It follows then quite easily that theflows �k

(s(k)

) = exp{s(k) Ak

}are symmetries for the original equation and that,

moreover, all these one-parameter groups pairwise commute.Whenever the characteristic polynomial of A coincides with the minimal polyno-

mial the powers of A yield all the infinitesimal symmetries of our equation, i.e., theygenerate the algebra of symmetries of A, which turns out then to be Abelian. That isthe case when the eigenvalues of A are non-degenerate. If instead A has degenerateeigenvalues, then additional symmetries will be provided by all the elements of thegeneral linear group acting on the corresponding eigenspace on which A acts as amultiple of the identity, and the symmetry group will be in general larger and nolonger Abelian (see, however, below Sect. 3.6 for more details).

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2.2.3 Linear Systems and Complex Vector Spaces

2.2.3.1 The Canonical Form of the Flow of a Linear Complex System

When the linear map A defining a linear system on the real vector space V is notdiagonalizable (i.e., the representative matrix associated to it is not diagonalizableover the reals) it is convenient to use complex numbers so that the fundamentaltheorem of algebra is available. Let then E be a complex n-dimensional vector space(wewill discuss its structure immediately after these remarks) and A a complex linearmap on E . Then the characteristic polynomial p(λ) = det(A − λI ) of A ∈ End (E)

factorizes as

p(λ) = (λ − λ1)r1 · · · (λ − λm)rm , (2.17)

where the complex numbers λ1, . . . ,λm, m ≤ n are the different eigenvalues of A, riis the algebraic multiplicity of the eigenvalue λi , i = 1, ..., m, and r1+· · ·+rm = n.We consider the operators (A − λk I )rk along with their kernels Ek . Each Ek hasdimension rk and E = ⊕

k Ek . This splitting leads to a decomposition of the identityby means of a family of projections Pi on each subspace Ei in such a way thatEk = Pk E ,

∑mi=1 Pi = idE and Pi Pj = δi j Pi . Then:

1. A leaves each Ek invariant, that is A(Ek) ⊂ Ek , therefore it makes sense to defineAk = APk . Note that A-invariance of Ek implies that Pk APk = APk . As thesupplementary subspace E ′

k = ⊕l �=k El is also A-invariant, Pk A = APk .

2. A = ∑k Ak, with [Ak, A j ] = 0, ∀k, j . that is a consequence of Ak A j =

APk APj = A2Pk Pj = δk j A2Pk . Also (Ak − λk I )rk = 0.

We now find the flow of A decomposes as:

et A = et A1 · · · et Am . (2.18)

If a vector x ∈ E is decomposed as x = ∑k Pkx = ∑

k xk, we get exp(t A)x =∑k exp(t Ak)xk . If we write et Ak as et Ak = et (Ak−λk I )+tλk I , and since the identity

I commutes with everything, we have also

et Ak = et (Ak−λk I )etλk I ;

but now, (Ak − λk I ) is nilpotent of index not higher than rk and we get

et Ax =∑k

etλkPk(t)xk, (2.19)

where Pk(t) is a polynomial in t with coefficients in End (E) and of degree lessthan rk.


We have shown that the general solution of x = A · x is the sum of vectors ofthe form etλk xk(t), with xk(t) a vector whose components are polynomials in t ofdegree less than the multiplicity rk of the eigenvalues λk .

Remark 2.1 This splitting of A into the sum of commuting operators is the prototypeof separability of differential equations.Wewill take back this themewhenwediscussthe notion of integrability of dynamical systems and we will relate this notion to theexistence of normal forms (see Chap.8).

Remark 2.2 We will come back to this discussion again when analyzing systemswith compatible generic Hermitean structures (see Sect. 6.2.5).4

2.2.3.2 Complexification of Linear Dynamical Systems

It is time now to discuss with some care a structure that is going to play a relevantrole in what follows: complex linear structures.

We recall that a complex linear space E is a vector space over the field of com-plex numbers C. Any complex vector space has a basis. We will assume in whatfollows that E is finite-dimensional with complex dimension dimC E = n. LetB = {u1, . . . ,un } be a basis for E . Thus, any vector v ∈ E can be written asv = ∑

k zkuk , with zk ∈ C. The map ϕB : E → Cn given by ϕB(v) = (z1, . . . , zn)

is an isomorphism of complex vector spaces where Cn is a complex vector space

with the natural action of C.We can also think of E as a real vector space by considering the action of R on

E given by λ · v = (λ + i 0)v, λ ∈ R. We shall denote this real vector space asER and we will call it the realification of E . It is clear that the vectors u and i u arelinearly independent on ER. Hence the set BR = { u1, . . . ,un, i u1, . . . , i un } is alinear basis for ER. We will call this a real basis adapted to the complex structure onE . It is clear that dim ER = 2 dimC E .

The realification ER of a complex vector space E carries a natural endomorphismJ verifying J 2 = −I defined by J (u) = i u. Conversely if the real vector space Vis equipped with a real linear map J such that J 2 = −I it becomes a complex linearspace with the action of C on V defined by

z · v = xv + yJ (v)

for all z = x + iy ∈ C and v ∈ V . We shall denote by V (J ) the complex vectorspace defined in this way. The realification of the complex space V (J ) is the originalreal vector space V and the endomorphism induced on it coincides with J . We see

4 In Quantum Mechanics a similar decomposition of the total space can be achieved by using acompact group of symmetries for A. The irreducible invariant subspaces of our group will be finite-dimensional and the restriction of the Hamiltonian operator A to each invariant subspace gives raiseto a finite-dimensional problem. Motions in central potentials are often studied in this way by usingthe rotation group and spherical harmonics. The radial part is then studied as a one-dimensionalproblem.

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in this way that a complex structure on a real vector space is defined by a linear mapwhose minimal polynomial is the simplest possible p(λ) = λ2 + 1.

Definition 2.3 A linear complex structure on the real vector space E is a real linearmap J : E → E such that J 2 = −I .

Exercise 2.2 Prove that E must be even-dimensional as a real space.

Remark 2.3 The canonical model for a complex structure J is provided by the realspace R2n and the endomorphism J0 given by

J0 =(

0 −In

In 0

). (2.20)

Thus, if we denote by (x1, . . . , xn, y1, . . . , yn) a generic point inR2n , multiplicationby J0 gives, (−y1, . . . ,−yn, x1, . . . , xn), hence that is equivalent to multiplicationby i if we identifyR2n withCn by (x1, . . . , xn, y1, . . . , yn) �→ (x1 + iy1, . . . , xn +iyn).

We consider nowV to be ann-dimensional real vector space. It is possible howeverto exploit the previous discussions leading to the structure of the flow of a complexlinear system by constructing a complex vector space out of V . Such construction iscalled the complexification of V and it proceeds as follows:

Consider the set E = V × V . We can endow E with the structure of a complexspace by defining [Ar73]:

1. (v1, w1) + (v2, w2) = (v1 + v2, w1 + w2),2. (a + ib)(v,w) = (av − bw, bv + aw).

Exercise 2.3 Check that V × V with the binary composition law + defined by (1)above and the action of C defined by (2) satisfies the axioms of a complex linearspace.

The set V × V when endowed with the new structure of complex space will bedenoted VC and it is said to be the complexification of V . The space V is embeddedas a real linear subspace in VC by means of the map j : V → V × V , j (v) = (v, 0).Moreover, every element (v,w) ∈ VC can be written as (v,w) = (v, 0)+ i(w, 0) =j (v) + i j (w), because i(w, 0) = (0, w).

A vector of VC is said to be real if it is of the form j (v) = (v, 0), and then it willbe denoted as v instead of j (v). An arbitrary element (v,w) ∈ VC can be written asa sum v + iw, with v and w being real vectors.

Notice that if B = {ei | i ∈ I } is a basis of the real space V , then, B = {(ei , 0) |i ∈ I } will be a basis of the complex space VC. Therefore,

dimR V = dimC VC (2.21)


Another important remark is that if the real linear space V was endowed with a realinner product (·, ·), that is a positive definite symmetric bilinear form on E , then VC

becomes a pre-Hilbert space (actually a Hilbert space in the finite-dimensional case)by means of the Hermitean product:

〈v1 + iv2, w1 + iw2〉 = (v1, w1) + (v2, w2) + i(v1, w2) − i(v2, w1) (2.22)

which satisfies 〈v,w〉 = (v,w). We will discuss Hermitean products in depth inSect. 6.2.

We introduce now the important notion of complex linear map.

Definition 2.4 A complex linear map ϕ between two complex linear spaces E1 andE2, is a map ϕ : E1 → E2 such that

ϕ(z1 · u1 + z2 · u2) = z1 · ϕ(u1) + z2 · ϕ(u2), ∀z1, z2 ∈ C, u1,u2 ∈ E1.

Equivalently, we have:

Proposition 2.5 A complex linear map ϕ between two complex linear spaces(E1, J1) and (E2, J2), is a real linear map ϕ : E1 → E2 such that

ϕ ◦ J1 = J2 ◦ ϕ.

The first example of complex linear maps is provided by the complexification ofa real linear map. Given a linear map A : V → V of the real linear space V , It ispossible to complexify A as

AC(v + iw) = (Av) + i(Aw) (2.23)

and this correspondence satisfies:

1. (λA)C = λ AC.2. (A + B)C = AC + BC.3. (AB)C = ACBC.4. (AT )C = (AC)†.

Moreover, if AC has a real eigenvalue λ, then λ is also an eigenvalue of A. Morespecifically, if v+iw is an eigenvector of AC corresponding to the eigenvalue λ+iμ,then,

Av + i Aw = AC(v + iw) = (λ + iμ)(v + iw) = (λv − μw) + i(λw + μv) ,

and as a consequence of the uniqueness of the splitting

Av = λv − μw, Aw = λw + μv

In particular, when μ = 0, we get:

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Av = λv, Aw = λw (2.24)

But v + iw was an eigenvector of AC and therefore v and w cannot vanish simulta-neously.

Notice that if A is a real linear operator, the complexification AC will define acomplex liner dynamical system:

dz

dt= AC · z; z ∈ VC (2.25)

We notice that a solution of our complexified equation stays real if the initialcondition is real. In fact the operator that associates to any vector z = x + iy inVC its complex conjugate z = x − iy, commutes with AC. That means that if z(t)is a solution of Eq. (2.25), then z(t) will be a solution too. Then by uniqueness ofthe existence of a solution for given Cauchy data, if the Cauchy data are real thesolution must be real for all values of t . This solution is also a solution of the realdifferential equation. It is also clear that a curve t �→ z(t) = x(t)+ iy(t) is a solutionof the complexified equation iff the real and imaginary parts are solutions of the realequation. All this follows from the fact that AC commutes with the multiplicationby i .

Now we can look for solutions of x = A · x with initial conditions x(0) bysolving z = AC · z with initial condition z(0) = x(0). Now a diagonalization ispossible, all our previous considerations apply and we get the desired solution. Itremains to express it into real form for it is the sum of vectors like etλk xk(t) wherexk(t) is a polynomial of degree at most rk . The eigenvalue λk is not necessarily real.What we need is to use combinations of the real and imaginary part of the solution.If λ = a + ib, a, b ∈ R we shall consider terms of the form eat cos(bt)x(t) andeat sin(bt)x(t) where x(t) is a polynomial in t with real coefficients.

The following proposition shows that all complex structures in a given linearspace are isomorphic, and isomorphic to the canonical model (R2n, J0).

Proposition 2.6 Let (E, J ) be a complex linear space. Then, there exists a linearisomorphism ϕ : E → R

2n such that J0 ◦ ϕ = ϕ ◦ J , i.e., (E, J ) is complexisomorphic to (R2n, J0).

Proof We give two proofs. Because E is a complex linear space of dimensionn, let u1, . . . ,un denote a basis for it. The real linear space ER is the realifi-cation of the complex linear space (E, J ). A linear basis of it is provided byu1, . . . ,un, Ju1, . . . , Jun . Then, we identify E with R

2n by means of the linearisomorphism ψ : R2n → E given by:

ψ(x1, . . . , xn, y1, . . . , yn) = xkuk + yk J (uk).

Then notice that

ψ(J0(x1, . . . , xn, y1, . . . , yn)) = ψ(−y1, . . . ,−yn, x1, . . . , xn) = −ykuk + xk J (uk)


and then,

ψ(J0(x1, . . . , xn, y1, . . . , yn)) = J (yk J (uk) + xkuk) = J (ψ(x1, . . . , xn , y1, . . . , yn)),

and the result is established.An alternative proof that will be useful for us later on, works as follows.Consider the complexification EC of the real space E and complexity J to EC.

Thus if we denote by uC = u1 + i u2 a vector on EC, the complexified map JC

acts as JC(uC) = J (u1) + i J (u2). Moreover (JC)2 = −I , hence EC carries twoalternative complex structures. We can diagonalize JC with respect to the complexstructure on EC defined by multiplication by i , then denote by K± = ker(JC ∓ i),the eigenspaces of ±i respectively. Denote by R± = Im(JC ± i). Then, because(JC + i)(JC − i) = 0 it is clear that R± = K±. But EC/K± ∼= R∓. Hence,

dim R+ = dim R− = dim K+ = dim K−,

and dimC EC is even (hence the real dimension of E is even). Let us call this dimen-sion 2n. The dimension of K+ is n and let wC

1 , . . . , wCn a complex basis of it. The

vectors wC

k have the form wC

k = uk + ivk , k = 1, . . . , n, and they satisfy

J (uk) = vk, J (vk) = −uk.

Thus we have found a basis such that the natural identification withR2n provided byit gives the desired map. ��

The set of complex linear isomorphisms ϕ : E → E defines a subgroup of thereal general linear group GL(E). We shall denote such group by GL(E, J ).

As it was shown in Proposition2.6 we can identify (E, J ) with (R2n, J0), hencethe group GL(E, J ) is isomorphic with a subgroup of the group GL(2n,R). Suchsubgroup will be denoted by GL(n,C) and is characterized as the set of matricesA ∈ GL(2n,R) such that

AJ0 = J0A

or, equivalently that A has the block form

A =(

X −YY X

),

with X, Y n × n real matrices. Notice that if we identify E with Cn , then GL(E, J )

becomes simply the group of invertible complex n × n matrices, i.e., the groupGL(n,C). The identification between these two representations of the same group(the fundamental one and the 2n-dimensional) is given by the map


Z = X + i Y �→(

X −YY X

).

The (real) dimension of the group GL(n,C) is 2n2.Notice that if we consider the realification of VC, then VC

Ris isomorphic to V ⊕V .

A basis for VC

Ris obtained from a basis B = {ei }i∈I of V by B = {(ei , 0), (0, ei )}i∈I .

In such a basis, the R-linear map J of VC

Rcorresponding to multiplication for the

imaginary unit i is represented by the standard model matrix J0 Eq. (2.20). Noticethat

V ∩ J V = {0}, VC

R= V ⊕ J V . (2.26)

2.2.4 Integrating Time-Dependent Linear Systems:Dyson’s Formula

We can consider again the problem of integrating a linear dynamical system. In theprevious section we found the flow of the system by integrating the equation onthe group and checking the convergence of the series found in that way. A similaridea can be used to integrate and find explicitly the flow of a time-dependent (i.e.,non-autonomous) linear system on E like

dx

dt= A (t) · x . (2.27)

To discuss the solution of this equation we reconsider first the time-independentcase and put it into a slightly different perspective.

We assume that the Eq. (2.27) has a solution,

x(t) = φ(t, t0)x(t0) (2.28)

i.e., φ(t, t0) is the evolution matrix of the Cauchy datum x0 = x(t0). Of course, dueto the previous results we know that if A is constant φ(t, t0) = eA(t−t0), however forthe time being we are arguing independently of it. Taking the time derivative of x(t)we get,

x(t) = φ(t, t0)x(t0) = φ(t, t0)φ−1(t, t0)x(t) (2.29)

therefore we have A = φ(t, t0)φ−1(t, t0). Because by assumption A is independentof time, the right-hand side can be computed for any time t . Thus we find that ourinitial equation on E can be replaced by the equation on GL(n,R),

d

dtφ = Aφ (2.30)


We use now our knowledge that φ(t, t0) = eA(t−t0) is a solution to consider theseries expansion,

φ(t, t0) = I + A (t − t0) + · · · + An (t − t0)n

n! + · · · (2.31)

We denote the term An (t−t0)n

n! in the previous expansion by Rn(t, t0) and wenotice that,

Rn+1(t, t0) =t∫

t0

ARn(s, t0) ds (2.32)

Therefore, we can consider the matrix,

Sn(t, t0) = I +n∑

k=0

Rk(t, t0) (2.33)

as providing us with an approximate solution of our initial equation. It is clear thatbecause of the independence of A on t we have,

Rn+1(t, t0) = An+1

t∫t0

(s − t0)n

n! ds (2.34)

Therefore, the sequence: R0(t, t0), . . . , Rn(t, t0) converges uniformly to the limit,

S(t, t0) = I +t∫

t0

A S(s, t0) ds (2.35)

Now, S(t, t0) is differentiable and satisfies:

d

dtS = A S (2.36)

and we have found our solution in terms of an integral.This new way of finding S = eA(t−t0) holds true in the time dependent case as

long as A : I ⊂ R → gl(n,R) is a continuous map with M an upper bound for||A(t)|| in I . Then we define again,

Rn+1(t, t0) =t∫

t0

A(s)Rn(s, t0) ds, R0(t, t0) = I (2.37)


and notice that ||Rn(t, t0)|| ≤ |t − t0|n Mn

n! . Therefore the sequence R0(t, t0) + · · · +Rn(t, t0) converges uniformly on I to the limit,

S(t, t0) = I +t∫

t0

A(s)S(s, t0) ds (2.38)

with S satisfying the differential equation,

d

dtS(t, t0) = A(t)S(t, t0) (2.39)

with initial condition: S(t0, t0) = I .The matrix we have found is called the resolvent or the resolvent kernel of the

equation x(t) = A(t)x(t).In general one can write for Rn(t, t0) the formula,

Rn(t, t0) =∫

t0≤s1≤...≤sn≤t

A(sn) · · · A(s1) ds1 · · · dsn (2.40)

where due care is required for the order of factors because in general [A(si ),

A(s j ] �= 0.Defining a time-ordering operation T as,

T{

A (s) A(s′)} =

{A (s) A

(s′) , s ≥ s ′

A(s′) A (s) , s < s ′ (2.41)

and similarly for products with more than two factors (i.e., T {·} will order factors indecreasing (or non-increasing) order of the time arguments from left to right), it isnot hard to convince oneself that,

Rn (t, t0) = 1

n!t∫

t0

ds1 · · · dsn T {A (s1) · · · A (sn)} (2.42)

and hence that S (t, t0) can be expressed a ‘time-ordered exponential’,5 quoted asDyson’s formula:

S (t, t0) = T

⎧⎨⎩exp

t∫t0

ds A (s)

⎫⎬⎭ (2.43)

5 Also-called a ‘product integral’ (see, e.g.: [DF79]).


The matrix S (t, t0) can be given a simple interpretation. Starting (in thetime-independent case, for the time being) from the linear equation, dx/dt = A · x ,we can consider a fundamental system of solutions, say {xα (t)} ,α = 1, . . . , n, suchthat any other solution can be written as: y (t) = cαxα (t) , y (0) = cαxα (t = 0).We can then construct an n × n matrix X (t) whose α-th column (α = 1, . . . , n) isgiven by xα (t), i.e., X jα (t) = x j

α (t). Then, for this ‘matrix of solutions’ we have

X (t) = et A X (0)

i.e.,

et A = X (t) ◦ X−1 (0) .

Therefore, from a fundamental set of solutions we can construct

S (t, t0) = X (t) ◦ X−1 (t0) . (2.44)

This relation holds true also for time-dependent equations and any solution can bewritten as

x (t) = S (t, t0) x (t0) . (2.45)

2.2.5 From a Vector Space to Its Dual: Induced EvolutionEquations

From the equations of motion on the vector space E it is possible to induce equationsof motion on any other vector space that we may build canonically out of E . Weconsider first the induced motion on the dual space E∗, the vector space of linearfunctions on E .

Starting with the linear system dx/dt = A · x , we consider its linear flow ϕt , thenwe define the linear flow ϕt on E∗ defined as:

(ϕtα)(x) = α(ϕ−t (x)), ∀α ∈ E∗, x ∈ E, (2.46)

It is a simple matter to check that:

ϕt+s = ϕt ◦ ϕs, ϕ0 = I,

and that the dependence on t of ϕt is the same as that of ϕt , i.e., the family of linearmaps ϕt defines a flow on E∗. Moreover, a simple computation shows:


(d

dtϕtα

)(x) = d

dtα(ϕ−1

t x) = α

(d

dtϕ−1

t x

)= −α(ϕ−1

t A · x)

i.e.,

d

dt(ϕtα) = −A∗α

where A∗ : E∗ → E∗ is the linear map defined as A∗α = α ◦ A and usually calledthe dual map to A. A simple computation shows us that the matrix representing thelinear map A∗ in the dual basis of a given basis in E is the transpose of the matrixrepresenting the linear operator A. Thus we conclude stating that the dynamics onthe dual vector space is induced by −A∗, the opposite of the dual map to A, or inlinear coordinates the opposite of the transpose matrix which defines the dynamicson E . If we consider now the curves x(t) = ϕt (x0) and α(t) = ϕtα0 on E and E∗respectively, it is trivial to check that:

d

dt[α (x)] = 0 (2.47)

that is the quantity α (x) = α0(x0) is constant under evolution.The particular requirement we have considered here to induce a dynamics on E∗

is instrumental to inducing an isospectral dynamics on linear maps or, equivalently, adynamics that is compatible with the product of linear maps. The same can be statedfor more complicated tensorial objects.

Once we have defined the induced flow on E∗ it is easy to extend it to any othertensor space. For instance the induced flow on linear maps B : E → E , that is ontensors of order (1, 1), we require the following diagram,

to be commutative, i.e., B (t) = ϕt ◦ B ◦ ϕ−1t .

From this we get,

d

dtB (t) = [A, B (t)] , (2.48)

and the evolution is isospectral. This last equation is the analog of Heisenberg equa-tions in a classical context.

Returning to dual dynamics on the dual vector space E∗, let us consider a lineartransformation of the most elementary type, namely: B = x ⊗ α, with x ∈ E andα ∈ E∗. By using the derivation property on the tensor product we find: d B/dt =x ⊗ α − x ⊗ α = A (x ⊗ α) − (x ⊗ α) A, i.e. once again Eq. (2.48).


Remark 2.4

1. If we require the evolution on linear maps �t to satisfy

�t (M ◦ N ) = �t (M) ◦ �t (N ) , (2.49)

we find that there exists a one-parameter group {φt | t ∈ R} of automorphismsof the vector space such that

�t (M) = φt ◦ M ◦ φ−1t . (2.50)

In one direction the statement is easily verified. In the opposite direction it relieson preservation of the ‘row-by-column’ productα (x) = �t [α (x)] = (

�∗t (α)

)◦(�t (x)) = α ◦ φ−1

t ◦ φt (x). Then, by writing only elementary blocks in M andN , say y ⊗ β, x ⊗ α, we find

( y ⊗ β) · (x ⊗ α) = β (x) y ⊗ α (2.51)

and,

�t (β (x) y ⊗ α) = �t (y) · �t (β (x)) ⊗ �t (α)

= �t (y) [(�t (β)) (�t (x))] ⊗ �t (α)

= (�t (y) ⊗ �t (β)) · (�t (x) ⊗ �t (α))

= [�t (y ⊗ β)] · [�t (x ⊗ α)]

and thus, Eq. (2.49) is satisfied.2. In describing the evolution of the so-called open quantum systems one has to give

up the requirement of the preservation of the product structure on linear maps (inwhich they represent density states) and we get a more general dynamics whichcannot be described in terms of a commutator bracket.

2.3 From Linear Dynamical Systems to Vector Fields

2.3.1 Flows in the Algebra of Smooth Functions

In the previous sections we have been discussing some aspects concerning the struc-ture and properties of linear systems using elementary notions from linear algebra.

Now we are going to present the basic tools from calculus and geometry neededto get a deeper understanding of them. For this purpose we are going to departfrom the presentation of these subjects found in most elementary textbooks. We willemphasize the fact that the linear structure of the carrier space is not relevant forconstruction of a differential calculus in the sense that any other linear structure willdefine the same differential calculus.

2.3 From Linear Dynamical Systems to Vector Fields 85

This will be made explicit by using systematically a description in the space of(smooth) functions, where the particular choice of a linear structure will not play anyrole. This will have also far reaching consequences when we discuss linearization ofvector fields and vice versa.

Together with the development of differential calculus we will find immediatelythe basic notions of differential geometry, vector fields, differential forms, etc. Thenwe will construct the exterior differential calculus and we will discuss its appealingalgebraic structure. Finally we will use all these ideas to present the elementarygeometry of dynamical systems, thus setting the basis for later developments.

We assume a basic acquaintance of the reader with the elementary notions ofdifferential calculus and linear algebra in R

n or in finite-dimensional linear spacesE , such as they are discussed in standard textbooks such as [HS74], etc. We wouldlike to discuss them again briefly here, to orient the reader in building what are goingto be the notions on which this book is founded. A systematic use of concepts likethose of rings, modules, algebras and the such will done throughout the text, thus thebasic definitions are collected for the benefit of the reader in Appendix A. Relatedconcepts, such as that of graded algebras, graded Lie algebras and graded derivationsare also discussed in the same Appendix.

We denote by E a real linear space, finite-dimensional for most of the presentdiscussion. In a finite-dimensional linear space all norms are equivalent, thus wewill not specify which one we are using in the underlying topological notions wewill introduce. Whenever we deal instead with infinite-dimensional spaces we willassume them to be Banach spaces with a given fixed norm ‖ · ‖, and often, morespecifically Hilbert spaces.

We know from elementary analysis that functions with good regularity propertiesdo not have to be polynomial; let us think of trigonometric or exponential functions.However these functions do share the property that they can be well approximatedpointwise by polynomial functions of arbitrary degree. In fact, we may say that afunction f is differentiable of class Cr at x if there exists a polynomial functionPr of degree r such that f (y) − Pr (y) is continuous at the point x , and goes tozero faster than ||y − x ||r when y → x . The class of smooth or C∞ functions isdefined as the family of functions which are of class Cr for every r at any point x inE . Thus, smooth functions are approximated by polynomial functions of arbitrarydegree in the neighborhood of any point and it is the class of functions that extendsmost naturally the properties of the algebra of polynomials.6 In this sense we can saythat the algebra F(E) (or F for short if there is no risk of confusion) of the smoothfunctions on E extends the algebra P of the polynomials.7

Exercise 2.4 Prove that if f is a function on E homogeneous of degree 1 anddifferentiable at 0, then it must be linear.

6 See however below, Remark 2.5.7 To be quite honest, the class of functions that extend more naturally the algebra of polynomialsis the algebra of real analytic functions. However in this book we will restrict our attention to thealgebra of smooth functions.


Exercise 2.5 Find examples of functions which are homogeneous of degree k andwhich are not k-tic forms, k ≥ 1.

Because linear functions generate the algebra of polynomials and polynomialsapproximate arbitrarily well smooth functions in the neighborhood of any point, wewill say that linear functions will ‘generate’ the algebra of smooth functions (wewill make precise the notion of ‘approximation’ when introducing the notion ofdifferentiable algebras in Sect. 3.2.2 and the meaning of ‘generating’ used here inSect. 3.2.3).

As we will see in the chapters to follow, the algebra F(E) will play a central rolein our exposition and we will come back to its structure and properties when needed.

Remark 2.5 All that has been said up to now (what we mean by linear functionsand so on) depends of course in a crucial way on the linear structure that is assumedto have been assigned on E , and a different linear structure (see below, Sect. 3.5.3for a discussion of this point) will lead to different notions of ‘linear functions’,‘polynomials’ and so on. On the other hand, being smooth is a property of a functionthat is independent of the linear structure. We can conclude therefore that F(E)

is obtained anyway as the closure (in the sense specified above) of the polynomialalgebras associated with the different linear structures.

2.3.2 Transformations and Flows

Invertible linear maps L from E to E are the natural transformations of E preservingits linear structure. They form a group, the group GL(E) of automorphisms ofE . If we fix a linear basis {ei | i = 1, . . . , n} on E , then E is identified with R

n ,n = dim E , and linear maps from E to E are represented by square matrices.Invertible linear maps correspond in this representation to regular matrices and thegroup GL(E) becomes the general linear group GL(n,R). Later on, we will discussin more detail this and other related groups of matrices (see Sect. 2.6.1). Using theprevious identification of E withRn , any map φ : E → E can be written as a n-tupleof component functions φ = (φ1, . . . ,φn). Thus the notion of smoothness translatesdirectly to φ via the components φi .

A smooth diffeomorphism of E is a smooth invertible map φ : E → E whoseinverse is also smooth. Clearly, if f is a smooth function, f ◦ φ is also smooth andwe obtain in this way a map φ∗ : F → F , called the pull-back map along φ, as

φ∗ ( f ) (x) = ( f ◦ φ) (x) (2.52)

which preserves the product structure and, as (φ2 ◦ φ1)∗ = φ∗

1 ◦ φ∗2, it is invert-

ible if φ is a diffeomorphism, (φ∗)−1 = (φ−1)∗. Thus the set of diffeomorphismstransforms smooth functions into smooth functions and they are the natural set ofmaps preserving the differentiability properties of functions, as we shall see shortly.

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The composition of two diffeomorphisms is again a diffeomorphism, hence theyconstitute a group, denoted as Diff (E), because of the associativity property of thecomposition of maps, and clearly contains the group GL(E). The group of diffeo-morphisms of E will also be called the group of transformations of the space E . Wemust remark that a transformation φ will destroy in general the linear structure on Ebut will leave invariant the class of smooth functions F . This means that the notionof smoothness is not related to any particular linear structure on the space E (seeRemark 2.5) and depends only on what is called the differential structure of E .

We can also consider local transformations, i.e., smooth maps defined only onopen sets of E and which are invertible on their domains. In what follows we will notpay attention to the distinction between local transformations and transformations inthe sense discussed above, because, given a transformation, we can always restrictour attention to an arbitrary open set on E .

2.3.3 The Dual Point of View of Dynamical Evolution

One of the aspects we would like to stress in this chapter is that all notions ofdifferential calculus can be rephrased completely in terms of functions and theirhigher order analogues, differential forms, i.e., we can take the dual viewpoint anduse instead of points in a linear space E , functions on it. In fact, let us consideragain the algebra F(E) of smooth functions on E . It is clear that F(E) contains thesame information as the set of points E itself. In fact, as we discussed earlier, wecan reconstruct E from F(E) by considering the set of homogeneous functions ofdegree one.

This attitude, one of the main aspects of this book, is very common in the physi-cal construction of theories where, implicitly, states and observables are used inter-changeably to describe physical systems. The states are usually identified with pointsmaking up the space E and the observables with smooth functions (or more generalobjects) in E . This means that they define the value taken by the observable on anypossible state. If the description of a system in terms of observables is complete, wecan reconstruct the states from the observables by taking appropriate measures onthem. This is the essence of Gelfand-Naimark theorem.

Evolution can then be described not in terms of howpoints (states) actually evolve,but in terms of the evolution of the observables themselves. This approach is oftentaken in physical theories where the actual description of states can be very compli-cated (if possible at all) but we know instead a generating set of observables. In fact,that is what it is usually done in elementary textbooks, where points are describedby their coordinates xi , thus if we say, for instance, that the position at time t of thepoint is given by xi (t), what we mean is that the basic position observables xi haveevolved and at time t they are given by the new functions xi (t) that turn out to be anew set of ‘basic’ position observables. Because the position observables xi generatethe full algebra F(E), describing how they evolve gives us the evolution of all otherobservables. An interesting observation is that the evolution described by means ofobservables or functions is always linear.


Ifwe are given twoobservables f , g and their sum f +g at a given time t0, then their‘evolved’ functions f (t), g(t) and ( f +g)(t) satisfy f (t)+g(t) = ( f +g)(t). It is alsoclear that evolutionmust be invertible, thus if we denote by�t the evolution operator,then�t : F(E) → F(E) is a linear isomorphism of the (infinite-dimensional) vectorspaces F(E). Using this notation we will write �t ( f ) for the observable f (t) andthe previous equation would be written as,

�t ( f ) + �t (g) = �t ( f + g) .

It is also clear that, when considering the (pointwise) product f · g, we have,

�t ( f · g) = �t ( f ) · �t (g) (2.53)

i.e., that �t preserves products as well. The family {�t }t∈R appears therefore as aone-parameter family of automorphisms of the algebra F(E) of functions on E .

Later on, see Chap.6, if E is a Hilbert space, we would describe the dynamics onexpectation value functions, a subspace of F(E).

Remark 2.6 Of course not all linear automorphisms of F(E) are also algebra auto-morphisms. For example the mapping: f �→ exp (λk) f with k a fixed function is alinear automorphism which is not an algebra automorphism.

Another example borrowed from Quantum Mechanics is given by the linear map�t ( f ) = exp(−i�2t)( f ) (� now denotes the Laplace operator). The family �t is aone-parameter group on square integrable functions but it does not define a group ofautomorphisms for the product because the infinitesimal generator is a second-orderdifferential operator, which is not a derivation.

Equation (2.53) or, otherwise stated, the requirement that time evolution of‘observables’ (i.e., functions in F(E)) should preserve both the linear structure andthe algebra structure, has some interesting consequences. The most relevant amongthem is that we can characterize the evolution operator �t in more mundane terms,and precisely as a transformation on E . With reference again to Eq. (2.53) we cango further and think of the effects of the iteration of evolution, i.e., of the result ofapplying �t and �s successively. In an autonomous world, i.e., the system has nomemory of the previous history of the state it is acting upon, then necessarily wemust have,

�t(�s( f )) = �t+s( f ). (2.54)

Obviously,

�0( f ) = f, (2.55)

and we find again (cfr. Eq. (2.50)) a one-parameter group. These properties may besatisfied even without the existence of φ−1

t ; in this ocurrence we would have a semi-group. Thus, evolution will be given by a one-parameter group of isomorphisms of

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the algebraF(E). Then, we can conclude this paragraph by postulating the axioms ofautonomous evolution as given by a smooth one-parameter group of automorphismsof the algebra F(E).

Notice that the axioms for autonomous evolution are satisfied for a smooth one-parameter group of diffeomorphisms ϕt on E :

�t ( f ) = f ◦ ϕ−t = ϕ∗−t ( f ). (2.56)

(In Sect. 3.2.2 it will be shown that this is the most general situation).

2.3.4 Differentials and Vector Fields: Locality

From their transformation properties, smooth functions are scalar quantities, i.e., theyverify a transformation rule that (cfr. Eq. (2.52)) can be stated as ‘the transformedfunction at a transformed point takes the value of the untransformed function at thegiven point’.

The usual definition of the differential of a given function f as

d f = ∂ f

∂xidxi

requires however the explicit introduction of a coordinate system, i.e., an identifi-cation of E with R

n . However, partial derivatives and differentials transform underchanges of coordinates (local transformations) in a contragradient manner. In otherwords, under a change of coordinates

xi �→ yi = φi (x) , (2.57)

we have

d f = ∂ f

∂xi dxi = ∂ f

∂yi dyi ,

or

dxi ∂

∂xi = dyi ∂

∂yi .

All that is rather elementary, but proves that the association of differentials tofunctions is an invariant operation. In more intrinsic terms, we can rewrite the aboveinvariance property as the commutation of the operator d and the pull-back along themap φ (see later Sect. 2.4.1):

φ∗ ◦ d = d ◦ φ∗,

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and will refer to this property by saying that “d” is a “scalar” operator, or that it is“natural” with respect to the group Diff (E) of diffeomorphisms of E .

It is well-known from elementary courses in calculus that the differential ofthe function f : E → R at the point x defines a linear map d f (x) : E → R

n

via d f (x)(v) = (∂ f (x)/∂xi )vi , the vi ’s being the components of v ∈ E in thegiven coordinate system. It is an easy exercise to prove that d f (x)(v) is actually acoordinate-independent expression. The differential d f (x) belongs therefore to thedual space E∗, i.e., to the space of (real) linear functionals on E , whose elements arecalled covectors. Thus, the differential of f at x is a covector. A basis of covectorsat x will be provided by the dxi ’s, which denote consistently the differentials of thelinear maps xi : E → R, xi (v) = vi , and a covector at x will be given, in localcoordinates, by an expression of the form α = αi dxi , with the αi ’s transforming inthe appropriate way.

The operator d does actually something more. Acting on the algebra F(E) ofsmooth functions it will produce a smooth field of covectors. Smooth fields of cov-ectors are called 1-forms and their space is denoted by �1(E). So, d is actually amap d : F(E) → �1(E), but more on this later in this chapter.

Another class of objects to be defined starting from a coordinate representationbut that has actually an invariant nature is that of vector fields. Any such an objectwill be defined, in a given system of coordinates, xi say, as a first-order differentialoperator of the form X = f i (x)∂/∂xi . If we require the set f i to transform under achange of coordinates like (2.57), as

f i (x) �→ gi (y) = f k(x)∂yi

∂xk, (2.58)

(i.e., just as the dxi ’s do in d f ), then X will acquire an intrinsic character as well.By using the chain rule we see immediately that there is associated, in a natural

way, to a vector field a first-order ordinary differential equation, namely,

dxi

dt= f i (x) . (2.59)

In this way we are ‘reading’ the components of the vector field as those of a velocityfield on E . We obtain therefore a definition of the action of X on functions as,

X (h) = dh

dt= ∂h

∂xi

dxi

dt= ∂h

∂xif i (x).

It is precisely the requirement that the evolution of a function be expressed in aninvariant manner (or equivalently, that the previous system of ordinary differentialequations be covariant with respect to changes of coordinates) that fixes the transfor-mation laws for the components of a vector field. As X ( f ) contains only the basicingredients defining both X and d f , we may read it also as an action of d f on Xitself,


X ( f ) = d f (X). (2.60)

Notice that, in this sense, we are identifying the value of X at a given point x witha vector dual to d f (x), thus X (x) is a vector on E . Then a vector field is simplywhat its name indicates, a field of vectors, i.e., a map X : E → E . More specifically,X : E → E × E , x �→ (x, X (x)); i.e., they are vectors along with their point ofapplication.

2.3.5 Vector Fields and Derivations on the Algebra of SmoothFunctions

As we have already pointed out, from an algebraic point of view, the set of smoothfunctions F(E) on a vector space is an algebra (over the field R in the presentcontext). Now, a derivation over a ringA is amap D : A → A, such that, D( f +g) =D( f ) + D(g) and

D( f g) = D( f )g + f D(g), (2.61)

for every f, g ∈ A. This equation is known as Leibniz’s rule. If, instead of beingsimply a ring, A is an algebra over a field K (we will consider always only K = R

or K = C) we can extend the requirement of linearity in an obvious manner as,

D(α f + β g) = α D( f ) + β D(g) , α,β ∈ K , f, g,∈ A. (2.62)

We can turn the set of derivations into an A-module by defining,

(D1 + D2)( f ) = D1( f ) + D2( f )

and

( f D)(g) = f D(g) , ∀ f, g ∈ A .

Furthermore, we can define a product of derivations as the commutator,

[D1, D2]( f ) = D1(D2( f )) − D2(D1( f )) , ∀ f ∈ A . (2.63)

One can check that if the algebra is associative, then [D1, D2] is again a derivation.Notice that, however, D1 ◦ D2 and D2 ◦ D1 separately are not derivations. It is alsoeasy to check that the Lie bracket [·, ·] above, Eq. (2.63), satisfies the Jacobi identity,

[D1, D2], D3] + [[D3, D1], D2] + [[D2, D3], D1] = 0 , ∀D1, D2, D3 . (2.64)


In this way, the set of derivations over a ring becomes actually a Lie algebra.Now we have the following:

Proposition 2.7 The derivations over F(E) are the vector fields on E. Explicitlythe Lie bracket of two vector fields X, Y is given by

[X, Y ]( f ) = X (Y ( f )) − Y (X ( f )).

Proof It is clear that vector fields are derivations. We shall consider now an arbitraryderivation D and prove that it defines a vector field. Let D(xi ) = Xi be the images ofa coordinate set of functions xi . Then let us consider the first-order Taylor expansionof f around a given point x0. Then,

f (x) = f (x0) +n∑

i=1

∂ f

∂xi

∣∣∣∣x ′

(xi − xi0), (2.65)

with x ′ lying in the segment joining x0 and x . Then,

(D f )(x) =n∑

i=1

(D

(∂ f

∂xi

∣∣∣∣x ′

)(xi − xi

0) + ∂ f

∂xi

∣∣∣∣x ′

Xi (x)

). (2.66)

If we take the limit x → x0 in the previous equation we get,

(D f )(x0) =n∑

i=1

Xi (x0)∂ f

∂xi

∣∣∣∣x0

, (2.67)

namely, D( f ) = X ( f ) for the vector field defined by the local components Xi . ��As the set ∂/∂xi , for i running from 1 to n, form a local basis for the module

F(E), we can easily compute the commutator of two derivations,

X = Xi ∂

∂xi, Y = Y j ∂

∂x j(2.68)

as follows:

[X, Y ] =(

Xi ∂Y j

∂xi− Y i ∂X j

∂xi

)∂

∂x j. (2.69)

We will denote as X (E) the set of vector fields on E .

Definition 2.8 The action of X ∈ X(E) on f ∈ F(E) defines the Lie derivativeLX

of f along X , i.e.,

X ( f ) = LX f. (2.70)


From the very definition of Lie brackets we obtain (always on functions),

LXLY − LYLX = L[X,Y ]. (2.71)

2.3.6 The ‘Heisenberg’ Representation of Evolution

The discussion in the previous sections has led us very close to what is known asthe Heisenberg representation of evolution in Quantum Mechanics. This approachconsists in translating the attention of the observer of the evolution of a given systemfrom the states to the observables and to consider that the states do not evolve butthat the observables change. In this sense, the states should be considered not asactual descriptions of the system at a given time, but as abstract descriptions of‘all’ possible descriptions of the system. Then the actual evolution takes place bychanging the observables while we measure the position, for instance, of the system.These considerations will sound familiar to those readers familiar with QuantumMechanics but they are not really related with a ‘quantum’ description of the worldbut only with the duality between states and observables sketched in the previoussection.

The postulate that evolution must also preserve the product of observables neces-sarily implies that infinitesimal evolution will be given by a derivation of the algebra.We notice immediately by differentiating Eq. (2.53) that,

d

dt�t ( f · g)

∣∣∣∣t=s

= d

dt�t( f )

∣∣∣∣t=s

· �s(g) + �s( f ) · d

dt�t ( f )

∣∣∣∣t=s

Thus the infinitesimal evolution operator � = d�t/dt |t=0 is a derivation in thealgebra F(E). But from the previous discussions we know that derivations of thealgebra of smooth functions on E are in one-to-one correspondencewith vector fieldson E , thus we can conclude that the axioms of evolution for observables discussedabove imply that the evolution is described by a vector field � on E whose flow isgiven by the (local) one-parameter group of diffeomorphisms ϕt . Then, we will have

�( f ) = d

dt�−t ( f ) = d

dtf ◦ ϕt , (2.72)

and,

�(x) = d

dtϕt (x) |t=0 .

This equation, relating a one-parameter group of diffeomorphisms to its infinitesimalgenerator, is strongly reminiscent, for those who are familiar with QuantumMechan-ics, of the Stone-von Neumann theorem, and can be taken actually to constitute its‘classical’ version. Note however that the above is valid under the assumption that f


be a smooth (or at least aC1) function, while the one-parameter group�t , per se, canact on more general classes of functions like, e.g., continuous or simply measurablefunctions. In the latter cases we cannot revert from the group to the infinitesimalgenerator acting on functions, and again that is the classical counterpart of the prob-lems with domains (of the infinitesimal generator) that are well known to occur inthe transition from the unitary evolution to its self-adjoint generator. A prototypicalexample of an operator for which this kind of problems arises is provided by theoperator d/dx , the infinitesimal generator of translations on the real line. A nicediscussion of the relation between completeness and self-adjointness may be foundin [ZK93].

Remark 2.7 A few remarks are in order here. First, not all vector fields on E arisein this form. The existence theorem for ordinary differential equations, states onlythe local existence of the flow ϕt , thus in general for an arbitrary vector field wewill not be able to extend the local solutions x(t) for all values of t , hence we willnot be able to define a one-parameter family of diffeomorphisms ϕt but only a localone-parameter group (We will discuss this issue in the next section).

An elementary example is provided, in one dimension, by the vector field:� (x) =αx2∂/∂x, α = const., x ∈ R, whose integral curves are of the form: x (t) =x0/ (1 − α x0t), x0 = x (0), that, for every x0 �= 0, will ‘explode’ to infinity in afinite time: t∗ = 1/αx0 and therefore is not complete.

A less elementary example is provided by the vector fields8: X(i) = εi jkx j∂/

∂xk, i = 1, 2, 3 onR3, where εi jk is the totally antisymmetric (Ricci or Levi-Civita)tensor (ε123 = 1), which close on the Lie algebra of SO(3) (or of SU (2)), i.e.:[X(i), X( j)

] = εi jk X(k) and generate the rotations in R3. They can be restricted to

the unit sphere S2, where they become, in spherical polar coordinates (φ, θ):

X1 = sin φ∂

∂θ+ cot θ cosφ

∂

∂φ,

X2 = cosφ∂

∂θ− cot θ sin φ

∂

∂φ,

X3 = ∂

∂φ. (2.73)

Spherical polar coordinates are, of course, a system of coordinates only for the spherewithout the poles and one meridian passing through them, i.e. for: θ ∈ (0,π) andφ ∈ (0, 2π). So, the vector fields (2.73) are actually globally defined on the cylinder:(0,π) × [0, 2π], and it is not difficult to convince oneself that, out of these threevector fields, only X (3) is complete.

An even more startling example is provided by the realization of the pseudo-rotation group on the real line;

8 These fields are not independent. In fact, denoting them collectively as: X = (X(1), X(2), X(3)

),

with: x = (x1, x2, x3

), it is obvious that: x · X = 0.


X1 = ∂

∂x, X2 = sin x

∂

∂x, X3 = cos x

∂

∂x.

The above notions can be carried over to the level of the algebra of smoothfunctions modifying in an appropriate way the axioms above but we will not doit here in order not to create an unnecessary complication in the definitions. Thusin what follows by evolution we will understand the (local) smooth one-parametergroup of transformations defined by a derivation� of the algebra of smooth functionsF(E) or equivalently by the vector field � on E .

Secondly, the autonomous condition on evolution introduced on the axioms abovecan be removed because systems can have at a given point, and in fact they often do,memory of their previous history. Then, evolution will be given simply by a (local)smooth one-parameter family of diffeomorphisms on E . That is equivalent to giving(local) smooth one-parameter family of derivations �t on E , which is usually calleda time-dependent vector field.

2.3.7 The Integration Problem for Vector Fields

Thus we have seen that a one-parameter group of automorphisms �t of F defines,at least formally, a derivation �. Derivations are identified with vector fields, thuswe have a way to recover the group of automorphisms by integrating the differentialequation defined by the vector field. Because we know the action of a vector field onfunctions, f �→ X ( f ) we could try to compute the flow ϕ of X on a function f by‘integrating’ the previous formula and we will get:

ϕ∗t ( f ) =

∑k≥0

tk

k!LkX ( f ). (2.74)

However the previous formula could raise a few eyebrows. When is the series on theright-hand side of equation (2.74) convergent? Even if it is convergent with respect tosome reasonable topology on some class of functions, would the family ϕt of mapsthus obtained be the ‘integral flow’ of X as in the linear case?

This is the first nonlinear integrability problem we are facing and its solutionprovides the key to predict the evolution of a given system. The answer to thisproblem is, of course, well known and it constitutes the main theorem in the theoryof ordinary differential equations. Before discussing it, we would like to elaboratefurther on the notion of tangent vectors and vector fields, both for the sake of thestament of the solution to this problem and for further use in the construction ofgeometrical structures associated to given dynamics.


2.3.7.1 The Tangent and the Cotangent Bundle

The interpretation of vector fields and 1-forms as fields of vectors and covectorsrespectively captures only a partial aspect of their use. The ordinary differentialequation (2.59), associated with a given vector field X , shows us an alternative inter-pretation of vector fields which is at the basis of the present geometrical construction.The components f i (x) of the vector field X are interpreted according to Eq. (2.59)as the components of the velocity of a curve γ(t) which is a solution of the ordinarydifferential equation defined by X . Thus, the value of the vector field X at a givenpoint x can be geometrically thought of as a tangent vector to a curve γ(t) passingthrough x at t = 0. Notice that two curves γ1(t) and γ2(t) having a contact of order1 at x define the same tangent vector, i.e.

dγ1(t)

dt

∣∣∣∣t=0

= dγ2(t)

dt

∣∣∣∣t=0

. (2.75)

Therefore a tangent vector at the point x can be thought as an equivalence classof curves9 passing through x with respect to the equivalence relation of having acontact of order 1 at x . We shall denote by Tx E the collection of all tangent vectorsto E at x . If vx denotes one of these tangent vectors we can define the variation of afunction f at the point x in the direction of vx as

vx ( f ) = d

dtf ◦ γ(t)

∣∣∣∣t=0

, (2.76)

where γ(t) is any representative in the equivalence class defining vx . Notice thatthe definition of the numerical value vx ( f ) does not depend on the choice of therepresentative γ we make. Thus vx is a linear map from the space of differentiablefunctions defined in a given neighborhood of the considered point x into R with theadditional property that

vx ( f1 f2) = f1(x)vx ( f2) + f2(x)vx ( f1)

This additional property is what characterizes vectors vx at a point x among all linearmaps. We can define an addition on the space of tangent vectors as

(vx + ux ) ( f ) = vx ( f ) + ux ( f ) (2.77)

Of course we need to guarantee that the object thus defined, ux + vx , correspondsagain to a tangent vector, i.e., we need to find a curve passing through x such that itstangent vector will be ux + vx . Because we are in a linear space E , that is actuallyvery easy. Given a vector v ∈ E there is a natural map identification v �→ vx

with a tangent vector at x , which is the equivalence class corresponding to the curve

9 See however below, Appendix C, Sect. C.2 for a similar discussion in a more general context.


γ(t) = x + tv. Such identification is clearly one-to-one and the vector correspondingto vx +ux is the vector corresponding to the curve γ(t) = x + t (u+ v).10 From thisperspective, we see that Eq. (2.77) actually defines the addition on the tangent spaceTx E and it shows that this addition does not depend on any linear structure of E .However, in this particular setting it is also true that we have a natural isomorphismbetween Tx E and E as linear spaces. Hence, we can think that the tangent space atthe point x is a copy of the background space E put at the point x . The tangent vectorscorresponding to the curves γi (t) = x + t ei , where {ei | i = 1, ..., n} is a given basisin E , are denoted by (∂/∂xi )|x . The notation is consistent with the operator definedon functions, because,

∂

∂xi

∣∣∣∣x( f ) = d

dt( f ◦ γi )(t)

∣∣∣∣t=0

= d

dtf (x + tei )

∣∣∣∣t=0

= ∂ f

∂xi

∣∣∣∣x. (2.78)

The union of all tangent spaces Tx E is collectively denoted by,

T E =⋃x∈E

Tx E (2.79)

and it is clearly isomorphic as a linear space to E ⊕ E . An element of T E is thus apair (x, v) where x is a point (vector) in E and v is a tangent vector at x . There isa natural projection τE : T E → E , defined as τE (x, v) = x . Such a structure, thetriple (T E, τE , E), is called the tangent bundle over E .

If φ : E → E denotes a smooth map and vx is a tangent vector at x , then φ∗(vx )

is a tangent vector at φ(x) defined as:

(φ∗(vx )) ( f ) = vx ( f ◦ φ) ,

for any f defined in an open neighborhood of φ(x). Thus we have defined a mapφ∗ : T E → T E called the tangent map to φ or, sometimes, the differential of φ. Itis clearly satisfied that (φ ◦ ψ)∗ = φ∗ ◦ ψ∗ which is just a consequence of the chainrule. Thus if φ is a diffeomorphism, then φ−1∗ = (φ−1)∗.

Turning back to the notion of a vector field, we see that a vector field consists of asmooth selection of a tangent vector at each point of x , the values of the vector fieldX being the vectors X (x) tangent to E at x . Thus, a vector field is a smooth mapX : E → T E such that it maps x into (x, X (x)) where X (x) ∈ Tx E .

The above maps satisfy τE ◦ X = idE and are called cross sections of the tangentbundle. Therefore, in this terminology a vector field is just a cross section of thetangent bundle. Moreover, if φ is a diffeomorphism, then we may define the push-forward φ∗ X of X along φ as follows: (φ∗ X)(x) = φ∗(X (x)).

10 However this idea will also work in a more abstract setting in the sense that it is possible toshow that there is a one-to-one correspondence between equivalence classes of curves possessing acontact of order 1 at x and linear first-order differential operators v acting locally on functions at x .


Because E is already a vector space, there is a distinguished vector field calledthe dilation (or Liouville) vector field �11:

� : E → T E; x �→ �(x) = (x, x) , x ∈ E . (2.80)

The Liouville vector field � allows us to identify the vector space structure on thetangent space with the vector space structure on E (see later on Sect. 3.3.1), i.e., �can be defined only if E itself is a vector space. The graph of � in T E ≈ E × E isa subspace of T E ⊕ T E , the diagonal vector subspace.

Together with the tangent bundle we do have its dual, the cotangent bundle. Wedescribe it briefly here. Again, as in the case of tangent vectors being identifiedwith equivalence classes of curves, we have a natural identification of functions ata given point by possessing the same differential (the actual value of the function isirrelevant).Any such equivalence class actually defines a covectorαx = d f (x) ∈ E∗,for some f . Thus the space of covectors at x (differentials of functions at x) defines thecotangent space denoted by T ∗

x E . Such a space is obviously naturally isomorphicto E∗ and is dual to Tx E . The set {dxi (p) | i = 1, . . . n} is the dual basis of{(∂/∂xi )p | i = 1, . . . n} at each point p ∈ E .

The pairing between both is given as follows: If γ is a curve representing thetangent vector vx and f is a function representing the cotangent vector αx , then,

〈αx , vx 〉 = d

dt( f ◦ γ)(t)

∣∣∣∣t=0

. (2.81)

The union of all cotangent spaces T ∗x E is denoted by T ∗E . Clearly, T ∗E is naturally

isomorphic to E ⊕ E∗ and it carries a natural projection πE : T ∗E → E , defined asπE (x,αx ) = x . The triple (T ∗E,πE , E) is called the cotangent bundle of E .

A smooth assignment of a covector at x to any point x ∈ E is called a 1-form.Thus a 1-form α is a smooth map α : E → T ∗E such that x �→ (x,α(x)), i.e.,again a 1-form is a cross section of the cotangent bundle, and therefore such thatπE ◦ α = idE .

2.3.7.2 Vector Fields and Local Flows

Thus given a vector field X on E , we want to determine the existence of a flow ϕt forX . In general, as it was pointed out before, this cannot be done globally, however it isalways possible to do it locally (see [Ar73] by a masterly exposition of the subject).

Theorem 2.9 (Fundamental theorem of ordinary differential equations) Given asmooth vector field X on E, for every x ∈ E there exists an open neighborhood Uof x and a number ε > 0 such that given any point y ∈ U and any t with |t | < ε,the solution ϕt (y) of the equation du/dt = X (u) satisfying the initial condition yat t = 0 exists, is unique, depends smoothly on y and t and satisfies:

11 Also-called the Euler differential operator.

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ϕt+s(y) = ϕt ◦ ϕs(y), |t | < ε, |s| < ε, |t + s| < ε.

Avector fieldwhose solutions can be extended from−∞ to+∞ so as to give raiseto a one-parameter group will be said to be a ‘complete’ vector field. As already said,generic vector fields need not be complete. However, if the vector field is defined ona compact set, or better, is different from zero only on a compact set, it is complete.This result is also true for smooth manifolds ([Ar73], Theorem 35.1), but we willstate it here just in the case of linear spaces.

Theorem 2.10 Let X be a smooth vector field different from zero only in a com-pact subset K of E. Then there exists a one-parameter group of diffeomorphismsϕt : E → E for which X is the velocity field:

d

dtϕt (x) = X (ϕt (x)), ∀x ∈ E .

Thus if � is a vector field on E , then we may find a one-parameter group ofdiffeomorphisms ϕt that describes the trajectories of it on a compact set, or in otherwords, there is a one-parameter group of automorphisms of F(E) that restricted toF(K ) for K a compact set satisfies:

d

dtϕt = � ◦ ϕt . (2.82)

The picture we can get from this situation is that given a vector field and choosing acompact set K neighborhood of a given point, there is a complete flow ϕt that actingon points on K will produce the trajectories of � but that a little bit after exiting Kthe flow will ‘freeze’ leaving the points fixed.

The idea to prove this is simple. Given a compact set K , we may construct (takingfor instance the a closed ball containing K ) a smooth ‘bump’ function ρ adapted toK , that is a function such that ρ = 1 on K and ρ = 0 in the complementary of theclosure of a ball containing K . Thus multiplying X by ρ we have a vector field towhich we may apply Theorem 2.10 and whose complete flow is a one-parametergroup of diffeomorphims satisfying the previous equation (2.82) on K .

The previous formula (2.82) provides a rigorous setting for Eq. (2.72) and makessense of the formal integration idea expressed by Eq. (2.74).

Thus, using the previous observation, we may assume that we have a one-parameter group of diffeomorphisms integrating a given dynamics (that is describingthe trajectories of our system on compact sets) and we will use this in what followswithout explicit mention to it.

We will close this collection of ideas by noticing that the Lie derivative of afunction f along a vector field � that was derived before represents the infinitesimalvariation of the function in the direction of the vector field and can be defined (byusing Eq. (2.82)) as:


L� f = �( f ) = d

dtϕ∗−t f |t = 0= d

dtf ◦ ϕ−t |t = 0 .

2.4 Exterior Differential Calculus on Linear Spaces

2.4.1 Differential Forms

Having defined the differentials as objects that behave as ‘scalars’ one may also saythat they are ‘natural’ under arbitrary (smooth) changes of coordinates. Thus wemayformF(E)-linear combinations of differentials of functions, i.e., sums ofmonomialsof the form f dg, f, g ∈ F(E). Then if we transform f and g by using a diffeomor-phisms φ of E , then f dg transforms accordingly. Themonomials f dg will generateanF(E)-module�1(E)whose elements are differential 1-forms over E . Notice that�1(E) is just the space of sections of the cotangent bundle T ∗E because any 1-formα can be written as α = αi dxi , once a linear coordinate system xi on E has beenchosen, i.e., a linear basis ei has been selected. Then it is clear that the dxi ’s are abasis of �1(E), i.e., any 1-form can be given uniquely as an F(E)-linear combina-tion of them. Equivalently, the dxi ’s are linearly independent. We may choose alsoany other basis, i.e., any other set d f i , provided they are also independent. Of course,the d f i ’s are linearly independent iff the f i ’s are functionally independent, i.e., nononconstant function � = �( f 1, . . . , f n) exists such that �( f 1, . . . , f n) = const.Later on, see Eq. (2.89), we will give a compact characterization of this condition).A 1-form α which is the differential of a function, i.e., α = d f will be called exact.

We may also wish to extend the action of d from functions, i.e., zero-forms, to1-forms. Let us start with a monomial like f dg, and let d( f dg) be defined by actingon the coefficients of the dxi ’s (which are functions, after all). We are immediatelyfaced with the problem of defining products of differentials. If we choose, e.g., tensorproducts, we obtain for d( f dg) the expression,

d( f dg) =(

∂ f

∂xi

∂g

∂x j+ f

∂2g

∂xi∂x j

)dxi ⊗ dx j .

Changing coordinates x �→ x ′, a tedious but straightforward calculation yields,

d ( f dg) =(

∂ f

∂x ′i∂g

∂x ′ j+ f

∂g

∂xk

∂2xk

∂x ′i∂x ′ j+ f

∂2g

∂x ′i∂x ′ j

)dx ′i ⊗ dx ′ j .

So, naturality of the operator d gets lost (except for linear changes of coordinates)unlesswe redefine the product of differentials in such away as to eliminate symmetricparts. This leads us to define the wedge (or exterior) product dxi ∧ dx j as theantisymmetrized product,

2.4 Exterior Differential Calculus on Linear Spaces 101

dxi ∧ dx j = 1

2(dxi ⊗ dx j − dx j ⊗ dxi ), (2.83)

and, by extension,

d f ∧ dg = ∂ f

∂xi

∂g

∂x jdxi ∧ dx j = 1

2

(∂ f

∂xi

∂g

∂x j− ∂g

∂xi

∂ f

∂x j

)dxi ∧ dx j ,

and to extend (by definition now) the action of d on monomials of the form f dg as

d( f dg) = d f ∧ dg. (2.84)

This definition has the advantage of retaining the naturality of the exterior differential,as d will be called from now on. It is remarkable that d is the only derivation whichis a ‘scalar operator’ with respect to the full diffeormorphism group [Pa59].

Remark 2.8 With theDefinition (2.83), thewedge product differs by a normalizationfactor 1/2 in the case of the product of two one-forms as in Eq. (2.83), 1/n! for theproduct of n one-forms, from the antisymmetrized product one would obtain usingEq. (10.6). This normalization turns out however to be more convenient, and we willuse it throughout whenever we will deal with differential forms.

The F(E)-linear space spanned by the monomials dxi ∧ dx j (or by the indepen-dent wedge products in any other basis) will be called the space of smooth two-forms,or just 2-forms for short, and will be denoted by �2(E). A general 2-form will havethe expression,

α = αi j dxi ∧ dx j , αi j = −α j i , αi j ∈ F(E) (2.85)

It is left to the reader to work out the transformation law of the coefficients αi j ’sunder arbitrary (smooth) changes of coordinates.

Because E is a linear space then theF(E)-module of 1-forms is finitely generated(a system of generators is provided by the differentials dxi of a linear system ofcoordinates xi for instance). Not only that, �1(E) is a free module over F(E) andthe 1-forms dxi provide a basis for it, then dimF (E) �1(E) = dim E .

Similarly the F(E)-module �2(E) is free and the 2-forms dxi ∧ dx j provide abasis for it. Then clearly if dim E = n, dimF (E) �2(E) = (n

2

). Here too, if α = dθ

for some θ ∈ �1(E), α will be called an exact two-form.As we discussed before, starting with the tangent bundle T E we can form the

cotangent bundle or the bundle of covectors or linear 1-forms over E , but we couldalso form the bundle of linear 2-forms (or skew symmetric linear (0, 2) tensors) overE . We shall denote such bundle as �2(T ∗ E), and it is just the union of all spaces�2(Tx E), x ∈ E . We also have as before that �2(T ∗E) ∼= E ⊕ �2(E). Notice that�1(T ∗E) = T ∗ E . Cross sections of �2(T ∗E) are smooth 2-forms ω ∈ �2(E). It

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is also customary to denote the space of cross sections of the bundles T E , T ∗ E ,�2(T ∗E) by �(T E), �(T ∗E), and �(�2(T ∗E)), etc., thus �(T ∗ E) = �1(E) andso on.

Let us recall that a linear 1-form α acts on a vector producing a number. Thus wemay also think that�1(E) is the dual with respect to the algebraF(E) of the moduleof vector fields X(E), i..e, a 1-form α is a F(E)-linear map α : X(E) → F(E).α(X) ∈ F(E), for all X ∈ X(E) and being defined as α(X)(x) = 〈α(x), X (x)〉where 〈·, ·〉 denotes as usual the natural pairing between a linear space and its dual.

A similar argument applies to 2-forms. A 2-form ω ∈ �2(E) can be considered asdefining a skew symmetric F(E)-bilinear map on the module X(E), that is ω(X, Y )

is a smooth function on E for any X, Y ∈ X(E).Now from the definition of the wedge product it follows at once that,

(dxi ∧ dx j )(X, Y ) = dxi (X)dx j (Y ) − dxi (Y )dx j (X)

for any pair of vector fields X and Y .In particular, having in mind the definition of the Lie derivative on functions, it is

not hard to see that, if α = dθ is exact, then,

dθ(X, Y ) = LX (θ(Y )) − LY (θ(X)) − θ([X, Y ]), ∀X, Y ∈ X(E) . (2.86)

It is left as an exercise to prove that, notwithstanding the differential nature of the Liederivative, the right-hand side of Eq. (2.86) is actually F (E)-linear in both X andY (besides being manifestly skew-symmetric). Together with Eq. (2.71), this tells usalso that, if θ = d f is an exact 1-form, then, dθ = d2 f = 0, i.e., that d2 = d ◦d = 0(on functions only, for the time being).

2.4.2 Exterior Differential Calculus: Cartan Calculus

If we extend the wedge product into an associative product, we can generate formsof higher degree by taking wedge products of forms of lower order. In a systematicmanner, if α1, . . . ,αn is a basis of 1-forms (e.g., αi = d fi for a set of functionallyindependent functions), then, monomials of the form,

αi1...ik = αi1 ∧ · · · ∧ αik , i1, . . . , ik = 1, . . . , n, (2.87)

will generate the F(E)-linear space of k-forms on E , denoted as �k(E). Alterna-tively, we may think that a k-form α is an F(E)-multilinear map

α : X(E) × k· · · × X(E) → F(E),


such thatα(X1, . . . , Xi , . . . , X j , . . . , Xk) = −α(X1, . . . , X j , . . . , Xi , . . . , Xk) forall i, j . Then, given any smooth map φ : E → E , we may define the pull-back ofany k-form along the map φ as

φ∗α(X1, . . . , Xk) = α(φ∗ X1, . . . ,φ∗ Xk), ∀Xi ∈ X(E). (2.88)

Remark 2.9 The previous formula should be understood pointwise, that is consid-ering the k-form α as a section of the bundle �k(T ∗E) → E . Then we will write:

(φ∗α)x (v1, . . . , vk) = αx (φ∗v1, . . . ,φ∗vk), ∀v1, . . . , vk ∈ Tx E .

However the formula above (2.88) makes perfect sense if φ is a diffeomorphism.

Remark 2.10 It is not hard to prove that the wedge product of two 1-forms (andhence of any number) vanishes iff the forms are linearly dependent. The conditionfor the linear independence for k-monomials will be then,

αi1 ∧ · · · ∧ αik �= 0.

Hence, if k ≤ n, dimF (E) �k(E) = (nk

)and there will be no room for forms of degree

higher than the dimension n of E , actually �n(E) = F(E). We note parentheticallythat functional independence of a set of k functions f1, . . . , fk ∈ F(E) will beexpressed by the condition

d f1 ∧ · · · ∧ d fk �= 0. (2.89)

Note that �•(E) = ⊕k≥0 �k(E) is an associative graded algebra (see Appendix A)

and the elements of �k(E) are said to be homogenous of degree k.

Wewish now to extend the action of the exterior differential d to forms of arbitraryrank. Let us start again with monomials in a basis generated by exact 1-forms. Tomake things simpler, let α = f dx i1 ∧ · · · ∧ dxik be a monomial of rank k (for afixed set of il’s, l = 1, . . . , k). Then, we define dα as the monomial of rank k + 1,

dα = d f ∧ dxi1 ∧ ... ∧ dxik . (2.90)

The exterior differential, with this obvious extension, is then defined on forms ofarbitrary rank and is anR-linear map d : �k(E) → �k+1(E), with�0(E) = F(E),and �n+1(E) = 0.

If a set of coordinates {x1, . . . , xn} has been chosen for E , then, � = dx1 ∧· · · ∧ dxn will be a basis for the (F(E)-one-dimensional) module �n(E), i.e., anyn−form will be expressible as f � for some f ∈ F(E). In view of the fact thatcoordinates are globally defined for a vector space, � can be thought of as well as abasis for the space�n(T ∗

x E) of the n-forms based at any point x ∈ E . As such, it will


be better denoted as �(x) although the notation may appear somewhat redundant inthis particular case. It enjoys the features that:

1. �(x) �= 0,∀x ∈ E ,2. If we perform a permutation of coordinates, xi �→ yi = xπ(i), π ∈ Sn the group

of permutations of n elements, then: �′ = dy1 ∧ · · · ∧ dyn = sign(π)�, wheresign(π) stands for the signature (the parity) of the permutation π. So, �′ = ±�according to the parity of the permutation.

In general, a nowhere vanishing form of maximal rank will be called a volumeform, and we have just seen that a volume form always exists on a vector space. Thismay not be so in more general situations in which we deal with spaces that can bemodeled on vector (and hence Euclidean) spaces only locally, in which case it maywell be that volume forms exist only locally, but this more general case is, for thetime being, outside our scopes. We have also seen that −� is an equally acceptablevolume-form if � is. Each choice will be said to define an orientation on E . Again,that is a globally defined notion as long as E is a vector space, but need not be so inmore general situations.

Let now φ be a linear map from E to E . In a given system of coordinates, φ :xi �→ yi = Ai

j x j , i.e., φ will be represented by the matrix A = (Aij ). Then, by

using the properties of the wedge product, it is not difficult to show that

(φ∗�)(x) = det(A)�(x).

More generally, if φ is a smooth map (not necessarily a linear one), the pull-backφ∗� will be again an n-form, and hence proportional to � itself, and this motivatesthe following:

Definition 2.11 Let φ : E → E be a smooth map and let� be a volume-form. Thenthe determinant of φ, det(φ), is defined by

φ∗� = det(φ)� . (2.91)

A straightforward calculation leads then to the result that, if φ1,φ2, are smoothmaps, then the determinant function enjoys the property we are familiar with in thelinear case, i.e., that,

det(φ1 ◦ φ2) = det(φ2 ◦ φ1) = det(φ1) det(φ2). (2.92)

Remark 2.11 If the volume-form is realized as: � = dx1 ∧ · · · ∧ dxn in a givensystem of coordinates, then det(φ) at point x is, of course, nothing but the familiarJacobian determinant of φ at x ∈ E .

Equation (2.90) defines also the action of d on a wedge product of 1-forms. Forexample, letα = g d f and β = h dk, f, g, h, k ∈ F(E), then,α∧β = (gh) d f ∧dkand one proves immediately that,


d(α∧β) = d(gh)∧d f ∧dk = (g dh+h dg)∧d f ∧dk = dα∧β−α∧dβ. (2.93)

Using bilinearity, if α were a 2-form (actually a monomial of rank 2) we would getinstead,

d(α ∧ β) = dα ∧ β − α ∧ dβ. (2.94)

Extending these results in an obvious way from monomials to forms we obtaineventually:

Proposition 2.12 If α ∈ �p(E) and β ∈ �q(E), then α ∧ β ∈ �p+q (E) and thegraded Leibniz rule is satisfied:

d(α ∧ β) = dα ∧ β + (−1)pα ∧ dβ. (2.95)

Morevoer, we have that d2 = 0.

Finally, if we consider a vector field, X = Xi ∂∂xi , the Lie derivative of a volume

form � is proportional to the volume form, LX� = f �. As

LX (dx1 ∧ · · · ∧ dxn) =(

∂X1

∂x1+ · · · + ∂X n

∂xn

)dx1 ∧ · · · ∧ dxn

when � = dx1 ∧ · · · ∧ dxn , the proportionality factor is called the divergence of X ,f = div (X), because using such global coordinate system,

LX (dx1 ∧ · · · ∧ dxn) = div (X) dx1 ∧ · · · ∧ dxn.

It is also possible to associate with any element X ∈ X(E) a derivation of degree−1 in the graded algebra of forms �•(E), called an inner derivation (or a contrac-tion). We set iX : �p(E) → �p−1(E), where α �→ iXα with iX f = 0, for f afunction (a 0-form) and,

(iXα)(X1, . . . , X p−1) = α(X, X1, . . . , X p−1). (2.96)

One finds that, as before, if α is homogeneous of degree |α|, then,

iX (α ∧ β) = (iXα) ∧ β + (−1)|α|α ∧ iXβ), (2.97)

and that, for any vector fields X and Y , the graded commutator of the associatedinner derivations vanishes,

[iX , iY ] = iX ◦ iY + iY ◦ iX = 0. (2.98)

Recall that the graded commutator of two derivations is defined by


[D1, D2] = D1 ◦ D2 − (−1)|D1| |D2|D2 ◦ D1.

As d and iX are (graded) derivations, of degree +1 and −1, respectively, theirgraded commutator: [d, iX ] = d ◦ iX + iX ◦ d is a derivation of degree zero. Wedenote it by LX and it will be called the Lie derivative with respect to X . From itsdefinition we have

LX = d ◦ iX + iX ◦ d . (2.99)

On functions, the Lie derivative coincides with the action of X on F(E), and itextends to general forms the action of derivations on F(E), i.e. LX f = X ( f ), and

LX (α ∧ β) = (LXα) ∧ β + α ∧ (LXβ) (2.100)

Together with d ◦ d = 0, this has the consequence that the exterior differential andthe Lie derivative commute,

d ◦ LX = d ◦ iX ◦ d = LX ◦ d (2.101)

Moreover,

[LX ,LY ] = L[X,Y ] (2.102)

(the graded commutator of two derivations of degree 0 is again a derivation of degree0) and finally,

LX ◦ iY − iY ◦ LX = i[X,Y ]. (2.103)

In particular, when X = Y , then LX ◦ iX = iX ◦ LX .With these ingredients at hand, one can prove (see e.g., [AM78] (Prop. 2.4.15)

and [Ne67] for details, [KN63] uses a slightly different normalization) that, withX ∈ X(E) and β ∈ �p(E),

dβ(X1, . . . , X p+1) =p+1∑i=1

(−1)i+1Xi (β(X1, . . . , Xi , . . . , X p+1))

+∑i< j

(−1)i+ jβ([Xi , X j ], X1, . . . , Xi , . . . , X j , . . . , X p+1).

(2.104)

(LXβ)(X1, . . . , X p) = LX (β(X1, . . . , X p) −p∑

i=1

β(X1, . . . , [X, Xi ], . . . , X p).

(2.105)


where the symbol means that the corresponding vector field should be omitted.This formula was used by R. Palais to provide an intrinsic definition of the exteriorderivative [Pa54].

In this way we have defined a sequence of maps d : �p(E) → �p+1(E), allof them denoted by d such that d2 = 0. The pair (�•(E), d) is called a gradeddifferential algebra.

Remark 2.12 More generally, the definition of d can be generalized to antisymmetricmultilinear maps φ : X(E)×· · ·×X(E) → M with M any vector space carrying anaction of X(E), i.e., a linear map ρ : X(E) → End (M). In this case we would have,

dρφ(X1, . . . , X p+1) =p∑

i=1

(−1)i+1ρ(Xi )(φ(X1, . . . , Xi , . . . , X p+1))

+∑i< j

(−1)i+ j φ([Xi , X j ], X1, . . . , Xi , . . . , X j , . . . , X p+1).

(2.106)

We find in an analogous way that dρ ◦ dρ = 0 iff ρ is a Lie algebra homomorphism.The exterior differential defined by equation (2.104) will be recovered when M =F(E) and ρ(X)( f ) = LX ( f ), the Lie derivative.

We wish to stress again that all our constructions rely only on the commutativealgebra structure of F(E). The linearity of E never played any role, therefore ourcalculus will be ‘insensitive’ to the kind of transformations we might perform on E .

Finally let us point out that the Lie derivative can be extended to the set of vectorfields, dual space of that of 1-forms, by requiring that, if X, Y ∈ X(E) and α ∈�1(E),

LX 〈α, Y 〉 = 〈LXα, Y 〉 + 〈α,LX Y 〉

and then we obtain the following definition for LX Y ,

LX Y = [X, Y ]

In fact (cfr. Eq. (2.105)), LXα was defined in such a way that

〈LXα, Y 〉 = LX 〈α, Y 〉 − 〈α, [X, Y ]〉

from where we find that LX Y is given by LX Y = [X, Y ].Once that LX has been defined on functions, on vector fields and on 1-forms, an

extension to the space of all tensors can be obtained requiring thatLX be a derivationof degree zero.


2.4.3 The ‘Easy’ Tensorialization Principle

It should be clear now that for any linear object associated with the abstract vectorspace E wemay think of it as realized in terms of ‘applied’ vectors at x , i.e., of tangentvectors, or covectors, or any other linear tensor constructed in the tangent space to Eat x . Then we can transform them into tensor fields and operations depending on x .This simple statement is what we call the ‘easy’ tensorialization principle. We willprovide now various examples of the effective use of this principle that will be ofuse along the rest of the text.

2.4.3.1 Linear Algebra and Tensor Calculus

We will start by geometrizing a linear map A ∈ End(E). First, we can geometrizeA by considering the associated (1, 1) tensor TA : T E → T E , defined as:

TA : (x, v) �→ (x, Av) ; x ∈ E, v ∈ Tx E ∼= E , (2.107)

or, dually T ∗A : T ∗E → T ∗E ,

〈T ∗Aα, v〉 = 〈α, T v〉; x ∈ E, v ∈ Tx E ∼= E, α ∈ Tx E∗ ∼= E∗ . (2.108)

Using linear coordinates xk induced by a given base {ek},

TA = A ji dxi ⊗ ∂

∂x j(2.109)

Then, Aik is given by

TA(∂/∂xk) = Aik ∂/∂xi .

The correspondence A �→ TA is an algebra homomorphism, i.e.,

TA·B = TA ◦ TB (2.110)

because,

(TA ◦ TB)

(∂

∂xk

)= TA

(Bi

k∂

∂xi

)= Bi

k A ji

∂

∂x j= (AB) j

k∂

∂x j(2.111)

Notice that once we have geometrized the linear map A on the vector space andpromoted it to a (1, 1) tensor on the space E , then we are not restricted to considerlinear coordinates or linear transformations. We can use any system of coordinatesto describe it (even if the underlying linear structure becomes blurred) and use theexterior differential calculus as was discussed in the previous section.


The tensorialization TA of A we have constructed retains all the algebraicproperties of A which are hidden in the fact that when we express TA in linearcoordinates, the tensor is constant. That is we can de-geometrize it by choosing theappropriate set of coordinates, returning to the algebraic ground. This property willbe instrumental in the general notion of tensorialization that will be discussed lateron.

It is also possible, as it was shown before, to tensorialize a linear map A byassociating to it a vector field X A defined as:

X A(x) = (x, Ax), x ∈ E, Ax ∈ Tx E ∼= E . (2.112)

Now the geometrized object is not constant anymore and in linear coordinates xk

takes the form:

X A = A ji xi ∂

∂x j.

In this case we have been using the additional feature offered by linear spaces thatTx E can be identified naturally with the base space E andwith the linear space whereA is defined. Notice that in the definition of TA we were simply using the fact thatTx E can be identified with the linear space where A is defined.

However on this occasion, the association A �→ X A fails to be a homomorphismofalgebras and is only a Lie algebra homomorphism because X AB( f ) �= X A(X B( f )),while,

X(AB−B A) ( f ) = X A (X B ( f )) − X B (X A ( f )) (2.113)

i.e.,

X[A,B] = [X A, X B] . (2.114)

Another interesting formula that transforms an algebraic identity into a geomet-rical operation is obtained by computing:

LX A TB = TAB,

We will see other formulae similar to the preceding one as we apply the tensorial-ization principle to other objects. The ‘easy’ tensorialization principle can also bestated as the thumb rule that transforms a tensor in a linear space E replacing ek (thevectors of a given base) by ∂/∂xk and the dual base elements e j by dx j . We canwrite it more precisely as follows:


The ‘Easy’ Tensorialization Principle

Given a tensorial object t in a linear space E , and given a linear base {ek}, by replacingek by ∂/∂xk and the dual base elements e j by dx j in the expression of t we willdefine a geometrical tensor Tt with the same structure as t.

The ‘easy’ tensorialization principle as stated before could seem to depend on thechoice of a system of linear basis or on a system of linear coordinates. However thatis not so. The choice of a base {e1, , . . . , en} on E (and {e1, . . . , en} its dual basis inE∗), provides an injection of the linear space E in the set of constant vector fields(i.e., homogeneous vector fields of degree −1) given by: v �→ vi ∂/∂xi (with vi

being the coordinates of the vector v with respect to the given basis). We also havethe injection: α = αi ei �→ αi dxi for the dual E∗.

This injection does not depend on the choice of the basis. Actually, we can orig-inate the previous association by the following construction: For a fixed x ∈ E wehave a linear map ξx : E → Ex = Tx E defined associating to v ∈ E the tangentvector to the curve γ : R → E given by t �→ x + vt , that is:

ξx (v) = d

dt(x + vt) |t=o (2.115)

In other words, when acting on functions,

ξx (v) f = d

dtf (x + vt)

∣∣∣∣t=0

= (d f )x (v) = vi ∂ f

∂xi

∣∣∣∣x, (2.116)

i.e.

v �→ vi ∂

∂xi

∣∣∣∣x.

Then, given a bilinear pairing b : E ⊗ E → R, say B = Bi j ei ⊗ e j , with ei abasis for E∗, we can apply the previous principle and associate to B the tensor,

TB = Bi j dxi ⊗ dx j . (2.117)

Of course we can evaluate it pointwise on vector fields like the dilation field� = X I whose expression in local coordinates is � = xi∂/∂xi and get fB(x) =τB(�,�) (x) = Bi j xi x j . By using TA we obtain also,

TA(�) = X A (2.118)

This shows that, while X A depends on the linear structure, TA depends only on theaffine structure of E , i.e., the tangent bundle structure.


Similarly we can geometrize an algebraic bivector. If � is an algebraic bivector,that is an algebraic skew symmetric contravariant tensor on E . Selecting a base {ei },� will take the form � = �i j ei ∧ e j , and we may construct a tensor field on E bymeans of:

� = �i j ∂

∂xi∧ ∂

∂x j(2.119)

In other words, bi-vectors like �i j ei ∧ e j , with �i j ∈ R, are to be identified withthe corresponding constant bivector fields.

We will discuss further extensions of this principle in Sect. 3.4.

2.4.4 Closed and Exact Forms

We say that a form β ∈ �p(E) is closed if dβ = 0, and exact if β = dθ for someθ ∈ �p−1(E). Quite obviously, by virtue of the fact that d ◦ d = 0, every exact formis closed. The converse is the content of the following:

Proposition 2.13 (Poincare’s Lemma for vector spaces) If E is a linear vector space,then every closed form in E is exact.

Proof For the proof, let us note preliminarily that, in view of the fact that d is’naturalwith respect to diffeomorphisms’ i.e., as already stressed previously,φ∗ ◦ d = d ◦φ∗,then: (i) Once a coordinate system has been fixed, E ≈ R

n , and we might as wellconduct the proof directly in R

n , and (ii) for the same reasons, the proof will holdfor any open set that is diffeomorphic to an open ball in Rn (and hence to Rn itself).

The theorem will be proved if we can construct a mapping, T : �p(E) →�p−1(E) such that, T ◦ d + d ◦ T = I d, for then, β = (T ◦ d + d ◦ T )β, and, inview of the closure of β, we see that β = dθ with θ = T β.

We claim then that the required mapping is provided by,

(T β)(x) =1∫

0

t p−1(i�β)(t x) dt, x ∈ Rn, (2.120)

where � denotes the dilation field in E . If β has the expression: β(x) =1p! βi1...i p (x) dxi1 ∧ · · · ∧ dxi p , the integrand has to be understood as, β(t x) =1p! βi1...i p (t x) dxi1 ∧ · · · ∧ dxi p .Also, for any vector field X , we have that

LX β = 1

p!

(Xk ∂

∂xkβi1...i p + p

∂Xk

∂xi1βk,i2...i p

)dxi1 ∧ · · · ∧ dxi p ,

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and

d

dtβ(t x) = d

dt

1

p! βi1...i p (t x) dxi1 ∧ · · · ∧ dxi p

= 1

t

(1

p! yi ∂

∂yiβi1...i p (y)

)y=t x

dxi1 ∧ · · · ∧ dxi p

= 1

t

(yi ∂

∂yiβ(y)

)y=t x

.

Therefore,

d

dt

(t pβ(t x)

) = t p−1

(p β(t x) +

[yi ∂

∂yiβ(y)

]y=t x

),

and hence,

d

dt

(t pβ(t x)

) = t p−1(L�β)(t x).

But then,

(T ◦ d + d ◦ T )β(x) =1∫

0

t p−1(i� ◦ d + d ◦ i�)β(t x)dt =1∫

0

t p−1(L�β)(t x)dt

=1∫

0

d

dt{t pβ(t x)}dt = β(x).

��Remark 2.13 (i) From the way the proof of Poincare’s Lemma has been constructed,it is quite clear that, if we consider a differential form defined on an arbitrary openset U , the relevant condition for the validity of the Lemma is that there is a point(taken above as the origin in R

n) such that any other point in U can be joined toit by a straight line segment lying entirely in U . That’s why, in an equivalent way,Poincare’s Lemma is often stated with reference to ‘star-shaped’ open sets.

(ii) As long as we consider forms that are differentiable over the whole of E(a vector space) there will be no distinction between closed and exact forms. Thesituation will change however as soon aswe consider topologically less trivial spaces.As a simple example, consider R2 − {0} with (Cartesian) coordinates (x, y) and the1-form,

β = x dy − y dx

x2 + y2. (2.121)


It is a simple computation to show that: β = dθ, with θ = tan−1(y/x), the polarangle. Hence: dβ = 0, i.e., β is closed, but it fails to be exact, because θ is not aglobally defined function.

2.5 The General ‘Integration’ Problem for Vector Fields

2.5.1 The Integration Problem for Vector Fields: FrobeniusTheorem

After the discussion in the last few sections we have arrived to the understanding thatour modeling of dynamical systems is done in terms of vector fields, i.e., first-orderdifferential equations, or equivalently, in the algebraic setting we have started todevelop, as derivations in an algebra of smooth functions. We have also emphasizedthat in the case of linear systems the system is described completely by means of aflow of linear maps, and in general—nonlinear—case, the uniqueness and existencetheorem of solutions of initial value problems for ordinary differential equations,guarantees the existence of local flows describing the dynamical behaviour of oursystem. The existence of such flows, globally defined in the case of linear systemsand only locally defined in the general case, does not imply that we have a simpleway of computing it.

We will say that the ‘integration’ problem for a given dynamics � consists indetermining its (local) flow ϕt explicitly.

Again a few remarks are in order here regarding what do wemean by the ‘explicit’determination of the flow. For instance, an ‘explicit’ solution of the dynamics couldbe an approximate numerical determination of the solution given for some initialcondition x0 and a time interval [0, T ]. Varying the initial condition x0 using somediscrete approximation on a given domain U would provide an approximate explicitdescription of the dynamics. Unfortunately that is the most that can be done inmany occasions when dealingwith arbitrary nonlinear equations, and often only afterdevising very clever numerical algorithms and solving a number of hard problemsregarding the stability and convergence of them.Even solving the integration problemfor linear systems could be a hard problem. As we know the flow of the system isgiven by ϕt = exp t A, so it seems that we have a closed expression for it, hence an‘explicit’ description of the dynamics. However that is not so. Of course the infinite-dimensional situation, like the oneswe facewhendealingwithMaxwell, Schrödinger,and other systems of interest, could be very hard to analyze because the structureof the linear operator A could be difficult to grasp, but also the finite-dimensionalcase could have interesting features which are not displayed in the simple mindedexpression for the flow above. Thuswe have seen that looking for constants ofmotionand symmetries is often quite helpful in discussing the structure of linear systemsoffering new and deep insights into its properties, recall for instance the discussionof the harmonic oscillator in Chap.1. That is the existence of structures compatiblewith the dynamics provides useful leads to analyze it even in the linear situation.

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Evenmore, it is a fact that many dynamical systems arising from physical theorieshave a rich structure (for instance symmetries, constants of motion, Hamiltonianand/or Lagrangian descriptions, etc.) and exploring such intricacies has been provedto be the best way to approach their study. Actually it happens that in some particularinstances, a judiciously and in some cases, extremely clever, use of such structuresleads to an ‘explicit’ description of the dynamics, where now ‘explicit’ means that theactual solutions to the dynamical equations can be obtained by direct manipulationof some algebraic quantities. In such cases the systems are usually called ‘integrable’with various adjectives depending on the context. For instance if the system is Hamit-lonian and it possesses a maximal number of independent commuting constants ofmotion, the system is called completely integrable, etc.

As it has already been stated, our interest in this work is focused in unveiling thesimplest and most significative structures compatible with a given dynamics but asit will be shown along these pages, anytime that there is a new structure compatiblewith the dynamics, we learn something on its integration problem. To the point thatin some cases we can actually integrate it. This process will be spread out along thebook and it will culminate in the last two chapter where the solution of the integrationproblem for various classes of dynamics will be discussed at length.

We would like to close this digression pointing out another twist of the ‘integra-tion’ problem that permeates some parts of this work and that has played a relevantrole in the development of modern geometry. Clearly if instead of having a singledynamics, let us say now, a vector field, we had two or more, we may ask againfor the determination of the solutions of all of them. To be precise, suppose that weare given vector fields X1, . . . , Xr in some vector space E (or on an open set in it).We can integrate them locally and obtain solutions x(1)(t1), . . . , x (r)(tr ) for a givencommon initial data x0. Nothing fancy so far, but we may ask, can be combine the rfunctions above in a single one, i.e., does there exist a function x(t1, . . . , tr ) such thatit reproduces the integral curves of the vector field X1 when we fix the parameterst2, . . . , tr and so on? Or, in other words, if we change the initial data x0 moving itfor instance along the solution of the flow of the first vector field the solutions wewould obtain now will be compatible with the ones obtained if we move the initialdata in the direction of any other vector field?

The answer to this question is the content of the so-called Frobenius theorem andprovides the backbone for the theory of foliations, but again we could raise the samequestions as in the case of a single vector field: can be describe ‘explicitly’ suchcollective solutions?

Theorem 2.14 (Frobenius theorem: local form)Let X1, . . . , Xr be a family of vectorfields on an open set U of a linear space E such that the rank of the linear subspacespanned by them at each point is constant. Then, this family can be integrated in thesense before, i.e., for each point x ∈ U there exist an open set V ∈ R

r and a smoothinjective function ϕ : U ⊂ R

r → U ⊂ E such that the local flows of the vectorfields Xi are given by the curves ϕ(c1, . . . , ci−1, t, ci+1, . . . , cr ) iff [Xi , X j ] can beexpressed as a superposition of the vector fields Xk .

2.5 The General ‘Integration’ Problem for Vector Fields 115

2.5.2 Foliations and Distributions

We can get a better grasp on the meaning of the local form of Frobenius theorem,Theorem 2.14, by ‘geometrizing’ the notion of ‘collective solutions’ used before.First we will define a smooth submanifold M ofRn of dimension r as a subset ofRn

such that at each point x ∈ S there is a neighborhood U of it which is the graph of asmooth function � : V ⊂ R

r → Rn−r , and open set V ⊂ R

r , that is, x = (u,ψ(u))

for all x ∈ U ⊂ M , u ∈ V (see a detailed account of the notions of manifolds andsubmanifolds in Appendix C),

In particular a submanifold M of dimension r of Rn can be defined in virtue ofthe Implicit Function Theorem (10.44), as a level set of a regular value c of a mapF : Rn → R

n−r . If we allow c to vary in a small neighborhood of Rn−r over therange of the map F , we will generate a family of submanifolds such that one andonly one such submanifold will pass through each point of Rn (or of the domainof F that, in case it is not the whole of Rn , will be always assumed to be an opensubset thereof), and we obtain what might be called a ‘slicing’ of Rn (or, again, ofthe domain of F) into a ‘stack’ of closely packed submanifolds. That is the basicidea of what ‘foliating a manifold’ will mean for us, an idea that we will try to makeslightly more precise here, together with the associated notions of ‘distributions’ andof ‘integrable distributions’.

So, let us begin with a more precise definition of what we mean by a ‘foliation’of a manifold.

Definition 2.15 Let U be an open set of a linear space of dimension n. A foliationL of U (of codimension m) is a family {Lα} of disjoint connected subsets Lα ofU (to be called from now on the leaves of the foliation), one passing through eachpoint of M , such that the identification mapping, i : Lα → U is injective, and foreach point x ∈ U , there exists a neighborhood V of x such that V is diffeomorphicto Lx ∩ V × B where Lx is the leaf passing through x and B is an open ball in Rm .

Then, L will consist of a family of connected submanifolds, each of dimensionn − m which stack together to fill up U .

A generalized notion of foliation including the possibility of having leaves of dif-ferent dimension could be introduced. Sometimes such foliations are called singularfoliations. In what follows we are going to discuss mainly non-singular foliations,even though throughout the text, examples of singular foliations will show up (andwill be discussed in its context). We will give now some simple examples of folia-tions, referring to the literature [AM78, AM88, BC70, MS85] for further details:

1. The simplest example of a foliation is provided by the set of the integral curvesof a non-vanishing vector field. The leaves of the foliation are its integral curves.They are all one-dimensional for non-vanishing vector fields, but could be alsozero-dimensional if we allow the vector field to vanish somewhere.For example, in: M = R

2 −{0} with the usual Cartesian coordinates, the (imagesof the) integral curves of the Liouville field:

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� = x∂

∂x+ y

∂

∂y(2.122)

(or: � = r∂/∂r in polar coordinates: x = r cos θ, y = r sin θ) are the rays fromthe origin: (r > 0, θ = const.). They are all one-dimensional and diffeomorphicto each other. If the origin were included, then 0 itself would be an integral curve,thus providing us with an example of a singular foliation, i.e., one whose leavesare not all of the same dimension.

2. A similar example is provided by the following construction. Consider the map,

ϕ : U = R3\ {0} → S2 (2.123)

which maps all the points (in spherical polar coordinates) (r, θ,φ) ∈ U to thepoint n = (θ,φ) in S2. Then: L = {

ϕ−1 ( n)}

n∈S2 will foliate U with leaves thatare rays through the origin.

3. Consider next in:

M = R3 − {(−1, 0, 0) ∪ (1, 0, 0)} (2.124)

again with the usual Cartesian coordinates, the foliation: � = {lb}b∈R whoseleaves lb are given by,12

[(x − 1)2 + y2 + z2

]−1/2 −[(x + 1)2 + y2 + z2

]−1/2 = b (2.125)

This foliation is depicted in the figure below:Being level sets, the leaves are regular submanifolds in the sense of Appendix C.They are all one-dimensional, compact and diffeomorphic to each other for b �= 0.However, the leaf corresponding to b = 0 is the y-axis, which is again one-dimensional but non-compact.

4. If we change the relative sign in the left-hand side of Eq. (2.125) and considerinstead,

[(x − 1)2 + y2 + z2

]−1/2 +[(x + 1)2 + y2 + z2

]−1/2 = b, b > 0 (2.126)

we obtain the foliation that is depicted in the figure below:

Now, the leave corresponding to b = 2, which contains the origin, is the ‘bubble-eight’ (8), which is not even a submanifold of R3 (with the induced topology). Ontop of that, the leaves are not connected for b > 2.

12 The leaves are essentially the equipotential surfaces of two opposite electric charges (a dipole)located at (−1, 0, 0) and (1, 0, 0) respectively.

2.5 The General ‘Integration’ Problem for Vector Fields 117

Fig. 2.1 Leaves of the foliation defined by equipotential surfaces of two charges of the same sign(left), and two charges of opposite sign defining a singular foliation (right)

The examples given above show that foliations can exhibit various kinds ofpathologies. In order to avoid them, we will always consider what are called regu-lar foliations [MS85], that, among other desirable features, are characterized by theleaves being all of the same dimension and all diffeomorphic to each other (Fig. 2.1).

2.5.2.1 Distributions and Integrability

Let � be a nonsingular foliation of M of dimension n. Then, a leaf lα of the foliationpasses through each m ∈ M . The tangent space Tmlα will be a vector space ofdimension n and will be spanned by a set of n vectors in Tm M . At least locally, i.e.,in a neighborhood U of m, it will be possible to single out a set of n vector fields:X1, ..., Xn ∈ X (U ) that span Tmlα for all m ∈ U .

AdistributionDwill be the assignment of a similar set of vector subspaces of Tm Mat each point m ∈ M , all of the same dimension n, spanned in each neighborhood Uby a set of smooth independent local vector fields X1, ..., Xn ∈ X (U ). The X j (m)’s,j = 1, ..., n will be called a basis for the distribution at the point m, that will bedenoted as D (m).

In the specific case of the distribution associatedwith a foliation�, the distributionwill be denoted as D�, and,

Tmlα = D� (m) . (2.127)

As a simple example, we may consider the foliation of M determined by the integralcurves (actually the images in M of the integral curves) of a vector field X havingno zeros, so that all the leaves of the foliation will be one-dimensional. Denoting asDX the one-dimensional distribution associated with this foliation, we will have,

DX (m) = span (X (m)) ≡ {aX (m) | a ∈ R} . (2.128)

It is clear that every foliation� defines a distribution, one that, moreover, satisfiesthe property expressed by Eq. (2.127). Whenever, vice versa, a distribution D isgiven satisfying the same property, i.e., we can find at every point m a submanifold l


passing through m and such that D (m) spans Tml or, stated otherwise, we know theright-hand side of Eq. (2.127) and we are able to solve for the left-hand side, we willsay that the distribution is integrable, and l will be called an integral manifold of D.

A distribution D will be said to be involutive if it is closed under commutation,i.e., if,

[X, Y ] ∈ D, ∀X, Y ∈ D. (2.129)

In the case of the distribution D� associated with the foliation �, the involutiv-ity property of Eq. (2.129) is granted by the fact that (cfr. Eq. (2.127)) lα is a(sub)manifold, and the tangent vectors to a (sub)manifold are obviously closed undercommutation. Therefore integrable distributions are involutive.

The converse of this constitutes the main content of Frobenius’ theorem [BC70,Wa71], which we will not prove here but simply state as:

Theorem 2.16 (Frobenius integrability theorem) A distribution is integrable if andonly if is involutive.

Not all distributions need to be involutive, as the following example shows.Consider, on R

3, the two-dimensional distribution D defined (globally) by thevector fields: X = ∂/∂x and: Y = ∂/∂y + x∂/∂z. As: [X, Y ] = ∂/∂z /∈ D,the distribution is not integrable. In fact, if it were, we could find a surface definedas a level function of a function, i.e., as: f (x, y, z) = b for some f ∈ F (R3

)and b ∈ R such that X and Y span the tangent space at every point, i.e., such that:LX f = LY f = 0. But it is immediate to see that the only solution to these equationsis: f = const., i.e., no level surfaces at all.

To complete this rather long digression, we state now the conditions of the Frobe-nius theorem in a dual way, i.e., in terms of one-forms. Let then θ1, ..., θm−n be a setof linearly independent one-forms, and let: ω = θ1 ∧ ... ∧ θm−n . The intersection ofthe kernels of the θ j ’s is a distribution that will be involutive if one of the followingequivalent conditions holds:

1. θi ∧ dω = 0, ∀i = 1, . . . , m − n.2. There exists a 1-form α such that: dω = α ∧ ω.3. There exist local one-forms αi

j , i, j = 1, ..., m − n such that: dθi = αij ∧ θ j .

4. There exist functions f i and gij such that: dθi = gi

j d f j .

2.6 The Integration Problem for Lie Algebras

In this section we will solve the problem of integrating the Lie algebra of symmetriesof a given dynamical system, rephrasing in this way the so-called Lie’s third theorem.In doing so we will move forward towards the notion of Lie group.

2.6 The Integration Problem for Lie Algebras 119

Let us recapitulate a situation we have found in the previous sections. Let �

be a vector field. We had seen that the collection of infinitesimal symmetries of adynamical system is a real Lie algebra (see Sect. 3.6.3). Let us suppose that the Liealgebra of infinitesimal symmetries of � is finite-dimensional generated by a familyof vector fields X1, . . . , Xr such that;

[Xi , X j ] = ci jk Xk ,

where ci jk are the structure constants of the Lie algebra. The analysis of such dynam-

ics will be done in full depth in Chap.9. Here we will try to understand the structureof the flows of the family of vectors Xi . Because the vector fields Xi do not commutewe cannot pretend to integrate their flows by using a single function ϕ(t1, . . . , tr )(remember the discussion on Sect. 2.5 about the simultaneous integration of a familyof vector fields). However it may happen that there is another space such that theflows of the vector fields are just curves on it. It happens that such space is what wecall a Lie group and we will devote the next few sections to this idea.

2.6.1 Introduction to the Theory of Lie Groups:Matrix Lie Groups

Wehave already found in the preceding sections someexamples of groups,GL(n,R),U (n), etc. Throughout the book many other groups are going to have a relevant role,like SU (2), SO(3), SL(2,C), H W (n), etc. A rigorous discussion of their propertiesand representations would require a detailed development of the theory of Lie groups(i.e., groups that are equipped with the structure of a smooth manifold). We will notattempt to do that in this book, referring the reader to the abundant literature on thesubject (see for instance [Wa71], etc.) even if we will provide an intrinsic definitionof the class of Lie groups in the next chapter.

However, most of the previous examples (and many more) are groups of matrices,or as we will call them, matrix Lie groups, and contrary to the general class of Liegroups, only elementary calculus is required to discuss some of their properties andstructure. This sectionwill constitute an approximation to the theory of Lie groups bymeans of the study of an important family of them, closed subgroups of the generallinear group.

Immediately after we will address the problem of integrating Lie algebras, theinfinitesimal trace of a Lie group, arriving to the main theorem in Lie’s theory thatestablishes a one-to-one correspondence between Lie algebras and connected andsimply connected (Lie) groups.

2.6.1.1 The General Linear Group and the Orthogonal Group

Consider the general linear group in Rn , that is the set of all invertible n × n real

matrices GL(n,R). It can be considered as an open subset in Rn2 by means of themap:

http://dx.doi.org/10.1007/978-94-017-9220-2_3

http://dx.doi.org/10.1007/978-94-017-9220-2_9


GL(n,R) → Rn2; A = (ai j ) �→ (a11, . . . , a1n, . . . , an1 . . . , ann) .

Obviously, the set GL(n,R) is a group, because if A, B ∈ GL(n,R), thenAB ∈ GL(n,R) and A−1 ∈ GL(n,R). The multiplication function is differen-tiable, because

(AB)i j =n∑

k=1

aikbk j ,

say, the elements (AB)i j are quadratic polynomial functions of the elements of Aand B, respectively.

In all that follows we will assume that GL/n,R) is a subset of Rn2 becauseof the previous identification. Because the determinant map det is continuous (is apolynomial of degree n), we get that the group GL(n,R) = det−1(R − {0}) is anopen subset of Rn2 .

If we consider now the group O(n) of orthogonal matrices, O(n) = {R ∈GL(n,R) | RT R = R RT = I} from the orthogonality condition Rt R = I, we get:

∑j

Ri j R jk = 0, i �= k;∑

j

R2i j = 1; i = k, (2.130)

showing that |Ri j | ≤ 1, for all i, j . The subset O(n) ⊂ Rn2 is closed because it is

defined by a set of algebraic equation (2.130). Notice that O(n) is F−1(I) whereF : GL(n,R) → GL(n,R) is the smooth map F(R) = RT R. Moreover O(n) isbounded because

∑i, j R2

i j = n, then O(n) is compact. Notice, however that O(n)

is not connected because in general we just have (det R)2 = 1 and O(n) has twoconnected components characterized by the sign of det R. The connected componentcontaining the neutral element is a normal subgroup:

SO(m,R) = {X ∈ GL(m,R) | X T X = Im, det A = 1}.

Wemay now compute its tangent space as a subset ofRn2 (it is actually a subman-ifold, see Appendix C). Let γ : (−ε, ε) → O(n) be a smooth curve passing throughthe identity matrix, i.e., γ(0) = I. Then γ(t)tγ(t) = I for all t and computing thederivative at t = 0 we get: γ(0)T + γ(0). Then the tangent vector γ(0) is a skewsymmetric matrix. Conversely any skew symmetric matrix A is the tangent vectorto a smooth curve in O(n). It is enough to consider the curve γ(t) = exp t A. Weconclude that the tangent space to O(n) at the identity can be identified with the setof skew symmetric matrices:

TI O(n) = {A ∈ Mn(R)|AT = −A}.


In a similar way we can compute the tangent space to O(n) at a given orthogonalmatrix R. It suffices to consider a curve γ(t) as before passing through the identityand multiply it by R on the right. Then the tangent space TR O(n) will be identifiedwith matrices of the form AR with A skew symmetric.

The set of skew symmetricmatrices n×n is a linear space of dimension n(n−1)/2andwewill say that the orthogonal group O(n) is amanifold of dimensionn(n−1)/2.

Definition 2.17 Wewill say thatG is amatrix Lie group if it is an algebraic subgroupof GL(n,R) and it is closed as a subset of Rn2.

Groups of n×n matrices with complex coefficients will be considered in a naturalway as subgroups of GL(2n,R) identifying C with R2. Thus the complex entry z jkwill be replaced by the 2 × 2 real matrix

(x jk −y jky jk x jk

)

with z jk = x jk + iy jk .It can be shown that because of the group law any closed subgroup of GL(n,R)

has a well defined tangent space at any point (that is, a smooth submanifold of Rn2 )[MZ55, Gl52].13 Thus the considerations we have made for O(n) can be extendedto any matrix Lie group.

Definition 2.18 The tangent space at the identity TI G of a matrix Lie group G willbe called the Lie algebra of the group and will be denoted as g.

Example 2.6 Not every subgroup of GL(n,R) is a matrix Lie group. Consider forinstance the subgroup of GL(2,C) ∼= GL(4,R) of matrices:

(eit 00 eiλt

)

where t ∈ R and λ is an irrational number. It is easy to check that such subgroup isnot closed in R

4.

2.6.1.2 The Lie Algebra of a Matrix Lie Group

Definition 2.19 A Lie algebra L is a linear space with a skew symmetric bilinearmap [·, ·] : L × L → L such that it satisfies Jacobi’s identity:

[[ξ, ζ],χ] + [[ζ,χ], ξ] + [[χ, ξ], ζ] = 0 ,

13 This is an application of a deep result in the theory of Lie groups, also-called Hilbert’s fifthproblem that shows that any finite-dimensional locally compact topological group without “smallsubgroup” is a Lie group [MZ55].


for all ξ, ζ,χ ∈ L .

Let L be a Lie algebra and B = {Ei } a linear basis, then we get:

[Ei , E j ] = cki j Ek. (2.131)

The constants cki j are called the structure constants of the Lie algebra L with respect to

the basisB. It is immediate to check that the structure constants cki j satisfy ck

i j = −ckj i

and

cli j c

mlk + cl

jkcmli + cl

ki cml j = 0, ∀i, j, k, m. (2.132)

Conversely, given a family of numbers cki j satisfying the previous conditions they

are the structure constants of a unique Lie algebra with respect to some linear basison it.

Example 2.7

1. the associative algebra Mn(R) of n ×n square matrices with real coefficients canbe endowed with the Lie product [A, B] given by the commutator of matrices[A, B] = AB − B A, and then Mn(R) is endowed with a Lie algebra structureof dimension m2. A basis is given by n2 matrices Ei j with elements given by(Ei j )kl = δikδ jl . Each matrix A = (ai j ) ∈ Mm(R) can be written in a uniqueway as a linear combination

A =n∑

i, j=1

ai j Ei j .

The structure constants in such a basis are

[Ei j , Ekl ] = δ jk Eil − δil Ek j ,

because Ei j Ekl = δ jk Eil .2. The cross product of vectors x × y defines a Lie algebra structure on R3.3. Let F be the linear space of smooth functions on R

2n . The bilinear map givenby the standard Poisson bracket:

{ f, g} =n∑

i=1

∂ f

∂xi

∂g

∂xi+n − ∂g

∂xi

∂ f

∂xi+n

defines a Lie algebra structure on F .

Let now G be a matrix Lie group and consider the map� : G ×g → g, defined as

�(g, ξ) = d

dtg · γ(t) · g−1 |t=0= g · ξ · g−1, (2.133)


where γ(0) = ξ and g ∈ G. This map defines an action of G on g, that is, it satisfies:�(g,�(h, ξ)) = �(gh, ξ), �(I, ξ) = ξ, called the adjoint action of G on its Liealgebra g.

We will denote by Adg : g → g the linear map Adg(ξ) = �(g, ξ) = g · ξ · g−1.Moreover Adg ◦Adh = Adgh . Then the adjoint action defines a linear representationof G as linear maps on its Lie algebra.

We have that the tangent space at the identity of a matrix Lie group is a Lie algebra(hence the name).

Proposition 2.20 Let G be a matrix Lie group. The tangent space at the identityg = TIG is a Lie algebra with respect to the commutator of matrices.

Proof Let ξ, ζ ∈ g and g(t) : (−ε, ε) → G be a smooth curve such that g(0) = ξ.Then the curve σ(t) = Adg(t)ζ is in g and σ(0) = ζ. Computing the derivative ofσ(t) at 0, we get:

σ(0) = d

dt(g(t).ζ.g(t)−1) |t=0= ξ · ζ − ζ · ξ = [ξ, ζ],

where we have used that d(g(t)−1)/dt = −g(t)−1 · (dg(t)/dt) · g(t)−1. ��Example 2.8 The following list describes some relevant groups of matrices:

1. SO(n) = {R ∈ GL(n,R)|R Rt = I, det R = 1}.2. U (n) = {U ∈ GL(n,C)|UU† = I }.3. SU (n) = { U ∈ GL(n,C)|UU† = I, detU = 1 }.4. SL(n,R) = { S ∈ GL(n,R)| det S = 1 }.

and their Lie algebras:

1. so(n) = { A ∈ Mn(R)|At = −A,Tr A = 0 }.2. u(n) = { V ∈ Mn(C)|V † = −V }.3. su(n) = { V ∈ Mn(C)|V † = −V,Tr V = 0 }.4. sl(n) = { A ∈ Mn(R)|Tr A = 0 }.

As a consequence we obtain that their dimensions are:

1. dim SO(n) = n(n − 1)/2.2. dimU (n) = n2.3. dim SU (n) = n2 − 1.4. dim SL(n,R) = n2 − 1.

We will devote the next few sections to work out in detail the Lie algebras andother properties of some groups that are of capital importance.


2.6.1.3 The Lie Algebra of SO(3) and SU(2)

The Lie algebra of SO(3) will be obtained by computing the tangent vectors tosmooth curves passing through I. We consider the rotations around the axis:

R(e1, θ) =⎛⎝ 1 0 00 cos θ − sin θ0 sin θ cos θ

⎞⎠ , R(e2, θ) =

⎛⎝ cos θ 0 sin θ

0 1 0− sin θ 0 cos θ

⎞⎠ ,

and

R(e3, θ) =⎛⎝ cos θ − sin θ 0

sin θ cos θ 00 0 1

⎞⎠ .

Denoting by Mi the tangent vector to R(ei , θ) at θ = 0, we get:

M1 =⎛⎝0 0 00 0 −10 1 0

⎞⎠ , M2 =

⎛⎝ 0 0 1

0 0 0−1 0 0

⎞⎠ , M3 =

⎛⎝ 0 −1 01 0 00 0 0

⎞⎠ , (2.134)

and we check immediately:

[M1, M2] = M3, [M2, M3] = M1, [M3, M1] = M2

that will be written as:

[Mi , M j ] = εi jk Mk. (2.135)

Conversely, if A is in the Lie algebra, i.e., is a skew-symmetric 3 × 3 matrix, weget:

A =⎛⎝ 0 −ζ3 ζ2

ζ3 0 −ζ1−ζ2 ζ1 0

⎞⎠ (2.136)

that can be written as: A = ζ1M1 + ζ2M2 + ζ3M3. It is clear that this constructiongeneralizes immediately to SO(n,R) with n ≥ 3.

To obtain the Lie algebra of SU (2) it is sufficient to consider the curves:

U (ek,ϕ) = cosϕ − iσk sinϕ (2.137)

where σk denotes Pauli’s sigma matrices:


σ0 = I2 =(1 00 1

), σ1 =

(0 11 0

), σ2 =

(0 i−i 0

), σ3 =

(1 00 −1

).

(2.138)Then we get:

U (e1,ϕ) =(

cosϕ −i sinϕ−i sinϕ cosϕ

),

dU (e1,ϕ)

dϕ|ϕ=0=

(0 −i−i 0

)= N1 = −iσ1

(2.139)

U (e2,ϕ) =(cosϕ − sinϕsinϕ cosϕ

),

dU (e2,ϕ)

dϕ|ϕ=0=

(0 −11 0

)= N2 = −iσ2

(2.140)

U (e3,ϕ) =(

e−iϕ 00 eiϕ

)dU (e3,ϕ)

dϕ|ϕ=0=

(−i 00 i

)= N3 = −iσ3 (2.141)

and the Lie algebra su(2) of SU (2) is given by:

[N1, N2] = 2N3, [N2, N3] = 2N1, [N3, N1] = 2N2 . (2.142)

A natural basis for the Lie algebra su(2) consists of the matrices −iσk tangent tothe curves (2.139). We realize immediately that the Lie algebra su(2) is isomorphicto the Lie algebra so(3) of SO(3); the isomorphism η : so(3) → su(2), is given by:

η(Mi ) = 1

2Ni = − i

2σi . (2.143)

However the Lie groups SU (2) and SO(3) are not isomorphic because theirtopological properties are different and if they were they should be homeomorphic,but it is easy to see that SU (2) is simply connected by identifying it with the 3-dimensional sphere while this is not the case of SO(3) (see below). However theyare locally isomorphic, they have the same Lie algebra.

The general theory of Lie groups shows that SU (2) is the universal covering (seeSect. 2.6, Theorem. 2.25) of all Lie groups with Lie algebra isomorphic to su(2). Anyother group possessing the same Lie algebra can be obtained as a quotient group ofSU (2) by a central discrete subgroup.

In our case because the center of SU (2) is Z2, the only two groups with the sameLie algebra are SO(3) and SU (2).

Exercise 2.9 Compute the center of SU (2) and SO(3). Prove that a central subgroupis a subgroup of the center of the group.

The covering map π : SU (2) → SO(3) is defined as follows: let x be a vector inR3 and x · σ the 2 × 2 Hermitean matrix:

x · σ = x1σ1 + x2σ2 + x3σ3 =(

x3 x1 + i x2x1 − i x2 −x3

).


The map x �→ x ·σ defines a one-to-one correspondence betweenR3 and the linearspace of traceless 2 × 2 Hermitean matrices. Then we define:

(π(U ) x) · σ = U ( x · σ)U†, ∀ x ∈ R3. (2.144)

Exercise 2.10 Check that ||π(U ) x || = || x ||, and det π(U ) = 1, hence π(U ) ∈SO(3).

2.6.1.4 More Examples: The Euclidean Group in 2-Dimensionsand the Galilei Group

If we consider the Euclidean group of transformations in two dimensions E(2),

x ′1 = x1 cosϕ − x2 sinϕ + a1,

x ′2 = x1 sinϕ + x2 cosϕ + a2, (2.145)

it is a Lie group of dimension three for which the composition law is

(a ′,ϕ′) · (a,ϕ) = (a ′ + R(ϕ′)a,ϕ′ + ϕ).

These transformations can be written in a matrix form as⎛⎝ x ′

1x ′21

⎞⎠ =

⎛⎝ cosϕ − sinϕ a1

sinϕ cosϕ a20 0 1

⎞⎠⎛⎝ x1

x21

⎞⎠ .

Hence, the infinitesimal generators are just the matrices

J =⎛⎝ 0 −1 01 0 00 0 0

⎞⎠ , P1 =

⎛⎝ 0 0 10 0 00 0 0

⎞⎠ , P2 =

⎛⎝ 0 0 00 0 10 0 0

⎞⎠ ,

with commutation defining relations for the Lie algebra:

[J, P1] = P2, [J, P2] = −P1, [P1, P2] = 0.

Another interesting example is the Galilei group. We can identify it with a sub-group of GL(5,R) (or the corresponding affine group in four dimensions)

⎛⎝ x′

t ′1

⎞⎠ =

⎛⎝ R v a

0 1 b0 0 1

⎞⎠⎛⎝x

t1

⎞⎠ .

The commutation relations defining the Lie algebra of Galilei group are then,


[J, J] = J, [J, K] = K, [J, P] = P, [J, H ] = 0 ,

[K, K] = 0, [K, P] = 0, [K, H ] = P ,

[P, P] = 0, [P, H ] = 0 . (2.146)

Here P are the generators of the one-parameter groups of space translations, H isthat of time translations, J are the generators of proper rotations and K those of pureGalilei transformations.

Finally, when considering one-parameter groups of transformations of an affinespace M , for instance et A, each point x ∈ M , transforms into x ′ = �(et A, x), andfor small values of the parameter t , which we will denote by ε,

x ′i = xi + ε ξi (x) + O(ε2) ,

and

ξ(x) =(

d�(eεA, x)

dε

)|ε=0 .

For instance, for the one-parameter group of translations in the x1 direction forthe case of the Euclidean group in two dimensions, ξ1 = 1, ξ2 = 0, while for theone-parameter group of translations in the other direction, ξ1 = 0, ξ2 = 1. For theproper rotation subgroup, ξ1 = −x2, ξ2 = x1.

In an analogous way, in the group of proper rotations in three dimensions, for thesubgroup of rotations around the axis determined by the vector n, ξi = εi jk n j xk .

2.6.1.5 Group Homomorphisms and Lie Algebras

Definition 2.21 Ahomomorphismbetween theLie algebras (g1, [, ]1) and (g2, [, ]2),is a linear map φ : g1 → g2 such that: φ([ξ, ζ]1) = [φ(ξ),φ(ζ)]2, for all ξ, ζ ∈ g1.If the homomorphism is bijective we will call it an isomorphism.

Example 2.11

1. The Lie algebras so(3) and su(2) are isomorphic, the isomorphism given byequation (2.143).

2. The Lie algebra sl(2,C) is isomorphic to the complexification sl(2,R)C of theLie algebra sl(2,R). (The complexification gC of a real Lie algebra g is thenatural Lie algebra structure obtained on the complexification of the linear spaceg by extending the bilinear map [·, ·] to a complex bilinear map.)

3. The Lie algebra sl(2,C) is isomorphic to the complexification of the Lie algebrasso(3) and so(1, 2) (where so(1, 2) is the Lie group of the linear isomorphismspreserving the metric with signature (− + +)).


Definition 2.22 Given two matrix Lie groups G1 and G2, a Lie group homo-morphism between them is a smooth group homomorphism ψ : G1 → G2 (i.e.,ψ(gh) = ψ(g)ψ(h) for all g, h ∈ G).

Notice that if ψ : G1 → G2 is a differentiable map, the differential of this mapat I is a linear map, dψ(I) : TIG1 → TIG2, that is, a linear map between the corre-sponding Lie algebras. We will denote in what follows the map dψ(I) as ψ∗ and wewill check that it is a Lie algebra homomorphism.

Actually, one-parameter Lie subgroups, described by curves γ : R → G whichare a group homomorphism, namely, such that

γ(t1)γ(t2) = γ(t1 + t2) ,

play a relevant role. Indeed, this last property means that γ(t) is determined by thetangent vector to the curve in the neutral element, γ(0) = e ∈ G. When G is asubgroup of GL(m,R), if A is the matrix

A = d

dtγ(t)|t=0 ,

then γ(t) = et A. In fact, it suffices to take into account the relation γ(t1)γ(t2) =γ(t1 + t2), and to take derivative with respect to t1 at t1 = 0, and then we findAγ(t) = γ(t), and as γ(0) = I , we obtain γ(t) = et A.

Thus, the matrices A obtained as tangent vectors to one-parameter subgroupsof GL(m,R) at the identity matrix, close on the Lie algebra gl(n,R), and thosecorresponding to Lie subgroups of GL(m,R) are Lie subalgebras, i.e. they are linearsubspaces stable under the Lie product.

By using exponentiation we can obtain the elements in a neighbourhood of I ∈ G,and these are generators of G when it is connected. For instance, the set of alltraceless matrices is a linear space and the commutator of two traceless matricesis also traceless. They determine a Lie subalgebra, usually denoted sl(n,R) and byexponentiation of these matrices we obtain the elements in the subgroup SL(m,R).

The exponential map for an arbitrary matrix Lie group G is defined as the mapexp : g → G given by the standard exponential function ofmatrices. Inmore abstractterms, we would use the correspondence above between one-parameter subgroupsand elements in the Lie algebra to define the exponential, that is, if ξ ∈ g and γξ(t)is the corresponding one-parameter subgroup, then exp tξ = γξ(t) for all t ∈ R.

It is not hard to see that the exponential map is surjective in any compact groupbut in non-compact groups is usually not surjective.

Exercise 2.12 Prove that the exponential map is not surjective for SL(n,R) but it issurjective for GL(n,C).

Because Ad exp tξζ = etξζe−tξ , computing the derivative with respect to t , we get:

d

dt|t=0 (Ad exp tξζ) = [ξ, ζ] = ad (ξ)ζ ,


with ad (ξ)ζ = [ξ, ζ]. Thus we get:

Ad exp ξ = exp(ad ξ), ∀ξ ∈ g .

Let us compute now the differential of the exponential map. Letξ : (−ε, ε) → g a smooth curve and we denote by δξ(t) = dξ(t)/dt . The differentialof exp on the tangent vector δξ(t) is by definition exp∗(δξ(t)) = d(exp ξ(t))/dt ,thus exp∗ : Tξg → Texp ξG. It is a simple exercise to check that:

d

dteξ =

1∫0

esξ dξ

dte(1−s)ξds.

Then

exp∗(δξ) = d exp ξ

dt=

1∫0

esξ dξ

dte(1−s)ξds =

⎛⎝

1∫0

esξ dξ

dte−sξ

⎞⎠ eξ

=⎛⎝

1∫0

es ad ξds

⎞⎠ dξ

dteξ =

∞∑k=0

1

(k + 1)! (ad ξ)kdξ

dteξ = F(ad ξ)(δξ)eξ

Then:

exp∗(δξ)e−ξ = F(ad ξ)(δξ). (2.147)

with F(x) = (ex − 1)/x . Notice that if ξ(0) = 0 and we evaluate the previousformula at t = 0, we get exp∗(0) = Id which shows that the exponential map is alocal diffeomorphism.

The next two propositions will provide more information on the relation betweenmatrix Lie groups and their Lie algebras.

Proposition 2.23 Let ψ : G1 → G2 be a homomorphisms of matrix Lie groups,then the differential at the identity ψ∗ : g1 → g2 at the identity is a homomorphismof Lie algebras.

Proof The proof is simple. Consider two vectors ξ and ζ in g1 with integral curvesg(t) and h(t) respectively. Then:

d

dtψ∗(Adg(t)ζ) |t=0= [ψ∗(ξ),ψ∗(ζ)]2.

On the other hand, because the differential is linear, we will get that the previousexpression is equal to:


ψ∗d

dt(Adg(t)ζ) |t=0

and computing it again we get: ψ∗([ξ, ζ]1). ��Thus, associated to any group homomorphism there is a homomorphism between

the corresponding Lie algebras. This relation can be qualified further because of thefollowing theorem that we establish without proof (see [Wa71]):

Theorem 2.24 Let ψ : G → H be a homomorphism of matrix Lie groups and letψ∗ : g → h, be the corresponding Lie algebras homomorphism. Then:

i. If ψ∗ is onto, then ψ is an onto on H0 (the connected component of H containingI).

ii. If ψ∗ is mono, then ψ is mono in a neighborhood of I in G.iii. If ψ∗ is bijective, then ψ is a local isomorphism between G0 and H0.

2.6.2 The Integration Problem for Lie Algebras*

Now we are ready to prove Lie’s third theorem that provides the solution for theintegration of a finite-dimensional Lie algebra of vector fields. The global object thatintegrates a Lie algebra is a Lie group.

Theorem 2.25 (Lie’s integration theorem) Let g be a finite-dimensional Lie algebra,then there exists a unique, up to isomorphisms, connected and simply connected Liegroup G whose Lie algebra is g.

It is interesting to notice that after more than one hundred years since Lie’s con-struction there is not an ‘easy’ proof of this theorem. The simplest way to addressit (and its meaning) is to use Ado’s theorem first [Ja79]. Ado’s theorem establishesthat any finite-dimensional Lie algebra can be seen as a subalgebra of the Lie algebraMn(R) of n × n matrices for some n. Then we can try to work inside the generallinear group GL(n,R). It is important to notice that Ado’s theorem does not extendto Lie groups, in other words, not every Lie group is a subgroup of the general lineargroup (recall Example 2.6). At the end of this section we will comment a bit on an‘intrinsic’ proof of Lie’s theorem without recurring to Ado’s theorem that will besignificant later on.

2.6.2.1 Proving Lie’s Theorem I: Using Ado’s Theorem

To address the proof of Lie’s theoremwithout usingmore sophisticated tools, wemayrely on Ado’s theorem stating that any (finite-dimensional) Lie algebra is isomorphicto a subalgebra of the Lie algebra of n × n real matrices for some n. We will denotesuch Lie algebra as gl(n), and Ado’s theorem can be restated saying that given a


finite-dimensional Lie algebra g there exist n and an injective homomorphism of Liealgebras i : g → gl(n). In what followswewill identify gwith its image i(g) ⊂ gl(n)

and we will not distinguish between the element ξ ∈ g and the n × n matrix i(ξ).We will follow now the arguments in [Mi83b]. We consider the collection of all

smooth maps:

ξ : [0, 1] → g ⊂ gl(n)

such that ξ(0) = ξ(1) = ξ ′(0) = ξ′(1) = 0. Given any such map ξ(t) we mayintegrate the time-dependent linear dynamical system defined on the space of n × nmatrices:

dϕ

dt= ξ(t)ϕ(t), ϕ(0) = 1. (2.148)

Aswe know from the general analysis of linear systems, such an initial value problemhas a unique solution ϕξ : [0, 1] → GL(n,R). The invertible matrices ϕξ(t) areconstructed from a family of fundamental solutions x(i) of the system dx/dt = ξ(t)xwith initial conditions x (i)(0) = ei (see Sect. 2.2.2). Then ϕξ(t) is just the matrixwhose columns are the vectors x(i)(t) solutions of the previous initial value problem.

We notice now that the elements ϕξ(1) ∈ GL(n,R) satisfy:

ϕξ(1)ϕζ(1) = ϕξ�ζ(1), (2.149)

where ξ � ζ is the concatenation of the paths ξ and ζ on g, that is:

ξ � ζ(t) ={2ζ(2t) if 0 ≤ t ≤ 1/22ξ(2t − 1) if 1/2 ≤ t ≤ 1

.

To prove that, check by direct substitution that the curve:

ϕ(t) ={

ϕζ(2t) if 0 ≤ t ≤ 1/2ϕξ(2t − 1)ϕζ(1) if 1/2 ≤ t ≤ 1

satisfies that dϕ/dt = (ξ � ζ)ϕ.Now consider the space of equivalence classes of smooth maps as before

ξ : [0, 1] → g where

ξ ∼ ζ iff ϕξ(1) = ϕζ(1). (2.150)

Let us denote by G such space. We will show now that G is the object integrating g.First we check that we can define a composition law in G as follows: if gξ , gζ denotetwo equivalence classes of paths on g with representatives ξ and ζ respectively, thenwe define:


gξ · gζ = gξ�ζ .

Notice that this composition law is well defined because of Eq. (2.149). This com-position law is associative. The proof requires some work because ξ � (ζ � η) is notequal to (ξ � ζ) � η, however we know that they are homotopic which is enough toguarantee that their equivalence classes with respect to ∼ are the same. There is aneutral element e = g0 corresponding to the trivial curve 0 on g and each elementgξ has an inverse element gξ−1

where ξ−1 is the opposite to the path ξ. Thus the setG becomes a group.

The set G inherits a natural topology from the topology of the space of paths ξ (forinstance that induced by the supremum norm ||ξ||∞ = {||ξ(t)|| | 0 ≤ t ≤ 1} and || · ||any norm in g), and the composition law as well as taking the inverse are continuouswith respect to this topology. Notice that G is trivially simply connected because gis. In this way G becomes a topological group, however we are not interested in thisapproach as we want to construct directly a local identification of G with g.

Given a matrix A we have defined exp A. Similarly we can define ln A (that willbe uniquely determined provided that A is close enough to the identity matrix).14 Ifthe map ξ(t) is small enough (that is ||ξ(t)|| < ε for some ε > 0 and a norm || · ||in g), then ϕξ(t) will be close enough to I for all t ∈ [0, 1]. We want to check thatnow lnϕξ(t) is in g for all t . Once we do that we have identified a neighborhood ofthe identity element e in G with a neighborhood of 0 in g.

To check that lnϕξ is in g we will compute its derivative. Taking A = lnϕξ inEq. (2.147) we obtain:

ξ(t) = F(ad A − I )(δA) ∈ g

as we wanted to show.

2.6.2.2 Proving Lie’s Theorem II: Extending Lie Algebra Homomorphisms

Once we have constructed G out of g we would like to understand how we canconstruct a homomorphism f : G1 → G2 of the groups G1 and G2 obtained bythe procedure above from the Lie algebras g1, g2, that ‘integrates’ a homomorphismα : g1 → g2,

If we consider the path g1(t) : [0, 1] → G1, we define ‘tangent’ path t �→ ξ1(t) =γ′1(t)g

−11 (t) ∈ g1. Now we take its image under α, i.e., t �→ ξ2(t) = α(ξ1(t)) ∈ g2.

Then we solve the differential equation on G2:

dg2

dt= ξ2(t)ϕ2, g2(0) = I.

14 The map exp : gl(n) → GL(n) is differentiable with differential the identity at I , hence by theinverse theorem there is local inverse of exp which is differentiable, that is the map ln we are using.


Thenwe define f (g(1)) = g2(1) and the proof finishes if we show that g2(1) does notdepend on the path g1(t) (we could have worked similar formulae and conclusionsusing a representative ξ1 for g1 in the space of paths in g1).

Now if we have two different paths g1(t) and g′1(t) ending in the same point

g ∈ G1, then because G1 is simply connected g1 and g2 are homotopic, that isthere exists a family of paths g(t, s) all from e to a fixed element g ∈ G1 such thatg(t, 0) = g1(t) and g(t, 1) = g2(t). Then we have the tangent vectors:

X1(t, s) = ∂g(t, s)

∂tg(t, s)−1, Y1(t, s) = ∂g(t, s)

∂sg(t, s)−1,

and after a simple computation we get:

∂X1

∂s− ∂Y1

∂t= [X1, Y1]. (2.151)

Then define X2 = α(X1) and Y2 = α(Y1). Then because α is linear and is a Liealgebras homomorphism we get for X2 and Y2:

∂X2

∂s− ∂Y2

∂t= [X2, Y2].

But these equations are just the compatibility conditions for the system of linearequations:

∂g′(t, s)

∂t= X2(t, s)g′(t, s),

∂g′(t, s)

∂s= Y2(t, s)g′(t, s)

hence this system has a solution g′ : [0, 1] × [0, 1] → G2 that shows that f (g) iswell defined (notice that Y1(t, 1) = 0, then Y2(t, 0) = 0 for all t , then g′(t, 1) isconstant.

Thus we may state:

Theorem 2.26 Let α : g1 → g2 be a homomorphism of Lie algebras, then if G1and G2 are two Lie groups with Lie algebras g1 and g2 respectively, there exists ahomomorphism of Lie groups ψ : G1 → G2 such that α = ψ∗.

2.6.2.3 Proving Lie’s Theorem III: The Hard Proof, Not UsingAdo’s Theorem

Now it is easy to devise how we can avoid Ado’s theorem in the proof of Lie’stheorem. We were using the realization of the Lie algebra g as a subalgebra of thealgebra of matrices gl(n) to define the equivalence relation (2.150) via the explicitintegration of the linear system Eq. (2.148). owever this can be replaced by simplyasking that the two paths ξ(t) and ζ(t) on g are equivalent if they can be joined


by a curve of paths X (s, t) such that it satisfies the compatibility equations aboveEq. (2.151). Then the quotient space of paths module this equivalence relation willgive us the group G as before. Again the hardest step in finishing the proof is toshow that locally G is like g. Again we have to compute lnϕξ and we can proceedalong similar lines as we did before (for that we need to show that the formula for thedifferential of the exponential still makes sense but we will not insist on this here).

Remark 2.14 It is pertinent to notice here that thisway of approaching the integrationof a Lie algebra has been continued in proving a much harder integration problem,that of integrating a Poisson structure solved by Crainic and Fernandez [CF04]. Thenthe compatibility condition is substituted by a more involved condition but the spiritis the same.

References

[MS85] Marmo, G., Saletan, E.J., Simoni, A., Vitale, B.: Dynamical Systems: A DifferentialGeometric Approach to Symmetry and Reduction. John Wiley, Chichester (1985)

[Ar73] Arnol’d,V.I.:OrdinaryDifferential Equations.MITPress,Cambridge, 4th printing (1985).[HS74] Hirsch, M., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra.

Academic Press Inc., New York (1974)[DF79] Dollard, J.C., Friedman, C.N.: Product Integrals. Addison-Wesley, Cambridge (1979)[ZK93] Zhu, J.C., Klauder, J.R.: Classical symptoms of quantum illness. Amer. J. Phys. 61, 605–

611 (1993)[Pa59] Palais, R.: Natural operations on differential forms. Trans. Amer. Math. Soc. 92, 125–141

(1959)[AM78] Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin, Reading

(1978).[Ne67] Nelson, E.: Tensor Analysis. Princeton University Press, New Jersey (1967)[KN63] Kobayashi, S., Nomizu K.: Foundations of Differential Geometry, vol. 2. Interscience,

New York (1963, 1969).[Pa54] Palais, R.: A definition of the exterior derivative in terms of Lie derivatives. Proc. Amer.

Math. Soc. 5, 902–908 (1954)[AM88] Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis and Applications, 2nd

edn. Springer, New York (1988)[BC70] Brickell, F., Clarke, R.S.: Foundations of DifferentiableManifolds (An Introduction). Van

Nostrand, Reinhold (1970)[Wa71] Warner, F.: Foundation of DifferentiableManifolds and Lie Groups. Scott Foresman, New

York (1971)[MZ55] Montgomery, D., Zippin, L.: Topological Transformation Groups, vol. 1. Interscience

Publishers, New York (1955)[Gl52] Gleason, A.M.: Groups without small subgroups. Ann. Math. 56, 193–212 (1952)[Ja79] Jacobson, N.: Lie Algebras (No. 10). Dover Publications, New York (1979).[Mi83b] Milnor, J.: Remarks on infinite-dimensional Lie groups. In: DeWitt B. (ed.) Proceedings

of Summer School on Quantum Gravity, Les Houches (1983).[CF04] Crainic, M., Fernandes, R.L.: Integrability of Poisson brackets. J. Diff. Geom. 66, 71–137

(2004)

Chapter 3The Geometrization of Dynamical Systems

A�E�METPETO� MH�EI� EI�ITOLet none but geometers enter here.

Frontispiece of Plato’s Academy

3.1 Introduction

In this chapterwewould like tomove a step forward and discuss the notions discussedin the previous chapter in such a way that they will not depend on being defined on alinear space. Such carrier space, as it was suggested before would be closely relatedto the notion of the space of ‘states’ of our system, and it is not always true that thereis a linear structure on it compatible with the given dynamics.

It is true however that in most dynamical problems related to physical theories,the measurable quantities that characterize the state of the system are related amongthem, at least locally, by regular transformations on some real linear space, thequantities themselves not determining a linear structure though.

The standard way to model mathematically such situation is by means of thenotion of smooth manifolds. A smooth manifold looks locally like a linear space, sotensorial objects can be defined and constructed locally, while globally it can differfrom a linear space and exhibit a non trivial topology. The consistency of the localconstructions is guaranteed by the transformations between different local picturesthat are required to be as regular as needed. Such transformations correspond to localchanges of coordinates in an operational description of our system and are implicitin the description of any dynamical system.

The algebra of smooth functions on a manifold provides an alternative way ofdescribing the carrier space of a given dynamics, this time focusing the attention onthe quantities that determine it. This point of view, equivalent to the previous one, issomehow more appealing from an algebraic point of view as some of the structuralproperties of the dynamics are reflected directly in terms of the algebraic structure of


135

136 3 The Geometrization of Dynamical Systems

this algebra and it is gaining more weight in describing the foundations of physicaltheories.

In this chapter we will take the point of view of developing the general notionof a dynamical system from the point of view of the algebra of their measurablequantities or observables. Of course the class of algebras that, at least in principle,could constitute the observables of a dynamical system, is extremely large, so someassumptions would have to be made on their structure. We will introduce the notionof differentiable spaces in the first part of the chapter by abstracting some of thefundamental properties of the algebra of smooth functions on a linear space dis-cussed in the previous chapter, essentially assuming that they provide a good ‘local’description of the observables of dynamical systems.

Differentiable spaces actually embrace the notion of smooth manifolds providingthe natural setting for a geometrical-algebraic description of their properties. Exam-ples of both circumstances will be exhibited, that is examples that just amount to thestandard description of smooth manifolds, and situations where spaces other thansmooth manifolds can be used. For the most part of this book however, either smoothmanifolds or just linear spaces are required.

We call the process of extending the definition and description of dynamicalsystems from the linear realm to the much broader class of dynamical systems ondifferentiable spaces the ‘geometrization’ of dynamical systems that provides thetitle for this chapter.

One of the first applications of such geometrization process will be to provide atensorial characterization of the linear structures, i.e., we will dissociate the carrierspace considered as a set of points equipped with a differentiable structure and thealgebraic structures making it into a linear space, i.e., addition and multiplicationby scalars. We will show that such algebraic structure is completely characterizedby a vector field satisfying a few properties. In this way we can formulate the firstcompatibility condition between a dynamical system and a geometrical structureleading to the notion of alternative linear structures for a given dynamical system.An exhaustive discussion on this subject, elementary but subtle, will be provided inSect. 3.3.1.

The geometrical viewpoint emphasized here will payback immediately as it willshow how the tensorial characterization of linear structures will lead right away to thenotion of vector bundles. We will discuss this in Sects. 3.3.2 and 3.3.3. The exteriordifferential calculus on manifolds introduced earlier will be extended to the newsetting as it will be discussed in Appendix E.

Of course it is very difficult to know in advance if given a differentiable space ora smooth manifold, there exists on it a tensorial object on it satisfying some givenproperties. In a linear space, all algebraic tensors are at our disposal, however indifferentiable spaces, even if locally they are like linear spaces, in the large that isnot necessarily so, thus it could happen that there is no a geometrical tensor withthe required properties, e.g., in general there will not exist a vector field with theproperties required to define a linear structure. However locally it is always possibleto construct tensorial quantities associated to algebraic tensors, extending the ‘easy’

3.1 Introduction 137

tensorialization principle stated in the previous chapter to the class of differentiablespaces. A detailed account of this idea will be given in Sect. 3.4.1.

In general we would ask for more. We would like to find out if it is possibleto construct (even locally) geometrical tensors that behaves like algebraic ones. Theexistence or not of suchgeometrical tensors, for a given class of algebraic ones,will becalled the holonomic tensorialization principle.Wewill discuss a few instances whenthe holonomic tensorialization principle could be applied, e.g., symplectic structures.It is clear that in most cases such principle does not apply, e.g., Riemannian metrics.This matters will be discussed in Sect. 3.4.

We will conclude this chapter reviewing the basic notions about dynamical sys-tems in the geometrical language provided by this chapter. This will be the content ofSect. 3.5, thus the arena to study further geometrical structures determined by themwill be set.

3.2 Differentiable Spaces*

As we have mentioned in the introduction, once we have devoted Chap.2 to developa comprehensive discussion on differential calculus on linear spaces, we would liketo extend it to make it suitable for spaces that are not linear. A first choice for thatwould be to introduce the notion of smooth manifold that captures exactly that.

The main idea behind the notion of a smooth manifold is that a smooth manifoldlocally it looks like an open set in a linear space. We can define all notions fromexterior differential calculus as we did in the previous chapter, using these open setsthen, in order to get a global consistent picture, we have to impose that wheneverwe are considering two different local pictures to describe the same object, the twopicturesmust be consistent, that is, itmust behave accordingly transforming smoothlywith the same specific algebraic rules determined by its tensorial nature.

The standard intrinsic approach to build the mathematical description of this ideais to introduce the notion of ‘local charts’ and ‘atlases’. An alternativeway to proceed,simpler in a sense, to build up the basic notions of differential calculus on manifoldsis provided by considering that our spaces are lying inside some finite-dimensionallinear space. Then introduce the objects we are interested in by taking advantage ofthat. Both approaches are well developed in the literature, even if the former one hasgained weight along the years because, even if more abstract, is less cumbersome inits development.

As it was commented in the introduction, we will depart from these approachesbecause, among other reasons, we want to emphasize the algebraic approach to themain notions discussed in this book. Thus we prefer to concentrate on the algebraof functions (and their associated structures) rather than the space itself. Then manyof the arguments become more transparent and they provide with the right intuitionwhen trying to use them in other circumstances, for instance when we try to studydynamical systems whose algebras of observables are non commutative.

http://dx.doi.org/10.1007/978-94-017-9220-2_2


The algebraic approach to describe smooth manifolds has a long history and canbe traced back to the work on singularity theory developed in the 50’s.1 Thus it turnedout to that the algebraic picture provided the tools needed to understand the analogof factorization properties of polynomials in the setting of smooth manifolds.

The notion of differentiable algebras was coined byMalgrange [Ma66] to indicatealgebras F with the properties of algebras of smooth functions of smooth manifoldsand the properties of such algebras were characterized intrinsically. It turns out thatsuch algebras are always quotients of the algebraC∞(RN ) by closed (with respect tothe appropriate topology) ideals J , thus in order to construct a differential calculusall that we have to do is to consider the differential calculus on R

N and restrict it tostructures that are compatible with the ideal J .

Wewill devote the next few sections tomake these notions precise and develop thebasic notions of exterior calculus onmanifolds from this perspective. It is remarkablethat this approach allows us to extend easily the well-known notions of exteriorcalculus in manifolds to more general spaces, for instance possessing some ‘mild’singularities, like orbifolds.

The family of spaces described in this way are called differentiable spaces, incontrast to differentiable or smooth manifolds, and there exists a substantial body ofwork around them. Because one of the main themes of this book is the systematicdescription of nonlinear systems obtained by reduction from simpler ones and takinginto account that in the reduction process singularities could arise (and they really do),the framework provided by differentiable spaces is particularly suitable to discusstheir properties. In any case, for the sake of the reader, Appendix C contains a conciseself-contained introduction to smooth manifolds using the standard language of localcharts, atlases, local coordinates, etc.

3.2.1 Ideals and Subsets

The main idea behind the notion of differentiable algebra is the description of asubset Y of a given set X by means of an ideal in the algebra of functions of the later.Thus, consider X be a set, i.e., a well specified collection of objects, and F(X) itsalgebra of real-valued functions f : X → R, that is,F(X) carries a natural structureof associative algebra by means of the operations,

( f + g)(x) = f (x) + g(x), (λ f )(x) = λ( f (x))

( f · g)(x) = f (x)g(x), ∀x ∈ X, f, g ∈ F(X), λ ∈ R .(3.1)

Notice that both, the pointwise addition and product, are induced from the additionand multiplication in R. Moreover the algebraF(X) has a unit given by the constantfunction 1 (1(x) = 1,∀x ∈ X ) and is obviously commutative or Abelian.

1 In a wider perspective this approach is foundational in what is called today Algebraic Geometry.

3.2 Differentiable Spaces 139

If Y ⊂ X is a subset of X , then the set of functions which vanish on Y define anideal of F(X) (bilateral because F(X) is Abelian). In fact, denoting by JY such set,i.e.,

JY = { f ∈ F (X) | f (y) = 0,∀y ∈ Y } , (3.2)

we have that if f ∈ F(X) and g ∈ JY , then clearly f · g ∈ JY . However such idealis not maximal if Y contains more than just one point. It is a simple exercise to checkthat maximal ideals are given precisely by the sets of functions of the form JY whenY = {y} consists ofly of one point y of X .

We may try to introduce a notion of distance on the space of functions F(X) byusing a norm, for instance, we may use the supremum norm given by: || f ||∞ =sup{| f (x)| | x ∈ X}. However such norm is not defined in general for all possiblefunctions on X . If we restrict our attention to the subalgebra B(X) of F(X) ofbounded functions on X , the norm above is well defined andB(X) becomes a Banachspace.2 It also becomes a Banach algebra in the sense that:

|| f · g||∞ ≤ || f ||∞||g||∞, ∀ f, g ∈ B(X).

If the set X carries some additional structurewewould like to restrict the algebra offunctionsF(X) and consider only those that are compatible with the given structure.For instance, if X is a topological space, thenwewould like to consider the subalgebraC(X) of F(X) of continuous functions on X . Again by considering the subalgebraof bounded continuous functions on X we can equip this space with a natural norm,the supremum norm as before. We just want to mention here that these algebras haveconstituted one of the main sources of inspiration for the theory of operator algebrasdeveloped along the XXcentury.

If Y ⊂ X now is a subset of the topological space X and we consider it with theinduced topology from X , sometimes called the trace topology on Y , then the idealJY of bounded continuous functions on X vanishing at Y is closed with respect tothe topology induced by the norm || · ||∞ above. Even more, it is easy to convinceoneself that the spaces B(Y ) of bounded continuous functions on Y and B(X)/JY

are homeomorphic. The quotient space B(X)/JY carries a natural topology definedby the quotient norm || f + JY ||∞ = sup{|| f + g||∞ | g |Y = 0}.

The proof is simple, we consider the map α : f + JY �→ f |Y from B(X)/JYto B(Y ). This map is well defined and has an obvious inverse, α−1 : fY �→ f + JY

where f is any continuous extension of the continuous function fY defined on Y .3

The map α is not only bijective but it also preserves the topologies. But there is morethen that. If we consider an arbitrary closed ideal J , then it is clear that the subset

2 It is not hard to show that the space B(X) is complete, i.e., any Cauchy sequence of functions isconvergent.3 We must impose some separability properties on our topological space X to guarantee that suchextension always exist but we do not insist on this here.


Y = {x ∈ X | g(x) = 0,∀g ∈ J } is such that JY = J and the correspondencebetween subspaces of X and closed ideals is one-to-one.

However as we keep introducing additional structures on X things become moreinsidious. Thus if we consider now a finite-dimensional linear space E and we con-sider the algebra of smooth functions F(E) = C∞(E) that we have been using sofar, there is no a single norm on such space that captures the “smoothness” of them,i.e., there will be always Cauchy sequences of smooth functions that will not beconvergent (that is whose limit would loose some of the regularity properties of theelements of the sequence).

Thus we will need stronger topologies to capture “smoothness” and, even if weare not going to need this in the future, we just want to point out that there is a naturaltopology on the algebra of differentiable functions that actually characterizes it. Thistopology is called the strongWhitney topology and then the algebraC∞(E) (E finite-dimensional) becomes a Fréchet space (that is a separable metrizable space), suchthat the algebra multiplication is continuous, sometimes called a Fréchet algebra.It suffices to say that a basis of open neighborhoods in this topology are providedby considering for each function f ∈ C∞(E) all smooth functions h such that hand all its derivatives a “close” to f and their derivatives; more precisely, considera covering of E by compact sets Ki and a family of positive numbers εi , then h willbe in the open neighborhood defined by ( f, Ki , εi ) iff ||∂k( f − h)/∂xk ||∞,Ki < εifor all i , where || · ||∞,Ki denotes the supremum norm restricted to functions definedon the compact set Ki .

Now again, if we have a subset Y ∈ X , we may consider the ideal JY of smoothfunctions vanishing at Y . Such ideal in general will not be closed with respect to thestrongWhitney topology described above. In otherwordswemay askwhat additionalstructure should be added to Y that guarantee that JY is closed and that the quotientspace C∞(E)/JY would be isomorphic to that structure. A partial answer is easy toprovide. Suppose that Y is a smooth regular submanifold of E , that is suppose thatthere is a map : E → R

r which is smooth, Y = −1(0) and 0 is a regular valueof (the rank of the differential map ∗ is maximal at all points x ∈ −1(0)), thenit is possible to show that the ideal of functions vanishing at Y is closed.

Exercise 3.1 Prove that the ideal of functions vanishing on a Y = −1(0) with 0 aregular value for : R

m → Rn is closed in the strong Whitney topology.

It can be shown that such ideal JY is the ideal in C∞(E)á generated by thefunction or, using the customary notation from algebra, JY = (). Now, it isclear that the quotient space C∞(E)/JY is isomorphic to C∞(Y ) where we will saythat a function on Y is differentiable if it is the restriction of a smooth function on E(or conversely if it can be extended to a smooth function on E).

For instance, if we consider R and : R → R, (x) = x , then −1(0) = {0}.Because 0 is a regular value of , then {0} is a regular submanifold (of dimension0) and the algebra of smooth functions on it C∞({0}) = R is just C∞(R)/(x).However if we consider the map : R → R, (x) = x2, then 0 is not a regularvalue and the space defined by the algebra F2 := C∞(R)/(x2) is the space of “two


infinitesimally closed points”. Notice that the algebraF2 is the two-dimensional realalgebra generated by elements 1, θ satisfying the relations 12 = 1, 1 · θ = θ · 1 = θ

and θ2 = 0.Thus the strategy to describe smooth manifolds will be to define their algebras of

smooth functions as quotient spaces of C∞(E) by closed ideals. We must remarkhere that because of Whitney’s embedding theorem [Wh44] this definition actuallyincludes all (separable paracompact) smooth manifolds (see Appendix C). Any suchmanifold can be embedded on R

N with N large enough, then its algebra of functionshas the form above. Taking into account all these comments we will proceed now toestablish the appropriate definitions.

3.2.2 Algebras of Functions and Differentiable Algebras

Definition 3.1 A differentiable algebra is a unital commutative associative Fréchettopological4 algebra isomorphic to C∞(RN )/J with J a closed ideal (with respectto the strong Whitney’s topology).

Remark 3.1 As itwas indicated above, because of theWhitney’s embedding theoremany smooth separable paracompact manifold M can be embedded into R

N for someN , hence F(M) ∼= C∞(RN )/J , where J is the closed ideal of smooth function onR

N vanishing on M (as a closed submanifold of it). Hence F(M) := C∞(M) is adifferentiable algebra too. Notice however that the class of differentiable algebrasis strictly larger than the class of algebras of differentiable functions on a manifold(for instance the example above, C∞(R)/(x2) is not an algebra of differentiablefunctions over a smooth manifold).

Given a real topological algebra F we define its spectrum SpecR(F) as the spaceof continuous R-morphisms ϕ : F → R, i.e., ϕ( f · g) = ϕ( f )ϕ(g), f, g ∈ F . It is asimple exercise to show such space can be naturally identified with the space of realmaximal ideals of F .

In the particular instance that F = C∞(E), E finite-dimensional linear space,then it is obvious that SpecR(F) ∼= E , where each point x ∈ E defines a maximalidealJ{x} (that accordingwith the uses of algebra is denoted asmx ). Conversely if wechoose a linear basis ei of E and the corresponding linear coordinate functions xi , anymorphism ϕ : F → R defines a collection of coordinates φi = ϕ(xi ). Then the pointx = φi ei is such thatmx = ker ϕ (actually notice that ϕ( f ) = ϕ( f (0)+ fi (x ′)xi ) =f (0) + fi (x ′)φi = f (x)).5

Notice that given an element f ∈ F we can define the function f : SpecRF → R

(sometimes called the Gel’fand transform)

4 A real algebra with a topology such that the operations of the algebra are continuous and suchthat it is separable and metrizable.5 It can be shown that if M a separable smooth manifold satisfying the second countability axiom,then Spec

R(F(M)) inherits a differentiable structure and it becomes diffeomorphic to M .


f (x) := x( f ), ∀x ∈ SpecRF . (3.3)

In particular if we consider F = C∞(RN ), then f coincides with f .6

The spectrum of a differentiable algebraF provides the local model for a class ofspaces called C∞-spaces (or, sometimes differentiable spaces). These spaces extendin a natural way the notion of smooth manifolds and allow us to work in a large classof topological spaces that share most of the natural notions of smooth manifolds andthe differential calculus attached to them.

Remark 3.2 We will not need the full theory of differentiable spaces to present thegeometrical notions discussed in this book but just the local model for them, thatis, differentiable algebras. In any case we can just say that a differentiable space isobtained by ‘patching together’ the spectra of differentiable algebras (technically itis a ringed space over a sheaf of differentiable algebras) thus, it is clear that once ageometrical notion has been discussed for differentiable algebras it can be extendedwithout pain to the category of differentiable spaces.

Differentiable spaces given by the real spectrum of a differentiable algebra arecalled affine differentiable spaces and again it can be proved that a differentiable spaceis affine if and only if it is Haussdorff with bounded dimension and its topologyhas a countable basis (see for instance [NS03], p. 67). Thus all spaces arising inour examples and applications are going to be affine differentiable spaces, that is,obtained from differentiable algebras.

Given a differentiable algebra F and a point x ∈ SpecRF , we can consider thelocalization Fx := Fmx of F , or the field of fractions of F at x , i.e., elements onFx are equivalence classes of fractions f/g where g(x) �= 0; i.e., g belongs to themultiplicative system Sx = F − mx , and f/g ∼ f ′/g′ iff f g′ = f ′g. Elements onFx are called germs of elements f of F at x and they can be also be considered asresidue classes of elements onF with respect to the ideal of elements ofF vanishingon open neighborhoods of x and will be denoted as [ f ]x .

Given a differentiable algebraF we can consider its space of derivations equippedwith its canonical Lie algebra structure [·, ·]. That is, a derivation X of F is a linearmap such that X ( f · g) = X ( f ) · g + f · X (g), ∀ f, g ∈ F and the Lie bracket

[X, Y ]( f ) = X (Y ( f )) − Y (X ( f )), ∀X, Y ∈ X(F) ,

is a derivation again (which is not true anymore for the anticommutator [X, Y ]+( f ) =X (Y ( f )) + Y (X ( f ))). It is clear that if F = C∞(E)/J and X is a derivation, i.e.,a vector field in E , such that X (J ) ⊂ J , then X induces a derivation on F by thesimple formula:

X ( f + J ) = X( f ) + J .

6 And similarly for C∞(M), M a separable smooth manifold.


It is again true, even if proving it requires some non trivial arguments, that all deriva-tions on F have this form. Thus we may define the space of derivations of F , whichwill be denoted by X(F) or Der(F), as the space of derivations of C∞(E) leavingJ invariant.

Notice that if X is a derivation then X (1) = 0. Moreover if f/g is a representativeof a germ [ f ]x at x , then the derivation X defines a derivation in the space of germsat x . The space of all these derivations is a linear space called the tangent space at xto the affine differentiable space M = SpecF(F) and denoted as Tx M .

The space of derivationsX(F) is aF-module (recall the properties of vector fieldson E) and we can consider in analogy as we did in the case of linear spaces E whenwe introduced 1-forms, the space Der(F)∗ of F-valued F-linear maps on Der(F),i.e., the space of 1-forms of the differentiable algebra F . The space Der(F)∗ is aF-module again and we usually denote it as �1(F) (or simply �1 if there is no riskof confusion).

Given an open set U in M all objects considered so far can be localized to it bystandard constructions, thus we can consider the differentiable algebraFU ,XU ,�1

U ,etc. Moreover differentiable spaces have the fundamental property of the existenceof partitions of unity (see [NS03], p. 52).

The differential map d : F → �1 is defined as in Sect. 2.3.4, Eq. (2.60),

d f (Y ) = Y ( f ) ,

for any Y ∈ X(F). In the case of F = C∞(M), the space �1 is just the module ofsmooth 1-forms �1(M) on the manifold M . Alternatively, we can consider for anypoint x ∈ SpecRF the linear space mx/m

2x denoted by T ∗

x X where X = SpecRF .Again, the space T ∗

x X is the localization of the module �1 with respect to the mul-tiplicative system defined by mx . Then given f ∈ mx , we have that d f (x) := [d f ]x

can be identified with the class of f in mx/m2x .

In what follows we will call the elements f of a differentiable algebra F “func-tions” unless there is risk of confusion.

3.2.3 Generating Sets

The main separability properties regarding ‘points’ and ‘functions’ on differentiablealgebras can be stated as follows:

Definition 3.2 Given a differentiable algebra F , we will say that a family of func-tions g1, . . . , gn separate points if they separate elements vx in Tx X , i.e., for anyvx �= wx there is gk such that vx (gk) �= wx (gk).

A family of functions g1, . . . , gn of the differentiable algebra F will be said toseparate derivations if X (gk) = 0, for all k, implies that X = 0. We will say that asubset S ⊂ F generates the algebra F if it separates derivations.

Notice that if S separates derivations, two derivations X , Y are equal iff X ( f ) =Y ( f ) for all f ∈ S.

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Definition 3.3 We say that the family of functions G = {gα} of the differentiablealgebraF is a generating set if they separate points in X = SpecRF and derivations.

It is not hard to see that smoothmanifolds (paracompact second countable) possessgenerating sets (all it is needed is to embed them inR

n and choose enough coordinatefunctions there). Again, it is not difficult to show that if G is a generating set for thealgebra F as an associative algebra then it is a differential generating set for F .

Notice that since derivations can be localized, generating sets also separate lo-cal derivations. In fact that is used to prove the following important property ofdifferential generating set that justifies its name.

Lemma 3.4 A set of functions G = {gα} of F(M) is a generating set iff the set of 1-forms dG = {dgα} generates the algebraic dual of the space of derivations Der(F)∗(that coincides with the space �1(M) of smooth 1-forms on M if F = F(M) withM a smooth manifold).

Proof We will write the proof for a smooth manifold, but the general case is similar.Consider first an open set U contained in the domain of a local chart of M . If therewere a local 1-form σU that could not be written as σU = σU,αdgα , then the span ofdgα would be a proper subspace of T ∗U and there would exist a vector field X lyingin the annihilator of such subspace. Hence dgα(X) = 0 for all α. This argument canbe made global by using partitions of unity: for any 1-form σ on M there must exista family of functions σα with compact support on M such that σ = σαdgα . �

In the subsequent analysis, we will assume the existence of a generating set G forF . Let us mention another direct consequence of the properties of a generating set Gin the case of smoothmanifolds. Since locally the differentials dgα generate T ∗M , wecan extract a subset dgαi that is locally independent, i.e., such that dgα1∧· · ·∧dgαm �=0 on a open neighborhood of any given point. Therefore, the differentials dgαi willdefine a local coordinate system. Later on we will exploit this fact to write explicitformulae in terms of subsets of generating sets.

A differentiable algebraF(M) is said to be differentiable finite if it admits a finitedifferential generating set G = {g1, . . . , gN }, N ∈ N. This more restrictive conditionis satisfied for instance if the manifold is compact or of finite type because in suchcase it can be embedded in a finite-dimensional euclidean space whose coordinatefunctions restricted to the embeddedmanifold would provide a differental generatingset. As a consequence of the previous discussion, it can be shown that a differentialgenerating set of a differentiable finite algebraF(M) provides, by restriction to smallenough open sets, and sieving out dependent functions, a set of local coordinatesystems, i.e., an atlas for the manifold M .

3.2.3.1 Derivations and Their Flows on Differentiable Algebras

Let � be a derivation of the differentiable algebra F , hence F ∼= C∞(Rn)/J .Denoting as before by X the real spectrum of F , the derivation � defines an element�x ∈ Tx X for each x ∈ X . Hence because X is a closed differentiable subspace of


Rn , the canonical injection i : X → R

n maps �x to a tangent vector i∗�x ∈ Ti(x)Rn ,

moreover we can extend the vector field i∗� along i(X) to a vector field � in Rn . Let

ϕt be the flow of �, hence by construction ϕt leaves i(X) invariant. Let us denote byϕt the restriction of ϕt to X (which always exists because of the universal propertyof closed differentiable subspaces [NS03], p. 60). We will call ϕt the flow of thederivation �. The flow ϕt will act on elements f ∈ F as:

ϕ∗t ( f ) = ϕ∗

t ( f ) + J , f + J = f,

and the flow ϕt can be integrated formally by using a close analogue to formula(2.74):

ϕ∗t ( f ) =

∞∑n=0

tn

n! �n( f ) + J . (3.4)

Notice that we have proved not only that derivations on differentiable algebras can beextended to derivations on C∞(Rn), but that they are continuous maps with respect tothe canonical topology onF because they are the quotient of derivations on C∞(Rn)

preserving the closed ideal J and they are continuous.Finally we will observe that formulae similar to those defining the local flow for

a vector field hold for derivations on differentiable algebras:

d f

dt= �( f ), f ∈ F . (3.5)

and the local flow ϕt satisfies:

dϕt

dt= � ◦ ϕt . (3.6)

3.2.4 Infinitesimal Symmetries and Constants of Motion

3.2.4.1 Infinitesimal Symmetries

Given a derivation � of the differentiable algebra F , we will define the space ofits infinitesimal symmetries as the Lie subalgebra of X(F) of derivations Z suchthat [�, Z ] = 0. More generally, if S is any subset of a Lie algebra L the space ofinfinitesimal symmetries of S is the commutant S ′ of S in L. We will define the nthcommutant of S recursively as:

S ′ = {ξ ∈ L | [ξ, x] = 0, forall x ∈ L}, S(k+1) = (S(k))′, k ≥ 1. (3.7)

Notice that by definition S ⊂ S ′′ where S ′′ is called the bicommutant of S. We willhave:

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Lemma 3.5 Let L be a Lie algebra and S ⊂ L a subset. Then:

S ⊂ S ′′ = S(4) = S(6) = · · · , S ′ = S ′′′ = S(5) = · · · . (3.8)

Moreover, if S is abelian, we have S ⊂ S ′′ ⊂ S ′ and the bicommutant S ′′ is anabelian Lie subalgebra of L.

Proof Notice that if S1 ⊂ S2, then S ′′2 ⊂ S ′

1. Because S ⊂ S ′, we have S ′′′ ⊂ S ′.On the other hand because for any set S ⊂ S ′′ we have that S ′ ⊂ (S ′)′′, we concludeS ′ = S ′′′ and (3.8) follows.

If S is abelian, i.e., [x, y] = 0 for all x, y ∈ S, then S ⊂ S ′, hence S ′′ ⊂ S ′.Moreover if ξ, ζ ∈ S ′′ ⊂ S ′, then ζ ∈ S ′, and [ξ, ζ ] = 0, which shows that S ′′ isabelian.

The bicommutant S ′′ is the minimal abelian Lie subalgebra of L containing S.More can be said when the Lie algebra is represented as an algebra of derivations. Letρ : L → X(F) be a morphism of Lie algebras, i.e., the Lie algebra L is representedas derivations of the algebra F . To each element ξ ∈ L we associate the derivationρ(ξ), satisfying ρ([ξ, ζ ]) = [ρ(ξ), ρ(ζ )].Remark 3.3 General derivations of a topological algebra F will not be in generalcontinuous, hence there is no a canonical topology on X(F).

3.2.4.2 Constants of Motion

Definition 3.6 Given the dynamics � on F , the subalgebra of constants of motionof �, denoted by C(�) (or simply by C) is defined by

C(�) = { f ∈ F(M) | �( f ) = 0}. (3.9)

More generally, we can consider as in the previous section a Lie algebra L whichis represented as derivations of the algebra F (in particular we can consider theLie algebra X(F) itself with the tautological representation). Then given any subsetS ∈ L we can define the subalgebra of its constants of motion C(S) defined as:

C(S) = { f ∈ F | ρ(x)( f ) = 0, for all x ∈ S}. (3.10)

We could use a notation reminiscent of the one used in the previous sectionby denoting the subalgebra of constants of motion by S ′

F ⊂ F and calling it the“commutant” of S inF with respect to the representation ρ. Similarly we can definethe bicommutant of S in F as the Lie subalgebra of L, denoted as S ′′

F , of elementsξ ∈ L such that ρ(ξ)( f ) = 0 for all f ∈ C(S) = S ′

F ⊂ F . We will define

recursivelyS(2k+1)F ⊂ F as the subalgebra of elements f ∈ F such that ρ(ξ)( f ) = 0

for all ξ ∈ S(2k)

F a and S(2k)

F ⊂ L as the Lie subalgebra of elements of L such that

ρ(ξ)( f ) = 0 for all f ∈ S(2k−1)F , with S(1)

F = S ′F = C(S), k ≥ 1.


Lemma 3.7 With the notations above,

S ⊂ S ′′F = S(4)

F = · · · , C(S) = S ′F = S ′′′

F = · · · . (3.11)

Moreover, if S is an abelian subset of L, then S ′′F is the minimal abelian subalgebra

of L containing S.

Notice that the kernel of any derivation of an algebra is a subalgebra. Since C isa subalgebra of F , if M is a F-module, M will be a C-module too. In particularF is a C-module. The space of C-linear maps from F to C is called the space ofsections of F over C (see later for the set theoretical interpretation of such set).The algebra of constants of motion for the derivation � is closed in F with respectto its canonical topology because � is a continuous map. However it is not true ingeneral that a closed subalgebra of a differentiable algebra is a differentiable algebra.We will assume in what follows that the algebra of constants of motion for � is adifferentiable algebra. As we will see in the various examples discussed later on, thisassumption is actually satisfied by a large class of derivations.

We will denote the real spectrum of the differentiable algebra C by C , i.e., C =SpecR(C). Hece the canonical inclusion morphism i : C → F induces a continuousmap π : X → C , such that i = π∗. The injectivity of i implies that the map π isa surjective projection map. Let c ∈ C be a point in the real spectrum of C, thenF/F i(mc) is the differentiable algebra of the fibre π−1(c) of π over the point c,Notice that the flow of � leaves invariant the fibres of π and projects to the trivialderivation on C .

3.2.5 Actions of Lie Groups and Cohomology

Lie groups have appeared so far from two different perspectives: on one side theyhave appeared, both matrix Lie groups and their Lie algebras, as symmetries ofdynamical systems and, on the other handLie groups are the global objects integratingLie algebras. The first perspective offers a ‘dynamical’ interpretation of Lie groupsbecause it presents them as ‘acting’ on the elements describing a dynamical system.

We want to elaborate further on this idea. We would like to consider groupsas transformations on a given space. Thus if we are given an abstract group Gand we provide for each element of the group g ∈ G a transformation ρ(g) onsome space such that the composition law of the group becomes the composition oftransformations we will say that ρ is a representation of the group. Let us make thisdefinition precise in the setting of differentiable algebras.

3.2.5.1 Lie Groups

As we have anticipated before, Lie groups are smooth manifolds G with a smoothgroup structure. We will write first the formal definition in the standard setting of


smooth manifolds and make some commentaries from the differentiable algebrasviewpoint afterwards.

Definition 3.8 A Lie group G is both a group and a smooth manifold such that theproduct map m : G × G → G, m(g, h) = gh for all g, h ∈ G, and the inverse maps : G → G, s(g) = g−1, g ∈ G are diffeomorphisms.

From the point of view of their algebra of functions F(G) the characterization ofsuch properties is natural but requires the dual notion of product, called a coproductthat will lead to the notion of aHopf algebra (see for instance [BM10] for a physicist’sintroduction to the subject).

Consider the map � : F(G) → F(G × G) such that �( f )(g, h) = f (gh) forall f ∈ F(G) and g, h ∈ G. The algebraic tensor product F(G) ⊗ F(G) is densein F(G × G) and we write its completion as F(G)⊗F(G). The map � is called acoproduct and satisfies the dual properties of the group product m:

i. Coassociativity: (� ⊗ I) ◦ � = (I ⊗ �) ◦ �.ii. Counit: The map ε : F(G) → R, given by ε( f ) = f (e) satisfies (ε ⊗ I)(� f ) =

f for all f ∈ F(G).iii. Antipodal map: The map S : F(G) → F(G) given by S( f )(g) = f (g−1),

satisfies that (I ⊗ S)� = (S ⊗ I)� = ε.

It is clear that from the point of view of algebras of functions, the natural wayto consider Lie groups is by means of the Hopf algebra structure on F(G). Thisapproach has proved to be extremely fruitful but we will not discuss the subjectfurther, only invite the reader to dwell on this approach.

Matrix Lie groups together with discrete groups, constitute the simplest familiesof Lie groups. Their smooth structure is inherited directly from the space of matriceson which are embedded (for matrix Lie groups).

We will denote by Lg the natural action of an element g ∈ G in the group Gby left translations, that is Lg : G → G, Lg(h) = gh and, similarly, Rg is theaction of g on G by right translations, Rg(h) = hg. In the case of matrix groups,these operations correspond to the standard multiplication of matrices on the leftor on the right respectively. On Lie groups, both operations Lg , Rg commute, i.e.,Lg ◦ Rh = Rh ◦ Lg for all g, h ∈ G, and they define diffemorphisms of G that maybe interpreted as the group acting on itself by transformations. If the group G is notAbelian, the effect of both could be different. These natural actions of G on itselfinspire a general notion of groups acting on other spaces, which is the natural wayof modeling the notion of group of transformations. We will devote the next sectionto discussing this idea.

3.2.5.2 Representations and Actions of Lie Groups

Definition 3.9 Let G be a group and F be an associative algebra. A representationof G on F is a map ρ : G → Aut(F) such that:


ρ(gg′) = ρ(g) ◦ ρ(g′); ρ(e) = idF ; ∀g, g′ ∈ G . (3.12)

Recall that the group of automorphisms of the differential algebra F consists of alllinearmaps : F → F which are invertible and such that( f · f ′) = ( f )·( f ′),f, f ′ ∈ F . If ρ is a representation of G onF , then ρ(g)( f · f ′) = ρ(g)( f )·ρ(g)( f ′).If F is the algebra of smooth functions on a manifold M , then it can be shown

immediately that any automomorphism of the algebra F is induced from a dif-feomorphism ϕ : M → M . In fact if mx is the ideal of smooth functions vanishingat x then (mx ) is another maximal ideal, hence we define ϕ(x) as the point suchthatmϕ(x) = (mx ). The fact that is bijective implies that ϕ must be bijective tooand, finally, we check that

( f ) = f ◦ ϕ−1 ,

using Gel’fand’s formula Eq. (3.3).Thus if we have a representation ρ of a Lie group G on the algebra of functions

F of a manifold (the same arguments apply for a differentiable algebra), then ρ(g)are automorphisms of the algebra, hence there will exist ϕg : M → M such that

ρ(g)( f ) = f ◦ ϕ−1g , ∀g ∈ G .

The conditions defining the representation imply that ϕg = idM and that ϕg ◦ ϕh =ϕgh . This allows as to define the notion of the action of a group G on a manifold Mas follows:

Definition 3.10 Let G be a Lie group and M a smooth manifold. An action of G onM (on the left) is a smooth map ϕ : G × M → M , such that:

ϕ(e, x) = x, ϕ(gh, x) = ϕ(g, ϕ(h, x)), ∀g, h ∈ G, x ∈ M . (3.13)

Given g ∈ G, we define ϕg : M → M by ϕg(x) := ϕ(g, x). Then propertiesEq. (3.13) imply that ϕg ◦ ϕh = ϕgh and ϕg defines a representation of G on thealgebra F(M). If there is no risk of confusion we will often write g · x (or simplygx) instead of ϕ(g, x) or ϕg(x), and we will call the element gx “the action of g onx”. We will say that the action of G on M is free if gx = x for some x ∈ M impliesg = e and the action will be called transitive if for any x, y ∈ M there exists at leastone element g ∈ G such that y = gx . There are completely analogous definitionsfor actions of groups on the right on a given manifold M . In this case, the action ofg on x will be written as xg instead.

The family of left translations Lg introduced above, define a free and transitiveaction of the Lie group G on itself (in the same way the family of right translationRg above define a right action of G on itself).

We have found actions of groups in various places, either as the definition ofthe groups themselves (think for instance of matrix Lie groups acting on the linearspace that we use to define them) or, for instance, when defining the adjoint action


of a matrix Lie group on its Lie algebra, Eq. (2.133). Exactly in the same terms wecan extend the definition of the adjoint action of a Lie group on its Lie algebra. Wewill denote it again as Ad.

Given a point x ∈ M , the subset of elements Gx = {g ∈ G | gx = x} are calledthe isot(r)opy group of x .7 The set Gx is actually a closed subgroup of G. Againgiven a point x ∈ M the collection of points � · x = {g · x | g ∈ G} is called theorbit through x of G. The set G · x is an immersed submanifold of M . It is easy toprove that G/Gx is diffeomorphic to G · x :

G · x ∼= G/Gx . (3.14)

This diffeomorphism is definedbygx �→ gGx and the inverse is given by gGx �→ gx .

Exercise 3.2 Given a differentiable algebra F , and ρ a representation of G on F ,establish the notions of isotopy group Gx an orbitG ·x of a given x ∈ X = SpecR(F)

and show that:

F(G/Gx ) = F(G)Gx , F(G · x) = F/mG·x ,

where F(G)Gx denotes the subalgebra of smooth functions in F(G) invariant underGx , i.e., f (gh) = f (h) for all g ∈ Gx , and mGx is the intersection of all idealsρ(g)(mx ), g ∈ G. Finally prove that:

F(G)G ∼= F/mG·x .

We would like to point out at this moment that the group of symmetries G of agiven dynamical system � defined on the manifold M acts in M . It is also clear thatbecause � in invariant under the action of G, then � induces a dynamical system inthe quotient space M/G.

3.2.5.3 Actions of Lie Groups and Cohomology

If we have a Lie group G acting on a manifold we can start to understand propertiesof this action by observing how G changes elements on the exterior algebra of M .Thus let us suppose that α is a k-form on M and consider the action g ·α := (g−1)∗α

of g on α (that is the pull-back of α with respect to the diffeomorphism defined byϕ−1g ). Then we define the k-form β(g) as:

g · α = α + β(g), g ∈ G . (3.15)

Now if we compute (gh) · α and g · (h · α) we will obtain:

β(gh) = β(g) + g · β(h) ,

7 In the past it was also-called the “little group” of x .

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that is, β : G → �k(M) is a 1-cocycle in G with values in the G-module �k(M).Recall (see Appendix F for more details) that a p-cochain on G with values in a

G-module � is just a map β : G × p· · · × G → �. The cohomology operator δ maps1-cochains into 2-cochains by δβ(g, h) = β(gh)−β(g)−g ·β(h) (see Eq. (10.282)).Thus the 1-cochain β defined in Eq. (3.15) satisfies δβ = 0, that is, β is a 1-cocycle.Moreover, if we change α adding an arbitrary k-form �, say α′ = α + ψ , then

g ·α′ = α′+β ′(g) = α+ψ+β ′(g) = g ·α−β(g)+ψ+β ′(g) = g ·α′−g ·ψ+ψ−β(g)+β ′(g) ,

that is

β ′(g) = β(g) + g · ψ − ψ,

but if ψ is a 0-cochain (i.e., just an element in �), then δψ(g) = g · ψ − ψ . Then

β ′ = β + δψ ,

and the family of k-forms�k(M) defines a cohomology class [β] ∈ H 1(G,�k(M)).Now, let us assume that dα = ω is such that g · ω = ω for any g ∈ G, that

is ω is G-invariant, then clearly d(g · α − α) = 0, hence dβ(g) = 0, thus β is a1-cocycle on G with values in closed k-forms. Taking now ψ to be closed, we getthat [β] ∈ H1(G, Zk(M)).

In a similar way wemay construct the Chevalley cohomology on the Lie algebra gwith values in ag-module�.Wedefine p-cochains ingwith values in�,C p(g,�), as

maps c : g× p· · · × g → �, and the cohomology operator dρ where ρ : g → Aut(�)

describes the representation of g as automorphisms of �. Then dρ : C p(g,�) →C p+1(g,�) defined as (eq. (10.286)):

dρc(ξ1, . . . , ξp+1) =p+1∑i=1

(−1)i ρ(ξi )c(ξ1, . . . , ξi , . . . , ξp+1)

+∑i �= j

(−1)i+ j c([ξi , ξ j ]ξ1, . . . , ξi , . . . , ξ j , . . . , ξp+1) .

(3.16)

Then a 1-cocycle c1 satisfies

c1([ξ, ζ ]) = ρ(ξ)c1(ζ ) − ρ(ζ )c1(ξ) . (3.17)

Similarly, if � = R and the action is trivial, we get that a 2-cocycle c2 satisfies:

c2([ξ1, ξ2], ξ3) + cyclic(1, 2, 3) = 0 .


It is clear (and the notation suggests it) that if ρ is an action of G on�, and we denoteby ρ the induced action of g on �, that is:

ρ(ξ)(w) = d

dsρ(exp(sξ)(w) |s=0 ,

each cochain β on G defines a cochain on g by derivation, that is, for instance for 1-and 2-cochains we get:

c1(ξ) = d

dsβ1(exp(sξ)) |s=0, c2(ξ, ζ ) = ∂2

∂s∂tβ2(exp(sξ), exp(tζ )) |s=0,t=0 .

Then if β(g) is the 1-cocycle defined by a k-form α, then c1(ξ) = LξM α.

Example 3.3 (Cohomology class defined by a 1-form) A particular instance that willbe of some use later on is provided by the following setting: let θ be a 1-form definedon a manifold M and let G be a Lie group acting on M , and such that ω = dθ isG-invariant. Consider now the 1-cocycle defined by θ , that is the cohomology class[β] ∈ H1(G, Z1(M)), that is g · θ = θ + βg. The corresponding 1-cocycle in g withvalues in Z1(M) will be defined as:

c1(ξ) = LξM θ, ∀ξ ∈ g .

We will assume now that either M is connected and simply connected, that isH1(M) = 0 (which in turn implies that Z1(M) = B1(M) or that βg is exact, that isβg = d Jg for all g ∈ G and Jg : M → R is a real function. Now we have the exactsequence of maps:

0 → R → F(M)d→ B1(M) → 0 .

Then a general theorem (see Appendix F) permits us to construct what is called thelong exact cohomology sequence for the cohomology groups of g with values in theg-modules R, F(M) and B1(M):

· · · → H1(g, R) → H 1(g,F(M)) → H 1(g, B1(M))�→ H2(g, R) → H2(g,F(M)) → · · ·

The connecting homomorphism � : H1(g, B1(M)) → H2(g, R) is defined as:

�c1(ξ, ζ ) = ξ(Jζ ) − ζ(Jξ ) − 2J[ξ,ζ ],

where c1(ξ) = d Jξ .

3.3 The Tensorial Characterization of Linear Structures and Vector Bundles 153

3.3 The Tensorial Characterization of Linear Structuresand Vector Bundles

In the previous sections we have introduced a formalism that allows us to deal withspaces that are not necessarily linear. In particular, we have discussed the geometriza-tion of the notion of dynamical system as a vector field on a differentiable space (orin a smooth manifold).

In what follows we are going to follow this pattern. First we will analyze thestructures compatible with a given dynamics at the linear level and afterwards wewill geometrize them, that is, we will study how these structures and their propertiescan be formulated in a purely geometrical context, or in other words, in generaldifferentiable spaces or smooth manifolds.

Thus each structure that we can describe as a linear algebraic object will be‘promoted’ to a geometrical structure. The first, and most fundamental example isthe geometrization of the linear space structure itself. In other terms we may ask,is there a natural tensorial object that characterizes geometrically a linear spacestructure on a smooth manifold?

The answer to this question is, of course, positive but subtle and we will devotethis section to answer it. Along the discussion we will see that a slight weakeningon their properties leads immediately to the notion of vector bundle, another of thefundamental building bricks of the language of differential geometry.

3.3.1 A Tensorial Characterization of Linear Structures

The ‘easy’ tensorialization of linear maps as linear vector fields, Eq. (2.112), sendingA �→ X A relies, as it was already pointed it out then, on the vector space structure ofE . Indeed, as we have seen, a vector space E carries a dilation operator, or Liouvillevector field �E , which is defined by the flow,

φt (v) = exp (t) v , (3.18)

and that in linear coordinates xi has the simple form,

� = xi ∂

∂xi(3.19)

because,

�(xi )(v) = dxi

dt(φt(v))

∣∣∣∣t=0

= dxi

dt(etv)

∣∣∣∣t=0

= d(et xi (v)

)dt

∣∣∣∣∣t=0

= xi (v) .

(3.20)

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The operator � creates a gradation on the ring F(E) of smooth functions on Eas follows:

Let F (k)(E) (or F (k) for short) be the subspace of homogeneous functions ofdegree k ∈ Z, ≥ 0, i.e.,

F (k) = { f ∈ F |L� f = k f } . (3.21)

Notice that in linear coordinates xi , the subspace F (k) is a finite dimensional spacegenerated by homogeneous polynomials of degree k on the variables x1, . . . , xn .This fact is an easy consequence of the following exercises:

Exercise 3.4 Prove that F (0) = R.

The argument in this exercise can generalized easily to show that:

Exercise 3.5 Prove that f ∈ F (k) if f (tv) = t k f (v) for all t ∈ R and v ∈ E .

Then we can easily prove:

Proposition 3.11 (Euler’s Theorem) The space F (k) is a linear space of dimension(n+k−1k

)of homogeneous polynomials of degree k denoted by Sk[x].

Proof Let f ∈ F k . Consider the function φ(t) as before for a fixed v �= 0. Thenφ(t) = t k f (v) andφ(t) is an homogeneous polynomial of degree k. Taylor’s formulafor φ(t) around t = 0, will give,

φ(t) =∑k≥0

1

k! φ(k)(0) tk (3.22)

and the computation of the k-th derivative of φ(t) gives,

φ(k)(0) = ∂k f

∂xi1 . . . ∂xik(0) xi1 . . . xik (3.23)

Then, taking t = 1, we obtain,

f (x) = 1

k!∂k f

∂xi1 . . . ∂xik(0) xi1 . . . xik (3.24)

thus f ∈ Sk [x]. �Proposition 3.12 The space F (1) of homogenous functions of degree one separatesderivations.

Proof In fact, using that the derivations ∂/∂xi form a basis of X(E), where xi arelinear coordinates on E defined by the choice of a basis of E , it follows immediatelythat if X = Xi∂/∂xi , then 0 = X (xi ) = Xi . �


We have seen to what extent the Liouville vector field�E determines the structureof the ring of functions of a vector space. Conversely, the existence of a vector fieldon a manifold possessing similar properties with respect to the ring of functions isenough to characterize a vector space structure on the manifold. Indeed:

Theorem 3.13 (Tensorial characterization of linear spaces) Let M be a smoothfinite-dimensional manifold of dimension m possessing a complete vector field �

with only one non-degenerate critical point such that:i. F0 = R,ii. The family of functions F1 separates derivations, where,

Fk = { f ∈ F |�( f ) = k f } , k ∈ Z . (3.25)

Then M can be given a unique vector space structure such that � is the vectorfield generated by dilations.

Proof Notice first that F(1) is a finite-dimensional linear space. Actually, F (1) isa linear subspace of the algebra F of smooth functions on M . Then two functionsf, g which are linearly independent define independent covectors d f and dg, butM is finite-dimensional, thus there can be at most a finite number n of linearlyindependent functions inF (1)) and then n ≤ m Then we can choose a linear basis forF (1) made up of the functions f 1, . . . , f n . Then they are functionally independent,i.e., d f 1 ∧ . . .∧d f n �= 0 everywhere (we use that they form a basis and condition i).

On the other hand, because F (1) separates derivations, then n ≥ m, thus n = mand we may use them as a global coordinate chart. It is better to define a family ofvector fields Xi by,

Xi ( f j ) = δji (3.26)

i.e.,

d f j

dsi= δ

ji (3.27)

These vector fields are well defined. If Xi and X ′i satisfy the same definition property,

then for an arbitrary f , (Xi−X ′i ) f = 0, then Xi = X ′

i . Ifwe compute the commutatorof � and Xi we obtain,

[�, Xi ]( f j ) = �(Xi ( f j )) − Xi (�( f j )) = �(δji ) − Xi ( f j ) = −δ

ji (3.28)

Then, [�, Xi ] = −Xi for all i = 1, . . . n. Besides,

[Xi , X j ]( fk) = Xi (X j ( f k)) − X j (Xi ( f k)) = 0 (3.29)

for all k, then [Xi , X j ] = 0. The vector fields Xi are complete, because each one ofthem has the following flow,


φs j

(f k

) = f k, k �= jφs j

(f j

) = f j + s j (3.30)

Then Eq. (3.30) define an action of Rn on M which is free and transitive.

Using the critical point x0 we can identify now Rn with M by means of the map

(s1, . . . , sn) �→ φs1 · · · φsn (x0). �Definition 3.14 A vector field � on a smooth manifold M satisfying conditions (i)an (ii) in Theorem3.13 will be called a Liouville vector field.

It is clear then that if amanifold carries aLiouville vector field itmust be connectedand contractible (because it is a linear space). In fact that is the only topologicalobstruction to the existence of a Liouville vector field on a manifold, because if M isa contractible finite-dimensional manifold, then it is diffeomorphic to a ball, henceto R

n and it can carry a Liouville vector field.It is now clear that a space carrying a Liouville vector field � allows us to define

the set of linear vector fields X as the set of all vector fields which commute with�, [X,�] = 0. From this definition and the Jacobi identity it follows that linearvector fields close in a Lie algebra. If we denote by F (1)(M) as before, the set oflinear functions on M , defined by L� f = f , we also find that for any linear vectorfield X :

LXF (1) ⊂ F (1) (3.31)

Indeed: L�LX f = LXL� f = LX f , then LX f is a linear function if it is not iden-tically vanishing. If F (1) is finite-dimensional and we select a basis f 1, f 2, . . . , f n ,we have,

LX f i = Aij (X) f j (3.32)

and we recover the matrix representation of X . It is clear that Aij (X) depends on

the basis we have chosen.It is possible to use this matrix representation for X to show that its flow exists

and in the given basis is represented by et A. Of course it is possible to write the flowin a way that is independent of the choice of the basis. Indeed, we can write etLX

and we could write,

f (x(t)) = (etLX f )(x(0)) = f (x(0))+ t (LX f )(x(0))+ t2

2!((LX )2 f

)(x(0))+· · ·

(3.33)When f is linear, let us say f = ai f i , it is clear that LX f = ai Ai

j f j and wesimply recover f (x(t)) = f (et Ax(0)), i.e.,

x (t) = et Ax (0) . (3.34)


One has to be cautious however because the right-hand side in terms of Lie seriesonly makes sense if f is real analytic. It should be noticed that as far as the flowis concerned it is enough to restrict to Cω functions because they are enough toseparate points and therefore a cancellation holds true. In other terms, from f (x(t)) =f (et Ax(0)), we can conclude that x(t) = et Ax(0) if the set of functions for whichthe result is true are enough to separate points on the space on which they are defined.

Remark 3.4 The formula we have written for the flow of a linear vector field X ,ϕt = etLX , might have additional limitations when X is not linear. In this case wealso need X to be a complete vector field.

3.3.2 Partial Linear Structures

Let us consider a derivation � in the algebra F(E) of differentiable functions in thelinear space E , i.e., a vector field on E . Then, the operator � may create a gradationon the ring F(E) of smooth functions on E as follows:

Let F (k)(E) be the subspace of homogeneous functions, with respect to �, ofdegree k ∈ Z, k ≥ 0, i.e.,

F (k) (E) = { f ∈ F (E) |L� f = k f } (3.35)

assumed to be not empty for k ≥ 0.We should denote better F (k)(E) as F (k)

� (E) to indicate the dependence on thederivation �, however we will not do it unless risk of confusion. We will often omitthe reference to the underlying space E in what follows.

We first remark that F (0)(E) is a subalgebra, because if f1, f2 ∈ F (0), thenL� f1 = L� f2 = 0, and thus, for each real number λ, L�( f1 + λ f2) = L� f1 +λL� f2 = 0, and L�( f1 f2) = f2 L� f1 + f1 L� f2 = 0. The subalgebra F (0)

will possibly be identified with the algebra of functions defined on a quotient spaceout of E . This quotient space is the collection of “linear fibres” identified by �.On the other side, F (1)(E) is not a subalgebra, but a F (0)(E)-module, because iff0 ∈ F (0)(E) and f1 ∈ F (1)(E), the conditions L� f0 = 0 and L� f1 = f1 implyL�( f0 f1) = f0 L� f1 = f0 f1.

The subalgebra F (0)(E) determines a subalgebra of vector fields, spanned by thevector fields X ∈ X(E) such that,

LXF (0)(E) = 0 (3.36)

Obviously if X, Y ∈ D, then LX f = LY f = 0, ∀ f ∈ F (0)(E), and therefore,L[X,Y ] f = LX (LY f ) − LY (LX f ) = 0, ∀ f ∈ F (0)(E), and therefore,

[X, Y ] ∈ D (3.37)


Before proceeding, let us digress briefly on some additional notions that are neededhere.

According then to what has been discussed in Sect. 2.5.2, the subalgebraD is (cfr.Eq. (3.37)) an involutive distribution, and it is integrable according to Frobenius’theorem [AM78].

Let us assume now that the differentials of functions ofF (0)(E) andF (1)(E) spanthe set of 1-forms �1(E) as a F (0)(E)-module.

Remark 3.5 If we consider � : E → T E , we may consider I : E → T E , I (x) =(x, x). The difference I − � : E → T E , defines another partial structure whichgives a “representative” of the quotient space associated with F (0)(E). This wouldbe a “tensorial version” of the decomposition of the vector space E as a direct sumE = E1 ⊕ E2.

In the present situation, it is actually possible to define a basis for dF (0)(E).Moreover we will assume that the set F (0)(E) is finitely generated by f (0)

i , i =1, . . . , r , and the restriction of � to the level sets of the functions f (0)

i defines linearstructures. Similarly, we will assume that F (1)(E) is finitely generated and we willdenote one such a basis of functions by f (1)

α , α = 1, . . . , k. Then we will say that �is a partial linear structure.

Definition 3.15 Let E be a linear space and let � be a complete vector field whichis a derivation on it. We will say that � is a partial linear structure if both F (0)(E)

and F (1)(E) are finitely generated and moreover dF (0)(E) and dF (1)(E) span theset of 1-forms �1(E) as an F (0)(E)-module. Moreover we require that the criticalset of � be a submanifold whose algebra of functions is isomorphic with F (0)(E).

In order to compare with the usual coordinate approach andwhat we know alreadyof linear structures, we will denote the functions f (0)

i as xi , while we denote the

functions f (1)α by yα Then the coordinate expression for � will be given by,

� = yα ∂

∂yα(3.38)

Definition 3.16 If � is a partial linear structure we will say that a vector field X ispartially linear (with respect to �) if,

[X,�] = 0 (3.39)

Its coordinate expression will be then:

X = f i(

x1, . . . , xk) ∂

∂xi+ aα

β (x) yβ ∂

∂yα(3.40)

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3.3.3 Vector Bundles

Given a partial linear structure � on E , we define a projection map π� : E → B,with the property π∗

�(F(B)) = F (0)� (E). In the particular case we are considering,

we may think of E as a product of vector spaces B × V , where V is a typical fibrerepresenting the linear subspace of E identified by �.

Given coordinates (x, y) ∈ B×V , we find that F : E → R ∈ F (1), i.e., satisfyingL�F = F , can be written in the chosen coordinates as F(x, y) = fr (x)yr . Thuswe can represent the map F : B × V → R as F : B → V ∗. In this way, functionswhich are ‘partially linear’ on E are in one-to-one correspondencewith vector valuedfunctions F : B → V ∗, which leads us to consider, together with the vector bundleB × V , the dual bundle B × V ∗.

It is appropriate to introduce here a notion of ‘projectability’ of vector fields withrespect to the projection π�.

Definition 3.17 A vector field Y ∈ X (E) will be said to be projectable with respectto the projection π� if there exists a vector field XY ∈ X (B) such that

T π� ◦ Y = XY ◦ π� . (3.41)

Equivalently, for every function f ∈ F (B),

LY(π∗

� f) = π∗

�

(LXY f)

. (3.42)

We will call XY the projected vector field associated with Y .It is now a natural question to inquire to what extent the differential calculus we

have elaborated on E can be put to work to give a differential calculus on vector-valued functions F : B → V ∗.

Let us consider first the Lie derivative and exterior derivative of functionsF : E → R satisfying the requirement L�F = 0, that is, of functions in the sub-algebra F (0)(E). It is clear that to remain within the same vector space of func-tions, we should consider vector fields Y which commute with �, in this wayL�LY F = LYL�F = 0. In the chosen coordinates this amounts to requiring,see Eq. (3.40),

Y = hi (x)∂

∂xi+ Aα

β(x) yβ ∂

∂yα. (3.43)

Thus, the projected vector field is XY = h (x) ∂/∂xi . On the other hand, if we wantto associate a vector field on B, say,

X = h (x)∂

∂xi(3.44)


with a vector field on E commuting with � and respecting linearity with projectedvector field X itself, we would set,

YX = hi (x)∂

∂xi+ hi Aα

β,i (x) yβ ∂

∂yα(3.45)

and it is clear that, in this association, we are using a chosen (1, 1) tensor field,namely,

TA = Aαβ,i (x)yβdxi ∂

∂yα(3.46)

so that, with some abuse of notation,

YX = X + TA (X) . (3.47)

Now we consider LY F in the chosen coordinate system. We find,

LY ( fα(x)yα) = h j(

∂ fα∂x j

+ Aβα j fβ

)yα . (3.48)

Therefore, we may conclude that for our function F : B → V ∗, we have a derivativeinduced from X = h j (x) ∂

∂x j given by,

DX F = h j(

∂ fα∂x j

+ Aβα j fβ

)yα (3.49)

or equivalently by,

DX fα = h j(

∂ fα∂x j

+ Aβα j fβ

)(3.50)

which is usually known as the covariant derivative (see below, Appendix E) of thevector-valued function F associatedwith the endomorphism defined by Aα

iβ yβ dxi ⊗(∂/∂yα). Along similar lines we can define now a covariant exterior differential offα by setting,

D fα = d fα + Aβα j fβdx j (3.51)

providing us with a differential 1-form associated with the vector-valued function fα .The expression:

∂ fα∂x j

+ Aβα j fβ

in the right-hand side of Eq. (3.51) shows that we arewriting the derivative in terms offirst-order differential operators (seeAppendixG)which are not derivations anymore


(i.e., they do not satisfy Leibnitz’s rule). For this reason we should not expect thecovariant exterior derivative D to be a derivation. On the other hand, if we go backto our starting point where we have considered the Lie derivative of F along Y weremark the possibility of describing covariant derivatives in terms of derivations,i.e., by going backwards, from F we define a function on the vector bundle, and wereplace the vector field X with the associated vector field YX .

Let us give now two simple examples.

Example 3.6 Consider first: E = R5 = R

4 × R, with coordinates (x, y), B = R4.

The partial linear structure is given by the vector field � = y∂/∂y. Thus F (0) ={f (x) |x ∈ R

4}. Moreover F (1) = {

y f (x) | f ∈ F (0)}. Consider now a basis of

vector fields along B, ∂/∂xμ. A vector field Y , commuting with � and satisfyingthe linearity requirement with respect to the vector fields along the base manifoldwill be,

Yμ = ∂

∂xμ+ Aμ(x)y

∂

∂y(3.52)

associated with ∂/∂xμ by means of the (1, 1) tensor field,

A = y Aμ (x) dxμ ⊗ ∂

∂y(3.53)

The covariant derivative of f : R4 → R

∗ will be expressed in terms of the associatedfunction y f on R

5 as,

LYμ(y f ) = y∂ f

∂xμ+ Aμ(x) f y = y

(∂ f

∂xμ+ Aμ(x) f

)(3.54)

It is now interesting to compute the commutator of two vector fields of the selectedtype,

[Yμ, Yν] =(

∂ Aν

∂xμ− ∂ Aμ

∂xν

)y

∂

∂y= Fμν y

∂

∂y(3.55)

i.e., if Aμ is interpreted as a vector potential on space-time, the commutator providesus with the electromagnetic field tensor Fμν .

Example 3.7 Another example along the same lines would be provided by E =R4+k = {

(x, y) |x ∈ R4, y ∈ R

k}. E = B × V , B = R

4, V = Rk . The par-

tial linear structure is � = ∑kα=1 yα∂/∂yα , the spaces of base functions F (0) ={

f (x) |x ∈ R4}and sections F (1) = {

yα fα (x)}.

Our selected fields Yμ to consider the Lie derivative of functions: F = yα fαsatisfying: L�F = 0, will be,


Yμ = ∂

∂xμ+ Aβ

μα yα ∂

∂yβ(3.56)

with,

TA = Aβμα yαdxμ ⊗ ∂

∂yβ(3.57)

and,

LYμ F = yα ∂ fα∂xμ

+ Aβμα fβ yα = yα

(∂ fα∂xμ

+ Aβμα fβ

)(3.58)

therefore,

Dμ fα = ∂ fα∂xμ

+ Aβμα fβ (3.59)

and, again, the commutator,

[Yμ, Yν] = (Fμν)βα yα ∂

∂yβ(3.60)

defines the Yang-Mills field strength.It is now appropriate to give a ‘coordinate free’ presentation of our covariant

derivatives. On E we consider a (1, 1)-tensor field T with the properties:

T (�) = �, L�T = 0, T ◦ T = T .

We will assume in addition that,

T · dF (0)� = 0 . (3.61)

Then for any vector field commuting with �, say [Y,�] = 0, we consider theassociated vector field YT = (I − T )(Y ) = A(Y ). The Lie derivative along thespecial family of vector fields we have considered earlier is given by,

LYT F = d F(YT ) = d F(A(Y )) (3.62)

which gives immediately DF = A(d F). By replacing F with F , the “vector valued”function, we obtain the usual exterior covariant differential. Thus the role of “vectorpotential” is played by the (1, 1) tensor field A = I − T as a “vector valued” one-form. Again, the curvature as a “vector valued” 2-form is obtained by considering,

[A(X1), A(X2)] − A([X1, X2]) = F(X1, X2) (3.63)


Then, to make contact with more traditional presentations of the covariant deriv-ative, we notice that the algebra of vector fields commuting with � is a subalgebraof the algebra of projectable vector fields. Therefore, starting with a projectable Y ,after projection we get a vector field X , the usual lifting of X to a horizontal vectorfield on E is provided by (I − T )(Y ) = A(Y ). Thus the choice of the (1, 1)-tensorfield T is equivalent to the choice of a ‘connection one-form’.

Example 3.8 The tangent bundle T E . Here the partial linear structure is providedby � = va∂/∂va . F (0)(T E) = F(E) and F (1)(T E) = {vα fα}.

A choice for the (1, 1) tensor field T could be,

T = ∂

∂vi⊗ (dvi − vk�i

k j dx j ) (3.64)

and,

I − T = dx j ⊗(

∂

∂x j+ vk�i

k j∂

∂vi

)(3.65)

It is now possible to consider vector fields on E as vector-valued functions. Theymay be obtained by projecting on E vector fields on T E which commute with �.

On the cotangent bundle T ∗ E , we have the partial linear structure � = pa∂

∂paand we may proceed exactly as for the tangent bundle.

In Appendix G we present the covariant derivative in more traditional terms.We hope that in this way the reader will make contact with the traditional way ofpresenting the covariant differential calculus.

3.4 The Holonomic Tensorialization Principle*

3.4.1 The Natural Tensorialization of Algebraic Structures

We are in a situation now of establishing on a firm ground some of the intuitiondeveloped previously about the relationship between linear algebra and geometry.In Sect. 2.4.3 we discussed the ‘easy’ tensorialization principle that states that anyalgebraic tensorial object gives raise to a geometrical tensor possessing the samestructure as the original tensor. Even as simple as this observation is it providesa useful hint in the geometrical structures of interest in analyzing the propertiesof dynamical systems. This point of view will be exploited systematically in whatfollows. There is however another aspect of this relation that we want to study now.If we are given a geometrical tensor in a given smooth manifold, say a (1, 1) tensorL for instance, then at each point x ∈ M , Lx : Tx M → Tx M defines a linearendomorphism of the linear space E = Tx M . The linear map Lx will change frompoint to point, however under some circumstances, such change is only apparent in

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the sense that, at least locally, that is in an open neighborhoodU of x , all Lx ′ , x ′ ∈ U ,are “equivalent”.

The simpler notion of equivalence for the (1, 1) tensor at different points of U ,would be that there exists a local frame {ei (x)} defined on U (that is a smooth choiceof a base for each tangent space Tx M) such that thematrices (L j

i (x)) associated to Lx

with respect to the base {ei (x)} are equivalent, or in other worlds, if we select a pointx0 ∈ U , then there exist a changeof base P j

i (x) such that L(x) = P(x)−1L(x0)P(x).Equivalently, we can find a local frame {ei (x)} such that the matrix representation ofL in this frame is constant (and equal to the matrix L(x0)). The specific form of thematrix L(x0) at a point x0 would be the algebraic model for the tensor L . We willsay in such case that the tensor L is locally constant.

In some cases the family of algebraicmodels or normal forms for a given algebraictensor t are known and we have nice structure theorems form them (see for instancethe structure theorem for a bivector or a linear symplectic form in Chaps. 4 and 5respectively). Then given a smooth manifold M of the same dimension as the linearspace E where the algebraic tensor t is defined, a geometrization of the tensor t isa locally constant tensor T on M such that, restricted to each tangent space, Tx M isequivalent to t. In most cases the tensorialization of a given algebraic tensor amountsto solving a topological problem in the sense that the existence of such tensorializationamounts to determining if there is a topological obstruction to the existence of sucha locally constant geometrical tensor.

We may consider for instance the simple tensorialization problem posed by agiven non-zero vector v ∈ E . Notice that given a manifold M of dimension dim E , atensorialization of the (1, 0) tensor v is just a vector field X in M which is non-zeroeverywhere. Notice that a non-zero vector field is locally constant. Actually a vectorfield which is non-zero at a given point x0 can be straightened out in a neighborhoodof that point. Even more it is possible to find a local system of coordinates in an openneighborhood of x0 such that X takes the simple form ∂/∂x1 in such neighborhood.So every non-zero vector field is locally constant and performing a change of baseit can be made into any vector v. Hence the existence of a tensorialization of avector v in a manifold M amounts to proving the existence of a nowhere zero vectorfield on M , however it is well-known that there are topological obstructions to thatdetermined by the Euler characteristic χ(M) of the manifold. For instance in thesphere S2 there exists no non zero vector fields as χ(S2) = −2.

Typically, the first step to study the global properties of a geometrical structureon a given manifold M is to determined its existence, that is, to study the existenceof a tensorialization on an algebraic model for it. For instance, if we wish to studythe global properties of a Riemannian metric on a manifold M , i.e., of a symmetricnon-degenerate (0, 2) tensor on M , we will start studying the existence of a tensorial-ization of an algebraicmodel for such structure. As it happens there is a uniquemodelfor such algebraic structures in any linear space equipped with a linear symmetricnon-degenerate bilinear form, i.e., a scalar product, has an orthonormal base. Then atensorialization of a scalar product always exists provided that M is orientable (andsecond countable and paracompact as we will assume all the time).

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3.4 The Holonomic Tensorialization Principle 165

Such a tensorialization problem is not always ‘easy’ in the sense that there is noguarantee of the existence of such tensorialization when the manifold M is not thelinear space E itself; however as we were pointing out, commonly the obstructionto the existence of such tensorialization is topological and can be solved by using“soft” techniques.Because this proposedgeometrization seeks to construct a tensorialobject along the fibres of the tangent bundle (or of any natural tensorial bundle) over amanifold M , we will call it a natural tensorialization of the given algebraic structure.

The example of the tensorialization of a scalar product brings to the fore a fun-damental issue regarding the structure of the geometrized object. By definition ageometrized object is locally constant but we use a general local frame to expresssuch constancy. For instance in the case of a metric, what we do is to construct alocally constant metric g = gi j ei (x) ⊗ e j (x), thus the non-constant character of g isabsorbed in the local frame components ei (x). We may ask ourselves if it is possibleto find a local frame of the natural form ei (x) = dxi for some local set of coordinatesxi or, in other worlds, if the “non-holonomic” local frame ei (x) can be integratedand be described by a function ϕ : U → R

n , ϕ = (x1, . . . , xn) of maximal rank, i.e.,by a local chart. Notice that if this were possible then our metric g will have locallythe form

g = gi j dxi ⊗ dx j . (3.66)

However as we will see in the next section, if this were the case, the curvature of gwould be zero and again for some manifolds this would be impossible because thereare tight relations between the topology of the manifold and the curvature tensorof any metric on it. For instance if M is an orientable closed8 connected Riemannsurface �, then the Gauss-Bonet theorem establishes that

∫�

κ = 2πχ(�) with κ

the scalar curvature of the metric. Then if χ(�) �= 0, then κ cannot vanish and thereare no metrics that can be written locally as in Eq. (3.66).

The existence of such integrated or holonomic form for a geometrized algebraicstructurewill be the subject of the forthcoming section. The statement of the existence(or not) of such geometrized structure will be called the holonomic tensorializationprinciple even if we may call it properly the holonomic tensorialization principle.

3.4.2 The Holonomic Tensorialization Principle

Before stating with precision the main idea of this section, a holonomic version ofthe natural tensorialization principle, we will explore two more examples.

Consider again a linear map A : V → V and a natural tensorialization of itL : TM → TM with M a smooth manifold of dimension dim V . Notice that the (1, 1)tensor L has the constant local form L = A j

i ξ j ⊗ θ i where {ξi } is a local frame

for M and {θ j } its dual frame, i.e., 〈θ j , ξi 〉 = δji . This situation can be achieved for

8 That is compact without boundary.


instance if M is parallelizable. In such case there exists a global frame ξi and L isdefined as TA with respect to the frame ξi and its coframe θ j .

Now if A is generic and has real spectrum, we can write A = ∑a λa Pa , with

σ(A) = {λa} ⊂ R and Pa the projectors on the corrresponding spaces of eigen-vectors Va (of rank one if A is generic), i.e., V = ⊕

Va , APa = λa Pa . Then, thegeometrized object will decompose as L = ∑

a λa TPa , and TM = ⊕a Da where

Da is the subbundle of TM defined by the range of TPa . In general the subbundlesDa will not define an integrable distribution and it will not be possible to find a holo-nomic expression for L , i.e., there could not exist a local system of coordinates μa

such that:

L =∑

a

λa∂

∂μa⊗ dμa.

If the linear map A is a projector, σ(A) = {0, 1}, then L2 = L and L(TM) = Dis just a distribution. In such case D is integrable iff the Nijenhuis torsion tensorNL = [L , L] vanishes and L can be written as its local model in a local coordinates.

Thus the existence of an holonomic natural tensorialization for a given projectorP of rank r is equivalent to the existence of an integrable distribution D of rank r(the dimension of the linear subspace Dx ⊂ Tx M). It actually happens that if thereexists a distribution (not necessarily integrable) D of rank r in the manifold M , thenif the manifold is open we can always deform D to an integrable distribution that willprovide the holonomic natural tensorialization of P . that is essentially the content ofHaefliger’s theorem [Ha70]:

Theorem 3.18 Let M be an open manifold, then any distribution D can be de-formed homotopically to an integrable distribution (i.e., into a foliation) of the samedimension.

This theoremcanbe completed by stating the topological conditions that guaranteethe existence of a distribution of rank r and that consists of the vanishing of thecohomology groups Hk+1(M), . . . , H n(M), n = dim M (notice that Hn(M) = 0imply that M must be open). Thus we may state now:

Corollary 3.19 A rank k projector P on a n-dimensional linear space has a holo-nomic tensorialization on an open manifold M iff Hk+1(M) = · · · = Hn(M) = 0.

3.4.2.1 Gromov’s h-Principle

Let us consider now as another example of the subtleties involved in finding a holo-nomic tensorialization of an algebraic structure, the problemof geometrizing constantPoisson structures (we are anticipating here material that will be discussed at lengthin Sect. 4.3).

Let � be a skew symmetric bivector on a linear space V . Choosing a linear basis{ei }, � can be written as: A = Ai j ei ∧ e j . We will say that � is regular if the

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induced linear map (denoted with the same symbol � : V ∗ → V is invertible. Insuch case, there will exists a skew symmetric bilinear form ω such that the naturalmap (again, denoted with the same symbol) ω : V → V ∗ is such that � ◦ ω = I .Then ω = ωi j ei ∧ e j , with {ei } the dual basis to the previous basis {ei }. In otherwords, �i jω jk = δi

k .The ‘easy’ tensorialization of� produces a geometric bisector, a section of T V ∧

T V , that will be denoted with the same symbol, and that will read as:

� = �i j ∂

∂xi∧ ∂

∂x j.

The range of the natural map � : T ∗V → T V defines a distribution S = �(T V ),that by definition is integrable (� is constant). In general, we may try to find a naturaltensorialization of � in a manifold M . The existence of a natural tensorializationhas the same obstructions as geometrizing a projector whose rank is the same as therank of the range of �, because if we find a distribution whose rank is the rank of� all we have to do is to construct an arbitrary non-degenerate bivector there andthen bring it to its normal form (which is unique as in the case of a positive definitemetric). Then we may conclude stating that provided that M is an open manifoldand Hr+1(M) = · · · = H n(M) = 0 with r = rank(�), then there exists a skewsymmetric bivectdor � on M such that locally it can be written as:

� = �i j ξi ∧ ξ j .

where ξi is a local frame of TM.We must notice that the distribution S = �(TM) defined by the an arbitrary biec-

tor � doesn’t have to be integrable. The integrability conditions for S are, applyingFrobenius theorem, that [X, Y ] ∈ S for any X, Y ∈ S, but it is easily checked that vec-tor fields in S are spanned by vector fields of the form X = �(d f ), hence if we com-pute [�(dxi ),�(dx j )] we get (taking into account that �(dx I ) = �i j (x)∂/∂x j )that Ran(�) = S is integrable iff

�il ∂�ik

∂xl+ cyclic(i, j, k) = 0 . (3.67)

The previous set of equations must be satisfied to find a holonomic tensorialization of�. As it will be discussed later on, a bisector� on M satisfying Eq. (3.67) is called aPoisson tensor, and it provides the geometrical definition of a Poisson structure on M .

The existence of such holonomic tensorialization has been elucidated recentlyby Crainic and Fernandes [LF12]. Let us analyze first the regular case, i.e., � non-degenerate.

Then a non-degenerate skew symmetric tensor� on a linear space can be thoughtof as a section of the bundle T E ∧ T E over E , and thus the holonomic conditionEq. (3.67) can be thought as a condition on the space of fields (xi ,�i j , �

i jk ), where

for a specific section �(x), then �[k i j] = ∂�i j/∂xk . The bundle before is called the


first jet bundle of E and the algebraic scripture of the holonomic condition above is:

∑(i, j,k) cyclic

�il�jk

l = 0 .

Thus, the holonomic condition can be written as a submanifold R of J1E , i.e.,

R = {(x,�,�) | �i j = −� j i , det�i j �= 0,∑

(i, j,k)cyclic

�il�jk

l = 0} ⊂ J 1E .

In the case of symplectic structures, we have a positive solution to the holonomictensorialization problem due to M. Gromov:

Theorem 3.20 (Gromov’s theorem) [Gr86a] Let M be an even-dimensional openmanifold. Then for each [α] ∈ H2(M) there exists a symplectic form ω cohomologousto α. Moreover if ω0, ω1 are two cohomologous symplectic forms, then there existsa symplectic homotopy ωt , t ∈ [0, 1] joining ω0 and ω1 and ωt = ω0 + dψt .

Therefore in the language we were developing before, we can state:

Corollary 3.21 A linear symplectic form on a vector space of dimension 2n has aholonomic tensorialization on any open manifold M of dimension 2n.

Proof Let M be an open manifold of dimension 2n. Then a simple transversalityargument shows that there is no obstruction to the existence of a non-degenerate2-form on M , i.e., a section of �2(T ∗M) of maximal rank. Gromov’s theorem statesthat there exists, not only the natural tensorialization of a linear symplectic form,but a closed one. But then because of Darboux’s theorem, such form is holonomicbecause there are local charts such that the form is constant and equal to its algebraicmodel (see Sect. 5.3.1 for a proof of Darboux’s theorem). �

The proof of Gromov’s theorem is based on Gromov’s h-principle [Gr86a].Given a bundle E → M , its first jet bundle J 1E → M and a first-order partial

differential relation R ⊂ J 1E . A section σ : M → J 1E of the first jet bundle is asolution of the partial differential relationR if σ(x) ∈ R for all x ∈ M . We will saythat σ is a holonomic solution if there exists φ : M → E such that σ = j1φ, wherej1φ(x) = (xi , φ(x), ∂φ/∂xi ).

As it was shown before, solutions of partial differential relations often exist (theobstructions for their existence is just topological), however, as it was also shown inthe previous examples, holonomic solutions are hard to find (and often do not exist).Thus R is said to satisfy the h-principle if for every solution σ there exists σt , ahomotopy of solutions such that σ1 is holonomic, i.e., σ0 = σ and σ1 = j1φ.

There is a parametric version of the h-principle. Given σ1, σ2 solutions ofR, theparametric h-principle is satisfied if there exists σt joining σ1 and σ2. Moreover ifσ1 = j1φ1, σ2 = j1φ2, then σt = j1φt .

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Then the main question regarding the existence of holonomic tensorialization fora given algebraic structure is when the partial differential relation R defined by itand its derived ‘integrability conditions’ satisfies the h-principle. There are varioustechniques to prove (if possible) the h-principle, e.g., removal of singularities ),flexible continuous sheaves and convex integration.Adiscussion of the ideas involvedin these methods fails completely out of the scope of this book, so we will refer thereader to the existing literature [Gr86b].

3.4.3 Geometric Structures Associated to Algebras

Let V be a vector space and V ∗ its dual. To any element v ∈ V , there is a corre-sponding element in the bi-dual v ∈ (V ∗)∗ given by:

v(α) = α(v) , ∀α ∈ V ∗ (3.68)

Thus anymultilinear functiononV ∗, f : V ∗×· · ·×V ∗ → Rdefines, by restrictingit to the diagonal, a polynomial function f ∈ F(V ∗), f (α) = f (α, . . . , α), whichcan be obtained from the “monomials of degree one”, v ∈ (V ∗)∗, on which one hasdefined the (commutative) product:

(v1 · v2)(α) := v1(α) v2(α) (3.69)

Suppose now that on V there is defined an additional bilinear operation:

B : V × V → V (3.70)

which induces a (in general noncommutative) product ×B on V ⊂ F(V ∗) by:

v1 ×B v2 = B(v1, v2) (3.71)

Then we can define a 2-tensor τB in F(V ∗), at the point α, by the relation:

τB(d v1, d v2)(α) := α(B(v1, v2)) (3.72)

which satisfies Leibniz’ rule:

τB(d v, d(v1 · v2)) = τB(d v, v1 · d v2 + d v1 · v2) = v1 · τB(d v, v2) + τB(d v, v1) · v2(3.73)

Thus, τB(d v, ·) defines a derivation on V ⊂ F(V ∗) with respect to the commutativeproduct (3.69).


In particular, suppose that B is a skew-symmetric bilinear operationwhich satisfiesthe Jacobi identity, so that g = (V, B) is a Lie algebra. The corresponding 2-tensor� := τB :

�(d v1, d v2) = B(v1, v2) (3.74)

is a Poisson tensor in F(V ∗) and �(d v, ·) is a derivation with respect to the com-mutative product (3.69). Moreover, �(d v, ·) is a derivation also with respect to theproduct (3.71). Indeed, by using the fact that B is antisymmetric and satisfies theJacobi identity, one has

�(d v, d(v1·v2)) = B(v, B(v1, v2)) (3.75)

= B(v1, B(v, v2)) + B(B(v, v1), v2)

= v1·�(d v, d v2) + �(d v, d v1)·v2Similarly, if on V one has a Jordan product B′, the corresponding 2-tensor G :=

τB′ is a metric tensor and G(d v, ·) is a derivation with respect to the commutativeproduct (3.69), but not with respect to the product (3.71).

If now V = A is a C∗-algebra, where we have defined both a Lie product and a

Jordan product as:

B(a1, a2) := [a1, a2] = 1

2i(a1a2 − a2a1) , ∀a1, a2 ∈ A (3.76)

and a Jordan product

B ′(a1, a2) := a1 ◦ a2 = 1

2(a1a2 + a2a1) , ∀a1, a2 ∈ A (3.77)

in F(A∗) we have defined both a Poisson tensor � and a metric tensor G such that�(da, ·) and G(da, ·) are both derivations with respect to the pointwise commutativeproduct, with the former being also a derivation with respect to the Lie product. It isalso not difficult to check that the subalgebra B ⊂ A composed of all real elements,when embedded in F(A∗), comes equipped with an antisymmetric and a symmetricproduct, denoted by [·, ·] and ◦ respectively, such that:

1. The Leibniz rule is satisfied: [a, b ◦ c] = [a, b] ◦ c + b ◦ [a, c],2. The Jacobi identity is satisfied: [a, [b, c]] = [[a, b], c] + [b, [a, c]], and3. The identity: (a ◦ b) ◦ c − a ◦ (b ◦ c) = [[a, c], b] holds.

This means that (B, [·, ·], ◦) is a Lie-Jordan algebra [Em84, FF13]. Finally, wenotice that the Hamiltonian vector fields:

Xa := �(·, da) = −[a, ·] (3.78)


are derivations with respect to the Jordan product, since, by using the propertiesabove:

Xa(d(a1 ◦ a2)) = −[a, a1 ◦ a2] = −[a, a1] ◦ a2 + −a1 ◦ [a, a2]= Xa(da1) ◦ a2 + a1 ◦ Xa(da2) (3.79)

3.5 Vector Fields and Linear Structures

3.5.1 Linearity and Evolution

In the previous chapter it was shown that the dynamics of a given system can bedescribed in various equivalent forms: for instance as a local flow ϕt on a carrierspace (Sects. 2.2.2 and 2.3.2), i.e., a vector field �, or as a one-parameter family ofautomorphismst on the algebra of observables (Sects. 2.3.3 and 2.3.6). This makesit clear that we must be careful about any statement of the type ‘linear evolution’ ofa given system because evolution from the point of view of observables is alwayslinear with respect to the linear structure on the algebra of observables. Somethingcompletely different happens when we ask the evolution of the system to preservesome linear structure on the carrier space E . This will be the case if the algebraicoperations done on states will be carried out onto themselves by the evolution definedby the diffeomorphisms ϕt ,

ϕt (u + v) = ϕt (u) + ϕt (v), ϕt (λu) = λϕt (u) (3.80)

for all u, v ∈ E , λ ∈ R. We will say then that the the dynamics preserves the linearstructure, that the dynamics and the linear structure given on E are compatible, orthat the linear structure on E is admissible for the given dynamics.

As we learned in Sect. 3.3.1 a linear structure on E is characterized by an Euleror Liouville vector field �, thus it can be easily checked that:

Proposition 3.22 A linear structure characterized by the Euler vector field � isadmissible for the dynamics � if and only if [�,�] = 0.

Proof Let ϕt the local flow defined by the vector field �. Then if the linear structure� is admissible for �, then because of Eq. (3.80) we have ϕt (esu) = es(ϕt (u)) forall s ∈ R, but multiplication by es is the flow of �, hence both flows commute andthis implies that [�,�] = 0.

Conversely, if [�,�] = 0, then both flows, the local flow ϕt of � and the flowϕ�

s = es commute. Let us recall that a linear function f on E satisfies that�( f ) = f .Moreover if f is a linear function we have f (u + v) = f (u) + f (v) (that is in factthe definition of the first symbol ‘+’ in the previous equation). Then it is trivial tocheck that if f is linear f ◦ ϕt is linear too. In fact we get:

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�( f ◦ϕt )(u) = d

ds( f ◦ϕt )(e

s(u)) |s=0= d

dsf (esϕt (u)) |s=0= d

dses( f ◦ϕt )(u) |s=0= ( f ◦ϕt )(u),

but then f (ϕt(u +v)) = ( f ◦ϕt )(u +v) = ( f ◦ϕt )(u)+ ( f ◦ϕt )(v) = f (ϕt (u))+f (ϕt (v)) for all linear f and the conclusion follows.

The reasons why one would like the evolution to preserve a given tensor, a linearstructure for instance, must come from the experiments performed on the system.Wemay be able to compose states of the system additively, then we will be extremelyinterested to see if the dynamics of the system is compatible with such composition.This will actually happen with the states of a harmonic oscillator described as pointson phase space orwith the states of amediumdescribed as the position of the elementsof it. A different example is provided by pure states of a quantum system describedas vectors on a Hilbert space. In all these cases we are able to compose the states ofthe system in a linear way.

3.5.2 Linearizable Vector Fields

In the previous chapter we saw that if the carrier state space of the system whose dy-namics wewant to describe has a differentiable structure then, vector fields constitutean appropriate way of describing such dynamics.

Thus we will consider a system whose dynamics is determined by a vector field �

on a smooth finite-dimensional manifold M . The vector field � defines a differentialequation on M , that is for each local chart xi on M we will have:

dxi

dt= �i (x), i = 1, . . . , n, (3.81)

where �i (x) are the local components of the vector field � in the given chart.Because of the theorem of existence and uniqueness of solutions for ordinary

differential equations, the previous equation provides, locally, a description of theevolution of the states of the system associated with the vector field X .

If M carries a Liouville vector field�, then there is a linear structure on M whosedilation vector field is �. We will denote by E the linear space (M,�) to emphasizethe choice of a linear structure on M defined by �. It is clear that if � is a linearvector field, then

[�,�] = 0 . (3.82)

Conversely, if � is a vector field on E commuting with �, then it is linear, becauseusing the expression of � in linear coordinates we have that if � = � j ∂/∂x j ,Eq. (3.82) becomes

xi ∂� j

∂xi= � j ,

3.5 Vector Fields and Linear Structures 173

i.e., the � j are homogeneous smooth functions of degree 1, and they are assumednot to have singularities, hence they are linear.

A vector field will be called globally linearizable if there exists a global diffeo-morphism φ : E → E such that φ∗� is a linear vector field, i.e., [�,φ∗�] = 0.Notice that in such case φ(0) = 0.

A globally linearizable vector field can be easily integrated by transforming itinto the linear form φ∗�, obtaining the linear flow for it ψt = exp t A, and changingback to old coordinates, φ−1 ◦ ψt ◦ φ, i.e.,

ψ Xt = φ−1 ◦ exp t A ◦ φ.

In other words we can think that we were looking at � in a wrong set of coordinatesand that after rectifiyng them by using the diffeomorphisms φ, the vector field �

looks linear.It is important to notice that [�,�] �= 0, but [φ−1∗ �,�] = 0. We can think of the

vector field �φ = φ−1∗ � as defining an alternative linear structure on E such that �is a linear vector field with respect to it.

So far the previous observations are rather trivial as the reader may think immedi-ately that if we have a linear vector field on a linear space and we perform a nonlinearchange of coordinates, the vector field in the new coordinates will not look linear atall, but that is only because are looking at it in wrong coordinates. But what wouldhappen if the vector field in the new coordinates will look linear! This situation canactually occur.

For that to happen it is only necessary to find a diffeormorphism φ such thatφ∗� = � and φ∗� �= �. In such case, what we will have in our hands would be asystem that possesses an alternative linear structure description, one given by � andthe other by �φ = φ∗�. Consider the following example in two dimensions:

� = y∂

∂x− x

∂

∂y,

and consider the change of coordinates φ:

X = φ1(x, y) = x(1 + x2 + y2), Y = φ2(x, y) = y(1 + x2 + y2),

which is obviously nonlinear. If we compute the vector field in the new coordinateswe find

φ∗� = Y∂

∂ X− X

∂

∂Y.

Thus it is linear with respect to the linear structure φ∗�.As another simple example, consider inR

2, with coordinates (x, y), the dynamicalsystem described by


⎧⎪⎨⎪⎩

x = y

cosh x

y = − sinh x

. (3.83)

By setting: ξ = sinh x, η = y, we find:

⎧⎨⎩

ξ = η,

η = −ξ

, (3.84)

i.e., the harmonic oscillator. Therefore, our original system, Eq. (3.83), is diffeomor-phic to a linear one by using the diffeomorphism

φ : R2 −→ R

2, φ (x, y) = (sinh x, y) = (ξ, η) (3.85)

Description of the linear structure φ∗�: In this particular example it is relativelyeasy to describe the new linear structure because of the symmetry of the transfor-mation. First notice that the map φ : R

2 → R2 maps circles of radius r into circles

of radius R = r(1 + r2) and sends the rays lines l passing trough the origin intothemselves, φ(l) = l. This would imply that if we denote by +φ the addition and by·φ the multiplication by scalars defined respectively by �φ we will have:

u +φ v = u + v, λ ·φ u = λ(1 + λ2)

2u,

for any u, v ∈ R2 and λ ∈ R.

This observation raises a number of interesting questions that have an importancebeyond the purely mathematical level. For instance, if a vector field is linear withrespect to two alternative linear structures, then it will have different eigenvalues inboth of them because the matrices defined by it using the two linear structures willnot be equivalent. Thus, if the eigenvalues of a given vector field play any relevantrole in the (physical, engineering, etc.) interpretation of such vector field, which oneshould we choose? In the particular instance of quantum-like systems that is a crucialquestion that demands a clear answer.

Given x0 ∈ E , we will say that � is linearizable (or locally linearizable) aroundx0 if there exists a local chart (U, ψ), x0 ∈ U , such that in the local coordinates yi ,the vector field � is linear, i.e.,

�U = A ji yi ∂

∂y j.

This notion is equivalent to claiming the existence of a local diffeomorphismφU : U → φ(U ) = V , where V is an open set of E , such that φ∗� commuteswith �.


We must point out that this notion is quite different from the straightening outproperty. In fact, a vector field could be (locally) straightened out if there exists a(local) diffeomorphism φ such that φ∗� = �0, where �0 is a constant vector field. Inother words, the vector field could be straightened out if there are local coordinatesyi (not necessarily linear ones) such that in these coordinates the vector field X takesthe form

� = ∂

∂y1.

It is well known (we will come back to this question later on) that any vector fieldcan be straightened out around a noncritical point x0. Thus, a vector field which islinearizable around x0 will always have a critical point at x0. Thus, the first normalform theorem will concern the linearizability of a vector field around a critical point.This problem has been discussed in a series of papers by a number of people sincePoincaré (Poincaré, Hermann, Guillemin, Sternberg, etc.)

3.5.3 Alternative Linear Structures: Some Examples

It is known that all finite-dimensional linear spaces of the same dimension are linearlyisomorphic.9 However, alternative (i.e., not linearly related) linear structures can beconstructed easily on a given set. For instance consider a linear space E with addition+ and multiplication by scalars ·, and a nonlinear diffeomorphism φ : E → E . Nowwe can define a new addition +(φ) and a new multiplication by scalar ·(φ) by setting

u +(φ) v = φ(φ−1 (u) + φ−1 (v)) (3.86)

andλ ·(φ) u = φ

(λφ−1 (u)

). (3.87)

These operations have all the usual properties of addition and multiplication by ascalar. In particular,

(λλ′) ·(φ) u = λ ·(φ)

(λ′ ·(φ) u

)(3.88)

and

(u +(φ) v

) +(φ) w = u +(φ)

(v +(φ) w

). (3.89)

9 The same is true for infinite-dimensional separable Hilbert spaces (even more, the isomorphismcan be chosen to be an isometry).


Indeed,

λ ·(φ)

(λ′ ·(φ) u

) = φ(λφ−1 (

λ′ ·(φ) u)) = φ

(λλ′φ−1 (u)

)= (

λλ′) ·(φ) u, (3.90)

which proves (3.88), and similarly for (3.89).Obviously, the two linear spaces (E,+, ·) and (E,+(φ), ·(φ)) are finite-

dimensional vector spaces of the samedimension andhence are isomorphic.However,the change of coordinates defined by φ that we are using to ‘deform’ the linearstructure is a nonlinear diffeomorphism. In other words, we are using two different(diffeomorphic but not linearly related) global charts to describe the same manifoldspace E .

Using then the two different linear structures defined in this way, one obtains alsotwo different (and, again, not linearly related) realizations of the translation groupR

n (n = dim E).As a simple (but significant) example of this idea consider the linear spaceR

2. Thiscan also be viewed as a Hilbert space of complex dimension 1 that can be identifiedwith C. We denote its coordinates as (q, p) and choose the nonlinear transformation

q = Q(1 + λR2),

p = P(1 + λR2), (3.91)

with R2 = P2 + Q2, which can be inverted as

Q = q K (r)

P = pK (r), (3.92)

where r2 = p2 + q2, and the positive function K (r) is given by the relation R =r K (r), satisfying the equation

λr2K 3 + K − 1 = 0 (3.93)

(hence, actually, K = K(r2

)and when λ = 0 then K ≡ 1). Using this transforma-

tion we can construct an alternative linear structure on C by using formulae (3.86)and (3.87). Let us denote by +K and ·K the new addition and multiplication byscalars. Then, with

φ : (Q, P) �→ (q, p) = (Q

(1 + λR2

), P

(1 + λR2

)),

φ−1 : (q, p) �→ (Q, P) = (q K (r) , pK (r)) ,


one finds

(q, p) +(K )

(q ′, p′) = φ

(φ−1 (q, p) + φ−1

(q ′, p′)) =

= φ((

Q + Q′, P + P ′)) = φ(q K + q ′K ′, pK + p′K ′) ,

(3.94)

where K = K (r), K ′ = K(r ′), i.e.

(q, p) +(K )

(q ′, p′) = S

(r, r ′) ((

q K + q ′K ′) ,(

pK + p′K ′)) , (3.95)

where

S(r, r ′) = 1 + λ

((q K + q ′K ′)2 + (

pK + p′K ′)2) . (3.96)

Quite similarly,

a ·(K ) (q, p) = φ(

aφ−1 (q, p))

= φ ((aq K (r) , apK (r)))

= S′ (r) (aK (r) q, aK (r) p) (3.97)

where

S′ (r) = 1 + λa2r2K 2 (r) . (3.98)

The two different realizations of the translation group inR2 are associatedwith the

vector fields (∂/∂q, ∂/∂p) and (∂/∂ Q, ∂/∂ P) respectively. The two are connected by

(∂

∂ Q∂

∂ P

)= A

(∂∂q∂∂p

), (3.99)

where A is the Jacobian matrix:

A = ∂(q,p)∂(Q,P)

=(1 + λ(3Q2 + P2) 2λP Q

2λP Q 1 + λ(Q2 + 3P2)

)

=(1 + λK (r)2(3q2 + p2) 2λK (r)2 pq

2λK (r)2 pq 1 + λK (r)2(q2 + 3p2)

).

In the sequel we will write simply A as

A =(

a bd c

), (3.100)

with an obvious identification of the entries. Then,


A−1 = ∂ (Q, P)

∂ (q, p)= D−1

(c −b

−d a

), D = ac − bd. (3.101)

Thus the 2D translation group R2 is realized in two different ways.

Going back to the general case, we have just proved that to every linear structurethere is associated in a canonical way a dilation (or Liouville) field � which is theinfinitesimal generator of dilations. Therefore, in the framework of the new linearstructure, it makes sense to consider the mapping

� : R × E → E (3.102)

via,

�(t, u) = et ·(φ) u = u (t) , (3.103)

where again, we are considering a transformation φ : E → E . The transformed flowtakes the explicit form

u (t) = φ(

etφ−1(u))

. (3.104)

Property 3.88 ensures that

�(t, u

(t ′)) = �

(t + t ′, u

), (3.105)

i.e., that (2.80) is indeed a one-parameter group of transformations of E . Then, theinfinitesimal generator of the group is defined as

�(u) =[

d

dtu(t)

]t=0

=[

d

dtφ

(etφ−1(u)

)]t=0

, (3.106)

or, explicitly, in components,

� = �i ∂

∂ui, (3.107)

�i =[∂φi (w)

∂w jw j

]w=φ−1(u)

. (3.108)

In other words, if we denote by �0 = wi∂/∂wi the Liouville field associated withthe linear structure (+, ·) on E ,

� = φ∗�0, (3.109)


where φ∗ denotes, as usual, the push-forward.It is clear that, if φ is a linear (and invertible) map, φi (w) = ai

j w j , then (3.108)yields: �i = ui , i.e.,

φ∗�0 = �0. (3.110)

Conversely it is simple to see that if a map φ satisfies (3.110) then, ∂φi/∂w j = aij

and φ is linear with respect to the linear structure defined by �0.The above scheme can be generalized to the case of a diffeomorphismφ : E → M

between a vector space E and amanifold M possessing ‘a priori’ no linear structureswhatsoever. This will require, of course, that M be such that it can be equipped witha one-chart atlas. Then it is immediate to see that Eqns. (3.86) and (3.87) (withu, v ∈ M , now) apply to this slightly more general case as well.

As a simple example of this sort, let us consider: E = R, M = (−1, 1) and

φ : E → M; x → X = tanh x . (3.111)

Then,

λ ·(φ) X = tanh(λ tanh−1 (X)

)(3.112)

and

λ ·(φ)

(λ′ ·(φ) X

) = λ ·(φ) tanh(λ′ tanh−1 (X)

) == tanh

(λλ′ tanh−1 (X)

) = (λλ′) ·(φ) X,

(3.113)

while

X +(φ) Y = tanh(tanh−1 (X) + tanh−1 (Y )

)= X + Y

1 + XY, (3.114)

which is nothing but the elementary one-dimensional relativistic law (in appropriateunits) for the addition of velocities. It is also simple to prove that

(X +(φ) Y

) +(φ) Z = = tanh(tanh−1 (

X +(φ) Y) + tanh−1 (Z)

)=

= tanh(tanh−1 X + tanh−1 (Y ) + tanh−1 (Z)

)(3.115)

i.e., that

(X +(φ) Y

) +(φ) Z = X +(φ)

(Y +(φ) Z

). (3.116)


Explicitly,

X +(φ) Y +(φ) Z = X + Y + Z + XY Z

1 + XY + X Z + Y Z. (3.117)

The mapping (3.104) is now

X (t) = tanh(

et tanh−1 (X))

(3.118)

and we obtain, for the Liouville field on (−1, 1),

�(X) =(1 − X2

)tanh−1 (X)

∂

∂ X, (3.119)

and �(X) = 0 for X = 0, while, as tanh−1 x = (1/2) ln ((1 + x) / (1 − x)), �(X)

tends to 0 for X going to ±1.

3.6 Normal Forms and Symmetries

3.6.1 The Conjugacy Problem

If we are given a vector field X A, we can try to find the simplest possible form ittakes in any linear structure we can construct on E , i.e., we can look for the formthat X A takes with respect to an arbitrary diffeomorphism φ. In other words, wewill be looking for the orbits of linear vector fields of the Lie algebra X(E) withrespect to the group Diff(E, 0) of diffeomorphisms of E fixing 0. An elementaryargument shows that it is not possible to simplify more a linear vector field by usingsmooth transformations. First we disregard the possibility of converting the vectorfield X A into a constant vector field because of the completely different nature oftheir flows. Then, we can consider whether we can transform the linear structureof the vector field into another simpler linear one, but that is impossible because asmooth transformation will preserve the spectrum of the matrix A. In fact denotingby P the Jacobian matrix of the transformation defined by φ at 0 we obtain after aneasy computation that

AP = P B ,

where B will denote the matrix defining the transformed linear vector field. Hence,B = P−1AP and the spectrum of A is unchanged. Thus, if we are looking fortransformations simplifying our vector field, they must transform our linear vectorfield into another linear vector field without adding spurious higher order terms toX A. Such a family of maps will contain linear transformations x �→ P · x and

3.6 Normal Forms and Symmetries 181

eventually more transformations (that can be called ‘linearoid’ with respect to X A),but the change they can produce on X A is at most the change that would be producedjust by linear transformations. Thus we are lead to look for the simplest form thatX A takes under linear transformations, i.e., under diffeomorphisms preserving thegeometry of the problem, in this case the vector field � (in further studies we willaddress the same questions with respect to geometrical structures different from �).It is clear now that such a subgroup of the group of diffeomorphisms is precisely thegroup of linear automorphisms of E . Thus looking for normal forms of linear vectorfields amounts to describing the orbits of linear vector fields under linear changesof coordinates. Under a linear change of coordinates φ(x) = P · x the vector fieldX ′

A transforms into φ∗ X A = X P AP−1 , thus classifying orbits of linear vector fieldsunder linear changes of coordinates is equivalent to understanding the structure ofthe set of matrices under conjugation A �→ P AP−1, i.e., the adjoint representationof GL(n, R). It is well-known from elementary courses in linear algebra that theclassification of conjugacy problem of real matrices is solved by Jordan’s NormalForm Theorem that we rephrase succinctly here.

Theorem 3.23 [Normal Form Theorem] For any real n × n matrix A there exists a(essentially unique) representative on each adjoint orbit under the group GL(n, R)

which has the block diagonal form

J =

⎛⎜⎜⎜⎝

J1 00 J2

. . .

Jk

⎞⎟⎟⎟⎠ , (3.120)

where the matrices Ji correspond to the i-th eigenvalue λi of the matrix A andare made up of elementary blocks whose structure depends on λi and have the form

Ji =

⎛⎜⎜⎜⎜⎝

J (1)i 00 J (2)

i. . .

J (ri )i

⎞⎟⎟⎟⎟⎠ . (3.121)

Here the J (l)i are elementary Jordan blocks corresponding to the same eigenvalue

which are listed below, such that the order of J (1)i is precisely the multiplicity of λi in

the minimal polynomial if λi is real, and twice the multiplicity of λi in the minimumpolynomial if λi is complex. The number ri of blocks is the geometric multiplicityof λi , i.e., the dimension of its eigenspace. The sum of all orders is the algebraicmultiplicity of λi , and the order of J (l)

i is decreasing with l.


The elementary real Jordan blocks are the following:

I. A single element (λ) (corresponding to a non-degenerate real eigenvalue).

II. A 2 × 2 matrix

(0 ν

−ν 0

), ν ∈ R (corresponding to a non-degenerate purely

imaginary eigenvalue λ = iν, ν �= 0).

III. A 2× 2 matrix

(a b

−b a

), ab �= 0, a, b ∈ R (corresponding to a non-degenerate

complex eigenvalue λ = a + ib).

IVa. If λ is a degenerate real eigenvalue,

Jλ =

⎛⎜⎜⎜⎝

λ 1 0 . . .

0 λ 1 . . .

0 0 λ . . ....

......

. . .

⎞⎟⎟⎟⎠ .

IVb. If λ = iν is a degenerate purely imaginary eigenvalue,

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 ν

−ν 01 00 1

0 . . .

00 ν

−ν 01 00 1

. . .

0 00 ν

−ν 0. . .

......

.... . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

IVc. If λ = a + i b, ab �= 0, is degenerate complex eigenvalue,

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a b−b a

1 00 1

0 . . .

0a b

−b a1 00 1

. . .

0 0a b

−b a. . .

......

.... . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The linear vector fields associated with any of the elementary blocks listed aboveare given by:

1. XI = λ x ∂/∂x .2. XII = ν (−x ∂/∂y + y ∂/∂x).3. XIII = (ax + by) ∂/∂x − (bx − ay) ∂/∂y.


4.XIVa = λ

(x1

∂

∂x1+ · · · + xn

∂

∂xn

)+ x2

∂

∂x1+ · · · + xn

∂

∂xn−1

XIVb = ν

(−x1

∂

∂x2+ x2

∂

∂x1

)+ · · · + ν

(−x2n−1

∂

∂x2n+ x2n

∂

∂x2n−1

)+

+ x1∂

∂x3+ x2

∂

∂x4+ · · · + x2n−3

∂

∂x2n−1+ x2n−2

∂

∂x2n

XIVc = (ax1 + bx2)∂

∂x1+ (ax2 − bx1)

∂

∂x2+ · · · +

+ (ax2n−1 + bx2n)∂

∂x2n−1+ (ax2n − bx2n−1)

∂

∂x2n+ x1

∂

∂x3+

+ x2∂

∂x4+ · · · x2n−3

∂

∂x2n−1+ x2n−2

∂

∂x2n

Thus any linear vector field can be brought, by a linear change of coordinates, toa sum of elementary vector fields as listed. In this sense the conjugacy problem iscompletely solved. As each one of these vector fields belongs to a different subspace,they all commute among themselves, so that in finding the flow associated with anydirect sum we simply take the product of the component flows.

We might comment very briefly on these elementary vector fields.

1. XI is an infinitesimal generator of dilations, it has a flow given by φt = eλ t .2. XII is the generator of rotations in the plane, it has a flow given by

φt =(

cos νt sin νt− sin νt cos νt

).

3. For XIII the two vector fields a(x ∂/∂x + y ∂/∂y) and b(x ∂/∂y − y ∂/∂x) com-mute, so we can write

φt =(

eat 00 eat

) (cos bt sin bt

− sin bt cos bt

).

4. For XIV:

a. The vector field X I V a is again the sum of two commuting vector fields, thatin dimension two are given by, λ (x ∂/∂x + y ∂/∂y) and y ∂/∂x , thereforewe obtain:

φt =(1 t0 1

) (eλt 00 eλt

).

b. The vector field X I V b is again the sum of two commuting vector fields, thatin dimension four are given by, λ(x1 ∂/∂x2 − x2 ∂/∂x1) + λ(x3 ∂/∂x4 −x4 ∂/∂x3) and x1 ∂/∂x3 + x2 ∂/∂x4.


⎛⎜⎜⎝

eat 0 0 00 eat 0 00 0 eat 00 0 0 eat

⎞⎟⎟⎠

⎛⎜⎜⎝1 0 t 00 1 0 t0 0 1 00 0 0 1

⎞⎟⎟⎠ .

c. Finally, the vector field X I V a is again the sum of two commuting vectorfields, that in dimension four are given by, (ax1 + bx2) ∂/∂x1 + (ax2 −bx1) ∂/∂x2 + (ax3 + bx4) ∂/∂x3 + (ax4 − bx3) ∂/∂x4 and x1 ∂/∂x3 +x2 ∂/∂x4.

⎛⎜⎜⎝

eat 0 0 00 eat 0 00 0 eat 00 0 0 eat

⎞⎟⎟⎠

⎛⎜⎜⎝

cos bt sin bt 0 0− sin bt cos bt 0 0

0 0 cos bt sin bt0 0 − sin bt cos bt

⎞⎟⎟⎠

⎛⎜⎜⎝1 0 t 00 1 0 t0 0 1 00 0 0 1

⎞⎟⎟⎠ .

3.6.2 Separation of Vector Fields

The conjugation problem discussed in the previous section can be cast in a moregeneral scheme which is suitable for discussing other situations of big interest. Wehave already noticed that E can be embedded inF(E∗) by the canonical identificationE ∼= E∗∗. In fact, if { ei | i = 1, . . . , n } is a linear basis for E , we will denote byxi ∈ F(E∗) the associated linear functions defined by

xi (μ) = 〈μ, ei 〉 , μ ∈ E∗.

We notice that a generic vector u = ui ei ∈ E will define a linear function fu on E∗by

fu(μ) = 〈μ, u〉 ,

i.e., fu = ui xi . A linear vector field X on E can be thought as a vector field X∗ onE∗ by means of

X∗( f )(μ) = f (X (μ)) ,

where X (μ) is the action of X in the linear function μ ∈ F(E). Notice that ifX is linear, X (μ) is linear too, thus X (μ) is an element of E∗. Hence, if X =A j

i xi ∂/∂x j , then X∗ = A ji x j ∂/∂xi . Notice that

L X∗ fu = A ji x j

∂

∂xi(uk xk) = A j

i x j ui ,

and if u is an eigenvector of A with eigenvalue λ, we have

L X∗ fu = λ fu , (3.122)


i.e., the linear function fu on E∗ is an eigenvector of the linear operator X∗ actingon functions on E∗ with the same eigenvalue λ, and conversely.

Up to here there is nothing new except for a change of language. However, becauseE is a subspace of the associative and commutative algebra F(E∗), we can considerthe eigenvalue problemEq. (3.122) as an equation directly onF(E∗). Therefore fromnow on we can perform any coordinate transformation and the eigenvalue problemwill be unchanged. Notice that

�∗( fu) = xi∂

∂xi(uk xk) = xi ui = fu ,

thus, we can reformulate our eigenvalue problem in the form

L X∗−λ�∗ f = 0 .

This implies that we are looking for constants of the motion of the linear vectorfield Xλ = X∗ − λ�∗. Notice that the family of vector fields Xλ1 , Xλ2 , . . . , Xλk ,commute, because [X,�] = 0. The space of constants of the motion for a vectorfield X is a subalgebra Fλ(E∗) of F(E∗). The first step in the conjugacy problem isequivalent to finding a decomposition of the algebra F(E∗) as generated by a directproduct of subalgebras: Fλ1(E∗), …, Fλk (E∗), i.e.

F(E∗) = Fλ1(E∗) ⊗ · · · ⊗ Fλk (E∗) .

The vector field results X∗ = X∗1 + · · · X∗

r where X∗i is the restriction of X∗ to

Fλi (E∗). Then [X∗i , X∗

j ] = 0, and

L X∗iFλ j (E∗) = 0, i �= j ,

L X∗iFλi (E∗) ⊂ Fλi (E∗) .

We can find the flow φt of X∗ by determining the flow of each X∗k on the associated

eigenalgebra Fλi (E∗), which in terms of the exponential map can be written as

φ∗t ( f ) = exp(t L X ) f =

∞∑n=0

tn

n! (L X )n f = f + t L X f + t2

2(L X )2 f + · · · ,

where the right-hand side makes sense for instance if f is real analytic and X is acomplete vector field. that is the case when both X and f are linear.

It is clear now that by replacing our eigenvalue problem on the vector spacewith an eigenvalue problem on the algebra of functions, Eq. (3.122), we can try todecompose a vector field on any manifold. This decomposition is usually known asa separation of the vector field or it is said that the vector field X is separable. Thenotion of separability of a dynamics will be used later on to analize the intregrability


properties of a dynamics (see Chapter 8) and in the foundations of reduction theory(Sect. 7.2).

3.6.3 Symmetries for Linear Vector Fields

Let us investigate now the existence of symmetries for a linear vector field. We mustnotice first that the algebra of symmetries of a vector field can be strictly larger thanthe algebra of linear symmetries. In fact, a linear vector field X A will be a symmetryof the dynamical vector field �, if

[�, X A] = 0.

But that is equivalent to the matrix equation

[M, A] = 0.

As it was discussed in Sect. 3.6.1, every matrix A can be written as a direct sum

A =

⎛⎜⎜⎜⎝

A1 0 · · · 00 A2 . . . 0...

.... . .

...

0 0 · · · Ak

⎞⎟⎟⎟⎠ ,

where each A j has the form

A j =

⎛⎜⎜⎜⎜⎝

J ( j)1 0 · · · 0

0 J ( j)2 · · · 0

......

. . ....

0 0 · · · J ( j)n j

⎞⎟⎟⎟⎟⎠ ,

with J ( j)r an elementary Jordan block corresponding to the eigenvalue λ j . The sizes

of the matrices J ( j)m do not increase as m increases. The matrix A j is an r j × r j

matrix, with r j the algebraic multiplicity of the eigenvalue λ j . The number n j isthe number of independent eigenvectors associated with the eigenvalue λ j , i.e., itsgeometric multiplicity. The first Jordan block in the matrix A j is a d j × d j matrix,where d j is the multiplicity of λ j as a root of the minimal polynomial. We recallhere that if we denote by P[x] the algebra of polynomials in the indeterminatex , with any polynomial p = a0 + a1x + · · · + an xn , we can associate a matrixp(A) = a0 I +a1A +· · ·+an An . This association is an algebra homomorphism andits image is the algebra associated to A. Thus, any polynomial p defines amatrix p(A)

commuting with A. The kernel of the homomorphism p �→ p(A) is an ideal in P .Theminimal polynomial for A is the (unique) monic generator of the ideal associated

http://dx.doi.org/10.1007/978-94-017-9220-2_8

http://dx.doi.org/10.1007/978-94-017-9220-2_7


with A, andwedenote it by pA. If pA has degree s, we havea0+a1A+· · ·+as As = 0.Therefore, there is no linear combination of A0, A, A2, . . . , As−1 which is zero. Itfollows that it is possible to generate, by taking powers of A, s independent matricescommutingwith A. By the previous decompositionwe can restrict our considerationsto A j . If A j contains only one elementary Jordan block, the minimal polynomial isof maximum degree (i.e., coincides with the characteristic polynomial) and we find anumber of independent matrices equal to the algebraic multiplicity of the eigenvalueλ j . If A j contains an elementary Jordan block two or more times, let say three times,we have

A j =⎛⎝ J ( j) 0 0

0 J ( j) 00 0 J ( j)

⎞⎠ ,

and, in addition to the powers of each elementary Jordan block taken separately, wehave additional commuting matrices of the form

C (1)j =

⎛⎝0 I 00 0 00 0 0

⎞⎠ , C(2)

j =⎛⎝0 0 00 0 I0 0 0

⎞⎠ , C(3)

j =⎛⎝ 0 0 I0 0 00 0 0

⎞⎠ ,

along with transposed matrices (C (a)j )t . For instance, for

Aλ =⎛⎝λ 0 00 λ 00 0 λ

⎞⎠ ,

the algebra of matrices we construct with the previous method is gl(3, R).We conclude therefore that for any matrix with minimal polynomial of maximum

degree the algebra of commuting matrices is Abelian and generated by powers ofthe matrix itself. For other matrices the algebra of symmetries is bigger and notcommutative.

We list now a few symmetries for vector fields in normal forms. For a singleelement Jordan block, � = λ x ∂/∂x we have X A = x ∂/∂x .

For a 2 × 2 matrix we have

(1) � = ν(

y ∂/∂x − x ∂∂y

), X A = y ∂/∂x − x ∂/∂y, X A2 = x ∂/∂x + y ∂/∂y.

(2) � = (ax + by) ∂/∂x − (bx − ay) ∂/∂y, X A = �, X A2 = ((a2 − b2)x +2aby) ∂/∂x + ((a2 − b2)y − 2abx) ∂/∂y.

(3) � = (x ∂/∂x + y ∂/∂y)+y ∂/∂x , X A = �, X A2 = x ∂/∂x +y ∂/∂y+2y ∂/∂x .

We conclude this section by saying that, apart from a more thorough analysisof the use of symmetry algebras in coming sections, we can use symmetries togenerate new solutions out of a given one. For the sake of illustration let us considera simple example on R

2. Let � = λ (x∂/∂x + y∂/∂y). Symmetries for � are given


by gl(2, R), generated by x∂/∂x , y∂/∂y, y∂/∂x and x∂/∂y. A particular element isX R = x∂/∂y − y∂/∂x with associated one parameter group

ψs(x, y) =(cos s − sin ssin s cos s

) (xy

).

A particular solution of the system

x = λ x , y = λ y ,

with initial conditions x = x(0), y = y0 = 0, is given by

x(t) = eλ t x(0) , y(t) = 0 .

By using our symmetry we have new solutions

(x(t)y(t)

)=

(cos s − sin ssin s cos s

)(eλt x(0)

0

)=

(eλt x(0) cos seλt x(0) sin s

).

Proposition 3.24 A generic linear vector field on an n-dimensional linear spacepossesses an n-dimensional Abelian symmetry algebra.

Non-generic vector fields define bigger symmetry algebras which are not nec-essarily Abelian. The space of linear vector fields supports a bundle of symmetryalgebras over the generic open dense submanifold. In fact, consider the subset ofMn(K) × Mn(K) given by

{ (M, A) | M ∈ Mn(K), [M, A] = 0 } .

The bundle over the generic part is trivial because we can choose the family ofsections I, A, A2, . . . , An−1. But the singular set has an involved structure. We willdiscuss some of these sets in the forthcoming sections.

3.6.4 Constants of Motion for Linear Dynamical Systems

As it was stated before a constant of motion for the vector field � is any functionf such that f (φt (x)) = f (x), where φt is the flow of the vector field �. If f isdifferentiable we get

�( f ) = 0 . (3.123)

Let us first search for a linear constant of motion, i.e., fa = ai xi . We find thatEq. (3.123) implies, Ai

j x j ai = 0 for all x , therefore, a = (a1, . . . , an) must be aneigenvector of At with zero eigenvalue, or in other words, a ∈ ker At .


If f1, f2, . . . , fk , are a maximal set of independent linear constants of motionfor � we can use them to define a map F : R

n → Rk by means of F(x) =

( f1(x), f2(x), . . . , fk(x)). By taking F−1(c) ⊂ Rn for any c ∈ R

k we find a familyof n−k dimensional vector spaces (affine spaces) inR

n parametrized by points inRk .

Remark 3.6 In fact the map F defines a (trivial) principal fibration of Rn over R

k ,the group R

n−k acting on the fibres.

Solutions of our dynamics are contained in Vc = F−1(c) if they started withinitial data x(0) such that F(x(0)) = c. It is clear that there exists a linear changeof coordinates such that the dynamical system X A can be brought into the form X Bwith

B =(

B0 00 0

)

and B0 being a (n − k) × (n − k) matrix on the space Vc. The change of coordinatesis constructed by fixing an invariant supplementary subspace to Vc and choosing andadapted linear basis. In other words, let (y1, . . . , yn) be coordinates which are linearcombinations of xi coordinates and such that yn−k+ j = f j , for j = 1, . . . , k. Theexpression of the vector field � in these new coordinates � = Bi

j y j ∂/∂ yi is suchthat for �(yn−k+ j ) = 0 implies Bn−k+ j

i yi = 0, for j = 1, . . . , k.It is clear that linear constants of motion are few because generically matrices do

not have zero eigenvalues. In fact, a linear dynamical systemwill have as many linearconstants of motion as the dimension of the kernel of its associated linear operator.In such situation the null degrees of freedom can be thrown away by an appropriatelinear change of coordinates and restricting the system to a subspace.

If we allow for quadratic constants of motion, the situation improves. However, asimilar map, that we will call “momentum map” for reasons that will become clearlater on,

F : Rn → R

k ,

can be constructed, where F = ( f1, . . . , fk), and the set { fi | i = 1, . . . , k} con-stitutes a maximal set of quadratic constants of motion. Now, however, there is no,in general, regular slicing for the map F . The solutions of � are forced to sit on thelevel set F−1(c) if the initial condition x(0) lies on it, but we cannot decompose �

linearly as we did before for the case of linear constants of motion.A more general analysis of this situation, i.e., foliations compatible with dynam-

ical systems, will be taken up later. Here we limit ourselves to a simple exampleshowing us some of the difficulties we will encounter in the analysis of this problem.


3.6.4.1 A Few Examples

On R3 we consider the linear system

y∂

∂z+ (x + z)

∂

∂y− y

∂

∂x,

corresponding to the matrix

A =⎛⎝ 0 −1 01 0 10 1 0

⎞⎠ .

Three commuting symmetries are provided by � itself,

� = x∂

∂x+ y

∂

∂y+ z

∂

∂z,

described by the identity matrix, and

X2 = x∂

∂z− z

∂

∂x+ z

∂

∂z− x

∂

∂x,

corresponding to

B = A2 =⎛⎝−1 0 −1

0 0 01 0 1

⎞⎠ .

There are two constants of motion. One is linear, given by f1 = x + z, correspondingto the null eigenvector (1, 0, 1)t of the matrix At . It is easy to check that another oneis f2 = x2 + y2 − z2 = y2 + (x − z) f1. Of course, these constants of motion canbe found directly from the adjoint system

dx

−y= dy

x + z= dz

y.

Only in the points (t, 0,−t) are the differentials d f1 and d f2 linearly dependent.This corresponds to the choice c1 = c2 = 0.

The map F : R3 → R

2 has one-dimensional slices given by f1 = c1, f2 =y2 + c1(x − z) = c2, except for c1 = 0, c2 = 0 when F−1(0, 0) is a straight line(t, 0,−t) and a point. For f2 we have a “slicing” by hyperboloids and f1 gives aslicing by planes.

Solutions for our vector fields are given by the intersection of these two slicings.It is clear that the “slicing” associated with f2 is not “regular”. For c2 > 0 we getconnected hyperboloids, for c2 = 0weget a cone, for c2 < 0weget twodisconnectedhyperboloids (space-like, light-like and time-like).


The existence of constants of motion may suggest us to use them as part of a newcoordinate system. In our example we can take f1, f2, η = x −z, as new coordinates,which define a compatible change of coordinates except for y = 0. Our dynamicalsystem has now the form f1 = 0, f2 = 0, η2 = 2( f2 − f1η), i.e., it is an implicitdifferential equation. For fixed values of f1 and f2 it looks like a parametric implicitdifferential equation. In the plane (η, η) we have the following situation:

Another example in the same spirit is provided on R3 by

� = (x + z)∂

∂y− y

(∂

∂x+ ∂

∂z

).

This time we have constants of motion f2 = x2 + y2 + z2 and f1 = 12 (x − z). By

using coordinates η = 12 (x + z), f1 and f2 we get a compatible coordinate system

except for y = 0. In the new coordinates our dynamical system is given by

f1 = 0 , f2 = 0 , η2 = f2 − 2( f 21 + η2) ,

or η2 + η2 = f2 − f 21 .Trajectories of our dynamical systemare provided by circles on concentric spheres

lying in planes parallel to the x = z plane. In the implicit equation, of course, f2 isthe square of the radius of the sphere and | f1| is the distance of the plane from thex = z plane. If f 21 = f2 the plane is tangent to the sphere and the trajectory reducesto a point. For f 21 > f2 there is no intersection. The two equations that we get innormal form are due to the fact that there are two planes at distance | f1| from thex = z plane.

Let us investigate now how many constants of motion we might find for a genericlinear vector field. Let us first discuss an example. On R

2 we consider

� = x∂

∂x+ y

∂

∂y.

It is easy to show that there are no global constants of the motion. Indeed anytrajectory contains the fixed point in its closure, therefore if a continuous constant ofmotion exists it has the same value on a given trajectory and on the “accumulationpoint” constituted by the fixed point. As this point is in the closure of any trajectoryit follows that the constant of motion is identically constant.

Now it is clear that ifO(m1) andO(m2) are two orbits with initial conditions m1and m2 respectively, a continuous constant of motion has the same value on O(m1)

and its closureO(m1), therefore ifO(m1)∩O(m2) is not empty, then any constant ofthe motion has the same constant value on both orbits. For a generic linear dynamicalsystem the representative matrix will have simple eigenvalues, therefore we find inR

n a situation similar to the one we have discussed inR2. We conclude that a generic

linear dynamical system has no globally defined constants of motion. We notice thatvarious eigenvalues will only change the component of the speed along the “star”


phase portrait. To have constants of motion for linear dynamical systems we shallconsider non generic families.

Proposition 3.25 A generic linear vector field has no constants of the motion.

Wemay conclude this section summarizing themost important properties we havefound for linear dynamical systems.

1. Every linear dynamical system � on a vector space E can be decomposed into adirect sum� = �0⊕�1⊕· · ·⊕�k . of irreducible commuting dynamical systems�k . The vector space E decomposes into �-invariant subspaces E1 ⊕ · · · ⊕ Ek

such that � j = �|E j and [�i , � j ] = 0. If A j is the representative matrix of� j we get additional commuting symmetries by considering linear vector fieldsassociated with powers of A j (there are as many independent ones as the degreeof the minimal polynomial).

2. Linear constants of motion correspond to zero eigenvalues, i.e., there existenceimplies that there is an invariant subspace on which � is identically zero.

3. Nonlinear constants of motion allow to “factor out” an identically zero part of �,the price to pay is that we have to introduce nonlinear coordinates.

4. By using symmetries we are able to generate new solutions out of a given one.5. Generic linear dynamical systems do not have global constants of the motion.

Non generic families can be selected by requiring that they preserve some non-degenerate (0, 2) tensor. This tensor can be used to raise and lower indices, allow-ing us to associate invariant covector fields with various infinitesimal symmetrieswe have found.

References

[Ma66] Malgrange, B.: Ideals of differentiable functions. Oxford University Press, Published bythe Tata Institute of Fundamental Research, Bombay (1966)

[Wh44] Whitney, H.: The self intersections of a smooth n-manifold in 2n-space. Annals Math. 45,220–246 (1944)

[NS03] Navarro González, J., Sancho de Salas, J.: ∞-Differentiable Spaces. Lecture Notes inMaths, 1824. Springer, Berlin (2003)

[BM10] Balachandran, A.P., Marmo, G.: Group Theory and Hopf Algebras: Lectures for Physi-cists. World Scientific, Hackensack (2010)

[AM78] Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin, Reading(1978)

[Ha70] Haefliger, A.: Homotopy and Integrability. Lecture Notes inMaths, vol. 197, pp. 133–164,Springer, Berlin (1970)

[LF12] Loja Fernandes, R., Frejlich, P.: A h-principle for symplectic foliations. Int. Math. Res.Not. 2012, 1505–1518 (2012)

[Gr86a] Gromov, M.: Soft and hard symplectic geometry. In: Proceedings of the InternationalCongress of Mathematicians at Berkeley, vol. 1, pp. 81–98 (1986)

[Gr86b] Gromov, M.: Partial Differential Relations. Springer, Berlin (1986)[Em84] Emch, G.G.: Mathematical and Conceptual Foundations of 20th Century Physics. North-

Holland, Amsterdam (1984)[FF13] Falceto, F., Ferro, L., Ibort, A., Marmo, G.: Reduction of Lie-Jordan Banach algebras and

quantum states. J. Phys. A Math. Theor. 46, 015201 (2013)

Chapter 4Invariant Structures for Dynamical Systems:Poisson Dynamics

The miracle of the appropriateness of the language ofmathematics for the formulation of the laws of physics is awonderful gift which we neither understand nor deserve. Weshould be grateful for it and hope that it will remain valid infuture research and that it will extend, for better or for worse, toour pleasure, even though perhaps also to our bafflement, towide branches of learning.

Eugene Wigner, The Unreasonable Effectiveness ofMathematics in the Natural Sciences, 1960.

4.1 Introduction

In this chapter we will start developing systematically one of the inspiring principlesof this book: all geometrical structures should be dynamically determined. In otherwords, given a dynamical system � we try to determine the geometrical structuresdetermined by �. The exact nature of the geometrical structure determined by �

that we will be interested in will depend on the problem we are facing, howeverthe simplest ones will always be of interest: symmetries and constants of motion asit was discussed in the previous chapter. Higher order objects like contravariant ocovariant tensors of order 2 tensorial will be discussed now. This problem will leadus in particular to the study of Poisson and symplectic structures compatible withour given dynamical system �.

Because of the special place that linear systems occupy in this work, we willcomplete the discussion on linearity started in the previous chapter and we willdetermine all linear structures compatible with a given dynamics. We will relate thiswith the problem of linearizability of a given dynamics and the study of normalforms. This will merge with another of the main themes of the book, the studyof linear systems and their geometrical structures. In particular we will study thefactorization problem for a linear dynamical vector field. This problem will lead usimmediately to the study of Poisson structures compatible with a given vector field.We will discuss the properties and structure of Poisson tensors and the particularinstance of symplectic structures that will be fully developed in the next chapter.


193

194 4 Invariant Structures for Dynamical Systems: Poisson Dynamics

We will solve the factorization problem for linear dynamics, that is under whatconditions a linear dynamical systemhas aPoisson structure.After solving the inverseproblem we will come back to the study of infinitesimal symmetries of a dynamicalsystem. When considering not one alone, but all the alternative ones, we will obtainthe notion of Lie algebra of symmetries of a dynamical system and their integratedform, a Lie group of symmetries of a dynamical system. We will establish the basicideas of this problem whose full description will be postponed to Chaps.6, 7 and 8.

The situation where the system has a compatible linear structure and we ask forthe existence not just of a Hamiltonian factorization but a gradient factorization willbe left to Chap.6 where Hermitean structures compatible with a given dynamics andthe geometry of quantum systems will be thoroughly discussed.

4.2 The Factorization Problem for Vector Fields

4.2.1 The Geometry of Noether’s Theorem

We have explored the basic properties of dynamical systems starting from the basicassumption that “dynamics is first” i.e., any description of a dynamical system has toconsider the dynamical evolution as themain conceptual ingredient and all additionalgeometrical structures used on its description should be ‘dynamically determined’.

In order to substantiate this program some additional mathematical ingredientsare needed. We have provided them with a minimum of additional assumptions, themain one, that the space of observables carries the structure of an algebra.

We have isolated linear structures as natural ingredients of the description of alarge family of dynamical systems emphasizing its non-uniqueness and the specialproperties derived of the compatibility of a given dynamical system with one partic-ular linear structure, i.e., its linearity and the explicit integrability of the system.

However, the nonexistence of linear structures compatible with a given dynamicsmakes necessary to use the full algebra of smooth observables, and thus we haveopened the door for the use of differential calculus ‘in the large’ in our explorationof dynamical systems.

A whole family of structures compatible with a given dynamics have alreadyemerged: smooth structures, linear structures, constants of the motion and symmetryalgebras. This is the first generation of compatible structures determined by dynam-ical systems and constitute the basic vocabulary of the theory. We would like to gobeyond this and discover additional structures compatible with a given dynamics.

In this book we will follow an “Occam’s razor” principle in the economy ofadditional structures used in the description of systems, thus we will always tryto search for structures that will emerge in simple situations, e.g., the existence ofregularities in the trajectories (like periodic behavior, trajectories easily mapped intoother trajectories, etc.). Then we have shown that we should explore the existence ofconstants of the motion and symmetry properties. The examples discussed in Chap. 1

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4.2 The Factorization Problem for Vector Fields 195

show that in many occasions both things come together: the system has constants ofmotion and has a rich symmetry algebra.

The existence of a link between constants of the motion and symmetries for agiven dynamical system will be called generically “Noether’s theorem”. We willanalyze first some implications of this situation. All these observations will lead usnaturally to the notion of Poisson structures, the first of the second generation ofstructures compatible with dynamical systems that are going to be discussed in thisbook.

4.2.1.1 Noether’s Theorem and 2-Tensors

The link between infinitesimal symmetries and constants of the motion is given byadditional invariant structures, and therefore this fact motivates the search for suchinvariant structures.

We have discussed some basic properties of infinitesimal symmetries and of con-stants of motion for a given dynamical system �. We are also used to the fact thatthese two notions are related. In fact, in Chap.1 we were able to show in concreteexamples how this link is established (see for instance Sect. 1.2.8 for a discussion ofsome examples).

Thus, we are used to think that symmetries are associated to invariants that canbe measured, i.e., to observables. However, it is not obvious how this connectionbetween symmetries and constants of motion is actually established.

Infinitesimal symmetries are described by vector fields X and constants of motionby functions f , thus, we will need an object that maps vector fields into functions.On the other hand, the particular value of a constant of motion at a given point isirrelevant because if f is a constant of motion for �, so it is f + c, where c is anarbitrary constant. Then as we discussed in previous sections, this implies that therelevant object is the 1-form d f rather than f and hence wewill consider the space ofclosed (locally exact) invariant 1-forms rather than the space of constants of motion.

The fact of looking for, instead of constants of the motion, their differentials, i.e.,1-forms, leads us to consider the natural geometrical objects mapping vector fieldsinto 1-forms, (0, 2)-tensors or, 1-forms into vector fields, that is (2, 0) tensors.

This is the road we are going to follow to understand the link between symmetriesand constants of the motion, looking for compatible (0, 2)-tensors or (2, 0)-tensorssuch that they transform infinitesimal symmetries into constants of motion or viceversa for a given dynamical system �.

4.2.2 Invariant 2-Tensors

As we have already discussed before, constants of motion for a given dynamicalsystem � close a subalgebra C(�) of the algebra of observables. But, as indicatedabove, we should consider the space of closed invariant 1-forms, rather than the space

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of constants of the motion. On the other side, the elements of the symmetry algebraof � close a Lie subalgebra X(�) of the Lie algebra of derivations.

A 1-form α will be said to be �-invariant if L�α = 0. The set of all such 1-formswill be denoted by �1(�). If f ∈ C(�) is a constant of motion for �, then d f is a�-invariant 1-form d f ∈ �1(�), because

L�d f = dL� f = 0 ,

and hence, dC(�) ⊂ �1(�). Conversely, if the 1-form α is exact, α = d f , the �-invariance of α implies that �( f ) is a constant, because d(L�( f )) = L�(d f ) = 0.Notice that this implies that such a function f can always locally be written at aneighbourhood of a regular point x of � (�(x) �= 0) as f (x1, x2, . . . , xn) = c x1,where x1 is the natural flow parameter of � (it suffices to choose local coordinatesstraightening out the vector field �).

Let us consider now a �-invariant 1-form α and a rule to assign to it a symmetryvector field Xα, α �→ Xα. We want such a map to be R-linear. Moreover, if we mul-tiply a �-invariant 1-form by a function which is zero in the neighborhood of a givenpoint then, when restricting to such a neighborhood, the corresponding symmetryhas to vanish identically. Then, as we discussed in Sect. 2.3.4, the correspondence hasto be local. If we assume, and that is the only assumption which could be relaxed,that such correspondence is established not only for invariant 1-forms, but for all1-forms, we conclude that it has to be given by a (2, 0)-tensor T on the state spaceE of the system, and then, there will exist a map T : T ∗E → T E , linear along thefibers of the space, defined by contraction with T , such that

Xα = T (α) .

The same argument could be reproduced in the converse direction. Thus, startingwith infinitesimal symmetries we could have argued in the same way, concludingthat there should exists a (0, 2)-tensor L such that

αX = L(X) ,

where now αX denotes the 1-form which is associated to the infinitesimal symmetryX . If one of the two tensors T or L is non-degenerate, then clearly they are the inverseone of each other, T = L−1, and that is what we will assume for the purposes of thediscussion now. Degenerate tensors can actually appear and they will do, but theirdiscussion will be postponed until we have advanced in our discussion.

Themere existence of such tensor T imposes some conditions on it. Thus, becauseXα is a symmetry of �, we have

0 = L�(Xα) = (L� T )(α) + T (L�α) = (L� T )(α) ,

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hence the tensor T itself must be invariant under � along the �-invariant 1-forms.Thus, in order to obtain such tensors, we can look, in particular, for tensors that areinvariant under �, that is tensors which are compatible with the given dynamics.

On the other side, we should remark that L�T = 0 if and only if L� T = 0,because for any pair of forms α and β,

(L�T )(α,β) = 〈β, (L� T )(α)〉 .

We shall consider now the �-invariant (0, 2) tensor L . We can decompose such atensor into its symmetric and skew-symmetric parts, L = S + A. Let us first considerthe case in which L is a (0, 2)-symmetric tensor, and we shall study later on the casein which L is skew-symmetric.

4.2.2.1 Invariant Symmetric Tensors

We study first the problem for symmetric (0, 2)-tensors. Following the linear par-adigm we will start first with a linear vector field X A and we would like to findsymmetric tensors g = gi j dxi ⊗ dx j , with gi j constants such that gi j = g j i , satis-fying

LX Ag = 0 . (4.1)

Wewill only consider the simplest case inwhich det G �= 0,whereG is the symmetricsquare matrix with elements gi j . We first remark that the invariance condition isequivalent to:

AT G + G A = 0 , (4.2)

because

LX g = gi j

[d(Ai

k xk) ⊗ dx j + dxi ⊗ d(A jl xl)

]=[(AT G)kl + (G A)kl

]dxk ⊗ dxl .

The above condition is equivalent to

(G A)T = −G A , (4.3)

then the matrix G A must be skew-symmetric. Moreover, as we assume that det G �=0, we see that

AT = −G AG−1 ,

and this implies that Tr AT = −Tr A = 0. By taking powers of our relation we get

(AT )k = (−1)k G Ak G−1 , (4.4)


and, therefore, all odd powers of A should be traceless.It is to be remarked that (4.4) means that the vector fields associated with odd

powers of A will also preserve g and the functions g(X Ak , X A j ), for any pair ofvalues k, j , are constants of the motion because

LX Ai [g(X Ak , X A j )] = (LX Ai g)(X Ak , X A j ) + g(LX Ai (X Ak ), X A j ) + g(X Ak ,LX Ai (X A j )) = 0 .

To have a better understanding of this situation, let us consider the two dimensionalcase with

X = (ax + by)∂

∂x+ (mx + ny)

∂

∂y,

and

g = α dx ⊗ dx + β dy ⊗ dy + γ (dx ⊗ dy + dy ⊗ dx) ,

i.e.,

A =(

a bm n

), G =

(α γγ β

).

Then,

AT G + G A =(

2 (a α + m γ) γ(a + n) + α b + m βγ(n + a) + b α + β m 2 (b γ + n β)

),

and the invariance conditions for g are⎧⎨⎩

a α + m γ = 0 ,

γ(a + n) + α b + m β = 0 ,

b γ + n β = 0 .

which is a homogeneous linear system for the unknown α,β, γ. The determinant ofthe matrix of coefficients is (a + n)(b m − a n) = −Tr A · det A and, therefore, ifdet A �= 0, the compatibility condition is Tr A = a + n = 0. Then, a solution forthese equations is provided by m = α, n = γ = −a, β = −b. Consequently, themost general constant non-degenerate (0, 2)-tensor invariant under X A is given by:

g = m dx ⊗ dx − b dy ⊗ dy − a (dx ⊗ dy + dy ⊗ dx) .

We see that if Tr A = a + n �= 0, there is no solution. Actually, if det A = 0 as well,there is a second solution with a degenerate tensor.

Exercise 4.1 Complete the previous computation finding out the constants ofmotionof the system and the algebra of symmetries of X A.


4.2.2.2 Invariant Skew-Symmetric Tensors

We next study the characterization of those skew-symmetric (2, 0)-tensors in thelinear space E which are invariant under a linear dynamics. We will start with a fixedlinear dynamics given by the vector field X A = Ai

j x j ∂∂xi . If the invariant bi-vector

field � we are looking for is a constant, i.e., �i j = constant, then because of therelations

LX A

∂

∂xi= −A j

i∂

∂x j, i = 1, . . . , n ,

which imply that

LX A� = �i j[−Ak

i∂

∂xk∧ ∂

∂x j− Al

j∂

∂xi∧ ∂

∂xl

]

= −(A�)k j ∂

∂xk∧ ∂

∂x j− (�AT )il ∂

∂xi∧ ∂

∂xl.

and because the matrix � is skew-symmetric, the condition for LX A� = 0 is

A � + � AT = 0 , (4.5)

which can be rewritten (A�)T = A� (compare with Eq. (4.3)). It is easy to show byiteration of (4.5) that A2k+1� = −�(AT )2k+1 while A2k� = �(AT )2k . Therefore,A2k� (and �(AT )2k ) is a skew-symmetric matrix, and A2k+1� is a symmetric one,for any positive integer number k. Therefore, �(AT )2k provide new solutions ofadmissible skew-symmetric matrices, because

A[�(AT )2k] = −�(AT )2k+1 .

Note, however, that when A is not invertible the rank of each power may change andwe would get a sequence of admissible skew-symmetric tensors with different rank.

When considering a general dynamics on a smooth manifold M , we have as ithappens in the symmetric case, much less control on the situation, however we canstill obtain a theoremconnecting the constant ofmotion and infinitesimal symmetries.

Theorem 4.1 Let � be a dynamical system possessing a (2, 0) �-invariant tensor�. Then,

i. If f is a constant of motion for �, then the vector field X f = −�(d f ) is aninfinitesimal symmetry for �.

ii. If X is a symmetry of � such that there exists a function f with �(d f ) = −X,and � is regular, then �( f ) is a constant.

Proof 1. Having in mind that

[X f , �] = L�(�(d f )) = (L��)(d f ) + �(L�(d f )) ,


as we have assumed that � is invariant under �, L�� = 0, and as f is a constant ofmotion, we also have L�(d f ) = d(L� f ) = 0. Therefore, [�, X f ] = 0.

2. If X is a symmetry of � such that there exists a function f with �(d f ) = −X ,then,

L�(�(d f )) = 0 = (L��)(d f ) + �(d(L� f )) ,

and therefore, when � is regular,

d(L� f ) = 0 ,

i.e., L� f is constant. �

4.2.3 Factorizing Linear Dynamics: Linear Poisson Factorization

After the previous discussion, given a dynamical system � we observe that � itselfis a symmetry of �, hence we would like to know if there exists a constant of motionH corresponding to it. For that we should find a skew-symmetric (2, 0)-tensor suchthat −�(d H) = �. Thus, we are led to solve the equation

� = �(α) ,

where � is given and �, and the exact form α are unknown. Notice that all vectorfields � can be written in this way. We will analyze this problem by thinking first inthe linear case.

Thus, let � be a linear vector field determined by the linear map A : E → E ,i.e., � = X A. Then, � is homogeneous of degree 0, because it is linear, L�� = 0.If we denote the degree of homogeneity of a (homogeneous) tensor T by h(T ), i.e.,L�T = h(t)T , then we have that if � = �(α), with � and α homogeneous, thenh(�) = h(�) + h(α). Thus, h(�) = −h(α) if we want � to be linear. It happensthat h(α) ≥ 1 and h(�) ≥ −2 if we want α and � to be smooth on E . Hence,−2 ≤ h(�) = −h(α) ≤ −1. Thus the only solutions for these conditions are givenby h(�) = −2 = −h(α) and h(�) = −1 = −h(α).

4.2.3.1 First Case: � Constant

The first case, means that the bi-vector � has the form

� = �i j ∂

∂xi∧ ∂

∂x j, �i j = −� j i ∈ R ,

with �i j constants, and α has the form

α = −Hi j xi dx j ,


with Hi j constants. Moreover if α is closed, then Hi j dxi ∧ dx j = 0 and thereforeHi j = Hji , i.e., H is a symmetricmatrix. Thenα = −d H with H being the quadraticfunction

H = 1

2Hi j xi x j .

Weare going to see that finding a constant ofmotion for a linear dynamics amountsin the first case to solve the factorization problem of finding a skew-symmetricmatrix� and a symmetric matrix H , which will provide us the quadratic constant of motion,such that

A = � H , (4.6)

because

−�(d H) = �il Hkl xk ∂

∂xi,

and therefore,A j

k xk = xk Hki � j i .

Conversely, if we suppose that the matrix A = (Aij ) can be factorized as a product

of matrices A = � H , i.e.Ai

j = �ik Hk j ,

with � being a skew-symmetric matrix and H being a symmetric one,

�i j = −� j i , Hi j = Hji ,

then the vector field X A is Hamiltonian with respect to the Poisson structure definedby the matrix � and the Hamiltonian function is the quadratic function defined bythe matrix H .

Example 4.2 (The 1-dimensional harmonic oscillator) As an example we can con-sider the 1-dimensional harmonic oscillator, described by the system:

q = p , p = −q ,

whose associated vector field is � = X A with A being the matrix

A =(

0 1−1 0

),

which can be factorized as a product A = � H , where

� =(

0 1−1 0

), H =

(1 00 1

),


which correspond to

� = ∂

∂q∧ ∂

∂ p, H = 1

2(q2 + p2) .

The function f is a constant of motion if, and only if,

p∂ f

∂q− q

∂ f

∂ p= 0 ,

whose general solution is given by an arbitrary function of q2 + p2. The symmetryof X A corresponding to the quadratic function H = 1/2 (q2 + p2) is X A itself.

A vector field X B commutes with X A if it takes the form

B =(

a b−b a

), a, b ∈ R ,

because that is the general form of a matrix commuting with the given matrix A.Then X B can be written as X B = a X I + b Xi σ2 , where σ2 is the second Pauli

matrix. We can see that X I is not a Hamiltonian vector field with respect to �,because there is no symmetric matrix such that

(1 00 1

)=(

0 1−1 0

)(α ββ γ

).

However, as we have seen above, Xi σ2 is the given dynamical system, which isHamiltonian.

Another remarkable property is that if the vector field X A is Hamiltonian, thenthe bivector field � is invariant under X A, because if A admits a factorization likein (4.6), then

� AT + A � = � H �T + � H � = � H (�T + �) = 0 ,

and therefore condition (4.5) holds.In the second case we mentioned before, when h(�) = −1, the bi-vector �

should have the form

� = ci jk xk ∂

∂xi∧ ∂

∂x j,

where ci jk are constants. The 1-form α (with h(α) = 1) looks like

α = ai dxi ,

with each ai being a constant. The inverse problem now for � is equivalent to deter-mining


A jk = −c ji

k ai ,

because

−i(al dxl)

(ci j

k xk ∂

∂xi∧ ∂

∂x j

)= ai c ji

k xk ∂

∂x j,

where c is a (2, 1) tensor, skew-symmetric in the superscripts and a is a constantvector. From now on we will concentrate on the first situation, leaving the secondcase for a further discussion that will bring us into the general theory of Lie algebrasand Lie groups (see Sect. 4.3.3).

4.2.3.2 The Factorization Theorem

Let assume that the vector field X A is Hamiltonian with respect to the constantPoisson structure � and with Hamiltonian the quadratic function defined by thematrix H . Therefore, A admits the factorization A = � H , with H T = H and�T = −�. Then, AT = H T �T = −H �, and consequently,

A � = � H� = −� AT . (4.7)

A few simple consequences of these facts are:

1. AT = −H �,2. H A = H � H = −AT H .3. Having in mind that the trace of a matrix coincides with that of its transpose, and

that the trace is invariant under circular permutation, we see that Tr (A) = 0.4. For any integer number k (if A is invertible, for k positive in general), a similar

argument works for A2k+1, and therefore,

Tr (A2k+1) = 0 . (4.8)

5. From the previous property it follows that the minimal polynomial and the char-acteristic polynomial of the matrix A only contains even degree powers. Asa consequence, both real and imaginary eigenvalues will appear in pairs; realeigenvalues ±λ with the same multiplicity, while other complex eigenvalues λare such that −λ and their conjugates ±λ∗ are also eigenvalues with the samemultiplicity.

6. On the other side, A3 = � H � H � H = �(−AT H A), shows that the vectorfield X A3 is Hamiltonian with respect to the same Poisson structure. A simpleiteration of the reasoning will lead to

A2k+1 = (−1)k �((AT )k H Ak) ,

and therefore, all odd powers of A give raise to Hamiltonian vector fields withrespect to the same Poisson structure. Of course, all of them pairwise commute,


but however they are not independent, because Cayley theorem establishes arelation among these powers: the matrix A satisfies its characteristic equation.

Moreover, if we recall that the correspondence between the set of linear homo-geneous vector fields and associated Hamiltonians is a Lie algebra homomorphism,we can conclude that any linear Hamiltonian vector field admits constants of motionin involution provided by

fk(x) = (−1)k((AT )k H Ak)i j xi x j .

We will study in another section what happens in the particular case of a regular �,i.e. when we require � to be invertible.

Summing up these properties we have that a necessary condition for A to befactorizable in the product of an skew-symmetric and a symmetric matrix is thatthe trace of all odd powers is zero. This implies that the characteristic and minimalpolynomials of A have only even degree terms. Thus the real and purely imagi-nary eigenvalues occur always in pairs (λ,−λ), the complex eigenvalues occurs inquadruples (λ, λ,−λ,−λ) and the zero eigenvalue has even degeneracy. If � is askew-symmetric factor for A, it is also a skew-symmetric factor for any odd power ofA. Even powers of A (A invertible) will permute admissible skew-symmetric factors.

The previous conditions are also sufficient to characterize linear systems that canbe factorized. We would like to end this section by stating it in the form of a theorem[Gi93].

Theorem 4.2 (Factorization theorem) Let � be a linear dynamical system on thereal vector space E with associated linear operator A. A necessary and sufficientcondition for the vector field � to be factorizable, i.e., that there exists a skew-symmetric (2, 0) tensor � and a symmetric (0, 2) tensor H such that � = � ◦ H, isthat all odd powers of A are traceless (Tr A2k+1 = 0, k ≥ 0) and:

i. No further condition if the eigenvalues of A are non-degenerate or purely imag-inary,

ii. For degenerate real or complex eigenvalues λ, the Jordan block belonging to λhas the same structure as the Jordan block belonging to −λ,

iii. Zero eigenvalues have even multiplicity.

In such case the flows corresponding to odd powers of A will leave invariant boththe skew-symmetric and the symmetric tensors � and H.

Moreover the linear flows ϕ(k)t = exp(t A2k), k = 1, 2, . . . corresponding to even

powers of A will transform the previous factorization of � in an alternative one � =�

(k)t H (k)

t with �(k)t = exp(t A2k)� exp(t A2k) and H (k)

t = exp(t A2k)H exp(t A2k).

Proof The necessity follows from the discussion before. We show now that theprevious conditions are sufficient.

The canonical real Jordan blocks for A are assume to be of the following types(see the classification after Theorem 3.23 and Eq. (3.121)): (simple Jordan blocks)type I, type II, type III, (non-simple Jordan blocks) type IVa, type IVb and type IVc.

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There are three cases to consider (of course in all themwe assume Tr A2k+1 = 0):

1. In the first case A has non-degenerate eigenvalues, real or complex. We considerfirst the simple case when the eigenvalues are all real and distinct. In this casean invertible matrix P will bring A to the canonical form:

A = P−1AP =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

λ1 · · · 0 0 · · · 0...

. . ....

.... . .

...

0 λn 0 00 0 −λ1 · · · 0...

.... . .

...

0 0 · · · −λn

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

which is obviously Hamiltonian with respect to the canonical Poisson struc-ture (the canonical form of a non-degenerate �) that in linear coordinatesx1, . . . , xn, xn+1, . . . , x2n is given by � = ∑

k∂∂x k ∧ ∂

∂x k+n . Then we use P−1

to transform � into a possible Poisson tensor for A.It is also possible to bring A to the form:

A′ = P ′−1AP ′ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

λ1 0 00 −λ1 0 00 λ2 0 00 0 −λ2 0 0...

. . ....

0 λn 00 0 −λn

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

which is Hamiltonian with respect to the Poisson tensor:

� =

⎛⎜⎜⎜⎝

�1 0 00 �2 0...

. . ....

0 �n

⎞⎟⎟⎟⎠

with �k =(

0 λk

−λk 0

). This Poisson structure shows that our spaceR

2n decom-

poses into 2-dimensional spaces Wk such that our starting vector field � decom-poses accordingly into the direct sum of Hamiltonian vector fields �k = � |Wk

for the subspace corresponding to the pair (λk ,−λk).2. We consider now the situation of complex non-degenerate eigenvalues. We can

construct a non-degenerate Poisson structure for the subspace Wk correspondingto the quadruple {λ} = (λ, λ,−λ,−λ) and then we consider the direct sum ofthe Poisson tensors, the direct sum of Hamiltonian forms and vector fields. We


can write the restriction of A to W{λ} in the form:

(Jλ 00 −J T

λ

)

with Jλ of type III. This matrix is Hamiltonian with respect to the canonicalPoisson structure.Therefore when A contains only simple Jordan blocks and real or complex non-degenerate eigenvalues, the system can be factorized by requiring that the traceof odd powers vanishes.

3. As a second case we consider the situation when there is degeneration in theeigenvalues of A, but eigenvalues are not purely imaginary or zero.For a Hamiltonian vector field � the Jordan block belonging to a real or complexeigenvalue λ has the same structure as the Jordan block belonging to lambda,that is, if in the almost diagonal form of A there is a certain Jordan block J , theremust be also J ′ conjugate to −J (see [Ar76], Appendices 6 and 7). Then whenthis is the case Aλ = A |Wλ can be brought to the form:

Aλ =(

J 00 −J T

),

which again is Hamiltonian with respect to the canonical Poisson structure. Thusfor degenerate eigenvalues (excluding imaginary eigenvalues and zero eigenval-ues) the requirement that the Jordan blocks belonging to λ and−λ have the samestructure gives a necessary and sufficient condition (along with the zero trace ofall odd powers of A) for A to be Hamiltonian.

4. We consider now imaginary eigenvalues. It is clearly sufficient to consider amatrix A with only one pair±iλ n times degenerate with one non-simple Jordanblock. The almost diagonal representation of A is:

A =

⎛⎜⎜⎜⎜⎝

J I

... J

.... . . I

0 · · · 0 J

⎞⎟⎟⎟⎟⎠ (4.9)

with J =(

0 λ−λ 0

)repeated n times. Let us first suppose that n is odd. For

n = 1 we have, of course, A = J and a possible � is given by the canonicalone:

�0 =(

0 1−1 0

).


For arbitrary (but odd) n, we may set:

� =

⎛⎜⎜⎜⎜⎜⎜⎝

0 0 · · · 0 �0

0... −�0 0

.... . .

......

0 −�0 0�0 0 · · · 0 0

⎞⎟⎟⎟⎟⎟⎟⎠

that is, the �0 are repeated on the counterdiagonal with alternating sign. Then:

�−1 A =

⎛⎜⎜⎜⎜⎜⎜⎝

0 0 · · · 0 −�0 J

0... �0 J �0

.... . .

......

0 �0 J −�0 0−�0 J �0 · · · 0 0

⎞⎟⎟⎟⎟⎟⎟⎠

Now the terms in the principal counterdiagonal, which are in even place, aftertransposition of �−1 A go in terms of even place too, and the sign is unchanged.Moreover�0 J is symmetrical, thus as for these terms �−1 A is symmetrical. Letus now consider the second non-null counterdiagonal. There is an even numberof them and the transposition sends the transpose of a term �0 into the placeoccupied by a former −�0 but this is just �T

0 and also for these terms �−1 Aturns out to be symmetric.For even n, a possible � is:

� =

⎛⎜⎜⎜⎜⎜⎜⎝

0 0 · · · 0 I

0... −I 0

.... . .

......

0 I 0−I 0 · · · 0 0

⎞⎟⎟⎟⎟⎟⎟⎠

(4.10)

and the proof goes along the same lines as before.5. Finally, we consider zero eigenvalues. When λ = 0 the expression Eq. (4.9)

becomes:

A =

⎛⎜⎜⎜⎜⎝

0 I

... 0

.... . . I

0 · · · 0 0

⎞⎟⎟⎟⎟⎠

with n even, and then the � given in Eq. (4.10) gives a Poisson structure for �. �


4.2.3.3 The Symmetric Case Again

At this point we compare the condition for X A to preserve a symmetric (0, 2)-tensorand a symplectic structure.Wehave found AT �+�A = 0 for the case, AT G+G A =0 for the symmetric case. By using �A = H symmetric and (G AT )T = B skew-symmetric, we have A = �−1H or A = G−1B respectively. Therefore if det A �= 0,any solution of the first problem also provides a solution for the second problem.

Actually, if AT � + �A = 0, then G defined by G = �A is such that

AT G + G A = AT �A + �AA = (AT � + �A)A = 0,

and conversely, if AT G + G A = 0, then � = G A−1 is such that

AT � + �A = AT G A−1 + G A−1A = (AT G + G A)A−1 = 0.

To put it differently, if the symmetric matrix associated with the Hamiltonian isnot degenerate we can use it to define a (pseudo) metric tensor. Vice versa, the skewsymmetric matrix G A can be used to construct a symplectic structure and G will beused to define the Hamiltonian function. Thus our problem amounts to find necessaryand sufficient conditions for A to be decomposed in the product of a symmetricmatrixby a skew-symmetric one with one or both factors non degenerate.

From AT � + �A = 0 or AT G + G A = 0, we find

(AT )3� = −(AT )2�A = AT �A2 = −�A3

(AT )3g = −(AT )2G A = AT G A2 = −G A3

Therefore all odd powers share the same property. In either one of the two problemswe have

A = S(−AT )S−1,

i.e., a necessary and sufficient condition for A to be decomposed into the productof a symmetric matrix by a skew-symmetric one is that the matrix is conjugatedto the opposite of its transpose. This has the consequence that the characteristicpolynomial and minimal polynomial only contain even powers. Both polynomialscan be considered as polynomials in the variable y = x2 and degree 1

2 of the startingones.

It is possible now tomake a fewcomments on the uniqueness of the decomposition.We start with AT � + �A = 0 and consider any symmetry transformation for A,i.e., an invertible matrix S such that S AS−1 = A. We notice that also ST AT =AT ST . Now, from AT � + �A = 0 we get by left multiplication by ST and rightmultiplication by S,

ST AT �S + ST �AS = 0

and using that S A = AS,


AT ST �S + ST �S A = 0,

i.e., ST �S is another alternative structure if S is not a canonical transformation.Therefore with any noncanonical transformation that is also a symmetry for A wefind a new structure. Another way to approach the same problem is to start with thedecomposition A = �−1H . If S is an invertible linear transformation we have

A �→ S−1AS

� �→ ST �S

H �→ ST H S

�−1 �→ S−1�−1(ST )−1.

Now from A = �−1H we find

S−1AS = S−1�−1(ST )−1ST H S

and for S−1AS = A we get the new decomposition.Notice that not all possible different decompositions of A can always be obtained

by the previous method. However notice that if A admits two different decomposi-tions if

A = �−11 H1 = �−1

2 H2

there exists a linear coordinate transformation taking �1 into �2 (the existence ofDarboux charts implies thatGl(n, R) acts transitively on linear symplectic structures,see Sect. 5.2.4). Let us say that

�1 = ST �2S ,

then �−11 = S−1�−1

2 (ST )−1 and we have H1 = ST H2S therefore

�−11 H1 = S−1�2(ST )H2S = S−1�2H2S = S−1AS.

Then if H1 = H1 we getA = �−1

1 H1 = S−1AS ,

i.e., S is a symmetry for A. As for noncanonical symmetries we know that all evenpowers of A will be infinitesimal generators of non canonical symmetries. For ageneric A, i.e., all eigenvalues are simple, these symmetries exhaust all noncanonicalsymmetries. Therefore a linearHamiltonian systemhas at least an n-parameter familyof alternative Hamiltonian descriptions. As a matter of fact, because odd powers, inthe generic case, generate a locally free action of R

n which is a canonical actionwe also find n independent quadratic functions which are pairwise in involution.

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Therefore a generic linear Hamiltonian system is completely integrable and admitsan n-parameter family of alternative Hamiltonian descriptions.

4.3 Poisson Structures

4.3.1 Poisson Algebras and Poisson Tensors

We will start the discussion of Poisson structures by considering first the simplestpossible case: constant Poisson structures on linear spaces. We have found suchstructures already in the discussion of the factorization and inverse problem forlinear dynamics Sect. 4.2.3, and in the analysis of the geometry of Noether’s theoremSects. 4.2.1 and 4.2.2. We will summarize here some of the properties that we havealready found and we will prepare to move to the general situation.

Let E be a finite-dimensional linear space of dimension n and E∗ will denoteits dual space. Consider a skew-symmetric bilinear form � on E∗. Remember thatbecause of the easy principle of tensorialization of algebraic structures Sect. 2.4.3,there exists a unique bivector field on E associated to� denoted by the same symbol.Let us recall that if we choose a linear basis {ek}, and we denote by {el} its dual basis,and �kl denotes the matrix representing � in such basis, i.e., �(ek, el) = �kl , thenthe bivector � takes the form:

� = �i j ∂

∂xi∧ ∂

∂x j. (4.11)

This allows us to define a Poisson algebra structure structure on the linear spaceF(E) of real functions in E , by setting

{ f, g} = �(d f, dg) = �i j ∂ f

∂xi

∂g

∂x j , (4.12)

and therefore the coefficients �i j are just

�i j = �(dxi , dx j ) ,

i.e.,

{ f, g} = �(d f, dg) = ∂ f

∂xi�(dxi , dx j )

∂g

∂x j= ∂ f

∂xi{xi , x j } ∂g

∂x j. (4.13)

The function { f, g} is called the Poisson bracket of f and g and the bilinear form onthe space of smooth real-valued functions on E given by ( f, g) �→ { f, g} is called thePoisson bracket associated with �. The Poisson bracket (4.12) is skew-symmetricand bilinear by construction, and it is easy to check that this bracket will also satisfy

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4.3 Poisson Structures 211

for any triplet of functions f, g, h the property:

{ f, {g, h}} + {h, { f, g}} + {g, {h, f }} = 0, (4.14)

which is called the Jacobi identity. This means that {·, ·} endows F(E) with a Liealgebra structure. Moreover, it is easy to check that it satisfies Leibniz identity too:

{gh, f } = {g, f }h + {h, f }g , (4.15)

and therefore, for every function f the map g �→ {g, f } is a derivation of theassociative algebra F(E) with the usual product of functions as composition law.The properties above define a Poisson algebra structure on the space of smoothfunctions on E . To be precise we have the definition:

Definition 4.3 A Poisson algebra is a real linear space F (finite or infinite-dimen-sional) which is an associative algebra with respect to a bilinear product ·, a Liealgebra with respect to a skew-symmetric bilinear product {·, ·} and both operationsare compatible in the sense of Eq. (4.15), i.e., the Lie bracket satisfies the Leibnizidentity with respect to the associative product.

Notice that because of the previous discussion, any constant bivector� on a linearspace E defines a Poisson algebra structure on the space of smooth functions on E .Conversely, if we have a Poisson structure on the algebra of smooth functions on alinear space E , we may define the bivector:

� = {xi , x j } ∂ f

∂xi

∂g

∂x j, (4.16)

where xi are linear coordinates on E .

Exercise 4.3 Prove that the object� defined by Eq. (4.16) actually defines a bivectorfield on E .

Notice that in general the bivector field� associated to a Poisson algebra structurewill not be constant. In fact we can extend the construction of � not just consideringthe algebra of smooth functions on a linear space E but to the algebra of smoothfunctions on a smoothmanifold M . Thus if we have a Poisson algebra structure on theassociative algebra of smooth functions on M , we may define a bivector field � byEq. (4.16) with xi any local chart on M . Because of Exercise 4.3 the local expressionprovided by (4.16) actually provides a globally well-defined bivector field on M .

Definition 4.4 Given a Poisson algebra structure (F(M), ·, {·, ·}) on the associativealgebra of smooth functions on a manifold M , we will call the bivector field �

defined locally by Eq. (4.16) the Poisson tensor associated to the given Poissonalgebra structure on F(M).


If� is the Poisson tensor of a Poisson algebra structure onF(M) it is an immediateconsequence of Leibnitz’s rule, Eq. (4.15), that the assignment:

X f (g) = {g, f } (4.17)

defines a vector field on M because it actually defines a derivation on F(M). Wewill denote this vector field by X f and it can also be written as:

X f = −�(d f )

where � denotes as before the natural bundle map � : T ∗M → T M defined as〈�(α),β〉 = �(α,β). The vector field X f will be called the Hamiltonian vectorfield associated to f .

When the bi-vector� is such that ker � = 0, we will say that it is non-degenerate.In this case the bundle map � will be invertible and we can construct a 2-form ω onM associated to � by means of:

ω(X, Y ) = �(�−1(X), �−1(Y )) .

We will show later that the 2-form ω defined in this way is closed (and non-degenerate). Such forms will be called symplectic (see Sect. 5.2.1, Eq. (5.4)).

Taking into account the explicit expression of the Poisson bracket given above,we see that the Hamiltonian vector field X f can be written as:

X f = {xi , x j } ∂ f

∂x j

∂

∂xi. (4.18)

It is easy to check that:

Proposition 4.5 LetF be Poisson differentiable algebra (that is,F is a differentiablealgebra), the assignment f �→ X f , with X f (g) = {g, f } is a Lie algebra anti-homomorphism:

[X f , Xg] = −X{g, f } f, g ∈ F

Proof Obviously, the correspondence is R-linear. Let f, g and h be arbitrary func-tions in F . Thus,

[X f , Xg]h = X f (Xgh) − Xg(X f h) = {Xgh, f } − {X f h, g} = {{h, g}, f } − {{h, f }, g} .

Using Jacobi identity the preceding relation can be written

[X f , Xg]h = {{h, g}, f } + {{ f, h}, g} = −{{g, f }, h} = −{h, { f, g}} = −X{ f,g}h ,

from which we obtain [X f , Xg] = −X{ f,g}. �

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Consequently, the distribution defined by the Hamiltonian vector fields isinvolutive.

A function h is a constant of the motion for X f if and only if { f, h} = 0. Due tothe Jacobi identity, the Poisson bracket of two constants of motion is still a constantof the motion, because if { f, g} = 0 and { f, h} = 0, then

{ f, {g, h}} = −{g, {h, f }} − {h, { f, g}} = 0 .

On the other side, when the Poisson structure is degenerate there can exist func-tions f such that { f, h} = 0 for any function h, i.e., d f ∈ ker�, and thereforethey will be constant for any possible motion. These functions are called Casimirfunctions. The associated vector field vanishes identically.

The Jacobi identity (4.14) can be expressed in terms of the components �i j ofthe bivector field �. This condition is equivalent to the quadratic relation:

�il ∂� jk

∂xl+ �kl ∂�i j

∂xl+ � jl ∂�ki

∂xl= 0 . (4.19)

Of course this condition is automatically satisfied when the components �i j areconstant. It is clear that the nondegeneracyof�playedno role in the stated properties,therefore we can also allow� to be degenerate and eventually of a non constant rank.Thereforewe have that a bivector field�, degenerate or not, defines a Poisson bracketif and only if,

0 = �(d f,�(dg, dh)) + �(dh,�(d f, dg)) + �(dg,�(dh, d f )) . (4.20)

An important remark is that this relation is not additive, and then, even if �1 and�2 satisfy this condition, it may be false for the sum�1+�2. When�1+�2 is alsoa Poisson structure we will say that the Poisson structures�1 and�2 are compatible,and then any linear combination λ1 �1 + λ2 �2 is a Poisson structure too.

Pairs of compatible bivectors are characterized by:

�il1

∂�jk2

∂xl+�kl

1∂�

i j2

∂xl+�

jl1

∂�ki2

∂xl+�il

2∂�

jk1

∂xl+�kl

2∂�

i j1

∂xl+�

jl2

∂�ki1

∂xl= 0 . (4.21)

An interesting way of expressing this property is by means of a commutator [ · , · ]introduced by Schouten [Sc53] and used by Lichnerowicz in this context [Li77]. Infact it was Lichnerowicz who showed that Jacobi’s identity is given by

[�,�] = 0 . (4.22)

We recall that the Schouten bracket (see Appendix D for other related brackets) isthe unique extension of the Lie bracket of vector fields to the exterior algebra of mul-tivector fields, making it into a graded Lie algebra (the grading in this algebra givenby the ordinary degree of the multivectors minus one). In fact, given a multivector V ,


the linear operator [V, · ] defines a derivation on the exterior algebra of multivectorfields on M . The degree of [V, · ] equals the ordinary degree of V . So, for any triplet(U, V, W ) of multivectors,

[U, V ∧ W ] = [U, V ] ∧ W + (−1)u(v+1)V ∧ [U, W ] ,

u and v being the ordinary degree of U and V respectively. In the particular case ofthe wedge product of vector fields,

[X∧Y, U∧W ] = [X, U ]∧Y ∧W+X∧[Y, U ]∧V +Y ∧[X, W ]∧U+X∧U∧[Y, W ] .

(4.23)Therefore if V = X ∧ Y is a monomial bivector, then

[V, V ] = 2X ∧ Y ∧ [X, Y ] . (4.24)

Now, it is clear that if �1 and �2 are Poisson structures, then �1 + �2 is a Poissonstructure if and only if [�1 + �2,�1 + �2] = 0, i.e., iff [�1,�2] = 0. Providedthis condition is met, any linear combination of �1 and �2 with real coefficients isalso a Poisson structure.

4.3.2 The Canonical ‘Distribution’ of a Poisson Structure

A Poisson structure provides a linear correspondence, f �→ X f , which is equivalentto giving for any point x ∈ M an R-vector bundle map �(x) : T ∗

x M → Tx M whichpreserves the base manifold. The rank of �(x) at the point x ∈ E was called therank of the Poisson structure at the point x . In order to simplify notation we will notdistinguish � from �.

Now, for each x ∈ M the image of Tx M under �x is a linear subspace�x (T ∗

x M) ⊂ Tx M . In general the family of subspaces �x (T ∗x M) ⊂ Tx M will

not define a distribution on T M because its rank can change.However because of Proposition 4.5, the family of Hamiltonian vector fields X f

are involutive. Then it is possible to extend Frobenius theorem to this setting to provethat there is a singular foliation of the manifold such that the tangent spaces to theleaves at each point are actually the subspace �x (T ∗

x M). This result is sometimesreferred to as Kirillov’s theorem and the singular foliation of a Poisson structurecalled Kirillov’s foliation [Ki62].

Actually it can be proved by using transversality arguments that the set of pointswhere the rank of � is maximal (hence constant) is an open dense set in M calledthe regular part of the Poisson tensor. If we consider now the restriction of � tothe regular part of M , it will define an involutive distribution and we can apply theFrobenius theorem, Theorem 2.16.

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We have started with Poisson algebra F and considered the set of Casimir func-tions

C(F) = {h ∈ F | { f, h} = 0,∀ f ∈ F} .

Obviously the Casimir functions define a Lie subalgebra of F . On the other sidewe recall that every such subalgebra has associated an equivalence relation, denoted≈, in the state space M by saying that two points of M , x and y, are equivalent iff (x) = f (y), for each function f ∈ C(F). Correspondingly we have a foliationwhose leaves are the level set of the map π : M → M/ ≈. Notice that in generalC(F) will not be a differentiable algebra thus the space of leaves M/ ≈ will not bea manifold.

The level sets of the Casimir functions describe the regular leaves of the Kirillov’sfoliation. However the singular part of M , that is, the points where the rank of � isnot maximal, is the union of smooth submanifolds whose tangent spaces are spannedby Hamiltonian vector fields, but in order to define them we need extra functionswhich are not Casimirs (we will see that when studying spaces of coadjoint orbits ofgroups Sect. 4.3.5).

Moreover, these leaves are invariant for any Hamiltonian dynamics because thevector field XH corresponding to a function H is by definition tangent to the foliation.This allows us to define a new Poisson structure on the leaves of the foliation whichis non-degenerate, that is a symplectic form ω, by considering:

ωx (v, u) = �x (d f (x), dg(x)) ,

with v = X f (x) and u = Xg(x). Notice that the definition of ω does not depend onthe choice of the functions f such that X f (x) = v. All these remarks together leadto the following theorem:

Theorem 4.6 (Kirillov’s symplectic foliation theorem) Let � be a Poisson structureon a smooth manifold M. Then there exists a singular foliation of M whose leavesare such that their tangent spaces are spanned by the Hamiltonian vector fields. Eachleaf carries a canonical symplectic structure induced by the Poisson brackets on M.

4.3.3 Poisson Structures and Lie Algebras

When looking for a solution of the inverse problem we can fix our attention in aconcrete family of Poisson tensors. The factorization problem of dynamics discussedin Sect. 4.2.3 targets constant Poisson tensors, i.e., Poisson tensors homogeneous ofdegree −2. There the only condition on the components of the tensor is the skew-symmetry of them, �i j = −� j i . An extremely interesting case is when �i j arelinear functions in the coordinates,

�i j (x) = ci jk xk . (4.25)


Then conditions for � to define a Poisson structure becomes the Jacobi identity forthe structure constants of a Lie algebra Eq. (2.132):

∑cyclic i, j,k

∑l

ci jl clk

k = 0 .

This shows that for any Lie algebra there should be a related Poisson structure. Wewill discuss this correspondence in detail.

Let g be a finite-dimensional Lie algebra (Definition 2.19), and let g∗ be its dualspace. The choice of a basis {ξk} for the linear pace g provides us a dual basis {θk},〈θk, ξl〉 = δk

l , for g∗ and the corresponding set of linear coordinates xk , μ = xkθ

k ,for all μ ∈ g∗. Let ci j

k denote the structure constants relative to the given basis[ξi , ξ j ] = ci j

kξk . The family of linear functions

�i j (x) = ci jk xk

define the components of a Poisson tensor in g∗. The corresponding Poisson algebrastructure on F(g∗) can be defined intrinsically as follows.

If f ∈ C∞(g∗), the differential (d f )μ at μ ∈ g∗ is a linear map (d f )μ : Tμg∗ →

R. There is a natural identification of g∗ with Tμg∗, and in this sense (d f )μ can be

thought of as an element of (g∗)∗ and therefore of g. Let us assume that δu f denotessuch an element, i.e.,

〈ν, δμ f 〉 = (d f )μ(ν) = d

dtf (μ + tν)

∣∣∣∣t=0

.

The canonical Poisson structure on g∗, called in what follows the Lie–Poisson struc-ture on g∗, is then defined by

{ f, g}(μ) = 〈μ, [δμ f, δμg]〉 . (4.26)

Exercise 4.4 Check that Eq. (4.26) defines a Poisson structure on F(g∗).

The linear subspace F (1)(g∗) of linear functions on F(g∗) is closed with respectto the Lie–Poisson algebra structure. In fact, for any ξ ∈ g we define the linearfunction xξ ∈ F (1)(g∗):

xξ(μ) = 〈μ, ξ〉 . (4.27)

It is quite easy to see that,δμxξ = ξ ,

because,

(dxξ)μ(ν) = d

dtxξ(μ + tν)

∣∣∣∣t=0

= 〈ν, ξ〉,

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and consequently, if ξ and ζ are elements of g,

{xξ, xζ}(μ) = 〈μ, [ξ, ζ]〉 = x[ξ,ζ](μ) . (4.28)

Therefore, the Poisson bracket of two linear functions is again a linear function. Thuswe have proved:

Theorem 4.7 Any Poisson algebra structure on the space of smooth functions of afinite-dimensional linear space E whose associated Poisson tensor is homogeneousof degree −1 defines a Lie algebra structure on E∗. Conversely, given a finite-dimensional Lie algebra g, there is a canonical Poisson structure on the space ofsmooth functions on g∗ called the Lie–Poisson structure on g∗. Moreover the originalLie algebra g is canonically isomorphic to the Lie subalgebra of linear functionson g∗.

Proof Regarding the first assertion we have already seen that if� is a Poisson tensorassociated to a given Poisson algebra structure on F(E) which is homogeneous ofdegree −1, then its component functions with respect to a chart of linear coordinatesxi have the linear form Eq. (4.25). Then we define the Lie algebra structure on E∗by:

[α,β] = αiβ j ci j

kθk,

where θk is the dual basis to the basis ei used to define the linear chart xi , i.e.,u = xi ei , u ∈ E .

Conversely, we have already seen that the Lie–Poisson bracket Eq. (4.26) definesa Poisson algebra structure onF(g∗). Moreover formula (4.28) shows that the linearsubspaceF1(g∗) is closed with respect to the Lie bracket defined by the Lie–Poissonbracket. Then if we consider the map x : g → F1(g∗), given by ξ �→ xξ , againformula (4.28) says that such a map is a Lie algebra homomorphism and trivially itis an isomorphism too. �

The Poisson tensor associated to the Lie–Poisson structure on g∗ is easily com-puted from (4.28) because if we denote by xi the linear coordinate functions in g∗associated to the linear basis θk then with the notations above we have:

{ f, g} = ∂ f

∂xi

∂g

∂x j{xi , x j } = ∂ f

∂xi

∂g

∂x jx[ξi ,ξ j ] = ∂ f

∂xi

∂g

∂x jci j

k xk .

and the Poisson tensor associated to the Lie–Poisson structure, sometimes calledthe Lie–Poisson tensor, has components given by (4.25). The previous constructionand theorem is another instance of the tensorialization principle stated in Sect. 2.4.3,showing that an algebraic structure, in this case a Lie algebra structure, can begeometrized, in this case by defining a Poisson tensor and all the algebraic propertiesare captured by it.

Recall that if A is an endomorphism of a vector space E of dimension n, thedeterminant of A is defined by choosing a basis {e1, . . . , en} and then Ae1 ∧ · · · ∧

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Aen = (det A) e1 ∧ · · · ∧ en . The choice of the basis allows us to introduce linearcoordinates in E and we can consider a volume element � = dx1 ∧dx2 ∧· · ·∧dxn .Given any automorphism A ∈ Aut(E), det A is then the factor in d A(x1)∧d A(x2)∧· · ·∧d A(xn) = (det A)�. The set of transformations with det A = 1, correspondingto those linear transformations preserving the volume form, is the subgroup SL(n) ⊂GL(n).

In terms of geometrical objects, the previous notions become:

LX A� = (Tr A) � ,

and then, if Tr A = 0, we get LX A� = 0.In fact, if we define the n − 1-form �A = iX A�, its local expression is:

�A =n∑

j=1

(−1) j+1A ji x

i dx1 ∧ · · · ∧ dx j ∧ · · · ∧ dxn ,

where dx j means that this term is omitted from the expression. Clearly,

LX A� = d�A = (Tr A) dx1 ∧ · · · ∧ dxn .

Therefore, we see that traceless endomorphisms A defining the Lie subalgebra sl(n)

are those such that �A is exact.Since L�� = n�, i.e., � is homogeneous of degree n, given any linear vector

field X A, we can decompose it in the following way

X A =(

X A − Tr A

n�

)+ Tr A

n�, (4.29)

which means that X A is decomposed into a traceless part and a part proportional tothe Liouville vector field �. This corresponds to the splitting

A =(

A − Tr A

nI

)+ Tr A

nI .

Because with any vector v we can associate the constant vector field Yv = vi ∂∂xi

and LYv� = 0, we can construct the inhomogeneous special linear algebra isl(n) byputting together traceless linear vector fields X A and constant ones. Since

[X A, Yv] = YAv ,

the vector fields Yv generate an ideal, and the set of vector fields X A define a sub-algebra whose intersection with the previous ideal reduces to the zero vector field.Therefore the set of vector fields {X A, Yv} span the Lie algebra isl(n) which is asemi-direct sum of both the subalgebra and the ideal.


Consider now an arbitrary vector field X = Xi (x) ∂∂xi . We have

LX� = d X1(x) ∧ dx2 ∧ · · · ∧ dxn + · · · + dx1 ∧ · · · ∧ d Xn(x) =

=(

n∑i=1

∂Xi

∂xi

)dx1 ∧ dx2 ∧ · · · ∧ dxn = (div X) �.

We can consider the Lie algebra g of vector fields X(M) and the cohomologycorresponding to its action on the set of functions in M (see Sect. 3.2.5 and AppendixF). In other words, we define �(X) f := LX f , which is a linear representation ofX(M) because of

[�(X)◦�(Y )−�(Y )◦�(X)] f = (LXLY −LYLX ) f = L[X,Y ] f = �([X, Y ]) f.

The elements of Z1(g,F(M)) are linear maps (not F(M)-linear) α : g → F(M)

satisfying (see Eq. (3.17)):

LXα(Y ) − LY α(X) = α([X, Y ]) .

We have seen that the divergence of a vector field is the generalization of thetrace and that the set of all divergence-less vector fields is an infinite-dimensionalLie algebra. We want to remark that the association div : X(E) → F(E), given byX �→ divX is a 1-cocycle, i.e.,

LXdivY − LY divX = div[X, Y ] , (4.30)

and therefore, the set ker div ⊂ X(E) is the Lie subalgebra which generalizes isl(n).We notice that in general it is not possible to decompose a vector field like in (4.29).However, it is possible to consider a new vector space E × R, and a new volume

� = dx1 ∧ dx2 ∧ · · · ∧ dxn ∧ ds = � ∧ ds

such thatwe can associate to any vector field X a newvector field X = X+(divX)s ∂∂s

which is divergenceless with respect to �. In fact,

LX � = (LX�) ∧ ds − (divX)� ∧ ds = 0 .

4.3.4 The Coadjoint Action and Coadjoint Orbits

We recall that if G is a Lie group and g its Lie algebra, for every g ∈ G the innerautomorphism ig = Lg ◦ R−1

g : G → G, given by ig : g′ �→ gg′g−1 induces a Liealgebra automorphism ig∗ : g → g, which gives raise to the adjoint representation

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of G on g, given by Adg = ig∗. Let g∗ denote the dual space of g, the coadjointrepresentation of G on g∗ is defined naturally by:

〈Coad g(μ), ξ〉 = 〈μ,Adg−1(ξ)〉, μ ∈ g∗, ξ ∈ g .

We will call coadjoint orbits the orbits of the coadjoint action and we denote byOμ the coadjoint orbit passing through the element μ ∈ g∗, that isOμ = {Coad gμ |g ∈ G}. Notice that g∗ is foliated (the foliation would be singular in general) by thecoadjoint orbits of G. We want to explore the properties and structure of them.

The fundamental vector fields Xξ corresponding to the coadjoint action of G ong∗, are given by

(Xξ f )(μ) = d

dtf [Coad (exp(−tξ))μ] |t=0 ,

where f ∈ C∞(g∗) and ad ≡ Ad ∗ is the adjoint representation of g, that is ad ξ(ζ) =[ξ, ζ]. Then it follows, exp(−t ad ξ) = Ad (exp(−t ξ)). Moreover we get that forevery ζ ∈ g,

〈Coad (exp(−tξ))μ, ζ〉 = 〈μ,Ad (exp tξ)ζ〉 = 〈μ, (exp(tad ξ))ζ〉 ,

then

〈Coad (exp(−tξ))μ, ζ〉 = 〈(exp tad (ξ))∗μ, ζ〉 = 〈exp t (ad (ξ))∗μ, ζ〉 ,

and henceCoad (exp(−tξ))μ = exp t (ad (ξ))∗μ .

Taking into account the preceding relation, the expression of the fundamental vectorfield Xξ reduces to

(Xξ f )(μ) = d

dtf [exp(ad tξ)∗μ]|t=0 = d f [(ad ξ)∗μ] .

When the function f is linear it coincides with its differential and then

(Xξ f )(μ) = f ((ad ξ)∗μ) .

If we consider now the linear functions xξ (see Eq. (4.27)) we get

Xξxζ(μ) = xζ [(ad ξ)∗μ] = 〈(ad ξ)∗μ, ζ〉 = 〈μ, (ad ξ)ζ〉 = 〈μ, [ξ, ζ]〉 = x[ξ,ζ](μ) .

More specifically, if {ξk} and {θk}, for k running from 1 to dim G, are dual basisof g and g∗ respectively, the expression in coordinates for the fundamental vectorfield Xξi , now written Xi is


Xi = ci jk xk

∂

∂x j,

where ci jk are the structure constants of g with respect to the basis {ξi }.

Let us remark that the vector field Xi is but the Hamiltonian vector field on g∗with xi as Hamiltonian function. Moreover this means that coadjoint orbits are levelsets of Casimir functions for the Lie–Poisson structure on g∗. In fact, if f is a Casimirfunction, then {xi , f } = 0 is equivalent to Xi f = 0. The level sets of the Casimirfunctions determine submanifolds in which the Poisson structure is non-degenerateand the Hamiltonian vector fields are tangent to such submanifolds, that is the orbitsof the coadjoint action.

All these remarks together, lead to the following remarkable theorem by Kostant,Kirillov and Souriau on the structure of the symplectic foliation of the Lie–Poissonstructure on g∗.

Theorem 4.8 (Kostant-Kirillov-Souriau) Let G be a connected Lie group. Thenthe symplectic leaves of the canonical Lie–Poisson structure on the dual of the Liealgebra g∗ of a Lie group G are the coadjoint orbits of the coadjoint action of G.Moreover the symplectic structures induced on them are given by

ωμ(X f , Xh) = 〈μ, [d f (μ), dh(μ)]〉, μ ∈ g∗, f, h ∈ F(g∗) .

Proof The proof is a direct consequence of the previous remarks. The symplecticfoliation (singular in general) defined by a Poisson structure is the integral foliationcorresponding to the distribution defined by the range of the Poisson tensor. But nowthe range of the Poisson tensor is spanned by the fundamental vector fields Xi before,hence the integral leaves are just the orbits of the connected Lie group. �

Notice that the symplectic structure ωμ defined on the coadjoint orbit Oμ takesthe following simple explicit form when evaluated in fundamental fields:

ωμ(Xi , X j ) = ci jk xk ,

with xk the linear coordinates ofμ induced by the basis used to define the fundamentalfields Xi .

We will discuss now a few examples of these results.

4.3.5 The Heisenberg–Weyl, Rotation and Euclidean Groups

4.3.5.1 The Heisenberg–Weyl Group

As a particularly interesting example we will study next the so called Heisenbergor Heisenberg–Weyl group. It is the group arising naturally in Quantum Mechanics


because position and momenta have the same commutation rules as the Lie algebraof this group.

Let H W (3) denote the 3-dimensional Heisenberg group,1 namely,

H W (3) :=⎧⎨⎩⎛⎝ 1 x z0 1 y0 0 1

⎞⎠ : x, y, z ∈ R

⎫⎬⎭ .

A (global) chart (or parametrization) φ for the group is given by:

φ

⎛⎝ 1 x z0 1 y0 0 1

⎞⎠ = (x, y, z) .

The composition law is:

(x1, y1, z1) · (x2, y2, z2) = (x1 + x2, y1 + y2, z1 + z2 + x1y2) .

The neutral element e is such that φ(e) = (0, 0, 0) and if g ∈ H W (3) is such thatφ(g) = (x, y, z) the differential of the left translation Lg at the point e ∈ H W (3) isgiven by

(Lg)∗(e) =⎛⎝ 1 0 00 1 00 x 1

⎞⎠ .

Thus the left-invariant vector fields determined by ∂/∂x |0, ∂/∂y|0, ∂/∂z|0 are

X = ∂

∂x, Y = ∂

∂y+ x

∂

∂z, Z = ∂

∂z,

which satisfy the commutation relations

[X, Y ] = Z , [X, Z ] = [Y, Z ] = 0 ,

from which we obtain

ad X =⎛⎝ 0 0 00 0 00 1 0

⎞⎠ , ad Y =

⎛⎝ 0 0 0

0 0 0−1 0 0

⎞⎠ , ad Z = 0 .

Let (α,β, γ) be the corresponding coordinates of a vector μ ∈ hw(3)∗ ∼= R3. Then

the Poisson structure in hw(3)∗ is given by

1 The extension to the 2n + 1-dimensional Heisenberg–Weyl group is straightforward.


{α,β} = γ , {α, γ} = {β, γ} = 0 ,

and the Hamiltonian vector fields for the coordinate functions are

Xα = −γ∂

∂β, Xβ = γ

∂

∂α, Xγ = 0 .

The vector fields Xα, Xβ and Xγ span a distribution of dimension two (except at thepoints of the plane γ = 0 where all fields vanish). Therefore the integral submanifoldof the distribution passing through a point (α,β, γ) with γ �= 0 is given by thelevel sets of a function f solution of the system of partial differential equationsXα f = Xβ f = 0, i.e., the generic integral submanifolds are planes γ = const. Onthe other hand, the orbits of points like (α,β, 0) reduce to one point.

To compute explicitly the fundamental vector fields of the coadjoint representationof the Heisenberg group, we note that

Ad exp a X = ead a X =⎛⎝ 1 0 00 1 00 a 1

⎞⎠ , Ad exp bY =

⎛⎝ 1 0 0

0 1 0−b 0 1

⎞⎠ , Ad exp cZ = I ,

and,

Coad exp aX =⎛⎝ 1 0 00 1 −a0 0 1

⎞⎠ , Coad exp bY =

⎛⎝ 1 0 b0 1 00 0 1

⎞⎠ , Coad exp cZ = I .

Using the above-mentioned coordinates (α,β, γ) in g∗ ∼= R3, we have

Coad exp aX (α) = α , Coad exp aX (β) = β − aγ , Coad exp aX (γ) = γ ,

Coad exp bY (α) = α + bγ , Coad exp bY (β) = β , Coad exp bY (γ) = γ ,

and, finally,

Coad exp cZ = Id .

Thus, the fundamental fields are:

X = γ∂

∂β, Y = −γ

∂

∂α, Z = 0 ,

namely, the opposite vector fields of Xα, Xβ and Xγ .The coadjoint orbits may also be seen directly from the explicit expression of

Coad exp aX , Coad exp bY andCoad exp cZ . If γ �= 0, thenOγ = {(α,β, γ) : α,β∈ R}, while for γ = 0 every point (α,β, 0) is an orbit.


Notice that the tensor� defining the Poisson structure is given by� = γ ∂∂α ∧ ∂

∂β .Here Casimir functions are arbitrary functions of γ, on the orbit Oγ , parameterizedby (α,β), and the non-degenerate two form corresponding to the Poisson tensor isω = 1

γ dα ∧ dβ.

4.3.5.2 The Coadjoint Orbits of the Group of Plane Motions E2

Consider now again the Euclidean group of plane motions E(2) (Sect. 2.6.1). TheLie algebra of the group E2 is generated by { J, P1, P2 } with the relations:

[J, P1] = P2 , [J, P2] = −P1 , [P1, P2] = 0 .

The Lie–Poisson structure in e(2)∗ is determined in the corresponding coordinates(s, p1, p2) by

{s, p1} = p2 , {s, p2} = −p1 , {p1, p2} = 0 ,

the Hamiltonian vector fields being

Xs = −p2∂

∂ p1+ p1

∂

∂ p2, X p1 = p2

∂

∂s, X p2 = −p1

∂

∂s.

The rank of the distribution generated by these vector fields is the rank of thematrix: ⎛

⎝ 0 −p2 p1p2 0 0

−p1 0 0

⎞⎠ ,

and therefore the rank is two except at the points (s, 0, 0). The generic orbits are there-fore determined by functions f solutions of the system Xs f = X p1 f = X p2 f = 0,namely, the orbits are cylinders p21 + p22 = const. Coordinates adapted to the distrib-ution will be ‘cylindrical’ coordinates (s, ρ, θ) given by p1 = ρ cos θ, p2 = ρ sin θ.

The tensor � defining the Poisson structure is given by

� = p2∂

∂s∧ ∂

∂ p1− p1

∂

∂s∧ ∂

∂ p2.

In such ‘cylindrical’ coordinates � is expressed as

� = − ∂

∂s∧ ∂

∂θ,

because {s, ρ} = 0, and from

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dθ = p1dp2 − p2dp1p21 + p22

and dρ = 2p1dp1 + 2p2dp2 we obtain that {ρ, θ} = 0 and {s, θ} = −1. Note thatin this example the Casimir functions are arbitrary functions of ρ.

The adjoint representation is given by

ad J =⎛⎝ 0 0 00 0 −10 1 0

⎞⎠ , ad P1 =

⎛⎝ 0 0 0

0 0 0−1 0 0

⎞⎠ , ad P2 =

⎛⎝ 0 0 01 0 00 0 0

⎞⎠ ,

Ad expφJ = ead φJ =⎛⎝ 1 0 00 cosφ − sin φ0 sin φ cosφ

⎞⎠ ,

Ad exp a1P1 = ead a1P1 =⎛⎝ 1 0 0

0 1 0−a1 0 1

⎞⎠ , Ad exp a2P2 = ead a2P2 =

⎛⎝ 1 0 0

a2 1 00 0 1

⎞⎠ .

Thus

Coad expφJ =⎛⎝ 1 0 00 cosφ − sin φ0 sin φ cosφ

⎞⎠ ,

and

Coad exp a1P1 =⎛⎝ 1 0 a10 1 00 0 1

⎞⎠ , Coad exp a2P2 =

⎛⎝ 1 −a2 00 1 00 0 1

⎞⎠ .

Using coordinates (s, p1, p2) in e(2)∗ ∼= R3, we find that if (0, p1, p2) lies in an

orbit O and p21 + p22 �= 0, then (s, p′1, p′

2) ∈ O for s ∈ R, p′12 + p′

22 = p21 + p22.

Hence these orbits are cylinders. If p1 = p2 = 0, the orbit of (s, 0, 0) reduces to apoint.

Since,

(Coad expφJ )( j, p1, p2) = ( j, p1 cosφ − p2 sin φ, p1 sin φ + p2 cosφ),

(Coad exp a1 p1)( j, p1, p2) = ( j + a1 p2, p1, p2),

(Coad exp a2 p2)( j, p1, p2) = ( j − a2 p1, p1, p2),

we obtain the fundamental vector fields:

X J = p2∂

∂ p1− p1

∂

∂ p2, X P1 = −p2

∂

∂s, X P2 = p1

∂

∂s.


Notice that

[X J , X P1 ] =[

p2∂

∂ p1− p1

∂

∂ p2,−p2

∂

∂s

]= p1

∂

∂s= X P2 ,

and

[X J , X P2 ] =[

p2∂

∂ p1− p1

∂

∂ p2, p1

∂

∂s

]= p2

∂

∂s= −X P1 .

In ‘cylindrical’ coordinates (s, ρ, θ) the fundamental vector fields become:

X J = − ∂

∂θ, X P1 = −ρ sin θ

∂

∂s, X P2 = ρ cos θ

∂

∂s.

On the orbit Oρ := { (s, ρ, θ) | s ∈ R,−π < θ ≤ π } the non-degenerate 2-formω corresponding to the Poisson bivector field � will be

ω = −ds ∧ dθ = dθ ∧ ds .

4.3.5.3 The Rotation Group SO(3)

The Lie algebra of the rotation group (see Sect. 2.6.1) is:

so(3) =⎧⎨⎩B =

⎛⎝ 0 x −y

−x 0 zy −z 0

⎞⎠⎫⎬⎭ ∼= R

3 ,

where we identify these matrices with the corresponding vectors x = (x, y, z) ∈ R3.

With this identification, the adjoint action is the natural action of SO(3) on R3. The

Lie algebra structure of so(3) is determined by [Ji , Jk] = εikl Jl . Furthermore so(3)is identified with so(3)∗ using the Killing–Cartan form 〈B1, B2〉 = −Tr B1B2.

The tensor � defining the Lie–Poisson structure is:

� = εikl xl∂

∂xi∧ ∂

∂xk.

The Hamiltonian vector fields corresponding to the coordinate functions are

Xi = −εi jk xk∂

∂x j,

which correspond, up to sign, to the fundamental vector fields. The coadjoint orbitsare spheres, except for (0, 0, 0), which is a point. Using coordinates in so(3)∗ ∼= R

3

adapted to the characteristic distribution, namely spherical coordinates (r, θ,φ), weobtain the fundamental vector fields on the orbit Or :

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X1 = sin φ∂

∂θ+cot θ cosφ

∂

∂φ, X2 = − cosφ

∂

∂θ+cot θ sin φ

∂

∂φ, X3 = − ∂

∂φ.

To compute the non-degenerate 2-form ω = f (θ,φ) dθ ∧ dφ on Or we observethat the Casimir functions are arbitrary functions of r and that {θ,φ} = 1/sin θbecause

dθ = 1

(x21 + x22 + x23)√

x21 + x22

(x3x1 dx1 + x3x2 dx1 − (x21 + x22 ) dx3

),

dφ = 1

x21 + x22(x1 dx2 − x2 dx1) ,

and therefore

{θ,φ} = �(dθ, dφ) = 1√x21 + x22

= 1

r sin θ,

from which we see that the restriction on every orbit is ω = r sin θ dθ ∧ dφ.

4.4 Hamiltonian Systems and Poisson Structures

4.4.1 Poisson Tensors Invariant Under Linear Dynamics

Given a vector field X A, we will be interested in looking for bivector fields invariantunder X A and which are not of the constant type. Here we are much better off, indeedwe can construct quite a few invariant bivector fields which define Poisson structures.Let us start with the vector field X A associated with the matrix A. We construct thesequence of vector fields associated with A0, A1, . . . , As−1, with s being the degreeof the minimal polynomial associated with A, and let X j denote the vector fieldX j = X A j . We claim that

�B = Bi j Xi ∧ X j ,

with Bi j = −B ji ∈ R, provides us with a Poisson bracket onF(Rn) whose compo-nents are quadratic in coordinates and which is invariant under X A. The invarianceof �B under X A is obvious, because the Bi j are constant and LX A Xi = 0. We needonly to prove the Jacobi identity for the bracket defined by B. We have to show thatgiven three arbitrary functions f1, f2, f3

{{ f2, f3}, f1} = {{ f2, f1}, f3} + { f2, { f3, f1}} ,

and having in mind that


LX f1(�B(d f2, d f3)) = {{ f2, f3}, f1} ,

and

�B(LX f1d f2, d f3) = {{ f2, f1}, f3} , �B(d f2,LX f1

d f3) = { f2, { f3, f1}} ,

we have to prove that X f1 = −�B(d f1) satisfies

LX f1(�B(d f2, d f3)) = �B(LX f1

d f2, d f3) + �B(d f2,LX f1d f3) ,

which is equivalent to

LX f1�B = 0 .

Now

LX f1�B = Bi j [X f1 , Xi ] ∧ X j + Bi j Xi ∧ [X f1 , X j ]

= −Bi jLXi X f1 ∧ X j − Bi j Xi ∧ LX j X f1 ,

and taking into account that

LXi X f1 = −LXi (�B(d f1)) = −(LXi �B)(d f1) + �B(LXi d f1) ,

we can write

LX f1�B = Bi j�B(LXi d f1) ∧ X j − Bi j Xi ∧ �B(LX j d f1),

and using that

�B(LXi d f1) = �B(d(Xi f1)) = Bkl [LXkLXi f1]Xl ,

we obtain

LX f1�B = −Bi j [Bkl(LXkLXi f1)Xl − (LXlLXi f1)Xk] ∧ X j

−Bi j Xi ∧ Bkl(LXkLX j f1)Xl − (LXlLXi f1)Xk] ,

and by relabeling some indexes and using skew-symmetry it reduces toLX f1�B = 0.

In a more general case in which we start with a given dynamics, i.e., a vector fieldX A, we will be interested in those bivector fields for which time evolution satisfies

d

dt{ f, g} = { f , g} + { f, g} ,

4.4 Hamiltonian Systems and Poisson Structures 229

or in more geometric terms, the vector field X A giving the dynamics satisfies

X A{ f, g} = {X A f, g} + { f, X Ag} . (4.31)

This corresponds in physical terminology to the fact that the evolution is given bycanonical transformations.

The preceding relation can be written as

LX A [�(d f, dg)] − �(dLX A f, dg) − �(d f, dLX Ag) = 0 ,

i.e.

(LX A�)(d f, dg) = LX A [�(d f, dg)] − �(LX A d f, dg) − �(d f,LX A dg) = 0 ,

for any pair of functions f and g, and therefore, the relation (4.31) is equivalent toLX A� = 0. Vector fields satisfying this property will be called infinitesimal Poissonsymmetries.

An interesting property of � is the following statement: The Jacobi identity forthe Poisson bracket defined by the tensor field � is equivalent to the fact that thebivector field � is preserved by X f = −�(d f ) for any function f , i.e.,

LX f � = 0 .

In fact,

(LX f �)(dg, dh) = LX f [�(dg, dh)] − �(LX f dg, dh) − �(dg,LX f dh) ,

and then,

(LX f �)(dg, dh) = {{g, h}, f } − {{g, f }, h} − {g, {h, f }} ,

from which we can see that LX f � = 0, for any function f , if and only if Jacobiidentity holds.

Of course, Jacobi identity suffices to prove that a Hamiltonian vector field, X f ,is such that

X f {g, h} = {{g, h}, f } = −{{h, f }, g} − {{ f, g}, h} = {g, X f h} + {X f g, h} .

It may turn out to be quite surprising that we are able to construct many indepen-dent Poisson structures for any linear dynamical system.

To help in visualizing the situation let us consider a few examples.We consider

A =(

1 0−1 1

),


i.e.,

X A = x∂

∂x+ y

∂

∂y− x

∂

∂y.

We have commuting vector fields provided by

X1 = X I = x∂

∂x+ y

∂

∂y, X2 = X A = −x

∂

∂y+ X1 ,

and we construct

� = X1 ∧ X2 = −x2∂

∂x∧ ∂

∂y,

i.e.

� =(

0 x2

−x2 0

),

or, in other words,{x, y} = −x2 .

We look for a 1-form α = αx dx + αy dy such that

�(α) = X A ,

and we get �(α) = −x2 αx ∂/∂y + x2 αy ∂/∂x , and then, if x2 �= 0,

α(x, y) = − y − x

x2dx + 1

xdy = +d

(log |x | + y

x

).

In this way we have found a non-globally defined constant of the motion as it wasto be expected because our vector field does not allow for globally defined constantsof motion.

Another example in the same spirit is provided by

A =

⎛⎜⎜⎝1 0 0 00 1 0 00 0 −1/2 1/20 0 −1/2 −3/2

⎞⎟⎟⎠ =⇒ A2 =

⎛⎜⎜⎝1 0 0 00 1 0 00 0 0 −10 0 1 2

⎞⎟⎟⎠ ,

then

X A = x∂

∂x+ y

∂

∂y− 1

2(−w + z)

∂

∂z− 1

2(z + 3w)

∂

∂w,


and

X A2 = x∂

∂x+ y

∂

∂y+ 2w

∂

∂w+(

z∂

∂w− w

∂

∂z

),

we construct one of the many bivector fields

� =(

x∂

∂x+ y

∂

∂y

)∧(

y∂

∂x− x

∂

∂y

)+(

−1

2(w + z)

∂

∂z+ 1

2(z − 3w)

∂

∂w

)

∧(

w∂

∂z+ (2w − z)

∂

∂w

).

We point out that X A is not compatible with any constant maximal rank Poissonstructure.

Remark 4.1 From our construction of � it is clear that all that matters for the Jacobiidentity is that it is constructed out of vector fields commuting between them andwith X A. We can show that if Y1, Y2, . . . , Ym are any algebra of commuting vectorfields, then

�B = Bi j Yi ∧ Y j , Bi j = −B ji ∈ R ,

will define a Poisson structure which is invariant under the action of each one of theYk (and of any linear combination of them with coefficients Casimir elements for�B).

4.4.2 Poisson Maps

Definition 4.9 A map φ : M1 → M2 between Poisson manifolds (M1, {·, ·}1) and(M2, {·, ·}2) is said to be a Poisson map when

{F ◦ φ, G ◦ φ}1 = {F, G}2 ◦ φ

for any couple of functions F, G in M2.

In particular it is easy to see that if φ : M1 → M2 is a diffeomorphism, then φ isa Poisson map if and only if φ∗�1 = �2. The flow φt of a vector field X in M ismade of Poisson maps iff LX� = 0, because of

φt ∗LX� = d

dtφt ∗� .

A vector field X will be said to be canonical if

LX� = 0 . (4.32)


Canonical vector fields are characterized as follows: A vector field X in M is canon-ical iff

X{F, G} = {X F, G} + {F, XG} , (4.33)

for any pair of functions F, G in M . Notice that in this case the vector field X is aderivation of the Lie algebra structure defined on M by {·, ·}. Equation (4.33) followsfrom

(LX�)(d F ∧ dG) = X (�(d F ∧ dG)) − �(LX d F ∧ dG) + �(d F ∧ LX dG),

namely,(LX�)(d F ∧ dG) = X{F, G} − {X F, G} − {F, XG} .

In particular, when X H is a Hamiltonian vector field, it is canonical, because of therelation

(LX H �)(d F ∧ dG) = LX H {F, G} − �(d(X H F) ∧ dG) − �(d F ∧ d(X H G))

= {{F, G}, H} − {{F, H}, G} − {F, {G, H}} , (4.34)

and then LX H � = 0 as a consequence of the Jacobi identity. Obviously the sameproofworks for a closed 1-form θ, forwhichLXθ� = 0with Xθ = −�(θ). However,not every canonical vector field is a Hamiltonian one. As an example, let us considerthe Poisson structure in R

3, parametrized by x1, x2, x3, defined by

{F, G} = ∂F

∂x2∂G

∂x1− ∂F

∂x1∂G

∂x2.

The tensor � defining the Poisson structure is � = ∂/∂x2 ∧ ∂/∂x1. The vectorfield Y = ∂/∂x3 is canonical, but it cannot be Hamiltonian, for if a correspondingHamiltonian H did exist, then Y (F) = {F, H}, for any function F . Taking F = x3

we will find that x3 being a Casimir function, leads to a contradiction: 1 = Y (x3) ={x3, H} = 0.

We can also consider the more general case of an arbitrary 1-form α in M . If wedenote by Xα the vector field: −�(α). Then taking into account that for any pair offunctions F, G ∈ F(M),

(LXα�)(d F ∧ dG) = Xα{F, G} − {Xα F, G} − {F, XαG} ,

and

�(d F ∧ α) = −Xα F = −α(X F ) .

we see that

(LXα�)(d F ∧ dG) = Xα{F, G} − XG(Xα F) + X F (XαG) .


Also if we recall that

α([X F , XG]) = −α(X{F,G}) = −Xα{F, G} ,

then

(LXα�)(d F ∧ dG) = −α([X F , XG ]) + X F (α(XG )) − XG(α(X F )) = (dα)(X F ∧ XG) .

Then we see that Xα is a derivation iff dα is closed along Hamiltonian vector fields.Thus for generic Poisson manifolds, there will be derivations that will not be locallyHamiltonian vector fields.

Definition 4.10 A submanifold S of M endowed with a Poisson structure is calleda Poisson submanifold of M when i : S → M is a Poisson map.

If S is a Poisson submanifold, the Hamiltonian vector fields in M are tangent to S.In fact, let φ be a constraint function for S, i.e., it vanishes onto S, φ ◦ i = 0. Then,for every F ∈ F(M),

{F,φ} ◦ i = {F ◦ i,φ ◦ i} = 0 ,

and hence 0 = {F,φ} ◦ i = −X F (φ) ◦ i , which means that the restriction of X Fonto S is tangent to S.

Conversely, if i : S → M is a submanifold such that the Hamiltonian vector fieldsin M are tangent to S, then we can define a Poisson structure in S in such a waythat it is a Poisson submanifold of M . It suffices to define the Poisson bracket ofF, G ∈ F(S), by

{F, G} = {F, G} ◦ i ,

where F, G are arbitrary extensions of F and G respectively, i.e., F = F ◦ i ,G = G ◦ i .

We remark that in the computation of {F, G} = XG F only the values of F alongthe integral curves of XG , which lie in S by the tangency condition, are relevant.The Poisson bracket is then well defined, as it does not depend on the choice of therepresentative G, nor does it depend on the representative of F .

4.4.3 Symmetries and Constants of Motion

In Sect. 1.5 we mentioned that contraction of invariant tensors are still invariant andLie algebraic products of invariant objects generate new invariants.

In this section we would like to consider a dynamical system � which is Hamil-tonian with respect to a Poisson structure � with Hamiltonian function H .

We will first make some general comments and then we further analyze the par-ticular cases in which � is linear and the components of � are either constant, orlinear, or quadratic. Our special interest is concentrated in the relationship between

http://dx.doi.org/10.1007/978-94-017-9220-2_1


symmetries and constants of motion on one side and on alternative Poisson descrip-tions and their relevance for the correspondence between constants of motion andsymmetries, on the other side.

We startwith general aspects. If� = −�(d H), thenL�� = 0.Whenα1, . . . ,αn ,are invariant 1-forms, say L�α j = 0, j = 1, . . . , n, by contracting with � wewill find vector fields X j = �(α j ) and obviously [�, X j ] = 0, for any indexj = 1, . . . , n, i.e., the vector fields X j are infinitesimal symmetries of �. We alsohave that X j , j = 1, . . . , n, preserve �, LX j � = 0.

Aswe already remarked, the commutator of two symmetries, Xi and X j , [Xi , X j ],is again a symmetry. Therefore, by taking all possible commutators of our symmetrieswe generate a (possibly infinite-dimensional) Lie algebra. One may ask the naturalquestion of a possible counterpart at the level of invariant 1-forms, i.e., is it possible toconstruct a binary operation which associates an invariant 1-form out of an invariantpair?

As a matter of fact that is possible with the following binary operation2:

{α,β} = LXαβ − LXβαi − d[�(α,β)] .

When α = d f and β = dg we see that

{d f, dg} = d{ f, g} .

Therefore, a Poisson structure allows us to define a Lie algebra product on 1-formsand if all ingredients are invariant under the vector field �, the result is still invariant.The following particular situations are of great interest.

1. Let us assume that the invariant 1-forms are associated with constants of themotion, i.e., αi = d fi , with � fi = 0. In this case we find that { fi , f j } is again aconstant of motion and we have a realization with the bracket defined on 1-forms

d{ fi , f j } = {d fi , d f j } .

2. Let assume that the vector fields Xi are such that there exist real constants ci jk

satisfying

[Xi , X j ] =n∑

k=1

ci jk Xk ,

or in other words, our symmetries close on a finite-dimensional real Lie algebraand will define (at least locally) an action of a Lie group G which preserves �

and �.3. The 1-forms α1, . . . ,αn , satisfy Maurer–Cartan relationships,

2 This composition law is just the Lie bracket arising in the algebroid structure in the cotangentbundle of a Poisson manifold.


dα j + 1

2

n∑i,k=1

b jikαi ∧ αk = 0 ,

where b jik are real numbers that are the structure constants of some finite-

dimensional real Lie algebra.In this case there will be a coaction of some Lie group G′, i.e., a map

� : E → G ′

such that our invariant 1-forms are generated by �∗(g−1 dg), g ∈ G ′. Our dy-namics � is related to a one parameter subgroup on G ′.We remark that this relation is not related to

{αi ,α j } =n∑

k=1

ci jk αk .

4. Let us now suppose that we are in a situation in which the conditions of (2) and(3) are satisfied, i.e., invariant vector fields closing on a Lie algebra are associatedwith invariant 1-forms that satisfy Maurer–Cartan relations. In general ci j

k andb j

ik are different structure constants and are related by a compatibility condition.5. We can assume now that the invariant 1-forms α j are closed along Hamiltonian

vector fields,dα j (X f , Xg) = 0 , ∀ f, g ∈ F(E) .

In this case, we haveLX j � = 0 .

If

[Xi , X j ] =n∑

k=1

ci jk Xk ,

we still have a local action of a Lie group G. This situation generalizes the mostfamiliar case we are going to discuss now as a next item.

Let us suppose that the invariant 1-forms are exact, α j = d f j , and that thecorresponding vector fields satisfy the relation

[Xi , X j ] =n∑

k=1

ci jk Xk ci j

k ∈ R .

Here f j is only defined up to addition of a constant function a j . Notice that becauseof the relation d{ fi , f j } = {d fi , d f j } we have, in general,


{ fi , f j } =n∑

k=1

ci jk fk + di j ,

which is a Lie algebra extension of that defined by the vector fields Xi .The ‘integration constants’ di j can be eliminated if and only if there exists a vector

with components ak such that

di j =n∑

k=1

ci jk ak ,

because, since { f, ai } = 0, for any function f , we see that it suffices to choosef j = f j + a j and then the corresponding relations for the functions f j become

{ fi , f j } =n∑

k=1

ci jk fk + di j =

n∑k=1

ci jk fk .

The correspondence φ : E → Rn associating with any point x on our carrier

space φ(x) = ( f1(x), . . . , fn(x)) is called the momentum map.

Remark 4.2 This situation considered from the point of view of (c) gives b jik = 0,

i.e., we are dealing with the coaction of an Abelian group Rn and the structure con-

stants ci jk arise because there is an action of the group G, associated with structure

constants ci jk , induced via the Poisson tensor.

We are now ready to consider situation (e) again. We consider the algebra ofCasimir functions on E , C ⊂ F(E). Our invariant 1-forms will have (locally) thestructure

α j =n∑

i=1

cij d fi + ak

j dck ,

where cij and ck are Casimir functions and fi are constants of motion for �.

If the associated vector fields close on a finite-dimensional real Lie algebra, the“structure constants” may change from one to another leaf. The momentum mapwould also depend on the particular orbit (of Hamiltonian vector fields).

Let us now study particular cases. We will always assume that the components ofthe vector field � are linear, i.e., the coordinate expression is

� =N∑

i, j=1

A ji x

i ∂

∂x j.

Let us first suppose that the components of the Poisson tensor � are constant. Itscoordinate expression will be


� =N∑

i, j=1

�i j ∂

∂xi∧ ∂

∂x j.

Therefore, if H is the Hamiltonian of the vector field �, it should be a quadraticfunction that we will write

H = 1

2

N∑i, j=1

Mi j xi x j ,

where Mi j is a symmetric matrix. The condition � = −�(d H), implies that

N∑j=1

�i j M jk = −Aik .

In this case, the search for linear Casimir functions C(x) = ai xi amounts todeterminating the vectors a such that

N∑i=1

ai �i j = 0, i = 1, . . . , N .

We can factorize the kernel of� and induce a dynamics on the quotient E∗/ ker� ≈V , where � is not degenerate anymore.

However, we continue to discuss the situation on E .As for symmetries, as already discussed in Sects. 1.3 and 3.5, the Lie algebra is

Abelian and its dimension is equal to the codimension of ker�.In this situation quadratic constants of motion will be associated with a symmetric

matrix N given by the equation

N∑k=1

�ik Nk j = (A2m+1)ij .

This equation may be solved on V and then associated quadratic functions may bepulled-back to E .

Let us now investigate the existence of quadratic invariant 1-forms. We set

α =N∑

i, j=1

ai j xi dx j .

By writing it as a symmetric part and a skew-symmetric part we have

http://dx.doi.org/10.1007/978-94-017-9220-2_1

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α =N∑

i, j=1

d[bi j x i x j ] +N∑

i, j=1

ci j (xi dx j − x j dxi ) .

Imposing the condition L�α = 0 gives that∑N

i, j=1 bi j xi x j should be a constantof motion and the additional matrix condition

MT C + C M = 0 ,

which is equivalent to the invariance of the 2 form

dα =N∑

i, j=1

ci j dxi ∧ dx j =N∑

i, j=1

ai j dxi ∧ dx j .

Therefore, the analysis of invariant quadratic 1-forms is encompassed by thesearch for quadratic constants of motion and invariant 2-forms.

If we want to associate infinitesimal canonical symmetries with these quadratic1-forms we see immediately that the composition of � with C should be vanishing.The skew-symmetric part of the quadratic 1-form must be in ker�.

The situation becomes more interesting if we drop the assumption of quadraticinvariant 1-forms and allow them to be arbitrary.

Example 4.5 Let be E = R4 with coordinates (xa, ya), a = 1, 2, and consider the

following dynamics

xa = ya

ya = −xa

which corresponds to the vector field

� =2∑

a=1

(ya ∂

∂xa− xa ∂

∂ya

).

A compatible Poisson bracket is provided by

{ya, xb} = δab , {xa, xb} = 0 = {ya, yb} = δab ,

with a Hamiltonian

H = 1

2

2∑a=1

((xa)2 + (ya)2

).

The following vector fields are infinitesimal symmetries


Z1 = 1

(x2)2 + (y2)2

(y1

∂

∂x1 − x1∂

∂y1

), Z2 = 1

(x2)2 + (y2)2

(y2

∂

∂x2− x2

∂

∂y2

).

Corresponding invariant 1-forms are given by

α1 = 1

(x2)2 + (y2)2(y1 dy1+x1 dx1) , α2 = 1

(x2)2 + (y2)2(y2 dy2+x2 dx2) .

These two 1-forms satisfy the Maurer–Cartan relations

dα1 + α2 ∧ α1 = 0 , dα2 = 0 ,

corresponding to the Lie algebra of SB(2, R). The coaction is provided by

� : (x1, y1, x2, y2) �→(

(x2)2 + (y2)2 (x1)2 + (y1)2

0 1

).

It is not difficult to show that

((x2)2 + (y2)2 (x1)2 + (y1)2

0 1

)−1

d

((x2)2 + (y2)2 (x1)2 + (y1)2

0 1

)

= α1

(0 10 0

)+ α2

(1 00 0

).

Vector fields Z1 and Z2 act trivially on α1 and α2, i.e., {α1,α2} = 0.As for alternative invariant Poisson tensors, it is clear form our previous analysis

that all even powers of the matrix associated with � will give raise to alternativePoisson tensors according to Sect. 2.1. Let us now consider the case of Poissontensor with linear coordinates. We have again

� =N∑

i, j=1

A ji xi ∂

∂x j,

and

� =N∑

i, j,k=1

Ci jk xk ∂

∂xi∧ ∂

∂x j.

and this time the Hamiltonian function must be linear, i.e.,

H =N∑

i=1

ai xi , ai ∈ R .

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We find

A ji =

N∑k=1

ak Cikj .

Of course, it is still true that powers of A will generate symmetries for �; thistime, however, we cannot conclude easily that with these powers (if they are notcanonical) we generate new Poisson tensors.

It is clear that if

�1 =N∑

i, j,k=1

Ci jk xk ∂

∂xi∧ ∂

∂x j, �2 =

N∑i, j,k=1

Bi jk xk ∂

∂xi∧ ∂

∂x j.

provide two Poisson descriptions for � we have

A ji =

N∑k=1

akCikj =

N∑k=1

bk Bikj

whereCikj and Bik

j are the structure constants of N -dimensional Lie algebras, i.e.,they satisfy some quadratic algebraic relations.

As we have already remarked, a Poisson tensor with linear components definesa Lie algebra structure on the vector space. The fact that � is Hamiltonian impliesthat the one-parameter subgroup it generates can be considered as a subgroup ofdifferent N -dimensional Lie groups. Linear Hamiltonian symmetries will be givenby the centralizer of � within the corresponding Lie algebra.

Before starting a general analysis let us consider an example:Let be E = R

3 = {(x1, x2, x3)} and � be the vector field

� = x1∂

∂x2− x2

∂

∂x1.

It is clear from our group theoretical considerations that � can be consideredas a subgroup of E(2), SL(2, R), SO(3, R) and SO(2, 1). It follows that thecorresponding Lie algebra, realized in terms of Poisson brackets will provide analternative Poisson description for �. We have:

1. E(2) , �1 = x1 ∂∂x3

∧ ∂∂x2

+ x2 ∂∂x1

∧ ∂∂x3

, H = x3.

2. SO(3, R) , �2 = x1 ∂∂x3

∧ ∂∂x2

+ x3 ∂∂x2

∧ ∂∂x1

+ x2 ∂∂x1

∧ ∂∂x3

, H = x3.

3. SL(2, R) , �3 = x1 ∂∂x3

∧ ∂∂x2

− x3 ∂∂x2

∧ ∂∂x1

+ +x2 ∂∂x1

∧ ∂∂x3

.

It is reasonable to try to use the previous idea of getting one Poisson tensor fromthe other via an invariant matrix. Because �1 and �2 are both homogeneous ofdegree −1, a connecting tensor must be homogeneous of degree zero. We consider


T = T ij dx j ⊗ ∂

∂xi.

If �1(X, Y ) and �2(X, Y ) stay for the commutator of vectors X and Y , respec-tively, we need

�1(T X, T Y ) = T �2(X, Y )

with

L�T = 0.

This last condition says that we are allowed to experiment with

T =N∑

i, j,k=1

akCikj dx j ⊗ ∂

∂xi

and try to solve for B the equation

N∑l,m=1

ClmnTl

r Tms =

N∑p=1

Tnp Brs

p ,

with the additional algebraic quadratic constraint on the components of B. Anotherpossibility would be to start with

T = eλM

and try to solve for M , this time by taking the λ-derivative at the origin of

T −1λ �1(Tλ X, TλY ) = �λ(X, Y ) .

It is convenient to define the Poisson bracket on 1-forms by setting

�1 =n∑

i, j,k=1

Ci jkdxk ⊗ ∂

∂xi∧ ∂

∂x j.

From eλM with

M = Mij dxi ⊗ ∂

∂x j,

the condition ford

dλ�λ


to be a Poisson bracket is given by

�1(M d f, M dg) + M2�1(d f, d f ) − M�1(M d f, dg) − M�1(d f, M dg) = 0 ,

i.e., M has to satisfy the zero Nijenhuis condition.Whenwe apply this condition to the previous example it is not difficult to show that

all previous Poisson tensors are mutually compatible (the sum of structure constantsstill satisfies the quadratic condition corresponding to Jacobi identity).

We may conclude that if a vector field � is an element of a Cartan subalgebra ofsome simple algebra, all real forms of the complex algebras to which � belongs willprovide alternative Poisson descriptions. Moreover, any combination of the structureconstants of the real form gives Lie algebra structures.

When going to the case of quadratic brackets, as we remarked before, the situationis simpler. A general result is the following: If � is any vector field and T any (1,1)-tensor field satisfying the Nijenhuis condition NT = 0 then setting

T k(�) = Xk

we find that

� =n∑

k,l=1

Bkl Xk ∧ Xl

is a Poisson tensor.When the components of� are linear and T is homogeneous of degree zero, previ-

ous construction provides linear vector fields Xk and compatible quadratic brackets.Let us consider the following example

� =∑

a

ωa(

xa ∂

∂ya− ya ∂

∂xa

)

and

T =∑

a

ωa(

dxa ⊗ ∂

∂ya− dya ⊗ ∂

∂xa

).

We find

�0 =∑a,b

[ωa

(xa ∂

∂ya− ya ∂

∂xa

)∧(

ωb)2(xb ∂

∂xb+ yb ∂

∂yb

)]δab

=∑

a

(ωa)3((xa)2 + (ya)2)∂

∂ya∧ ∂

∂xa

It should be noticed that even if�0 is invertible on some open dense submanifold,linear canonical symmetries are not associated with smooth Hamiltonian functions.


The reason being that Hamiltonian functions would be homogeneous of degree zeroand therefore could not be smooth on a vector space.

4.5 The Inverse Problem for Poisson Structures: Feynman’sProblem

As it was pointed out before, it is natural to ask when a dynamical system � can begiven a description in terms of some Poisson bracket and someHamiltonian function.That is, given a vector field � = �k(x)∂/∂xk defined on a manifold M with localcoordinates xk , the hamiltonian inverse problem, or general factorization problem,for � consists in finding a Poisson bracket {·, ·} and a smooth function H such that

x k = �k(x) = {H, xk} . (4.35)

In other words, the hamiltonian inverse problem for � is equivalent to determiningwhether or not � is a Hamiltonian vector field with respect to some Poisson brackets.

In this formulation, neither the Poisson brackets nor the Hamiltonian are deter-mined a priori, that is, if they carry physical information it has to be determined byfurther information not contained in the vector field � alone. Needless to say, thisproblem in its full generality has not been solved.

The Hamiltonian inverse problem can be considered locally or globally, i.e., wecan discuss the existence of a Poisson structure� in the neighborhood of an ordinarypoint for the vector field� such that� = �(d H), and in a second step,we can discussthe existence of a globally defined Poisson structure. We will not pursue this aspectof the problem here (see [Ib90] for a discussion of the related issue of global aspectsof the inverse problem in the Lagrangian formalism).

The inverse problem of Poisson dynamics stated in such general terms it is hope-less, difficult, or useless because of the lack of uniqueness of solutions and the lackof understanding of their physical meaning.

One way to improve the problem is to make some further assumptions on thePoisson manifold M . For instance, we can assume that M = R

2n and that theclassical system is described by a second order differential equation. In this way weobtain new problems that could be called restricted inverse problems for Poissondynamics. Obviously attached to these problems is the question of uniqueness ofthe Poisson description for the given classical system. This was already raised byWigner in [Wi50].

If the inverse problem of Poisson dynamics is solved for a given dynamical system� on M and a Poisson tensor � is found, we can ask if there exists a symplecticrealization for it, that is, a symplectic manifold (P,�) and a Poisson map � : P →M . Such a symplectic realization exists locally, but in general there are obstructionsto obtaining globally defined ones [We83, We87].


Together with the problem of finding symplectic realizations for Poisson mani-folds, arises the problem of finding Lagrangian realizations. That is, does there existsa Lagrangian for a given dynamical system? Once a symplectic realization has beenfound, and if the symplectic manifold is a tangent bundle, we will apply the inverseproblem of the calculus of variations to look for a Lagrangian describing the system.The conditions assuring the existence of such Lagrangians, both local and global[Mo91, Ib90], are well known and can be systematically applied to the problems athand.

We will address some of these questions in Sect. 5.6.

4.5.1 Alternative Poisson Descriptions

Thus given �, we should look for a bivector field � = �i j (x) ∂∂xi ∧ ∂

∂x j , and afunction H such that

� = X f = �(d H) ,

i.e., such that the coordinate expression of � is,

� = X f = �i j ∂ f

∂x j

∂

∂xi.

Note, however, that there are vector fields X satisfying the derivation property (4.31)and which are not Hamiltonian vector fields. In other words, not every vector fieldX such that LX� = 0 is a Hamiltonian one.

As an instance, let us consider the Poisson structure in R3 defined by

{ f, g} = ∂ f

∂x2

∂g

∂x1− ∂ f

∂x1

∂g

∂x2.

The tensor � defining the Poisson structure is

� = ∂

∂x2∧ ∂

∂x1.

In this case the vector field Y = ∂/∂x3 is an infinitesimal Poisson symmetry, butit cannot be Hamiltonian; if a Hamiltonian H does exist, then Y f = { f, H}, for anyfunction f . It is remarkable that the obstruction for the existence of the function fis not only local, but global, due to the degeneracy of the Poisson bivector.

Taking f = x3 we will find that x3 being a Casimir function, it would be

1 = Y (x3) = {x3, H} = 0 ,

which is impossible.

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4.5 The Inverse Problem for Poisson Structures: Feynman’s Problem 245

Of course, in case of existence of a Poisson bracket such that � is Hamiltonian,we can then ask whether such a Poisson bracket is unique and, in the contary case ofexistence of different Poisson brackets for which � is Hamiltonain what additionalinformation we have. In fact, we shall see that the knowledge of alternative Poissonstructures will provide us some constants of motion.

Let us recall that if (L, [·, ·]) is a Lie algebra, then for each x ∈ L the mapad x : L → L defined by ad x(y) = [x, y], is a derivation. These derivations ad x aresaid to be inner derivations. However there can be derivations of (L, [·, ·]) that arenot inner. For the case of Poisson manifolds, for any function f , the inner derivationad f is X f .

We can also consider the more general case of an arbitrary 1-form α in E . If wedenote by Xα the vector field: Xα = −�(α), then taking into account that for anypair of functions f, g ∈ F(E),

(LXα�)(d f ∧ dg) = Xα{ f, g} − {Xα f, g} − { f, Xαg} ,

and

�(d f ∧ α) = −Xα f = −α(X f ) ,

we see that

(LXα�)(d f ∧ dg) = Xα{ f, g} − Xg(Xα f ) + X f (Xαg) .

Also if we recall that

α([X f , Xg]) = −α(X{ f,g}) = −Xα{ f, g} ,

then

(LXα�)(d f ∧dg) = −α([X f , Xg])+X f (α(Xg))−Xg(α(X f )) = (dα)(X f ∧Xg) .

Consequently, we see that Xα is a derivation of the Poisson algebra iff dα is closedalong Hamiltonian vector fields. Thus, for generic Poisson manifolds, there will bederivations that are not locally Hamiltonian vector fields.

Given a linear vector field X A we have analyzed the question of whether or notthere exists a constant bivector field invariant under X A. We will also be interestedin the uniqueness problem or in finding alternative bivectors. We should remark thatevery infinitesimal symmetry Y for X A that is not canonical, will take from oneadmissible Poisson structure to another.

In fact, from [Y, X A] = 0 and LY � �= 0, but LX A� = 0, we see that

LX A (LY �) = 0 .


We can say more however. As the Jacobi identity is equivalent to some tensorfield being identically zero, we have that this property is preserved under a generalchange of coordinates. Therefore if � is a compatible Poisson structure for X A, anysymmetry for X A, if it is non canonical for � will define a new compatible Poissonbracket. It is clear however, that the rank of � may change.

We are also interested in vector fields X A which are bi-Hamiltonian, what meansthat there exist two Poisson structures�1 and�2 and two Hamiltonian functions H1and H2 such that

X A = �1(d H1) = �2(d H2) .

This corresponds to different factorizations of the matrix A as products of a skew-symmetric matrix times a symmetric one. A constructive method for finding suchbi-Hamiltonian structures is the following: If S is any invertible matrix and A can befactorized as before, A = � H , then

S−1 A S = S−1 � H S = (S−1 �(S−1)T ) (ST H S) ,

and then the vector field corresponding to S−1 A S is Hamiltonian with respect tothe Poisson structure defined by the skew-symmetric matrix S−1 �(S−1)T with aquadratic Hamiltonian function defined by the symmetricmatrix ST H S. This showsthat if the linear map S commutes with A but is not canonical, i.e., S−1 � �= � ST ,it will carry from one Hamiltonian description to another one.

Remark 4.3 As our � is given by

� = �i j ∂

∂xi∧ ∂

∂x j,

from [∂/∂xi , ∂/∂x j ] = 0 we find that we can add any two compatible Poissonstructures without spoiling the Jacobi identity. Therefore if f is any even functionwe can consider � · f (A)t and obtain a new compatible Poisson structure.

If B is anymatrix commutingwith A, degenerate or not, we get another admissiblePoisson structure by considering �(1) = Bt� + �B. Indeed,

(�(1))t = �t B + Bt�t = −(Bt� + �B)

and in the basis ∂∂xi it will satisfy the Jacobi identity because of an earlier remark. In

the vector field language the result is quite trivial, from L�� = 0, and [X, �] = 0we get L�(LX�) = 0. What is non trivial in general, and in our case depends on thelinearity of X and the constancy of �, is the assertion that LX� satisfies the Jacobiidentity.


4.5.2 Feynman’s Problem

An alternative to the inverse problem for Poisson dynamics as discussed above, con-sists in asking as Feynman did, which dynamical laws are compatible with certainbasic physical assumptions. One of Feynman’s aims was to obtain dynamical lawsbeyond those in the Lagrangian formalism. Starting with just a few postulates for theQuantum Mechanics of a single particle, Feynman was able to recover the standardinteraction with an electromagnetic field. The postulates which we will here reviewinclude assuming the existence of position and velocity operators with commuta-tion relations consistent with particle localizability. (In order to avoid unnecessarycomplication due to operator ordering we shall be concerned only with the classi-cal analogue of Feynman’s procedure.) This program failed, as described by Dyson[Dy90], because the conditions imposed on the Poisson brackets were restrictiveenough to show that only electromagnetic forces were compatible with them.

Feynman’s problem deals with a full family of dynamical systems, those beingsecond-order differential equations (SODE’s) on a fixed velocity phase space T E(we will come back to the general discussion of Feynman’s problem for Lagrangiansystems in Sect. 5.6). The classical Feynman problem can then be stated in a moregeneral setting as the problemof finding all Poisson tensors on T E such that they haveHamiltonian vector fields which correspond to SODE’s and such that the systemsare localizable.

Let us now recall Feynman’s derivation of the Lorentz force for a point particle.For the choice of phase space variables we take (ξa) = (xi , vi ) for i = 1, 2, 3, xi

denoting position and vi velocity.The analogue of Feynman’s postulates gives some information about the Poisson

tensor � and the equations of motion. The assumptions on the Poisson brackets arethat

{xi , x j } = 0 (4.36)

andm{xi , v j } = δi j . (4.37)

In terms of Poisson tensors �, we are postulating that � has the following form onR6:

� = 1

m

∂

∂xi∧ ∂

∂vi+ f i j (x, v)

∂

∂vi∧ ∂

∂v j, (4.38)

where the form of f i j is to be determined. Equation (4.36) is the classical analogueof the condition of localizability, while m in (4.37) refers to the particle’s mass. Theequations of motion were postulated to have the form

d

dtxi = {xi , H} = vi (4.39)

d

dtvi = {vi , H} = 1

mFi (x, v). (4.40)

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Notice that Eq. (4.39) implies that the Hamiltonian dynamical system is a SODE.For convenience we will at first assume that Fi has no explicit time dependence.

The problem which was then posed was to find dynamics compatible with as-sumptions (4.36), (4.37), (4.39) and (4.40). Alternatively this means that one shoulddetermine theHamilton function H and the remaining Poisson brackets defining�ab ,which here are just {vi , v j }. A unique solution to this problem gives the coupling toelectromagnetic fields, as we now review.

One first defines the magnetic field Bi according to f i j = 1m2 ε

i jk Bk , or

m2{vi , v j } = εi jk Bk . (4.41)

It is then easy to show that Bi is a function of only the spatial coordinates xi . Forthis, one writes down the Jacobi identity

{xi , {v j , vk}} + {v j , {vk , xi }} + {vk, {xi , v j }} = 0 , (4.42)

and applies (4.37) and (4.41) to find that {xi , B j } = 0.Hence Bi must be independentof the particle velocity vi . It also has zero divergence, which follows from the Jacobiidentity involving vi , v j and vk , and thus we recover one of the Maxwell equations.

Next one takes the time derivatives of Poisson brackets (4.37) and (4.41). Theformer gives

{vi , v j } + {xi , v j } = 0 ,

or

{xi , F j } = − 1

mεi jk Bk(x) , (4.43)

from which it follows that Fi is at most linear in the velocities

Fi = Ei (x) + εi jk v j Bk(x) . (4.44)

Equation (4.44) is the Lorentz force law (where the electric charge is set equal toone) and it here defines the electric field Ei (x).

From the time derivative of (4.41) one gets

m εi jk{Fi , v j } = Bk . (4.45)

Next apply (4.44) and the divergenceless condition on Bi to simplify (4.45) to

m εi jk{Ei , v j } = Bk − m{Bk, vi }vi . (4.46)

But the right-hand side of (4.46) vanishes because we have assumed Bi to have noexplicit time dependence, and so we are just left with

εi jk{Ei , v j } = 0 , (4.47)


implying that Ei has zero curl, as is the case for time independent magnetic fields.Upon repeating this calculation for time dependent fields, one instead recovers

the Faraday law and thus both of the homogeneous Maxwell equations. In this caseboth the velocity vi and the force Fi are allowed to have explicit dependence on t ,so that Eq. (4.40) is generalized to

d

dtvi = {vi , H} + ∂vi

∂t= 1

mFi (x, v, t) . (4.48)

The analysis then proceeds as in the above with the static fields Ei (x) and Bi (x)

replaced by Ei (x, t) and Bi (x, t). Now the right hand side of (4.46) is ∂Bi/∂t andhence the Faraday law is recovered.

The Hamilton function for the above system is

H = m

2v2 + φ(x, t) ,

where φ(x, t) is the standard scalar potential. From it we construct the electric fieldaccording to,

E = −∇φ + m∂v∂t

.

4.5.3 Poisson Description of Internal Dynamics

4.5.3.1 A Spinning Particle Interacting with a Magnetic Field

Let us go back to discuss the inverse problem for Poisson dynamics. For this weconcentrate on one particular system—that of a spinning particle interacting with amagnetic field. In addition to giving the standard Hamiltonian formulation of thissystem, we will exhibit examples of nonstandard formulations.

In this section we shall ignore the spatial degrees of freedom of the particle andrather only study the spin variables. The spatial coordinates will be included in thefollowing section.

Let us denote the spin variables by Si , i = 1, 2, 3. They span some 3-manifold M .(We initially impose no constraints on Si .) Since Si correspond to internal degreesof freedom (as opposed to the spatial coordinates discussed in the previous section)we should not make the same assumptions as Feynman used for getting dynamics onMinkowski space. That is, we should not in general assume localizability (i.e., thatthe Poisson brackets of Si with themselves are zero), nor should we assume that theequations of motion are second order in time derivatives. So let us instead start withthe most general Poisson tensor

� = εi jk Fk(S)∂

∂Si∧ ∂

∂S j. (4.49)


Then, the Jacobi identity requires that F · (∇ × F) = 0, where the gradient ∇ is withrespect to S. If we introduce the 1-formα = Fi d Si , the condition reads:α∧dα = 0.Locally, α can then be written α = f0 d f1, f0 and f1 being functions of Si and wehave

{Si , S j } = f0(S) εi jk∂ f1∂Sk

.

As a result f1 has zero Poisson bracket with all Si and it therefore defines a classicalCasimir function. If we now require that the dynamics be such that there is aHamiltonfunction H = H(Si ), then f1 is also a constant of motion and therefore

Si∂ f1∂Si

= 0 . (4.50)

For dynamics, which is non-trivial (i.e., Si �= 0), Eq. (4.50) implies a conditionon f1. To see what that is, let us now specialize to the dynamical system of a spininteracting with a magnetic field B. The standard equation of motion for such asystem gives the precession of the spin Si ,

Si = μ εi jk S j Bk , (4.51)

where μ denotes the magnetic moment. From (4.50) and (4.51) one finds that

∂ f1∂Sk

= ρ1Sk + σ1Bk , (4.52)

and so f1 must be of the form

f1 = f1(S2, S · B) . (4.53)

The Hamilton function H(Si ) must have an analogous form. To see this, substitute(4.51) into Si = {Si , H} to get:

f0(S) εi jk∂H

∂S j

∂ f1∂Sk

= μ εi jk S j Bk . (4.54)

It follows that∂H

∂Sk= ρH Sk + σH Bk ,

and thus that the most general solution for H is of the form

H = H(S2, S · B). (4.55)

From (4.54) we get the following restriction on the derivatives of f1 and H :


f0(ρH σ1 − ρ1σH ) = μ . (4.56)

The functions f1 and H given by (4.53) and (4.55) subject to the constraint (4.56)define a family of Poisson brackets and Hamiltonian functions all of which leadto the same equations of motion (4.51) for the spin. They correspond in general tocanonically inequivalent descriptions for the dynamics. We first review the standardHamiltonian description.

4.5.3.2 Standard and Alternative Formalisms

The standard canonical formalism for a spinning particle is recovered when we write

f0 = 1

2, f1 = S2 , H = −μS · B .

For this choice the Poisson bracket algebra (4.49) corresponds to the su(2) Liealgebra. Reducible representations of su(2) then arise upon performing a canonicalquantization of this system. Alternatively, to obtain irreducible representations forsu(2) one must impose a constraint on the variables Si . Namely, one must requirethat the classical Casimir function f1 takes on certain constant values (and as a resultSi will then span S2) [Ba82].

There exist alternative Hamiltonian formulations associated which yield the clas-sical equation of motion (4.51), which are canonically inequivalent to the standardone. One such formulation results from the choice [So92, St93],

f0 = 1

2, f1 = S21 + S2

2 + 1

2λ

[cosh 2λS3sinh λ

− 1

λ

](4.57)

andH = −μλS3 , (4.58)

where with no loss of generality we have taken the magnetic field to be in the thirddirection. Here λ is a ‘deformation’ parameter as the standard formalism is recoveredwhen λ → 0. For non zero λ, (4.58) leads to

{S2, S3} = S1 , {S3, S1} = S2 , (4.59)

{S1, S2} = 1

2

sinh 2λS3sinh λ

. (4.60)

Unlike in the standard formulation, the Poisson bracket algebra of observables Sidoes not correspond to su(2) or any Lie algebra.3

3 These brackets are a classical realization of the quantum commutation relations for generators ofthe Uq (sl(2)) Hopf algebra [Ta89, Ma90, Tj92].


Let us denote the latter generators by I+, I− and I0. Their commutation relationsare

[I0, I±] = ±I± , [I+, I−] = q2I0 − q−2I0

q − q−1 . (4.61)

These commutation relations reduce to the su(2) Lie algebra relations in the limitq → 1. To obtain the above classical system, one can replace the quantum operatorsI+, I− and I0 by S1 + i S2, S1 − i S2 and S3, respectively, q by expλ, and thecommutation relations (4.61) by i times Poisson brackets.

There is another choice for f0 and f1 (which is known to correspond to theclassical limit of the Uq(sl(2)) Hopf algebra):

f0 = λ

4S3 , f1 = (S1)

2 + (S2)2 + (S3)

2 + (S3)−2 . (4.62)

It leads to the following brackets for Si :

{S2, S3} = λ

2S1S3 , {S3, S1} = λ

2S2S3 ,

{S1, S2} = λ

2

((S3)

2 − (S3)−2)

, (4.63)

or if we define S± = S1 ± i S2, these relations can be expressed by

{S±, S3} = ± iλ

2S±S3 , {S+, S−} = −iλ

((S3)

2 − (S3)−2)

. (4.64)

Now in order to obtain the equations of motion for the spin (4.51) one can choosethe Hamiltonian

H = −2μB

λln S3 ,

where again we have chosen the magnetic field to be along the third direction.To show that the Poisson structure defined in (4.64) has something to do with the

classical limit of the algebra generated by I+, I− and I0, let us define new operatorsS+, S− and S0 according to

S± = −√q(q − q−1)I± , S3 = q I0 .

Then the commutation relations (4.61) can be expressed according to

S3S± = q±1S±S3 , [S+, S−] = q(q − q−1)((S3)

2 − (S3)−2)

. (4.65)

Next we introduce Planck’s constant � by setting q = exp λ�

2 and expand (4.65) inpowers of �. The result is:


[S3, S±] = ±λ�

2S±S3 + O(�2) , [S+, S−] = λ�

((S3)

2 − (S3)−2)

+ O(�2) .

(4.66)The lowest order terms in the commutation relations (4.66) for S+, S− and S0 cor-respond with i� times the Poisson brackets (4.64) of S+, S− and S0.

4.5.4 Poisson Structures for Internal and External Dynamics

We adapt Feynman’s procedure to examine the dynamics of particles which trace outtrajectories in space-time, aswell as in some internal space M . Physically, coordinatesof the latter space can be associated with an extrinsic spin of the particle (as discussedin the preceding section) or with some internal degrees of freedom like isospin orcolor. In the following general treatment we let the manifold M be of arbitrarydimension. Further, the algebra of observables spanning M need not be associatedwith any Lie algebra (as was the case in the alternative formulations of spin givenin the previous section). The treatment given here thus allows for the possibility offinding new kinds of particle dynamics. We recover some familiar systems uponspecializing to the case where the Poisson bracket algebra for the internal variablescorresponds to a Lie algebra; for instance, the Wong equations [Wo70] describingthe coupling of particles to a Yang–Mills field. We can also recover the magneticdipole coupling of a spinning particle with a magnetic field when we take the Poissonbracket algebra on M to be su(2). We will postpone the discussion of the generalsetting for these questions as well as the general results to Sect. 5.6.2. Of coursethe general treatment will include particles on curved space-time [Ta92]. Again inSect. 5.6.4 the discussion will again be specialized to the case of M being a Lie groupand a cotangent bundle of a group.

4.5.4.1 Free System

We first consider free motion on R3 × M . The manifold M now denotes a d-

dimensional internal space which we parametrize by Ia , a = 1, . . . , d. Free motionon R

3 corresponds to

x i = 0 , i = 1, 2, 3 , (4.67)

while on M one standardly writes

Ia = 0 . (4.68)

We parametrize the corresponding phase space of the system by xi , vi and Ia . Inwriting down a Poisson description on T R

3 × M we let

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m{xi , v j } = δi j ; {Ia, Ib} = Cab (4.69)

define the non zero Poisson brackets for the system, where Cab may be any generalfunction on R

3 × M . Notice that the zero brackets {xi , x j } = {xi , Ia} = 0 reflectthe localizability of the system and the simultaneous measurability of inner variableswith external coordinates. From Jacobi identity involving Ia , Ib and xi , we get that{xi , Cab} = 0 and consequently Cab can have no dependence on vi .

The Poisson tensor defined by (4.69) reads as:

� = 1

m

∂

∂xi∧ ∂

∂vi+ 1

2Cab

∂

∂ Ia∧ ∂

∂ Ib. (4.70)

The Jacobi identity [�,�] = 0, implies that [�I ,�I ] = 0, �I being the secondterm in (4.70). As a result Cab satisfies the following condition:

0 = Ca f∂Cbc

∂ I f+ Cbf

∂Cca

∂ I f+ Ccf

∂Cab

∂ I f, (4.71)

and �I defines a Poisson tensor itself on R3 × M . For the case where the Pois-

son brackets of Ia define a Lie algebra, we have Cab = cabf I f , cab

f being thecorresponding structure constants. Then the Jacobi identity is satisfied due to

0 = ca fdcbc

f + cbfdcca

f + cc fdcab

f .

Finally, the free motion (4.67) and (4.68) results from (4.69) along with the Hamiltonfunction

H = m

2> v2 .

4.5.4.2 Interactions

What kind of interaction can be introduced into the system by generalizing the pre-vious Poisson brackets and Hamilton function? In order to better define this questionwe generalize Feynman’s assumptions (4.36–4.40). We assume equations of motionof the form

x i = vi ,

vi = 1

mFi (x, v, I ) ,

Ia = fa(x, v, I ) , (4.72)

while for the Poisson brackets we take

{xi , x j } = 0 , {xi , Ia} = 0 , m{xi , v j } = δi j , (4.73)


and{Ia, Ib} = Cab , m2{vi , v j } = F i j , m{vi , Ia} = Ai

a , (4.74)

where the functions Fi , fa , F i j ,Aia and Cab are initially undetermined. This corre-

sponds to defining a Poisson tensor which is not a direct sum, since it will containcross terms. After the brackets have been determined, we once again demand thatthe equations of motion are Hamiltonian. Free dynamics is recovered if all functionsexcept for Cab vanish.

It is clear that we have two problems here. The first one is to find conditions on thefunctions Cab, F i j andAi

a such that the Poisson brackets satisfy the Jacobi identity.The second one is to ensure that the functions are compatible with the requirementthat the dynamics defined by (4.72) be Hamiltonian.

We begin by taking the time derivative of (4.73). This can be done under theassumption that the evolution equation is Hamiltonian with respect to the Poissontensor defined by the Poisson brackets (4.73) and (4.74). Upon using (4.72) we get

{xi , v j } + {vi , x j } = 0 ,

{vi , Ia} + {xi , fa} = 0 ,

m{vi , v j } + {xi , F j } = 0 . (4.75)

Of course the first equation is identically satisfied. From the second equation wenotice that if fa depends non trivially on velocity we need Ai

a �= 0. Similarly,from the third equation, if Fi depends non trivially on velocity we need F i j �= 0.Now using the Jacobi identity one finds that the coordinates xi have zero bracketswith the functions Cab, Ai

a and F i j , and consequently none of these functions canhave a dependence on velocity. The Poisson tensor will then be said to be velocityindependent. (This will be discussed further in Sect. 5.51.)

For F i j �= 0 or Aia �= 0, we must modify the free Poisson tensor � (4.70) to

� = 1

mDi ∧ ∂

∂vi+ 1

2Cab

∂

∂ Ia∧ ∂

∂ Ib+ 1

2m2F i j ∂

∂vi∧ ∂

∂v j, (4.76)

where

Di = ∂

∂xi− Ai

a(x, I )∂

∂ Ia

defines the covariant derivative associated to the connection A (see Eq. (3.51) andAppendix E).

The Hamiltonian vector field associated with dxi is �(dxi ) = ∂/∂vi . BeingHamiltonian then implies that L∂/∂vi � = 0, and we once again have the result thatCab, Ai

a and F i j can have no vi dependence.This result further implies that functions Fi and fa appearing in the equations of

motion (4.72) are at most first-order in the velocity. From (4.75)

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Fi = F i j (x, I )v j + F i0(x, I ) , fa = −Aia(x, I )vi − A0

a(x, I ) . (4.77)

Here we have introduced new functions F i0 and A0a which are by definition inde-

pendent of velocity.By expanding the Jacobi identity

[�,�] = 0 , (4.78)

we obtain additional conditions on Cab, Aia and F i j and consequently additional

conditions on the functions Fi and fa . From (4.78) we obtain

1

m2 [Di ∧ ∂

∂vi,D j ∧ ∂

∂v j] + [�I ,�I ] + 2[Di ∧ ∂

∂vi,�F ] + 2

m[Di ∧ ∂

∂vi,�I ]

+ 2[�I ,�F ] = 0 . (4.79)

�I and �F denote the second and third terms, respectively, in � (4.76). Using

[Di ,∂

∂vi] = 0 ,

we can reduce (4.79) to

1

m2 [Di ,D j ] ∧ ∂

∂v j∧ ∂

∂vi+ [�I ,�I ] + 2[Di ,�F ] ∧ ∂

∂vi+ 2

m[Di ,�I ] ∧ ∂

∂vi

+ 2[�I ,�F ] = 0 . (4.80)

Collecting our results we find the following conditions on the functions Fi j , Aia

and Cab:

Di Cab = −Ca f∂Aib

∂ I f+ Cbf

∂Aia

∂ I f,

D[iA j]a = Cab∂F i j

∂ Ib,

D[iF jk] = 0 , (4.81)

in addition to the condition (4.71) on Cab. Of the three conditions in (4.81), the thirdis the most familiar as it resembles the Bianchi identity. Its connection to the Bianchiidentity will be clarified in the examples which follow.

The conditions (4.81) more generally apply if we replace the spatial indicesi, j, . . . , going from 1 to 3 by space-time indices μ, ν, . . . , going from 0 to 3.As usual 0 denotes time, with F i0 = −F0i and A0

a defined in (4.77). The moregeneral result follows upon taking the time derivative of Poisson brackets (4.74). Forexample, the time derivative of {Ia, Ib} = Cab yields


{ Ia, Ib} + {Ia, Ib} = ∂Cab

∂ I fI f + ∂Cab

∂xix i . (4.82)

Then after substituting (4.72) and (4.74) into (4.82) one obtains an equation which isat most linear in velocity vi . The linear terms however cancel due to the first equationin (4.81) and we are left with

D0Cab = −Ca f∂A0b

∂ I f+ Cbf

∂A0a

∂ I f. (4.83)

Equation (4.83) is the first equation in (4.81) with i replaced by 0. Similarly, bytaking the time derivative of m{vi , Ia} = Ai

a and m2{vi , v j } = F i j one obtains

D[0Ai]a = Cab∂F0i

∂ Ib(4.84)

andD[0Fi j] = 0 , (4.85)

respectively, thus proving the result.In the above system the particle interacts with the “fields” Cab(x, I ), Aμ

a (x, I )and Fμν(x, I ). Since these fields are functions on the internal space as well as onspace-time, they resemble Kaluza–Klein fields. To reduce the theory to one wherethe coupling is to fields defined on space-time only, it will be necessary to makecertain ansätze for the form of the functions Cab(x, I ), Aμ

a (x, I ) and Fμν(x, I ).These ansätze must be consistent with conditions (4.71) and (4.81). In the followingwe give examples which lead to some standard particle interactions.

4.5.4.3 Wong Equations

Here we let the Poisson bracket algebra for Ia correspond to a Lie algebra; i.e.,Cab = cab

f I f , cabf being structure constants. As stated earlier, condition (4.71)

is then identically satisfied. The standard coupling of a particle to a Yang–Mills fieldis recovered upon choosing

Aμa (x, I ) = cab

f I f Abμ(x) , μ = 0, 1, 2, 3, (4.86)

where Abμ is independent of the internal variable Ia and is to be identified with theYang–Mills potential. This choice identically satisfies the first equation in (4.81).Upon substituting (4.86) into the second equation of (4.81), we get

∂Fμν

∂ Ia= ∂ Aaν

∂xμ− ∂ Aaμ

∂xν+ cbf

a Abμ A f ν . (4.87)


The right-hand side of (4.87) is independent of the internal variables Ia and is to beidentified with the Yang–Mills field strength Faμν(x). Therefore,

Fμν(x, I ) = Faμν(x)Ia , (4.88)

plus terms which are independent of the internal variables. Finally, the third equationin (4.81) corresponds to the usual Bianchi identities, while the equations of motionfollowing from (4.77) are the usual Wong particle equations,

mxμ = Faμν(x)Ia xν ,

Ia = −cabf Abμ(x)I f xμ . (4.89)

These equations of motion are obtainable using the Hamiltonian

H = m

2> v2 − Aa0(x)Ia . (4.90)

The above system of equations can be generalized to the case where Ia does notspan a Lie algebra. In this case (4.86) is replaced by

Aμa (x, I ) = Cab Abμ(x) , (4.91)

Equation (4.91) satisfies the first condition in (4.81) provided Cab satisfies (4.71) andit is independent of x . Upon substituting (4.91) into the second equation of (4.81),we get

Cab(I )

(∂Fμν

∂ Ia− ∂ Aaν

∂xμ+ ∂ Aaμ

∂xν− ∂Cbf

∂ IaAbμ A f ν

)= 0 . (4.92)

Ignoring the Ia-independent terms, (4.92) is solved by

Fμν(x, I ) =(

∂ Aaν

∂xμ− ∂ Aaμ

∂xν

)Ia + Cbf (I )Abμ(x)A f ν(x) . (4.93)

Thus Fμν is not linear in I if Cab is not. The third equation in (4.81) is identicallysatisfied forAμ

a andFμν of the form (4.91) and (4.93), and thus all conditions (4.81)hold for this case. The equations of motion which follow from (4.91) and (4.93)are a generalization of the Wong particle equations (4.89) (although their physicalmeaning is not clear when Ia does not span a Lie algebra). These equations of motionare also obtainable from the Hamiltonian (4.90).

4.5.4.4 Magnetic Moment Coupling

We next show how the standard Hamiltonian formulation for a spinning particleinteracting with a magnetic field can be obtained from the above treatment. For this


we take to Ia denote spin. Its Poisson brackets should define an su(2) Lie algebra onR3,

{F, G} = εabc Ic∂F

∂ Ia

∂G

∂ Ib, (4.94)

and thusCab = εab f I f . (4.95)

Further we set Aia = 0, from which it follows using (4.75) that fa has no velocity

dependence. Since Di Cab = 0, the first condition in (4.81) is satisfied, while thesecond one implies that F i j is independent of Ia . If we then set

F i j = εi jk Bk , (4.96)

the third equation in (4.81) tells us that B has zero divergence and hence can beidentified with a magnetic field. It remains to specify A0

a and F0i . They must bechosen to be consistent with conditions (4.83–4.85). We do not identify A0

a with anelectromagnetic potential but rather take

A0a = −μ εabc Ib Bc , (4.97)

where μ represents the magnetic moment. Then (4.83) is identically satisfied while(4.84) gives

μ εabc Ib∂i Bc = −εabc Ib∂F0i

∂ Ic(4.98)

and thusF0i = −μ Ic∂i Bc , (4.99)

plus some function of Ia Ia . Here we see that F0i is not interpreted as an electricfield. Condition (4.85) is satisfied only for static magnetic fields.

Upon substituting the above results into the equations of motion (4.72) and (4.77)we recover the standard dynamics for a spinning particle,

mxi = εi jk x j Bk + μ∂i B j I j , (4.100)

Ia = μ εabc Ib Bc . (4.101)

These equations of motion are obtainable from the Hamilton function

H = m

2v2 − μBa Ia . (4.102)

The above dynamical system can be generalized by replacing the su(2) Poissonbracket algebra for the spin by an arbitrary algebra. Thus instead of (4.95) we cantake Cab = Cab(I ), where we assume that condition (4.71) holds. Here we still take


Aia = 0, from which it again follows that the first equation in (4.81) is satisfied, the

second one implies that F i j is independent of Ia , and the third equation gives that Bhas zero divergence. For A0

a , we can generalize (4.97) to

A0a = μ Cab(I )Bb , (4.103)

which satisfies (4.83). Conditions (4.84) and (4.85) are once again fulfilled by (4.99)with static magnetic fields B. Now the equation of motion (4.100) for the spatialcoordinates is still valid,

Ia = −μ Cab(I )Bb .

These equations are obtainable from the Hamilton function (4.102).

4.6 The Poincaré Group and Massless Systems

Massless systems must be dealt with repeatedly. In order to understand them betterwe should discuss first their underlying symmetry group which is none other thanthe Poincaré group.

4.6.1 The Poincaré Group

We will devote this section to the study of the Poincaré group that constitutes themathematical model of Special Relativity.

4.6.1.1 Minkowski Space-Time

The collection of physical events could be modeled approximately by an affinespace,4 that we call Minkowski space-time and will be denoted by M, over a four-dimensional linear space V carrying a metric 〈·, ·〉 of signature (− + ++).

The points A of M are called events, and the interval sAB between two eventsA, B ∈ M is defined as s2AB = 〈v(A, B), v(A, B)〉, where v(A, B) denotes theunique vector in V such that B = A + v(A, B). The vectors in V can be classifiedas:

i. Temporal: vectors x ∈ M such that 〈x, x〉 < 0.ii. Spatial: vectors x ∈ M such that 〈x, x〉 > 0.iii. Light or isotropic: vectors x ∈ M such that 〈x, x〉 = 0, x �= 0.iv. x = 0.

4 Recall that an affine space is a pair (M, V ) where V is a linear space that as an abelian groupacts freely and transitively onM.

4.6 The Poincaré Group and Massless Systems 261

Two events are said to be temporal, spatial or light related if the associated vector inV is temporal, spatial or light respectively.

The collection of affine invertible maps of M5 form a group know as the affinelinear group AL(M). The translation of vector u ∈ V is defined as the transformationthat maps any point A into B = A + u. The associated linear map in V is theidentity and, conversely, any affine linear map whose associated linear map is theidentity is a translation. The set of all translations defines a normal subgroup andAL(A) = T4�GL(V ), where� denotes the semidirect product of theAbelian groupof translations T4 ∼= R

4 and the general linear group GL(V ) (see later Sect. 4.6.1 fordefinitions).

4.6.1.2 The Poincaré and Lorentz Groups

Observers describe events. In our extremely simplified picture, an observer wouldbe a point O in Minkowski space-time M and the choice of a pseudo-orthonormalbasis in V , i.e., a basis eμ of V such that 〈eμ, eν〉 = gμν , with gμν = (−1)δ0ν δμν .6

In other words, the only non zero elements are: g11 = g22 = g33 = −g00 = 1.The point O ∈ M allows us to identify M with V , and the basis of V provides

an identification of V with R4. Thus an observer would associate to each event

coordinates xμ, μ = 0, 1, 2, 3. Another observer would associate to the same eventdifferent coordinates x ′μ and the two observers would be (inertially) equivalent ifboth sets of coordinates are related as:

x ′μ = aμ + �μνxν . (4.104)

where �μν satisfy:

gμν �μα�ν

β = gαβ . (4.105)

The set of transformations of the form (4.104) satisfying (4.105) define a groupcalled the Poincaré group that will be denoted in what follow as P .

Transformations such that aμ = 0 are called (pure) Lorentz transformations anddefine a subgroup of the Poincaré group called the homogeneous Lorentz group L.Lorentz transformations can be written in matrix form as:

⎛⎜⎜⎜⎝

x ′0

x ′1

x ′2

x ′3

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎝

�00 �0

1 �02 �0

3

�10 �1

1 �12 �1

3

�20 �2

1 �22 �2

3

�30 �3

1 �32 �3

3

⎞⎟⎟⎠

⎛⎜⎜⎝

x0

x1

x2

x3

⎞⎟⎟⎠ . (4.106)

Then condition (4.105) is written as:

5 A map g : M → M is called affine if there exists a linear map g : V → V such thatv(g(A), g(B)) = g(v(A, B)).6 We will always assume Einstein’s convention of sum over repeated indices.


�T G� = G, (4.107)

and we get immediately that:| det�| = 1 .

Lorentz transformations such that det� = +1 define a unimodular subgroup inL denoted by L+.

Because Lorentz transformations preserve the metric then, in particular |�00| ≥

1, and as a topological spaceL is not connected, but it has four connected componentscharacterized as follows:

L0 = L↑+ Proper orthocronous, det� = 1, �0

0 ≥ 1 , (4.108)

L↑− Improper orthocronous, det� = −1, �0

0 ≥ 1 , (4.109)

L↓+ Proper antiorthocronous, det� = 1, �0

0 ≤ −1 , (4.110)

L↑− Improper antiorthocronous, det� = −1, �0

0 ≤ −1. (4.111)

Among all of them only the component that contains the identity L0, is a subgroup.Other important subgroups of L, are:

L↑ = L↑+ ∪ L↑

−, L+ = L↑+ ∪ L↓

+, L↑↓+− = L↑

+ ∪ L↓−.

The matrix form of a pure Lorentz transformation in the Z direction is:

⎛⎜⎜⎜⎝

x ′0

x ′1

x ′2

x ′3

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎝

c 0 0 d0 1 0 00 0 1 0a 0 0 b

⎞⎟⎟⎠

⎛⎜⎜⎝

x0

x1

x2

x3

⎞⎟⎟⎠ . (4.112)

and it must satisfy:

a2 − c2 = 1, d2 − b2 = 1, ab − cd = 1,

fromwhichwegeta2 = d2 andb2 = c2.Wecanwrite themso that the transformationbelongs to L0, as

a = d = cosh β, b = c = sinh β . (4.113)

Let us recall that the physical meaning of β is given by:

tanh β = v

c. (4.114)

where v is the relative velocity of the two systems.


4.6.1.3 Semidirect Products: P0 as a Semidirect Product

Given a group H and an action of H on an Abelian group N by automorphisms,that is a group homomorphism ρ : H → Aut(N ), where ρ(h) : N → N is a groupisomorphism, we can define a group structure on the product space G = N × H ina natural way and the resulting group will be called the semidirect product of N andH . The semidirect product will be denoted as H � N and the group law is given by:

(n, h) · (n′, h′) = (n + ρ(h)n, hh′), ∀n, n′ ∈ N , h, h′ ∈ H .

We will denote, as usual ρ(h)n by hn. We check immediately that the compositionlaw is associative and that (0, e) is the identity element. Finally (−h−1n, h−1) is theinverse element of (n, h).

Notice that: (0, h) · (n, e) · (0, h)−1 = (hn, h) · (0, h−1) = (hn, e), which showsthat N is a normal subgroup of N � H , and N � H/N ∼= H .

If h and n denote the Lie algebras of H and N respectively, then the Lie algebraof N � H is, as a linear space n ⊕ h, and the Lie bracket is given by:

[(α, ξ), (β, ζ)] = (ρ(ξ)β − ρ(ζ)α, [ξ, ζ]), (α, ξ), (β, ζ) ∈ n ⊕ h ,

where ρ(ξ) is the representation of the Lie algebra h induced by ρ, this is ρ(ξ)n =ddt ρ(exp(tξ)n |t=0.

4.6.1.4 The Poincaré Group as a Semidirect Product

The iterated application of Eq. (4.104) allows us to obtain the group law of Poincaré’sgroup. Thus if,

x1μ = aμ

1 + �1μ

νxν, x2μ = aμ

2 + �2μ

νx1ν,

then

x2μ = (�2�1)

μαxα + �2

μνaν

1 + aμ2

and in condensed form:

(a2,�2) · (a1,�1) = (a2 + �2a1,�2�1)

that is the composition law of the semidirect product of the Lorentz group L and theAbelian group of translations T4 with the natural action of L on T4 given by matrixmultiplication. Thus,

P = T4 � L .


In particular, if we restrict ourselves to the connected components we have P0 =T4 � L0. Notice the normal subgroup relation for T4:

(0,�)(a, I)(0,�−1) = (�a, I) . (4.115)

4.6.1.5 The Lie Algebra of the Poincaré Group

An infinitesimal Lorentz transformation will be written as

x ′μ = (gμν + εμ

ν)xν , (4.116)

and must satisfy the isometry condition:

gμν x ′μx ′ν = gμν(gμ

α + εμα)(gν

β + ενβ)xαxβ

= gαβxαxβ + gμν(gμ

αενβ + gν

βεμα) + O(ε2),

that is,εαβ + εβα = 0 .

We will denote by Mαβ the infinitesimal generators of L0, then we will rewrite(4.116) as:

x ′μ = (gμν + 1

2εαβ[Mαβ]μ ν)xν .

and the generators Mαβ can be defined as:

(Mαβ)μ ν = gμαgνβ − gν

αgμβ , (4.117)

Notice that, as a consequence of equation (4.117), we get:

Mαβ + Mβα = 0 .

The commutation relations that define the Lie algebra structure of L0 are obtainedeasily from (4.117):

[Mαβ, Mγδ] = gβγ Mαδ + gαδ Mβγ + gαγ Mδβ + gδβ Mγα, (4.118)

Notice that if we define M = (M1, M2, M3) by

M1 = M23, M2 = M31, M3 = M12 ,

from Eq. (4.118) we find the commutation rules:


[M1, M2] = [M23, M31] = g33M21 = −M3 , (4.119)

and in a similar way:

[M2, M3] = −M1, [M3, M1] = −M2 . (4.120)

If we define now N = (N 1, N2, N 3) as:

N i = M0i , i = 1, 2, 3 , (4.121)

again, using Eq. (4.118), we find the corresponding commutation rules:

[N 1, N2] = [M01, M02] = g00M21 = M3 , (4.122)

together with[N 2, N 3] = M1, [N3, N1] = M2 . (4.123)

Moreover Mi satisfy the commutation rules with N j

[M1, N2] = [M23, M02] = M30 = −N 3 , (4.124)

and[M2, N3] = −N 1, [M3, N1] = −N 2 ,

[M2, N1] = N 3, [M1, N3] = N 2 , [M3, N2] = N 1 . (4.125)

Notice that if we define K± = 1

2(M ± iN), from Eqs. (4.120)–(4.125), we get

the commutation rules:

[K i+, K j+] = −εi jl K l+ [K i−, K j

−] = −εi jl K l− [K i+, K j−] = 0 , (4.126)

that shows that the complexification of the Lie algebra of L0 is the Lie algebra ofSO(3) × SO(3).

The Lie algebra of the Poincaré group includes the generators of translations Pμ,satisfying [Pμ, Pν] = 0. Recall that because of Eq. (4.115), we get:

[Mαβ, Pμ] = [Mαβ]μ ν Pν (4.127)

and taking into account the explicit form of Mαβ , we finally get:

[Mαβ, Pμ] = (gμαgνβ − gν

agμβ)Pν = gμα Pβ − gμβ Pα . (4.128)


4.6.1.6 The Universal Covering SL(2, C) of L0

In a similar way as we did with the unitary group SU (2) and the group of rotations(see Sect. 2.6.1, Eq. (2.144)) wewant to show that the group SL(2, C) is the universalcovering of the group L0. We denote by H(2) the linear space of 2 × 2 Hermiteanmatrices.

Let us consider the map χ : R4 → A given by:

χ(x) = x0I + x · σ , (4.129)

Clearly χ is an isomorphism of linear spaces. Moreover the determinant of χ(x) is−〈x, x〉.

The image of the standard basis in R4 is the basis {σ0 = I,σ} of H(2). The

pre-image of a matrix X ∈ H(2) is the vector x with coordinates:

xμ = −1

2Tr (X σμ) , (4.130)

whereσ0 = I, σk = −σk, k = 1, 2, 3 .

Then, because:

χ(x) =(

x0 + x3 x1 − i x2

x1 + i x2 x0 − x3

),

we get:det χ(x) = (x0)2 − x · x = −〈x, x〉 .

Proposition 4.11 i. For any A ∈ SL(2, C) and X ∈ H(2), then AX A† ∈ H(2).ii. The transformation �A = χ−1 ◦ p(A) ◦ χ is a Lorentz transformation, where

p : H(2) → H(2) is the linear map given by p(A) : X �→ X AX†, for allA ∈ SL(2, C).

Proof i. It suffices to check (AX A†)† = AX†A† = AX A†.ii. Notice that 〈�Ax,�Ax〉 = − det p(A)χ(x) = − det AX A† = − det X =

〈x, x〉, thusthe transformation �A preserves the Minkowski metric. �

Proposition 4.12 The map � : SL(2, C) → L0, defined by A �→ �A = χ−1 ◦p(A) ◦ χ is a group homomorphism with kernel Z2.

Proof It suffices to check that p(AB)X = AB X (AB)† = AB X B†A† = p(A)

p(B)X . Then�AB = χ−1◦p(AB)◦χ = χ−1◦p(A)◦χ◦χ−1◦p(B)◦χ−1 = �A�B .The kernel is given by matrices A such that p(A)X = X, ∀X , but this implies

that [A,σi ] = 0, and then A = ±I. �

http://dx.doi.org/10.1007/978-94-017-9220-2_2

http://dx.doi.org/10.1007/978-94-017-9220-2_2


Notice that the elements of the matrix �A can be written as:

(�A)μ ν = −1

2Tr (σμ Aσν A†) .

4.6.2 A Classical Description for Free Massless Particles

The description of massless elementary systems with non-zero spin and/or helicity,like gluons, gravitons, neutrinos or photons, when thought of as classical particlesshould have some features inherited from our quantum description of them. In par-ticular:

a. If p denotes the particle 3-momentum, the value p = 0 should not be part of ourcarrier space for obvious reasons.

b. Denoting with x the position, a description in terms of Poisson brackets shouldbe such that

{xi , x j } �= 0, i �= j,

to take into account that massless particles are not localizable.c. The angular momentum of a massless particle should be non-zero along the

direction of the momentum of the particle, if its spin is nonzero.d. The translation group should state that the momentum p and the total angular

momentum J are conserved, i.e.,

dpdt

= 0; dJdt

= 0. (4.131)

Starting from these requirements,wemaywish to build aHamiltonian description,i.e., to define a carrier space, endow it with a Poisson structure and specify theHamiltonian function.

Let us make a few considerations on our requirements. They will consists of,

1. The carrier space should be R3 × (R3 − {0}).

2. The dynamical vector field on this space is � = pi∂/∂xi .3. The rotation group generated by

Ri = εi jk

(x j

∂

∂xk+ p j

∂

∂ pk

)

and the translation group generated by Ti = ∂∂xi

, should act canonically.

Let us start with a bivector field of the general form

� = Fi j∂

∂xi∧ ∂

∂x j+ Gi j

∂

∂ pi∧ ∂

∂ p j+ Ai j

∂

∂xi∧ ∂

∂ p j,


satisfying that the Schouten bracket of � with itself vanishes:

[�,�] = 0 ,

and such that:

L�� = 0; LTi � = 0; LRi � = 0; i = 1, 2, 3 . (4.132)

Invariance under translations, Eq. (4.132), implies that Fi j , Gi j , Ai j are indepen-dent of x . The invariance under dynamical evolution implies that Gi j is a constantof motion, moreover, Fi j = Ai j − A ji , and Gi j = Ai j .

Because A is a function only of p, it follows that A = 0, G = 0. By the sameargument F = 0 an we get Ai j = A ji .

Because of invariance under rotation Ai j = a(p2)δi j + b(p2)pi p j , but alsoFi j = f (p2)εi jk pk . Putting all this together we obtain that � must be of the form:

� = f (p2)εi jk pk∂

∂xi∧ ∂

∂x j+ a(p2)δi j

∂

∂xi∧ ∂

∂ p j+ b(p2)pi

∂

∂xi∧ p j

∂

∂ p j.

(4.133)Let us make a few more considerations. The last term in the right-hand side

of Eq. (4.133) defines a Poisson structure on its own. The compatibility condition[�,�] = 0 is equivalent to the differential equation:

3a f + p2(b f + 2da

dp2f + 2bp2

d f ′

dp2) = 0 .

Hence, for a = 0 we get:

f = f0 exp−1

2

∫(1

b+ 1)dp2 .

Finally, for a �= 0 we find:

b = a′12 − p2(log |a|)′ ; f = a2n

p2,

where a′ = da/dp2 and n is a constant of integration.Then the Poisson brackets defined by �, become:

{xi , x j } = na(p2)

p3εi jk pk , {pi , p j } = 0 , {xi , p j }

= a(p2)

[δi j + (log |a(p2)|)′ pi p j

12 − p2(log |a(p2)|)′

].


Upon replacing pi by Pi = pi/a(p2) we find the new brackets

{xi , x j } = n

P3 εi jk Pk , {Pi , Pj } = 0 , {xi , Pj } = δi j .

For n = 0, xi , Pi are the standard canonical variables, while for n �= 0, we mustexclude the origin of the momentum P = 0 from the phase space. The latter casecorresponds to the phase space description of a single helicity particle.

The significance of Pi , as opposed to pi , is that it corresponds to the generator ofspatial translation. The generators of the rotation subgroup are given by (i = 1, 2, 3):

Ji = 1

aεi jk x j pk − n

pi

|p| = εi jk x j Pk − nPi

|P| .

The generator of time translation can be taken to be |P|. Of course as we are dealingwith massless particles, a manifest relativistic description of our systems seemsdesirable. We shall proceed therefore to provide a relativistic description.

References

[Gi93] Giordano, M., Marmo, G., Rubano, C.: The inverse problem in the Hamiltonian formal-ism: integrability of linear Hamiltonian fields. Inv. Probl. 9, 443–467 (1993)

[Ar76] Arnol’d, V.I.: Méthodes mathématiques de la mécanique classique. Ed. Mir, 1976. Math-ematical methods of classical mechanics. Springer (1989)

[Sc53] Schouten, J.A.: On the differential operators of first-order in tensor calculus. Conv. Int.Geom. Diff., Italia (1953) (Ed. Cremonese, Roma 1954, pp. 1–7)

[Li77] Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Diff.Geom. 17, 253–300 (1977)

[Ki62] Kirillov, A.A.: Unitary representations of nilpotent Lie groups. Russ. Math. Surv. 17,57–110 (1962)

[Ib90] Ibort,A., López-Lacasta,C.:On the existence of local andglobals for ordinary differentialequations. J. Phys. A: Math. Gen. 23, 4779–4792 (1990)

[Wi50] Wigner, E.P.:Do the equations ofmotion determine commutation relations and ambiguityin the Lagrangian formalism? Phys. Rev. 77, 711–712 (1950)

[We83] Weinstein, A.: On the local structure of Poisson manifolds. J. Diff. Geom. 18, 523–557(1983)

[We87] Weinstein, A.: Symplectic groupoids. Bull. Am. Math. Soc. 16, 101–04 (1987)[Mo91] Morandi, G., Ferrario, C., Lo, G., Vecchio, G.M., Rubano, C.: The inverse problem in the

calculus of variations and the geometry of the tangent bundle. Phys. Rep. 188, 147–284(1991)

[Dy90] Dyson, F.J.: Feynman’s proof of theMaxwell equations. Am. J. Phys. 58, 209–211 (1990)[Ba82] Balachandran, A.P., Marmo, G., Stern, A.: A Lagrangian approach to the non-interaction

theorem. Nuovo Cim. 69A, 175–186 (1982)[So92] Soni, S.K.: Classical spin and quantum algebras. J. Phys. A: Math. Gen. 25, L837–L842

(1992)[St93] Stern, A., Yakushin, I.: Deformed Wong particles. Phys. Rev. D 48, 4974–4979 (1993)[Ta89] Takhtajan, L.A.: Lectures on quantum groups. Introduction to quantum group and inte-

grable massive models of quantum field theory. In: Ge, M.-L., Zhao, B.-H. (eds.). WorldScientific (1990)


[Ma90] Majid, S.: On q-regularization. Int. J. Mod. Phys. 5, 4689–4696 (1990)[Tj92] Tjin, T.: Introduction to quantized Lie groups and Lie algebras. Int. J. Mod. Phys. A 7,

6175–6214 (1992)[Wo70] Wong, S.K.: Field and particle equations for the classical Yang-Mills field and particles

with internal spin. Nuovo Cim. 65A, 689–694 (1970)[Ta92] Tanimura, S.: Relativistic generalization and extension to non-Abelian gauge theory of

Feynman’s proof of the Maxwell equations. Ann. Phys. 220, 229–247 (1992)

Chapter 5The Classical Formulations of Dynamicsof Hamilton and Lagrange

On ne trouvera point de Figures dans set Ouvrage. Les méthodesque j’y expose ne demandent ni constructions, ni raisonnementsgéométriqus ou méchaniques, mais seulement des opérationsalgébriques, assujetties à une march réguliere et uniforme.

Joseph-Louis Lagrange, Mécanique Analytique, Avertissement dela premiére édition, 1788.

5.1 Introduction

The present chapter is perhaps the place where our discourse meets more neatly theclassic textbooks on the subject. Most classical books concentrate on the descriptionof the formalisms developed by Lagrange and Euler on one side, and Hamilton andJacobi on the other and commonly called today the Lagrangian and the Hamiltonianformalism respectively. The approach taken by many authors is that of postulatingthat the equations of dynamics are derived from variational principles (a route whosehistorical episodes are plenty of lights and shadows [Ma84]).

Such a procedure is almost unquestioned because of its spectacular success inbuilding the foundations of many modern physical theories. However, the currentstate of affairs is not satisfactory, as variational principles are rather poor whentrying to explore the foundations of the theories of dynamics; our goal is to lookeven further forward to incorporate eventually quantum dynamics.

Thus we will approach this chapter as a continuation of our previous effort, that istrying to understand the properties of dynamical systemswhen they possess invariantstructures, that are (completely or not) determined by it. Hence our first taskwill be to

‘The reader will find no figures in this work. The methods which I set forth do not require eitherconstructions or geometrical or mechanical reasonings, but only algebraic operations, subjectto a regular and uniform rule of procedure’.


271

272 5 The Classical Formulations of Dynamics of Hamilton and Lagrange

explore non-degenerate Poisson structures, a question that is left from the previouschapter, and that we will conclude here arriving to the ‘standard’ presentation ofHamiltonian system. We will take a further step by exploring the consequences ofa dynamics possessing not just a cotangent bundle description but a tangent bundleone. In this way we will arrive to the standard Lagrangian description of dynamicalsystem and to a solution of the inverse problem for Lagrangian systems. It is mostremarkable that we will do that without relinking in any way to variational principlesor other ‘metaphysical principles’. We will only look for more and more exigentphysical properties to our systems until ttheir class is seen as being “simple”.

5.2 Linear Hamiltonian Systems

In the previous chapter we introduced the Hamiltonian formalism by looking forPoisson tensors � compatible with a given dynamics, in the sense that � (= XA ifthe dynamics is linear), is a Hamiltonian vector field with respect to such dynamics,that is there is a function H such that � = �(dH).

We will analyze now the particular case in which the Poisson structure consideredis non-degenerate, that is, the only Casimir functions are constant functions or, if youwish, there is a one-to-one correspondence between exact 1-forms and Hamiltonianvector fields. Such particular instances of Poisson structures have been playing amost relevant role in the understanding of the structure of physical theories so theydeserve a more detailed analysis. We will devote the first part of this chapter to them.

If the Poisson structure � is defined on a 2n-dimensional linear space E and isconstant, then it defines a non-degenerate bilinear form ω on E. Notice that since �

is non-degenerate, the linear isomorphism associated to it, � : E∗ → E, given by〈β, �(α)〉 = �(α, β), is invertible and �−1 : E → E∗ defines a bilinear form ω onE by means of:

ω(u, v) := �(�−1(u),�−1(v)) = 〈�−1(v), u〉, u, v ∈ E,

where 〈·, ·〉 denotes as usual the natural pairing between E and E∗. By selecting alinear basis {ei | i = 1, . . . 2n} on E, if �ij denotes the corresponding componentsof �, then the components ωij of the bilinear form ω above satisfy:

�ijωjk = −δik . (5.1)

It is also common to use the notation �−1 to denote −ω (or ω−1 to denote −�).Because of the geometrization principle, the bilinear formω defines a non-degenerateclosed differential form on E, denoted with the same symbol ω, and very often thissetting constitutes the entrance door to the study of Hamiltonian systems.

5.2 Linear Hamiltonian Systems 273

We will start our analysis by a quick review of the structure and properties of thisframework and will continue afterwards by providing dynamical systems possessingthese invariant structures, the same treatment as in former chapters.

5.2.1 Symplectic Linear Spaces

Definition 5.1 Let E be a real linear space and σ a non-degenerate skew symmetricbilinear form on it, that is, σ(u, v) = 0 for all v ∈ E implies that u = 0. The pair(E, σ ) is then said to be a (real) symplectic linear space.

A slightly more general concept is that of a presymplectic linear space, that is apair (E, σ ) such that σ is a skew-symmetric, but maybe degenerate, 2-form in E.

Because of the way we were introducing the notion of symplectic linear spaces,it is clear that the canonical leaves of linear Poisson structures are automaticallysympletic linear spaces. Thus all we have to do is go back to the catalogue of linearPoisson structures discussed in Sect. 4.3.3 and extract their leaves. We will not dothat though.We will proceed by describing a class of examples that exhibits the mainstructure of symplectic linear spaces.

Let V be a linear space of dimension n, and V∗ its dual space. We can define askew-symmetric bilinear form σ in E = V ⊕ V∗ by

σV ((u, α), (v, β)) = 〈α, v〉 − 〈β, u〉 , u, v ∈ V , α, β ∈ V∗, (5.2)

that can easily be shown to be non-degenerate: if σV ((u, α), (v, β)) = 0, ∀(v, β) ∈V ⊕ V∗, it suffices to consider the subspace of vectors of the form either (v, 0) or(0, β), respectively, to convince ourselves that u and α should be the zero vector andthe zero covector, respectively.

As a particular instance, if V = Rn, then E = R

n ⊕ Rn∗ ∼= R

2n, where the lastidentification is provided by choosing a linear basis in R

n. Let us consider just thecanonical basis {ei | i = 1, . . . , n} ofR

n and denote the corresponding coordinates asqi, namely, if v ∈ R

n , v = qi(v)ei. The corresponding dual basis {ei | i = 1, . . . , n}will also be denoted {ei = dqi | i = 1, . . . , n}, because dqi(v) is the i-th componentof the vector v. Then, the coordinates of covectors in R

n∗ with respect to the dualbasis will be denoted pi, and this will allow us to rewrite the differential 2-formsdefined by σ on R

2n as:

σ =n∑

i=1dqi ∧ dpi .

According to the tradition we will call it the canonical 2-form on R2n.

Another interesting instance of symplectic linear space is provided by the follow-ing construction. If (V1, ω1) and (V2, ω2) are symplectic linear spaces, then V1⊕V2can be endowed with a skew-symmetric bilinear 2-form ω1�ω2, by means of

http://dx.doi.org/10.1007/978-94-017-9220-2_4


(ω1 � ω2)((v1, v2), (u1, u2)) = ω1(v1, u1)−ω2(v2, u2) , va, ua ∈ Va, a = 1, 2 ,

or, in other words, if πi : V1 ⊕ V2 → Vi, i = 1, 2, are the canonical projections oneach factor, ω1�ω2 = π∗1ω1 − π∗2ω2.

The formω1�ω2 is non-degenerate, because ifweassume that (ω1�ω2)((v1, v2),

(u, 0)) = 0 for all u ∈ V1, as ω1 is non-degenerate, v1 must be zero, and the samereasoning works for the other argument.

Notice that the same arguments work for the standard direct sum bilinear formω1 ⊕ ω2, defined as:

(ω1⊕ω2)((v1, v2), (u1, u2)) = ω1(v1, u1)+ω2(v2, u2) , va, ua ∈ Va, a = 1, 2 .

For reasons that will be obvious later on, it is the ‘direct difference’ ω1�ω2, of twosymplectic forms that is going to be relevant for us now.

5.2.1.1 Symplectic Linear Spaces and Symplectic Structures

Because of the geometrization principle, we may consider the geometrical structurecorresponding to a symplectic linear space.

Thus given a symplectic linear space (E, σ ), we begin by thinking that σ definesa smooth differential 2-form on E. Actually because it is constant, σ is closed. Is thislast property an integral part of the geometrization principle for symplectic linearspaces?

To elucidate this question, we recall that the Poisson structure � = −σ−1 deter-mined by σ defines a structure of Poisson algebra in the space of functions on E, inparticular the Poisson bracket {f , g} thus defined satisfies the Jacobi identity. Noticethat the Poisson tensor � is defined as:

�(α, β) = σ(σ−1(α), σ−1(β)

),

where α and β are 1-forms on E and σ is the natural isomorphism σx : TxE → T∗x E,〈σx(u), v〉 = σ(u, v). In particular, when α = df and β = dg we find the expressionfor the Poisson bracket:

{ f , g } = �(df , dg) = σ(σ−1(df ), σ−1(dg)

).

But now, if we denote by Xf the vector field on E defined by σ−1(df ), it satisfiesthat �(Xf ) = −df , hence it is just the Hamiltonian vector field defined by f (withrespect to �). For this reason we will call also in this context Xf the Hamiltonianvector field defined by f . With this notation we conclude:

{ f , g } = σ(Xf , Xg) , f , g ∈ F(E), (5.3)

and then Xf = {·, f }.


It is clear that if we try to define a Poisson bracket in the algebra of smoothfunctions on E by using Eq. (5.3), in general it would not define a Poisson algebra.Actually the only property that we have to check is the Jacobi identity because theLeibnitz rule (or derivation property will be guaranteed as Xf is a vector field, hencea derivation). Thus the geometrization principle for symplectic linear spaces shouldbe completed by asking for that the bracket defined by (5.3) should be a Poissonbracket.

Now a simple computation shows that:

{f , {g, h}} + {h, {f , g}} + {g, {h, f }} = 2 dσ(Xf , Xg, Xh) . (5.4)

Of course, if σ is constant then dσ = 0 and the bracket above defines a Poissonbracket as we already know. However if we allow σ to be non constant, then the onlyrequirement that σ must satisfy to define a Poisson bracket in the space of functionsF(E) is that σ be closed. In this case Jacobi identity for the bracket follows andfurthermore such identity implies that

[Xf , Xg] = Xg,f .

Thus the implementation of the extended geometrization principle for symplecticlinear spaces leads us to the following definition:

Definition 5.2 We will say that a linear space E is endowed with a symplecticstructure ω when ω is a non-degenerate closed 2-form on E.

Moreover, we will say that the symplectic structure is exact when the closed2-form ω is exact, that is, there exists a 1-form θ such that ω = −dθ .

When ω is closed and of a constant rank, but maybe degenerate, we will say thatE is endowed with a presymplectic structure.

Recall that the non-degeneracy of ω is equivalent, by definition, to the propertythat the maps ωx : TxE → T∗x E, given by:

〈ωx(v), v′〉 = ωx(v, v′), v, v′ ∈ TxE ,

have maximal rank. They induce a mapping between sections which, with a slightabuse of notation, we will also write ω or again ω. Then, a non-degenerate Poissonstructure � in E is equivalent to the symplectic structure ω so defined [Jo64].

Given a symplectic structure ω, we can also do conversely what we did in theintroduction of this section: the regularity of the map ω may be used to define askew-symmetric contravariant tensor � by means of

�(α, β) = ω(ω−1(α), ω−1(β)

),

where α and β are 1-forms on E. In particular, when α = df and β = dg we againfind the expression:


{ f , g } = �(df , dg) = ω(ω−1(df ), ω−1(dg)

)

for the Poisson bracket defined by the symplectic structure, and likewise the expres-sion for the Hamiltonian vector field Xg .

The point emphasized here is that a symplectic structure is nothing but a, maximal,constant rank Poisson structure. That is, if { f , g } = 0 for any function g, then thefunction f must be locally constant. On the other hand, for the case of a degeneratePoisson structure on an n-dimensional linear space E, there will exist (at least locallydefined) non-constant Casimir functions C, such that { C, g } = 0 for any g ∈ F(E).Actually, the image of � defines an involutive distribution, and therefore integrable.If the rank of� is r in a neighborhood of the point x ∈ E, then it is possible to chooseadapted coordinates in a neighborhood of the point x ∈ E, {y1, . . . , yn}, in such away that { ∂

∂y1, . . . , ∂

∂yr } generate the distribution, while {yr+1, . . . , yn} are constantsalong the maximal integral submanifolds of the distribution and, therefore, they givelocally defined Casimir functions.

Finally, we also recall that in case of a symplectic structure, ω, the conditionLX� = 0 is equivalent to LXω = 0. This is also equivalent to the property ofiXω being a closed 1-form. Hamiltonian vector fields correspond to exact 1-forms,iXH ω = dH, and those X’s for which iXω is closed, but may be non-exact, are saidto be locally Hamiltonian.

5.2.2 The Geometry of Symplectic Linear Spaces

In this section we will study different properties of symplectic and presymplecticlinear spaces that will be used in the following sections.

Let (E, σ ) be a presymplectic vector space. Given any subspace V ⊂ E, it ispossible to define an associated vector space V⊥, called the reciprocal, symplecticpolar or symplectic orthogonal of V , by

V⊥ = {u ∈ E | σ(u, v) = 0, ∀v ∈ V} . (5.5)

In other words, V⊥ is the linear subspace of E whose image under σ is in theannihilator V0 = {f ∈ E∗ | f (u) = 0, ∀u ∈ V} of V .

Let iV denote the canonical injection iV : V → E, and i∗V : E∗ → V∗ the dualmap of iV . Then the kernel of the linear map i∗V is just the annihilator V0 of V andthen E∗/V0 ∼= V∗.

If E is finite-dimensional, then from the previous identifications we get

dim V0 = dim E − dim V .

Notice that with these notations, the characteristic space of σ , that is ker σ is justE⊥, thus if σ is non-degenerate E⊥ = {0}.


Let σV be the restriction of σ to V , that is σV = i∗V σ . Then, ker σV = V ∩ V⊥,because

ker σV = {u ∈ V | σ(u, v) = 0, ∀v ∈ V} = V ∩ V⊥.

Another remarkable property is that if σ is degenerate the quotient space E/ ker σcan be endowed with a symplectic form ω such π∗ω = σ , where π is the naturalprojection π :E → E/ ker σ . The 2-form ω so constructed is said to be the reductionof σ . In order to show this property, it suffices to observe that ω(v + ker σ, u +ker σ) = σ(v, u) is well defined and satisfies the desired conditions.

There is an important relation among the dimension of a subspace V and those ofV⊥ and V ∩ E⊥:

dim E + dim(V ∩ E⊥) = dim V + dim V⊥ , (5.6)

because taking into account that V⊥ = {v ∈ E | σ (v)u = 0, ∀u ∈ V}, and using theidentification of E with its bidual, V⊥ = [σ(V)]0, we find that

dim V⊥ = dim E − dim σ (V).

On the other hand, when considering the restriction of σ onto the subspace V weobtain a homomorphism σV : V → E∗, with kernel V ∩ E⊥, and then, from theisomorphism

σV (V) ∼= V/(V ∩ E⊥) ,

we get the following relation

dim σV (V) = dim V − dim(V ∩ E⊥) ,

that together with the first one gives us the mentioned relation (5.6). Notice that whenσ is non-degenerate,

dim V⊥ = dim E − dim V . (5.7)

It is easy to check that in a symplectic inear space the association V �→ V⊥ isnilpotent of order two, i.e., (V⊥)⊥ = V , and that the following properties are satisfiedby symplectic polars:

1. V ⊂ W ⇒ W⊥ ⊂ V⊥,2. V⊥⊥ = V ,3. (V +W)⊥ = V⊥ ∩W⊥,4. (V ∩W)⊥ = V⊥ +W⊥,

Regarding (1) if V ⊂ W and z ∈ W⊥, then σ(z, w) = 0, for all w ∈ W , and inparticular, σ(z, v) = 0, for all v ∈ V , which proves it.

To prove (2), notice that V ⊂ V⊥⊥. Moreover, as we are in the finite-dimensionalsymplectic case, E⊥ = {0}, and therefore dim V + dim V⊥ = dim E. Similarly,


dim V⊥ + dim V⊥⊥ = dim E, from which we see that dim V = dim V⊥⊥, whatimplies that V = V⊥⊥.

Property (3) follows because if ω(v, u1) = 0 and ω(v, u2) = 0 for any pairof vectors u1 of V and u2 of W , then also ω(v, λ1u1 + λ2u2) = 0 holds for anypair of real numbers λ1 and λ2, and then V⊥ ∩ W⊥ ⊂ (V + W)⊥. Conversely, ifv ∈ (V +W)⊥, then v must be in V⊥ ∩W⊥.

Finally, for proving (V ∩W)⊥ = V⊥ +W⊥ it suffices to take into account

(V ∩W)⊥ = (V⊥⊥ ∩W⊥⊥)⊥ = V⊥ +W⊥ ,

where use has been made of V⊥⊥ = Vas well as the preceding relation (V +W)⊥ =V⊥ ∩W⊥. �

So far the symplectic polar looks very much like considering the orthogonal to agiven subspace in an inner product space, however since ω is alternating we mighthave V ∩ V⊥ �= 0. Indeed, for dim V = 1, we have always V ⊂ V⊥.

According with the intersection properties of a subspace V and its symplecticpolar V⊥, there are four natural classes of subspaces in a symplectic linear space.We say that:

Definition 5.3 Let (E, ω) be a symplectic linear space and V a linear subspace1:

1. V is isotropic if V ⊂ V⊥, then dim V ≤ 12 dim E.

2. V is coisotropic if V⊥ ⊂ V , then dim V ≥ 12 dim E.

3. V is Lagrangian if V = V⊥, dim V = 12 dim E.

4. V is symplectic if V ∩ V⊥ = 0.

We remark that V is called symplectic when ω|V is non-degenerate, but becauseker σV = V ∩ V⊥, that justifies to call V a symplectic subspace. Notice that for anysubspace, V , E = V ⊕ V⊥.

If (E, σ ) is a presymplectic linear space, i.e., E⊥ �= {0}, then because of Eq. (5.6)we obtain the following proposition:

Proposition 5.4 Let (E, σ ) be a presymplectic linear space. Then:

1. If W is an isotropic subspace, dim E + dim E⊥ ≥ 2 dim W2. If W is a coisotropic subspace, then dim E + dim E⊥ ≤ 2 dim W.3. Finally, if W is a Lagrangian subspace, dim E + dim E⊥ = 2 dim W.

Proof In fact, if W is isotropic,W ⊂ W⊥, then dim W ≤ dim W⊥, and then 2 dim ≤dim W + dim W⊥ = dim E + dim W ∩ E⊥ ≤ dim E + dim E⊥.

When W is coisotropic, W⊥ ⊂ W , and E⊥ ⊂ W⊥, from which we see thatW ∩ E⊥ = E⊥. Therefore,

2 dim W ≥ dim W + dim W⊥ = dim E + dim W ∩ E⊥ = dim E + dim E⊥.

1 The same definitions hold for an infinite-dimensional symplectic linear space without the dimen-sion relations.


Finally, if W = W⊥, then W ∩ E⊥ = E⊥. Moreover,

2 dim W = dim W + dim W⊥ = dim E + dim W ∩ V⊥ = dim E + dim E⊥. �

As a corollary of he previous proposition we obtain:

Corollary 5.5 If (E, ω) is symplectic linear space, an W ⊂ E is linear subspacethen: if W is isotropic, then dim E ≥ 2 dim W ; if W it is coisotropic, then dim E ≤2 dim W and, if W is Lagrangian, dim E = 2 dim W.

If ω is a linear symplectic form in E, then each isotropic space W is containedin a Lagrangian one, because if we assume that W is contained in W⊥ and W �=W⊥, then there would exist a vector v ∈ W⊥ such that v �∈ W . In this case thesubspace W + 〈v〉 will also be isotropic and contains W . In the finite-dimensionalcase, by iterating the procedure we will arrive to a Lagrangian subspace containingW .2 Notice that every one-dimensional subspace is isotropic and therefore therealways exist Lagrangian subspaces. The proof of our assertion also shows that aLagrangian subspace is nothing but a maximal isotropic subspace. Therefore anisotropic subspace is Lagrangian if and only if it is coisotropic.

Our first example of a symplectic linear space was E = W ⊕ W∗ with thecanonical simplectic form σW , Eq. (5.2). It is easy to check that in this case both Wand W∗ are Lagrangian subspaces of E. We will show in the next few lines that thisis actually the general structure of a symplectic space.

A linear map φ : E → F between the symplectic spaces (E, ωE), (F, ωF) iscalled symplectic if ωF(φ(u), φ(v)) = ωE(u, v), for all u, v ∈ E. It is simple tosee that a symplectic linear map must be injective because ker φ ⊂ E⊥. If a linearsymplectic map is an isomorphism it will be called a symplectic isomorphism, orsymplectomorphism for short.

The importance of the previous construction relies on the following proposition:

Theorem 5.6 (Structure theorem for symplectic spaces) Let (E, ω) be a linear sym-plectic space and W be a Lagrangian subspace of E. Then, there exists anotherLagrangian subspace W ′ supplementary of W. Moreover there is a symplectic iso-morphism between (E, ω) an (W ⊕W∗, σW ).

Proof Let us remark that W = W⊥ because W is Lagrangian. There will be a vectorv1 such that v1 �∈ W and as W ∩ 〈v1〉 = 0, we find that (W ∩ 〈v1〉)⊥ = E, i.e.,W + 〈v1〈⊥= E. Let choose v2 ∈ 〈v1〉⊥ such that v2 �∈ W + 〈v1〉; once again wecan say that as W ∩ 〈v1, v2〉 = 0, the space E is a sum E = W + 〈v1, v2〉⊥, andfurthermore the subspace 〈v1, v2〉 is isotropic, by construction.We iterate the processby choosing v3 ∈ 〈v1, v2〉⊥ with v3 �∈ W +〈v1, v2〉. Finally we will arrive at a linearisotropic subspace W ′ such that E = W + W ′⊥ and W ∩ W ′ = 0, and there is novector v ∈ W ′⊥ which is not contained in W +W ′. Now, as W +W ′⊥ = E, we canconclude that W +W ′ = E, and therefore W ′ is Lagrangian.

2 The argument will work unchanged in the infinite-dimensional instance applying Zorn’s Lemma.


Lagrangian subspace with a supplementary Lagrangian subspace W ′. Let now φ

be the map φ : W ′ → W∗ defined by φ(v) = ω(v) |W . This map is an isomorphismbecause if φ(v) = 0 then w(v, u) = 0, ∀u ∈ W , and consequently v ∈ W ′ ∩W⊥ =W ′ ∩W = 0. Thus, the map idW ⊕φ : E = W ⊕W ′ → W ⊕W∗, is an isomorphismand a symplectic map, because

ω0((v1, φ(u1)), (v2, φ(u2)) = φ(u1)v2 − φ(u2)v1 = ω(u1, v2)− ω(u2, v1)

= ω(u1 + v1, u2 + v2) . ��It is clear from the construction that for a given Lagrangian subspace V we have

many supplementary Lagrangian subspaces and therefore many symplectic isomor-phisms E ∼= V ⊕ V∗.

GivenV andW wecanfind all other supplementary spaces by considering (vi, wj+Ajkv

k) with Ajk = Akj.Another remarkable property of Lagrangian subspaces is the following:

Theorem 5.7 An isomorphism φ : V1 → V2 between symplectic linear spaces issymplectic if and only if the graph of the map is a Lagrangian subspace of (V1 ⊕V2, ω1�ω2).

Proof It suffices to observe that the graph of φ, Graph(φ) = {(v, φ(v)) | v ∈ V1},is a linear subspace of dimension dim V1 and that

(ω1 � ω2)((v, φ(v), (u, φ(u))) = ω1(v, u)− ω2(φ(v), φ(u)) = ω1(v, u)− (φ∗ω2)(v, u).

Then ω1 � ω2 |Graph(φ)= 0 iff ω1 − φ∗ω2. ��Another simple consequence of the previous results is the following statement:

Corollary 5.8 Given a presymplectic linear space (E, ω), there is a basis B = {ei |i = 1, . . . , n} of E such that the expression of σ is

σ =r∑

j=1εj ∧ εj+r ,

where B∗ = {εi | i = 1, . . . , n} is the dual basis of B and 2r is the rank of σ . Inparticular, for a symplectic structure ω in a linear space of dimension 2 n, it shouldbe n = 2r, and then

ω =n∑

j=1εj ∧ εj+n .

Such linear basis will be called canonical (also sometimes standard).


5.2.3 Generic Subspaces of Symplectic Linear Spaces

A generic subspace V of a symplectic linear space (E, ω) will be neither isotropic,Lagrangian, coisotropic nor symplectic. We will analyze in this section, the structureinduced from the symplectic structure ω on an arbitrary linear subspace V ⊂ E.

The restriction of ω to V defines ωV . If V is symplectic, i.e., V ∩ V⊥ = 0, thenalso V⊥ is symplectic because V⊥ ∩ (V⊥)⊥ = V⊥ ∩ V = 0, and E = V ⊕ V⊥. Thesymplectic structure ω decomposes into

ω = ωV ⊕ ωV⊥ . (5.8)

In terms of the associated Poisson brackets we have that the previous decompositiondefines the subspace of linear functions F (1)

V ; that is, any α ∈ V∗ defines the linearfunction fα given by fα(u) = α(v)where u = v+v⊥, v ∈ V and v⊥ ∈ V⊥. Similarly(V⊥)∗ defines FV⊥(1) , and because of Eq. (5.8), it follows that

{F (1)V , F (1)

V⊥} = 0 (5.9)

Notice that we can also consider:

FV = { f ∈ F(E) | df (V⊥) = 0 } (5.10)

FV⊥ = { f ∈ F(E) | df (V) = 0 }, (5.11)

of course these subalgebras are generated by FV (1) and FV⊥(1) respectively.

When V is not a symplectic subspace we get a degenerate 2-form ωV . We canconsider now the quotient space V/ ker ωV = V . By construction, this quotientspace inherits a symplectic structure that we denote as ωV . If we consider V⊥ thenthe restriction ωV⊥ will be degenerate again with characteristic subspace ker ωV⊥ .On the other hand, because kerωV = V ∩ V⊥, we get:

dim ker ωV = dim ker ωV⊥ . (5.12)

Let us denote by Vσ the subspace image of any section σ : V → Vσ ⊂ E. We canconsider (Vσ )⊥ ⊂ E. Clearly (Vσ )⊥ is symplectic and contains V⊥, therefore V⊥ isa coisotropic subspace of the symplectic linear space (Vσ )⊥. Considering now theanalogue construction, starting with V⊥, i.e.,

(V⊥) = V⊥/ ker ωV⊥ , (5.13)

and now, if μ : (V⊥)→ V⊥μ ⊂ E is any section, the image V⊥μ is a symplectic sub-space of E. If we consider the associated polar subspace (V⊥μ )⊥ we find a symplecticsubspace of E which contains V as a coisotropic subspace. From this construction


it is clear that symplectic subspaces of E which contain V as a coisotropic subspacewill depend on the section μ.

By putting together Vσ and V⊥μ , we get a symplectic decomposition of E

E = Vσ ⊕ V⊥μ ⊕ (ker ωV ⊕ ker ωV⊥). (5.14)

It is clear that (Vσ )⊥ ∼= V⊥μ ⊕ ker ωV ⊕ ker ωV⊥ and (V⊥μ )⊥ ∼= Vσ ⊕ ker ωV ⊕ kerωV⊥ . It is also possible to rewrite this decomposition in terms of Poisson subalgebras.

5.2.4 Transformations on a Symplectic Linear Space

Consider an isomorphism φ ∈ GL(E) of a symplectic linear space (E, ω). In generalthe pull-back φ∗ω of ω under φ is different from ω. When φ∗ω = ω, φ will becalled a symplectic isomorphism (or symplectomorphism for short). We will denoteby Sp(ω) the group of symplectic isomorphisms of the linear symplectic structureω.

Because of Corollary5.8, in a canonical basis the symplectic form is representedby the matrix

J =(

0 I−I 0

)(5.15)

and thematrixA representing a symplectic isomorphismφ in a canonical basis shouldsatisfy

AT J A = J. (5.16)

Henceforth matrices A satisfying Eq. (5.16) will be called symplectic matrices andthe collection of all of them form a group called the symplectic group of order 2n,Sp(2n, R). It is clear that Sp(2n, R) is a closed subgroup of GL(2n, R), hence it is aLie group according with the analysis of matrix groups done in Sect. 2.6.1.

Thus, out of a given symplectic structure ω we get a new one φ∗ω unless φ is asymplectic isomorphism. Given any two different symplectic structures, say ω1 andω2, it is always possible to find a transformation φ that takes ω1 into ω2. Indeed, ifwe choose a canonical basis for ω1 and another canonical basis for ω2 then the linearmap determined by these two linear basis is the required map (which is not unique).Therefore GL(E) acts transitively on the space of non-degenerate 2-forms. BecauseSp(ω) is a subgroup ofGL(E), we have that the family of symplectic linear structureson E are in a one-to-one correspondence with the quotient space GL(E)/Sp(ω).

Notice that in the simplest case, that is when dim E = 2, the group of linearisomorphisms of E is isomorphic to GL(2, R) which has dimension four. On theother hand Sp(2n, R) is isomorphic to SL(2, R), then it has dimension three, andthe homogeneous space of non-equivalent symplectic forms in the two-dimensionalspace is one dimensional; in fact, they are all the nonzero multiples of the 2-form

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ε1 ∧ ε2. Note that if A =(

a bc d

)then the simplecticity condition Eq. (5.16),

A =(

a cb d

) (0 I−I 0

) (a bc d

)= (ad − bc)

(0 1−1 0

), (5.17)

shows that ad − bc = 1, which is just the condition det A = 1.To classify various subgroups in Sp(ω), it is convenient to exploit the structure

theorem, Theorem5.6, and use the isomorphism E ∼= V ⊕ V∗ for some chosenLagrangian subspace V . By using linear coordinate functions for V and V ∗, say(xi, pi) and inducing them on E via the given isomorphism, we can write ω as

ω = dxi ∧ dpi. (5.18)

This shows that the symplectic form is not only closed but exact too, i.e., we canwrite ω = −dθ with θ = pi dxi.

Wenotice at this point that ifφ is a symplectic isomorphism, itwill takeLagrangiansubspaces into Lagrangian subspaces, therefore we go from the isomorphism withV ⊕ V∗ to some other isomorphism with V ′ ⊕ V ′∗. In terms of coordinate functionsω will be written ω = dxi ∧ dpi. If we denote by θ ′ the pull-back of θ with respectto φ, then because φ is symplectic d(θ − θ ′) = 0. But in a linear space any closedform is exact, then there must be a function S such that

dSφ = θ − θ ′ = pi dxi − pi dxi ,

that will be called a generating function for the symplectic map φ. It is more con-venient to think of the symplectic map φ : V ⊕ V∗ → V ′ ⊕ V ′∗ as a Lagrangiansubspace in the direct sum of V ⊕ V∗ and V ′ ⊕ V ′∗. Then if we think of the differ-ence θ − θ ′ as defining a 1-form on (V ⊕ V∗) × (V ′ ⊕ V ′∗) such 1-form (actuallyπ∗1 θ−π∗2 θ ′) will be closed (hence exact) when restricted to the graph of φ. The graphof φ is a 2n-dimensional space and can be parametrized by an appropriate choiceof coordinates. This is the idea behind the construction of generating functions ofdifferent types, Types I, II, III and IV, as one usually finds in classical textbooks (seefor instance [Go81, Sa71]).

Thus, e.g., we can choose coordinates {xi, pi} and express θ−θ ′ |graph(φ) in termsof the generating function of Type II, SII

φ as follows:

pi dxi + xi dpi = dSIIφ (xi, pi) .

The transformation is then given by

pi = ∂Sφ

∂xi, xi = ∂Sφ

∂ pi. (5.19)

In order for the transformation to be well defined it should be:


det

(∂2S

∂xi ∂ pj

)�= 0 . (5.20)

We should remark that the generating function depends on bothφ and the variableswhich appear as differentials on the left-hand side, i.e., the Type. For instance, anotherchoice would lead to

pi dxi − pi dxi = dSIφ(xi, xi) . (5.21)

We will only develop the Type II case but other approaches follow similar lines.As our transformation φ is linear by assumption, then for

Sφ = 1

2Aijx

ixj + 1

2Bijpipj + Ci

jpjxi

we find

pi = ∂Sφ

∂xi= Aijx

j + Cijpj , xi = Bijpj + Cj

ixj .

By assuming that the matrix C = (Cij) is invertible we find

pj = (C−1)jipi − (C−1)j

kAkixi ,

xj = Ckjxk + Bjm(C−1)m

ipi − Bjm(C−1)mkAkix

i .

If the Lagrangian subspace described by xi is required to be invariant under φ thenB = 0, and our transformation is represented by

(xp

)=

(CT 0−C−1A C−1

) (xp

). (5.22)

If the same space is to be preserved pointwise we get C = I and the representativematrix becomes (

I 0−A I

)(5.23)

with A a symmetric matrix.If C is not invertible, the condition (5.20) is not satisfied and the generating

function must be expressed in different coordinates. We consider, for instance, oneof the Type I, Eq. (5.21), and now the transformation is defined by

pi∂S

∂xi; pj = − ∂S

∂ xj,

and it is well defined if

det

(∂2S

∂xi ∂ xj

)�= 0 .


We consider

Sφ = Cijxixj + 1

2Aijx

ixj + 1

2Bijx

ixj

to findpi = Cijx

j + Aijxj; pj = −Cijx

i − Bjixi .

Therefore,xj = (C−1)jipi − (C−1)jiAikxk ,

pj = −Cjixi − Bji(C

−1)ikpk + Bjm(C−1)miAikxk .

It is possible to consider subalgebras of sp(ω). Here it is convenient again to usethe vector field XS of a matrix S associated to an element in the Lie algebra of Sp(ω).We recall that if (

xp

)= S

(xp

)=

(A BM N

) (xp

),

we set

XS = (xT , pT )

(A BM N

) (∂x∂p

),

that is

XS = (xT A+ pT M)∂

∂x+ (xT B+ pT N)

∂

∂p. (5.24)

By using −d(pT dx) = ω and imposing on the vector field the condition thatLXS ω= 0 we will find

LXS (pT dx) = dfS(p, x) ,

where∂fS∂p= pT MT ,

∂fS∂x= xT B+ pT (AT + N) ,

becauseLXS (pk dxk) = dxj(xiB

ij + piNi

j + pkAjk + dpj pkMkj .

Therefore, very much like finite transformations, infinitesimal transformations canbe characterized by quadratic functions

fS(p, x) = Aijpipj + Bijxixj + Ci

jxipj .

Therefore the group of symplectic isomorphisms has dimension n(2n+ 1).If we consider now a one-parameter group of transformations, say

φt = etZ ,


thendSφ(t)

dt= fZ

gives the relation between generating functions of finite symplectic isomorphism φ

and those of the infinitesimal ones. This gives us the opportunity to comment on thecorrespondence between transformations and generating functions. If Z is the vectorfield Z = xT A∂x − pT AT ∂p we have

LZ(pT dx) = pT d(AT x)− pT AT dx = 0 ,

i.e., for any choice of the matrix A the transformations of this form correspond to theconstant generating function. We can consider the 1-form−xT dp, now LZ(xT dp) =xT Adp − xT d(Ap) = 0. We can take now any quadratic function F and consider a1-form like pT dx + dF in this case we will find

LZ(pT dx + dF) = dLZF .

Therefore the infinitesimal generating function can bewritten as the Lie derivativeof another function. This phenomenon is rooted in the cohomology of Lie algebras.In particular, as the 1-form θ such that ω = −dθ is only defined up to addition of aclosed 1-form, one may ask if for a given vector field Z it is always possible to find a1-form θ such that dθ = dxT ∧dp andLZθ = 0. For vector fields with representativematrix having degenerate purely imaginary eigenvalues the answer is negative.

Let us recall now (see Sect. 3.2.5 and Appendix F for more details) that an F-valued 1-cocycle on the group G is a map c1 : G→ F such that

δc1(g, g′) = gc1(g

′)− c1(gg′)+ c1(g) .

We can show in general that S : φ �→ Sφ is a 1-cocycle from Sp(ω) → F(E)

while f : sp(ω)→ F(E), Z �→ fZ is a 1-cocycle for the corresponding Lie algebra.The question we are asking concerns the possibility that such cocycle becomes a

coboundary when restricted to subgroups or subalgebras of the symplectic algebra.

5.2.5 On the Structure of the Group Sp(ω)

We can use this result to show that Sp(ω) is a closed subgroup of GL(E). Indeed,symplectic forms are a subset of skew-symmetric matrices, R

n(2n−1), which are avector space, therefore a manifold. The map det : R

n(2n−1) → R is continuous,therefore det−1(R−{ 0 }) is an open submanifold because non zero values of det areregular.

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Proposition 5.9 Every isotropic subspace is contained in a Lagrangian subspace.Every two Lagrangian subspaces can be mapped into each other via a symplectictransformation.

Necessary and sufficient conditions for the existence of a symplectic transforma-tion connecting two subspaces is that they have the same rank.

Proposition 5.10 Necessary and sufficient conditions for a symplectic transforma-tion to preserve the decomposition V = W⊕W⊥, into Lagrangian subspaces is thatit has the form (

A 00 (AT )−1

)(5.25)

Symplectic transformations which preserve all elements of a given Lagrangiansubspace have the form (

I S0 I

)(5.26)

with S = ST .

These transformations form an Abelian group, isomorphic to the additive groupof symmetric matrices. It is possible to enquire about the equivalence of these trans-formations under symplectic conjugation. One finds that a necessary and sufficientcondition is that xi(S1)ijxj is equivalent to xi(S2)ijxj.

Symplectic Transvections

Any hyperplaneH (of codimension 1) can be represented as kerω(a) for some vectora. The symplectic transformation V → V which preserves every element of H hasthe form x �→ x − λω(x, a)a.

Symplectic transvections are symplectically conjugate iff λ1/λ2 is a square in K.Thus for K = R, λ1λ2 < 0 will not provide conjugated transformations.

Proposition 5.11 Every symplectic transformation is the product of transvections.(Indeed it is at most the product of 2n symplectic transvections if dim E = n).

We have x �→ x+∑qi=1 λiω(x, ai)ai. It would be interesting to find the minimum

for q.

Proposition 5.12 The centre of Sp(n, K) contains only two elements, x �→ x andx �→ −x. There are no additional normal subgroups.


5.2.6 Invariant Symplectic Structures

Existence of Invariant Symplectic Structures

A generic linear dynamical system has a commuting algebra of symmetries butno constants of the motion. On the other hand, we will see that the existence ofinvariant structures on the carrier space of our dynamical systemallowsus to associatesymmetries with constants of the motion and other properties that are not shared bygeneric ones (like Liouville’s theoremon phase space [Ab78], energy-period theorem[Go69], etc.).

We would like to consider the restriction to symplectic structures of the problemof determining the existence of invariant Poisson structures for a given dynamicalproblem. More specifically, given a linear dynamical system, determine all, if any,linear symplectic structures ω = �ij dxi ∧ dxj which are preserved by our system.

The study is similar to that we did for Poisson structure (see Sect. 4.2.2).As usual we will consider first a very simple example so that the ideas involved

can be easily grasped. Let us consider the vector field

XA = (mx + ny)∂

∂x+ (px + qy)

∂

∂y,

corresponding to the matrix

A =(

m np q

).

The most general constant symplectic structure in R2 is

ω = α dx ∧ dy , α ∈ R ,

with associated matrix � = αJ (see Eq. (5.15)).We have that LXAω = 0 is equivalent to m = −q because

LXAω = −α(q dy ∧ dx + m dy ∧ dx) = α(m + q)dx ∧ dy .

In other words, XA is Hamiltonian if and only if the trace of the matrix A vanishes.Setting q = −m we compute the Hamiltonian function. Then

iXω = dH = −α(px dx − my dx − mx dy− ny dy) = −d

(1

2px2 − 1

2ny2 − mxy

)α ,

and the Hamiltonian H of the vector field XA is given by a quadratic function definedby the matrix G = −�A, which is symmetric because m = −q. Conversely, given aquadratic function defined by a symmetric matrix G, the corresponding Hamiltonianvector field is a vector field XA defined by the matrix A = −�−1G. We can considerour quadratic form

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2mxy + px2 − ny2

and reduce it to canonical form. We will get, depending on the value of the discrim-inant m2 + np:

1. 12 (p

2 + q2): with G = 12

(1 00 1

), A =

(0 1−1 0

).

2. 12 (p

2 − q2): with G = 12

(1 00 −1

), A =

(0 11 0

).

3. 12p2 or 1

2q2: with G = 12

(0 00 1

), A =

(0 10 0

), or G = 1

2

(1 00 0

), A =

(0 01 0

)

respectively .

The associated vector fields will be in each case:

1. X1 = p ∂∂q − q ∂

∂p .

2. X2 = q ∂∂p + p ∂

∂q .

3. X3 = p ∂∂q or q ∂

∂p .

Let us try to read in algebraic termswhat we have done. First a simple computationshows that LXAω = 0 if and only if

AT�+�A = 0, (5.27)

where �T = −� is the skew-symmetric matrix associated with ω as it was shownin Sect. 4.2.3, Eq. (4.7). Then we have that (�A)T = �A, that is �A is symmetricwhich is equivalent to

AT = −�A�−1 . (5.28)

This equation says that the matrix associated with the Hamiltonian function is sym-metric (our previous quadratic form). The corresponding Hamiltonian function Hwill be given by the quadratic function

H = −1

2(�A)kj xkxj .

By the same token, Eq. (5.28) implies that Tr A = 0. we also find

(A2k+1)T = −�(A2k+1)�−1 ,

therefore (see again Eq. (4.8))Tr A2k+1 = 0 , (5.29)

and because of the preceding equation the vector field associated with any odd powerof A is also Hamiltonian with respect to the same symplectic structure. On the otherhand,

A2k� = �A2k .

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This shows that the quadratic functions fk = (�A2k+1)ij xixj are constants of motionfor XA, and moreover, for XA2j+1 . In fact, if Nk = �A2k+1, the invariance condition

(A2j+1)T Nk + NkA2j+1 = 0

is a consequence of A2k� = �A2k , because

(A2j+1)T Nk + NkA2j+1 = (−1)2j+1�A2j+1�−1�A2k+1 +�A2k+1A2j+1

= −�A2(j+k)+1 + A2(j+k)+1� = 0 .

This means that the quadratic functions fk = 12 (�A2k+1)ij xixj satisfy

{fk, fj} = 0. (5.30)

with respect to the Poisson bracket associated with �.Going back to our original problem it looks like that everything is contained in

the starting relation Eq. (5.27) with det� �= 0. As a matter of fact it can be shownthat:

Theorem 5.13 A necessary and sufficient condition for the matrix A without zeroeigenvalues or even degenerate eigenvalues to define a Hamiltonian vector field withrespect to some symplectic structure is that A is similar to the opposite of its transpose−AT , i.e.,

A = −�AT�−1,

with � being a skew-symmetric matrix.

In one direction the statement is obvious, in the other direction can be provedusing reduction to normal form.

5.2.6.1 On the Uniqueness of Invariant Symplectic Structures

Having considered the existence problem of admissible symplectic structures, weconsider in this section how many different Hamiltonian descriptions are possiblefor the same linear vector field. In other terms we are looking for the set of allsolutions for � in the Eq. (5.27).

The problem of finding alternative symplectic structures for the vector fielddefined by the matrix A is equivalent to looking for all different decompositionsof A into the product of a skew-symmetric invertible matrix �, the opposite to theinverse of �, times a symmetric one H: A = � · H, that we discussed at length inSect. 4.2.3, Theorem4.2. Given such a decomposition, if A undergoes the transfor-mation A �→ P−1AP, we have

P−1AP = P−1�HP = P−1�(PT )−1PT HP ,

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where PT HP is symmetric and P−1�(PT )−1 is skew-symmetric. It is now clear thatif P−1AP = A we will get new decompositions of A provided that either P�PT �= �

(i.e., P−1�P−1T , or PT HP �= H. Therefore, symmetries for our system will take usfrom one symplectic description to another.

Turning back to the symmetries of a vector field XA, it is now clear that evenpowers of A will be associated with noncanonical transformations. The exponentialexp sA2k will take us from one symplectic description to another. For a genericHamiltonian system, i.e., all eigenvalues are simple, we generate in this way allconstant symplectic structures admissible for �; they will be parametrized by theparameters s0, s1, . . . , s2(n−1) appearing in the exponentials.

Let us give, as it is customary by now, an example. Consider the linear system:

� = ω1

(x

∂

∂y− y

∂

∂x

)+ ω2

(z

∂

∂w− w

∂

∂z

)(5.31)

with ω1 �= ω2. The associated matrix A is given by:

A =(

ω1J 00 ω2J

)(5.32)

with J the standard 2× 2 symplectic matrix.An admissible symplectic structure for � is provided by ω0 = dx∧ dy+dz∧ dw,

which is obtained from the factorization(

ω1J 00 ω2J

)=

(ω1I 00 ω2I

) (J 00 J

), (5.33)

where I denotes the identity matrix in two dimensions.Even powers of the representative matrix are given by

A2k = (−1)k

⎛⎜⎜⎝

ω2k1 0 0 00 ω2k

1 0 00 0 ω2k

2 00 0 0 ω2k

2

⎞⎟⎟⎠ . (5.34)

Therefore the most general constant invariant symplectic structure is a dx ∧ dy +b dz ∧ dw, with the condition ab �= 0.

It should be noticed that it is possible to obtain more general symplectic structuresbymaking a to be a function of x2+y2 and b a function of z2+w2, as long as ab �= 0.This is coherent with our earlier statement that our linear symmetries can be madenon linear by allowing the entries to depend on constants of the motion. Let us pursuethe computations for the present example. A general symmetry for our vector fieldis provided by


B =

⎛⎜⎜⎝

a b 0 0−b a 0 00 0 r s0 0 −s r

⎞⎟⎟⎠ (5.35)

with det B �= 0, i.e., (a2 + b2)(r2 + s2) �= 0.If we substitute a, b, r, s by constants of motion while preserving the condition

det B �= 0, we have

⎛⎜⎜⎝

xyzw

⎞⎟⎟⎠ =

⎛⎜⎜⎝

a b 0 0−b a 0 00 0 r s0 0 −s r

⎞⎟⎟⎠

⎛⎜⎜⎝

q1p1q2p2

⎞⎟⎟⎠ =

⎛⎜⎜⎝

aq1 + bp1−bq1 + ap1

rq2 + sp2−sq2 + rp2

⎞⎟⎟⎠ , (5.36)

therefore,

ω = d(aq1 + bp1) ∧ d(ap1 − bq1)+ d(rq2 + sp2) ∧ d(rp2 − sq2) (5.37)

with a, b, r, s arbitrary constants of motion.For the n-dimensional isotropic harmonic oscillator with phase space R

2n, sym-metries are given by the real form of Gl(n, C). Entries of each matrix can be made tobe any function of zaz∗b , i.e., any element of F(CPn×R+). It should be noticed thatsome of these transformations may be canonical and therefore they do not changethe symplectic structure to which we apply it.

5.2.7 Normal Forms for Hamiltonian Linear Systems

From the general reduction to normal forms we have that a linear vector field �

with a representative matrix that does not have imaginary or null eigenvalues can bereduced to the standard form

� = Aijx

j ∂

∂xi− (AT )j

iyj∂

∂yi(5.38)

with Hamiltonian function H = xjAijyi with respect to the natural symplectic struc-

ture.For the part which corresponds to imaginary eigenvalues or to an even number

of zero eigenvalues, the situation is different. For imaginary eigenvalues, a typicalelementary Jordan Block of Hamiltonian type will be


⎛⎜⎜⎜⎜⎜⎜⎝

0 λ

−λ 01 00 1

0

00 λ

−λ 01 00 1

0 00 λ

−λ 0

⎞⎟⎟⎟⎟⎟⎟⎠

(5.39)

with associated linear vector field

� = λ

(y1

∂

∂x1− x1

∂

∂y1+ y2

∂

∂x2− x2

∂

∂y2+ y3

∂

∂x3− x3

∂

∂y3

)

+ x2∂

∂x1+ y2

∂

∂y1+ x3

∂

∂x2+ y3

∂

∂y2. (5.40)

This vector field is Hamiltonian with respect to the symplectic form

� = dx1 ∧ dy3 + dy2 ∧ dx2 + dx3 ∧ dy1 (5.41)

with associated matrix

[�] =

⎛⎜⎜⎜⎜⎜⎜⎝

0 00 1−1 0

00 1−1 0

0

0 1−1 0

0 0

⎞⎟⎟⎟⎟⎟⎟⎠

(5.42)

and Hamiltonian function

H = −λ

2(x22 + y22 − 2x1x3 − 2y1y3)+ (x2y3 − x3y2) . (5.43)

For even degeneracy, say n = 2, we have:

� = λ

(y1

∂

∂x1− x1

∂

∂y1+ y2

∂

∂x2− x2

∂

∂y2

)+ x2

∂

∂x1+ y2

∂

∂y1, (5.44)

with symplectic form:� = dx1 ∧ dx2 + dy1 ∧ dy2 , (5.45)

and Hamiltonian:

H = λ(y1x2 − x1y2)+ 1

2(x22 + y22) . (5.46)

For null eigenvalues n must be even and we simply put λ = 0 in the vector fieldand Hamiltonian corresponding to the imaginary case with even degeneracy.


According to the present classification, linear Hamiltonian systems fall in threemain families.

1. The representative matrix does not have null or imaginary eigenvalues. In thiscase the carrier space can be given a structure of cotangent bundle and our vectorfield is a cotangent lift.

2. The representative matrix has only imaginary eigenvalues with odd degeneracy(minimal polynomial equal to the characteristic polynomial).

3. The representativematrix has only imaginary eigenvalues or nullwith even degen-eracy (minimal polynomial equal to the characteristic polynomial).

According to our general formula for the flow of a linear vector field

etAx =∑

k

etλk Pk(t)xk,

in the last two cases we have an oscillatory part due to purely imaginary eigenvaluesand a polynomial part of degree equal to the index of nilpotency minus one.

As an example we will consider the case of null eigenvalue which has evendegeneracy.

Consider the kth order system of differential equations:

x(k) = 0, y(k) = 0 . (5.47)

We can associate with them a first-order equation by setting

x = x1, x′ = x2, . . . , x(k−1) = xk , (5.48)

y = y1, y′ = y2, . . . , y(k−1) = yk ,

and we get the first-order system of differential equations:

dx1dt= x2, . . . ,

dxk−1dt= xk,

dxk

dt= 0

dy1dt= y2, . . . ,

dyk−1dt= yk,

dyk

dt= 0 .

The associated Hamiltonian system will be

� = x2∂

∂x1+ x3

∂

∂x2+ · · · + xk

∂

∂xk−1+ y2

∂

∂y1+ · · · + yk

∂

∂yk−1(5.49)

with symplectic structure

� =∑

j

(−1)j+1dxj ∧ dxk+1−j +∑

j

(−1)j+1dyj ∧ dyk+1−j , (5.50)


and Hamiltonian

H =1+k/2∑

j=1

((−1)j+1xj−1xk+1−j

)+ 1

2(−1)k/2x2k/2

+1+k/2∑

j=1

((−1)j+1yj−1yk+1−j

)+ 1

2(−1)k/2y2k/2 . (5.51)

5.3 Symplectic Manifolds and Hamiltonian Systems

As it was discussed in Sect. 4.2.3, integral leaves of the canonical distribution definedby a Poisson structure inherit a non-degenerate Poisson structure, that is, a symplecticstructure. If � is a constant Poisson structure on a linear space, its integral leavesare affine spaces and the induced symplectic structures are constant, however, evenin the case of linear Poisson structures, i.e., homogeneous Poisson tensors of degree−1, the symplectic leaves are coadjoint orbits of Lie groups (see Sect. 4.3.3) andthe induced symplectic structure is called the canonical Kirillov-Kostant-Souriausymplectic form. These symplectic manifolds are the fundamental model for theclass of Hamilltonian systems with symmetry as will be discussed later on.

Thus we observe that in discussing the structure of Hamiltonian systems (withrespect to the Poisson tensor), i.e., factorizable dynamics, we have to cope withsymplectic structures on manifolds and not just linear spaces. We will devote thissection to exploring the aspects of Hamiltonian dynamical systemswhich are specificto symplectic structures with respect to what was already discussed on the propertiesof Hamiltonian systems in the previous chapter. In particular we will discuss underwhat conditions a Hamiltonian system has the structure of a mechanical system, thatis, the symplectic manifold is the cotangent bundle of a given configuration space.

5.3.1 Symplectic Manifolds

As discussed earlier, a symplectic manifold is a smooth manifold M together witha symplectic form, that is a non-degenerate closed 2-form ω on it. Any symplecticmanifold is in particular a Poisson manifold with Poisson tensor � = −ω−1. Avector field � in M will be called Hamiltonian if it is Hamiltonian with respect to thePoisson tensor � or, in other words, if there exists a function H such that i�ω = dH.In such case, H will be called the Hamiltonian of the dynamical system �.

Notice that in a symplectic manifold (M, ω) the Poisson bracket of two functionsf , g, defined by the Poisson tensor � = −ω−1 becomes:

{ f , g } = ω(�(df ),�(dg)) . (5.52)

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As indicated above, the Jacobi identity satisfied by the Poisson brackets is equivalentto dω = 0. In fact, taking into account that for any triplet of functions f , g, h ∈ F(E),

Xf (ω(Xg, Xh)) = Xf {g, h} = {{g, h}, f } ,

and using the above mentioned relation [Xf , Xg] = X{g,f }, we find that ω([Xf , Xg],Xh) can be rewritten as

ω([Xf , Xg], Xh) = ω(X{g,f }, Xh) = {{g, f }, h},

(along with the corresponding expressions obtained by cyclically permuting theindices), we find that

dω(Xf , Xg, Xh) = 2 ({{g, h}, f } + {{h, f }, g} + {{f , g}, h}) .

Therefore, as a local basis for the space of vector fields on M can be built fromHamiltonian vector fields, the Jacobi identity for the Poisson bracket is equivalent toω being closed.

Notice that because of Gromov’s theorem, Theorem3.20, any open even-dimen-sional manifold possesses a symplectic structure. Thus symplectic structures areabundant. Not only that, symplectic structures satisfy the holonomic geometrizationprinciple, which in this situation, is an involved way of saying that they are locallyconstant. More precisely:

Theorem 5.14 (Darboux Theorem) Let (M, ω) be a symplectic manifold. For anyx0 ∈ M there exists a local chart (ϕ, U) such that x0 ∈ U and

ω |U= dqi ∧ dpi , (5.53)

where ϕ : U ⊂ R2n and ϕ(x) = (q1, . . . , qn, p1, . . . , pn).

Proof Because ω is closed, it is locally exact. Let x0 ∈ M and let U be a contractibleopen neighborhood of x0 small enough such that ω |U= dθ and U is contained inthe domain of a chart centered at x0 (ϕ(x0) = 0). Let ω0 be the constant form on Udefined by the value of ω at x0, i.e., ω0(x) = ω(x0) (in other words, ω0 is the pull-back of the canonical symplectic structure on R

2n with respect to ϕ). Then we showthat there exists a family of diffeomorphisms ψt , t ∈ [0, 1], on U, and an isotopy ofsymplectic 2-forms ωt defined on U such that

ψ∗t ωt = ω0 .

The family ψt is such that ψ0 = idU and ω1 = ω. Thus ψ∗1ω = ω0 and ω has thedesired structure on the chart ϕ ◦ ψ−11 .

Then, to show that there exists such a family of diffeomorphisms, let us considerθt = θ0 + t(θ − θ0) where dθ0 = ω0, and ωt = dθt . Let us compute the derivativeof the previous equation. Then,

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5.3 Symplectic Manifolds and Hamiltonian Systems 297

ψ∗1ω1 − ω0 =1∫

0

d

dtψ∗t ωt dt

=1∫

0

ψt∗(LXt ωt + ω − ω0) dt =1∫

0

ψt∗(d

(iXt ωt − (θ − θ0)

))dt.

Again, we may choose U small enough such that the equation

iXt ωt = θ − θ0 ,

has a unique solution Xt for each t ∈ [0, 1] (it suffices to consider U small enough sothat ωt = ω0 + t(ω− ω0) is invertible, i.e., such that the difference ω− ω0 is smallenough in U). Then the family of diffeomorphismsψt can be obtained by integratingthe differential equation:

dψt

dt= Xt ◦ ψt ,

in U (again, if necessary we can ‘squeeze’ U to guarantee that there is a flow boxaround x0 for the time-dependent vector field Xt , for t ∈ [0, 1]). ��

Local coordinates (qi, pi) on a symplectic manifold (M, ω) such that ω takes thecanonical form Eq. (5.53) will be called canonical (or Darboux) coordinates. Theexistence of local coordinates around any point on a symplectic manifold allows usto choose coordinates such that the Hamiltonian vector field XH corresponding to afunction H takes the canonical form:

qi = ∂H

∂pi, pi = −∂H

∂qi, i = 1, . . . , n . (5.54)

The previous equations are called Hamilton’s equations.Thus, locally all symplectic manifolds are ‘equal’, that is, symplectomorphic.

However this is not so globally. Notice that because they are locally constant andholonomic there are no local invariants characteristic of symplectic manifolds inacute difference with Riemannian or pseudo-Riemannian manifolds.

On the other hand, it is easy to provide examples of symplectic manifolds ofthe same dimension which are not symplectomorphic. For instance any compactorientable Riemann surface carries a natural symplectic structure, its volume form,but any two Riemann surfaces with different genus cannot be diffeomorphic, hencethey cannot be symplectomorphic either. Moreover, it is also possible to show thateven in a given manifold it is possible that there exist symplectic structures which arenot symplectically equivalent. Again, Gromov’s theorem guarantees that there existsa symplectic form for each cohomology class on an open manifold. Thus symplecticforms corresponding to different cohomology classes cannot be symplectomorphic.


Another simple instance of this phenomenon is provided by M = CP2 the com-plex projective 2-dimensional space. Then there is a family of symplectic structureslabelled by an integer k which are not symplectically equivalent (see Sect. 6.4.5 fora detailed study of the complex projective space in relation with the geometry ofquantum evolution).

We do not pretend to discuss here the state of the art concerning global invari-ants of symplectic manifolds, a subject that has driven an intense research lately.Instead we will characterize a class of symplectic manifolds, cotangent bundles, thatplay a significant role in the description of dynamical systems and that will lead useventually to characterize systems that can be described using a variational principle.

Before solving the previous question, let us discuss with more care the symplecticstructure of a cotangent bundle. Thus if πQ : T∗Q → Q is the cotangent bundle ofQ, there is a canonical 1-form θQ defined on it and that is defined by the formula:

θQ(q, p)(U) = 〈p, TπQ(U)〉, U ∈ T(q,p)(T∗Q) . (5.55)

Notice that this definition is equivalent to the tautological property of θQ:

α∗θQ = α,

where α ∈ �1(Q) is an arbitrary 1-form on Q, i.e., a section of T∗Q and the left-handside of the formula above indicates the pull-back of θQ along α : Q → T∗Q. Thecanonical symplectic structure on T∗Q is defined as

ωQ = −dθQ ,

and in local bundle coordinates (qi, pi) we have the expressions:

θQ = pi dqi, ωQ = dqi ∧ dpi ,

hence local bundle coordinates are canonical coordinates for ωQ and provide localDarboux trivializations of ωQ.

Exercise 5.1 Prove that the graph of any closed 1-form is a Lagrangian submanifoldin T∗Q with respect to the canonical symplectic form ωQ. Find local canonicalcoordinates for (T∗Q, ωQ) which are not bundle coordinates.

Now given a diffeomorphism φ : Q → Q, we may lift it to T∗Q as followsϕc : T∗Q→ T∗Q, ϕc(q, p) = (ϕ(q), (Tϕ−1)∗p). Then clearly

(ϕc)∗θQ = θQ , (5.56)

as the simple computation below shows:

〈(ϕc)∗θQ(q, p), U〉 = 〈θQ(ϕc(q, p), (ϕc)∗U)〉 = 〈(ϕ−1)∗p, (πQ)∗(ϕc)∗U〉= 〈(ϕ−1)∗p, ϕ∗(πQ)∗U〉 = 〈p, (πQ)∗U〉 = 〈θQ(q, p), U〉 .

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Moreover ϕc is a symplectic diffeomorphism because

(ϕc)ωQ = ωQ . (5.57)

Given a vector field X on Q we can lift it to T∗Q as follows. Consider the localflow ϕt of X, then we may define the family of local diffeomorphisms ϕc

t of T∗Q thatdefine a vector field Xc called the complete cotangent lift of X defined as:

d

dtϕc

t = Xc ◦ ϕct .

Notice that, by definition, the vector field Xc is πQ-projectable on (πQ)∗Xc = X,thus Xc and X are πQ-related. It is a simple exercise to check that in local bundlecoordinates we have the following expression for Xc:

Xc = Xi ∂

∂qi− pi

∂Xi

∂qj

∂

∂pj,

provided that X = Xi(q)∂/∂qi.Another characterization of the complete lift Xc is that it is the only vector field

in T∗Q projectable onto X and preserving the canonical 1-form θQ, i.e., such that

LXcθQ = 0 . (5.58)

In fact, the local diffeomorphisms ϕct preserve the canonical 1-form θQ and therefore

LXcθQ = 0. Conversely, if the local coordinate expression of X is X = Xi(q) ∂∂qi ,

then a vector field projecting on X must be of the form X = Xi(q) ∂∂qi + fj(q, p) ∂

∂pj,


LX(pi dqi) = (Xpi) dqi + pi d(Xqi) = fi(q, p) dqi + pi dXi,

and therefore the conditionLXc = 0 fixes the values of fi to be given by the coordinateexpressions indicated above for the complete lift Xc:

fi(q, p) = −pj∂Xj

∂qi.

The complete lifts of vector fields satisfy:

[Xc, Yc] = [X, Y ]c, ∀X, Y ∈ X(Q) . (5.59)

Finally, we must remark that because the flow ϕct is symplectic (Eq. (5.57)) the

complete lifts Xc are (locally) Hamiltonian vector fields, even more because LcX

θQ = 0 (Eq. (5.56)) we conclude that


iXcωQ = d(iXcθQ) , (5.60)

and consequently Xc is Hamiltonian with Hamiltonian the ‘momentum’ function PX

associated to X defined by:

PX(q, p) = 〈θQ(q, p), Xc(q, p)〉 = 〈p, X(q)〉 .

Then a simple computation shows that:

Proposition 5.15 The family of momentum functions PX , X ∈ X(Q), close a Liealgebra with respect to the Poisson bracket isomorphic to X(Q). That is:

{PX , PY }T∗Q = P[X,Y ], ∀X, Y ∈ X(Q) . (5.61)

and the map P : X(Q)→ F1(T∗Q) is a Lie algebra homomorphism.

Proof We first realize that P[X,Y ] is the Hamiltonian of [Xc, Y c], because of Eq.(5.59) hence

i[Xc,Yc]θQ = LXc iYcθQ = Xc(PY ) = {PX , PY }T∗Q . ��

5.3.2 Symplectic Potentials and Vector Bundles

5.3.2.1 Tensorial Characterization of Symplectic Linear Spaces

We want to characterize tensorially a symplectic linear space (E, ω), in a similarway as we did with the linear structure in terms of the Liouville vector field � inSect. 3.1.3. Thus the linear structure on E is characterized by the vector field � andwe get L�ω = 2ω, or better:

d(i�ω) = 2ω . (5.62)

Moreover ω is always exact in a linear space E, so there exists a symplectic potential1-form θ such that dθ = ω. Hence because Eq. (5.62) i�ω = 2(θ + df ). Redefiningthe symplectic potential θ as θ + df , we get that the Liouville vector field definingthe linear structure satisfies:

i�ω = 2θ ,

for some symplectic potential θ . Again because of the structure theorem for a sym-plectic linear space, there exists a Lagrangian subspace L such that θ |L= 0.

To show this we can argue as follows. Notice that W = ker θ is a coisotropicsubspace because codim W = 1, then W/W⊥ is a symplectic linear space. Chooseany symplectic subspace L inW/W⊥ and define L = π−1(L)withπ : W → W/W⊥the canonical projection. The subspace L is a Lagrangian subspace with the requiredproperties.

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If we choose now linear canonical coordinates (qk, pk) on E adapted to L, then wewill get that in these coordinates θ = pk dqk − qkdpk (notice that i�θ = 0). We arenow ready to prove the following theorem that characterizes tensorially symplecticlinear structures.

Theorem 5.16 Let (M, ω) be a symplectic manifold. Then there exists a linear struc-ture on M such that ω is a linear symplectic structure iff there exists a Liouville vector� field on M such that θ = 1

2 i�ω is a symplectic potential.

Proof We have to prove the converse implication. Then suppose that there exists aLiouville vector field �, then M is a linear space. Now if 2θ = i�ω is a symplecticpotential, then L�ω = 2ω. Notice that we also have that L�θ = 2θ and i�θ = 0.Hence θ is linear on a set of linear coordinates on M, i.e., θ = �ijxidxj , but then�ij = −�ij and ω = dθ = �ijdxi ∧ dxj. ��

5.3.2.2 Compatible Partial Linear Structures and Lagrangian Distributions

Cotangent bundles provide one of the most important examples of symplectic man-ifolds. Can we characterize them tensorially in a similar way as we did for linearspaces? The answer to this question is easily obtained after the following consid-erations. Suppose that M = T∗Q is a cotangent bundle symplectic manifold withthe canonical symplectic structure ωQ. The cotangent bundle T∗Q is a vector bundleover Q, hence it is characterized tensorially be a partial linear structure, or a partialLiouville vector field �T∗Q. Let us recall that such a vector field has as zero set thezero section of T∗Q and the fibers are the unstablemanifolds of any zero point.More-over the fibers are Lagrangian submanifolds (actually Lagrangian linear subspaces)and L�T∗QωQ = ωQ. Hence di�TQ ωQ = ωQ and θQ = i�T∗QωQ is a symplecticpotential. Notice that in addition L�TQ θQ = θQ and θQ |T∗q Q= 0, that is, the verticalLagrangian distribution of T∗Q is in ker θQ.

The vertical Lagrangian distribution of the cotangent bundle T∗Q is unperturbedif we add to the symplectic form ωQ terms of the form π∗QF with F a closed form onQ. That is, define the ‘charged’ symplectic form ωF in T∗Q to be:

ωF = ωQ + π∗QF ,

or, in local bundle coordinates (qi, pk), we have ωF = dqi ∧dpi+Fij dqi ∧dqj. Thisis the situation when we describe a charged particle in a magnetic field A. Then theHamiltonian system corresponds to consider the symplectic form ωF with F = dA.See more about this in the examples below.

Suppose that we have a symplectic manifold (M, ω). In general our manifold Mwill not carry a linear structure. However it can carry a partial linear structure, i.e.,a partial Liouville vector field � (recall Sect. 3.3.2). Then we have:

Theorem 5.17 (Characterization of cotangent bundles) Let (M, ω) be a symplecticmanifold, then (M, ω) is symplectomorphic to a natural alternative cotangent bundle

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T∗Q iff there exists a partial Liouville vector field � such that the zero manifold of� is diffeomorphic to Q (hence dim Q = 1

2 dim M), and L�ω = ω.

Proof It remains to prove the converse. Thus if M carries a partial Liouville vectorfield �, then it has the structure of a vector bundle π : M → Q, with Q the zero setof �. Moreover, the fibres of the map π carry the structure of a vector space whoselinear functions are just the solutions to the equation �f = f . The rank of the bundleπ : M → Q, i.e., the dimension of the standard fibre, is n, because dim Q = n anddim M = 2n. Finally, from the homogeneity condition for ω, L�ω = ω, we get thatω has a symplectic potential θ = i�ω.

Then, locally, if we choose a system of bundle coordinates, (qi, wi) for the bundleπ : M → Q, it is easy to show that ω has to have the local form:

ω = wiFi

jkdqj ∧ dqk +�kj dqj ∧ dwk , (5.63)

because of the homogeneity condition. Actually if we compute L�

(ω(∂/∂wj,

∂/∂wk)) we get that it must be, if different from zero, a function of degree −1on the coordinates wk , but it must be a smooth function at 0 (the zero point of �)which is impossible. Moreover, repeating the argument with

(ω(∂/∂qj, ∂/∂wk

)), we

get:

L�

(ω(∂/∂qj, ∂/∂wk

)) = 0

then, it is a function �kj depending just on the base coordinates qi. Finally, the same

computation for(ω(∂/∂qj, ∂/∂qk

)), leads to:

L�

((ω(∂/∂qj, ∂/∂qk

)) = ω(∂/∂qj, ∂/∂qk) ,

that shows that it must be a homogeneous function of degree 1, i.e., linear on wk’s.Moreover because of Eq. 5.63 the form ω will be non-degenerate iff the matrix �k

jis invertible.

Now, we have that dω = 0, thus computing dω with the expression given in Eq.(5.63), we get immediately that Fi

jk = ∂�kj /∂ql.

Then, defining for each α ∈ �1(Q), the vector fields αV as

iαV ω = π∗α ,

we can check immediately (because � is non-degenerate) that αV must be vertical,i.e., π∗αV = 0. Then the vertical distribution of the bundle M → Q is Lagrangian,i.e., the fibres of π are Lagrangian submanifolds of M. Notice that the functionω(αV , βV ) is homogeneous of degree -1 with respect to �, hence 0, and that thevertical vectors αV generate the tangent space to the fibres.

Then we define a map � : T∗Q → M given by �(q, αq) = αVq (0) where we

identify the vertical subspace of the vector bundle at 0q with the fibre π−1(q). This


map clearly is an isomorphism of vector bundles. However, this map is not a sym-plectomorphism with respect to the canonical symplectic structure on T∗Q. Noticethat the symplectic potential θ = i�ω is just θ� , i.e., the pull-back under � of thecanonical 1-form θQ under the bundle morphism � : T∗Q→ T∗Q. Thus (M, ω) issymplectomorphic to (T∗Q, ω�).

Moreover, if θ − �∗θQ is closed, this means that θ� − θQ is closed, thenω� = ωQ. ��

Thus we want to point out that the existence of a cotangent bundle structure for agiven Hamiltonian dynamics XH on a symplectic manifold (M, ω) is determined justby the existence of a partial Liouville vector field � satisfying the properties of theprevious theorem. We will complete this discussion incorporating in the argumentthe tangent bundle formulation of dynamical systems in Sect. 5.6.

Notice that we are not imposing that the dynamics have to leave invariant such apartial Liouville vector field. If this were the case, this would mean that XH must belinear in the momenta, or in other words, that H must be quadratic in the momenta.This is exactly the situation we are going to discuss now.

5.3.3 Hamiltonian Systems of Mechanical Type

Aswewere commenting, for a vector field� to possess an invariant cotangent bundlestructure, it is necessary and sufficient, first that it has a Hamiltonian description, thatis, it admits an invariant symplectic structure and in addition it must commute witha partial Liouville vector field � as described in Theorem5.17. If this is the case,the Hamiltonian must be a quadratic function on the momenta, thus homogeneous ofdegree 2 with respect to �. Hence the most general expression for such HamiltonianH : T∗Q→ R, is:

H(q, p) = 1

2gijpipj + V(q) .

Or using some pedantic notation, H(q, p) = 12g(p, p)+π∗QV , with g a contravariant

symmetric (2, 0) tensor on Q and V a function on Q. Such Hamiltonian systems willbe said to be of mechanical type. The term K(q, p) = 1

2g(p, p) is called the kineticenergy of the system. Notice that the kinetic energy defines a quadratic form alongthe fibres of T∗Q (that could be degenerate). The term V(q) is called the potentialenergy of the system and even if we now assume it not to be regular, this is not thecase in general (recall the Kepler problem Sect. 7.4.3).

Wewill not enter here a discussion of the specific properties of systems ofmechan-ical type, as this subject has been masterfully treated in many classical textbooks.

There are some relevant considerations regarding the integrability and separabilityproperties of such systems and its relation with the existence of alternative cotangentbundle structures for it. See a discussion of this phenomena at the end of Chap. 8.

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Before entering a discussion of the existence and properties of symmetries forHamiltonian systems in general, let us conclude this section by briefly commentingon the simplest of all Hamiltonian system: free motion on Riemannian manifolds.

5.3.3.1 Geodesic Motion and Sprays

As it was said before, free motion on a Riemannian manifold is the simplest classof homogeneous Hamiltonian systems. The dynamical system � is described on acotangent bundle T∗Q, where Q is a Riemannian manifold with metric η, or in localcoordinates η = ηijdxi ⊗ dxj. The Hamiltonian of the system is the ‘kinetic energy’Kη defined above, i.e.

Kη(q, p) = 1

2ηijpipj ,

where ηij ηjk = δik .

It is now clear that the Hamiltonian vector �η with Hamiltonian Kη is homoge-neous of degree 1 and in coordinates the Hamilton equation take the form:

qi = ∂Kη

∂pi= ηijpj, pi = −∂Kη

∂qi = −1

2

∂ηjk

∂qi pjpk .

The vector field �η is called the geodesic spray of the metric η. Because of itshomogeneity property, the geodesic spray satisfies:

�η(q, λp) = λ�η(q, p) .

The projection γ (t) to Q of integral curves of �η are called geodesics of the metricη. They represent the projection on the ‘space of configurations’ Q of the actualmotion of a system moving freely (no external forces acting upon it). Notice that ageodesic curve γ (t) is characterized by a point on Q, for instance q0 = γ (t0), andp0 ∈ T∗q0Q, but because of the homogeneity condition, if we substitute p0 �→ λp0,then the projection of the corresponding integral curve of �η is γ (λt), the originalgeodesic parametrized now by λt instead of t.

Geodesics play an important role both in Geometry and Physics. It is worth topoint out here that the space of geodesics carries a canonical symplectic structure.The description of such structure is left to this chapter where some related notionswill be discussed.

5.4 Symmetries and Constants of Motion for Hamiltonian Systems 305

5.4 Symmetries and Constants of Motion for HamiltonianSystems

5.4.1 Symmetries and Constants of Motion: The Linear Case

As we said earlier, a generic linear dynamical system has a commuting algebra ofsymmetries but no constants of the motion. On the other hand, if our vector fieldpreserves a symplectic structure we have seen that some of the previous symmetries(those associated with odd powers of the representing matrix A) are associated withconstants of the motion. It is reasonable therefore, once a symplectic structure hasbeen chosen, to consider those linear vector fields which preserve the given structure,i.e., LXAω = 0. Because of this relation, it is easy to see now that ω(XAp , XAq) = fpqis a constant of motion for any XA2k+1 . It should be expected that some of these fmn

are actually vanishing, indeed ω(XA2k+1 , XA2r+1) = 0. We can examine closer thisphenomenon.

FromLXA2k+1ω = 0,

we getd(iXA2k+1ω) = 0,

i.e., if we set α2k+1 = iXA2k+1ω, we have a closed 1-form. Because Rn is contractible

there will be a function fk such that dfk = α2k+1, thus

iXA2k+1ω = dfk .

Now we take the Lie derivative of this relation with respect to XA2r+1 and find thatd(LXA2r+1 fk) = 0, i.e.,

LXA2r+1 fk = ck ∈ R.

As fk is quadratic and XA2r+1 is linear, the constant ck must vanish. Thus wehave shown that for vector fields XA which preserve ω we can find a constantof motion directly. We have also shown that half of the symmetries of XA (thoseassociated with odd powers) are associated with constants of motion and moreoverω(XA2k+1 , XA2r+1) = 0.

If we consider submanifolds obtained by considering the exponentiated actionof XA, XA3 ,... etc., we get in general a cylinder of dimension s, Tr × R

s−r wherer ≤ s ≤ n. If the minimal polynomial of A is of maximum degree, we have s = n.If all eigenvalues of A are purely imaginary our cylinder becomes a torus. As ourdynamical vector field is a member of this action, we find that each trajectory withinitial conditions on a cylinder will stay on the same cylinder.

As we have shown in the previous section, the constants of motion fk that weobtained before for a linear Hamiltonian vector field commute among themselves,this is:


{fk, fl} = 0, ∀k, l,

because {fk, fl} = Xl(fk) = 0. However if we consider a Lie algebra g of infinitesimallinear symmetries of a given dynamics, then we will have that if they preserve thesymplectic structure, that is:

LXC ω = 0,

then arguing as before we get that:

iXC ω = dfC,

where fC is a quadratic function. Notice that now, what we can show is that:

{fCk , fCl } = f[Ck , Cl] + σ(k, l),

where the Ck’s generate the Lie algebra g and σ(k, l) is a constant. Of course theJacobi identities imply a 2-cocycle condition for σ .

5.4.2 Symplectic Realizations of Poisson Structures

Before continuing the discussion on symmetries it is worth addressing a different,though intimate related, problem, that will be of interest for us in the context ofthe general integration problem we have already addressed for Lie algebras. Oncewe have realized the relevance of the existence of Poisson structures for dynamicsand the fact that symplectic manifolds provide a simpler description because ofthe existence of normal forms as discussed above, we may ask about the existenceof ‘symplectic realizations’ for Poisson dynamics, that is, given a dynamics � on amanifold P possessing an invariant Poisson structure described by the Poisson tensor�, does there exist a symplectic manifold (M, ω) and a submersion π : M → P thatis a Poisson map? If there exists a Hamiltonian vector field XH on M such that �

and XH are π -related, Tπ◦XH = � ◦ π , we will say that XH provides a symplecticrealization of the dynamics � and that the projection map π : M → P is a symplecticrealization of the Poisson structure (P,�). Notice that in such case integral curves ofXH project down onto integral curves of�, and then a way to describe integral curvesof our dynamics will be achieved by selecting for each initial condition x ∈ P a pointm ∈ M such that π(m) = x, then integrate the dynamics XH finding the integralcurve γ (t) passing through m and then π(γ (t)) is the desired integral curve of XH .Thus we may summarize the previous observations in the following definitions.

Definition 5.18 Let (P,�) be a Poisson manifold. A symplectic manifold (M, ω)

and a submersion π : M → P such that

{f ◦ π, g ◦ π}ω = {f , g}� ◦ π, f , g ∈ C∞(P), (5.64)


is called a symplectic realization of (P,�).If � is a Hamiltonian vector field on P, i.e.,−�(df ) = �, then XF with F = f ◦π

is a symplectic realization of the dynamics �.

Given a symplectic realization (M, ω, π) of a Poisson manifold (P,�), there isa natural map sending Hamiltonian vector fields �f = −�(df ) on P, into Hamil-tonian vector fields Xf = ω−1dπ ◦ f . Let us denote this map by s : Ham(P,�) →Ham(M, ω). Then it is clear that because of Eq. (5.64),

s([�f ,�g]) = [s(�f ), s(�g)], ∀f , g ∈ C∞(P),

or, in other words, the symplectic lifting map s is a Lie algebra homomorphism.We may ask ourselves if the map s will lift to a homomorphism of the groups

underlying the Lie algebras of Hamiltonian vector fields on P and M respectively,Ham(P,�) and Ham(M, ω), as it happens in the case of finite-dimensional Lie alge-bras and Lie groups that we discussed in Sect. 2.6.2. The answer is that in general thisis not possible because of the existence of topological obstructions due to the struc-ture of the group of symplectomorphisms, however we can solve this question rathereasily if we restrict our map s to a finite-dimensional subalgebra g of Ham(P,�).In such a case, h := s(g) ⊂ Ham(M, ω) will be again another finite-dimensionalsubalgebra of the Lie algebra of Hamiltonian vector fields on M, hence because ofLie’s theorem, Theorem2.26, there will exist a group homomorphism S : G → Hwhere G and H are the unique connected and simply connected Lie groups whoseLie algebras are g and h respectively. The groups G and H are coverings of groupsof diffeomorphisms on P and M respectively and they can be interpreted as groupactions on the manifolds P and M. We will not pursue these questions here but weconcentrate on the map sg : g→ Ham(M, ω) obtained by restricting the map s to anon-trivial finite-dimensional Lie algebra of Hamiltonian vector fields (wherever itexists).

Notice that if such a finite-dimensional subalgebra g exists, then we can definethe dual map of s:

J : M → g∗, 〈J(m), ξ 〉 = s(ξ)(m), ∀m ∈ M, ξ ∈ g. (5.65)

This map J will be called a momentum map and is called on to play a central rolein the study of symplectic manifolds, hence, dynamical systems, with symmetry.Before, continuing the discussion of it, let us introduce a natural equivalence notion.

Given two symplectic realizations (Ma, ωa, πa), a = 1, 2 for (P,�) a morphismof symplectic realizations is a smooth map φ : M1 → M2 such that π1 ◦ φ = π2and φ is symplectic. Two symplectic realizations (Ma, ωa, πa), a = 1, 2, will beequivalent if there is a morphism φ between them which is a symplectomorphisms.Notice that because the map φ defining a morphism between symplectic realizationsis symplectic, then it must be locally injective, thus it makes sense to talk about aminimal symplectic realization (if it exists) for a given Poisson structure.

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If the Poisson structure is constant, i.e., homogeneous of degree −2, the task offinding a symplectic realization is rather simple.

Exercise 5.2 Describe the natural symplectic realizations of constant Poisson struc-tures.

More interesting is the case of homogeneity of degree −1, or if you wish, Lie-Poisson structures. Let us recall that a Lie–Poisson structure is the canonical Poissonstructure on the dual g∗ of the Lie algebra g of a Lie group G. The answer to this isvery illustrative and we will devote the following section to its study.

5.4.3 Dual Pairs and the Cotangent Group

Consider as indicated before a Lie group G with Lie algebra g, which is just thetangent space at the identity of G, g = TeG. Then we may consider the cotan-gent bundle T∗G equipped with its canonical symplectic structure ωG and Liouville1-form θG. Notice that the group G acts on G by left or right translations. Let usconsider left translations in what follows. Thus to any element g ∈ G we associatethe diffeomorphism Lg : G → G defined by Lg(h) = gh, h ∈ G. This action liftsto T∗G by �g : T∗G → T∗G, with �g(h, αh) = (gh, (TL∗g )−1(αh)). Notice thatTL∗g : T∗ghG → T∗h G by TL∗gαgh(vh) = αgh(Lgvh), with vh ∈ ThG, αh ∈ T∗h G andαgh ∈ T∗ghG, and the definition of�g acting on covectors by using the inverse of TL∗gis required to get an action, that is, a representation of G on the group of diffeomor-phisms of T∗G, or in other words to satisfy �g ◦ �g′ = �gg′ and not the order offactors exchanged.

On the other hand any Lie group is a parallelizable manifold, that is TG ∼= G× g,hence T∗G ∼= G× g∗. There are two natural ways of achieving such trivializationsof the (co-) tangent bundle: using left or right translations. We will use a similarnotation as before to indicate the trivialization of TG, that is:

L : TG→ G× g; L(vg) = (g, TLg−1(vg)), vg ∈ TgG ,

and similarly with using right translations. The induced trivializations for T∗G aregiven by:

R∗ : T∗G→ G× g; R∗(αg) = (g, TR∗g(αg)), αg ∈ T∗g G .

It is worth noticing that TR∗g maps T∗g G into T∗e G = g∗. Now, we may compose R∗with the projection into the second factor to get the map:

J : T∗G→ g∗; J(αg) = TR∗g(αg), ∀αG ∈ T∗g G .

Proposition 5.19 The map J is a momentum map and J : T∗G→ g∗ is a symplecticrealization of the Lie–Poisson structure on g∗.


Proof To check that J is a momentum map, we have to prove that the map s : g→Ham(T∗G), s(ξ) = Xξ , is a Lie algebra homomorphism. The vector field Xξ is theHamiltonian vector field whose Hamiltonian is Jξ = 〈J, ξ〉, for any ξ ∈ g. Noticethat this is equivalent to the map J being a Poisson morphism, Eq. (5.64):

{F ◦ J, H ◦ J}T∗G = {F, H}g∗ ◦ J, F, H ∈ F(g∗) . (5.66)

Notice that to prove the previous equation all is needed is to prove it for linearfunctions, that is, if ξ ∈ g, we define the linear function on g∗ by Fξ (μ) = 〈μ, ξ 〉,with any μ ∈ g∗.

Then, by definition (see Eq.4.28) we have that

{Fξ , Fζ }g∗ = F[ξ,ζ ]. (5.67)

On the other hand we see that if ξG denotes the vector field defined on G by theleft action of G on itself or, in other words, the right-invariant vector field defined byξ ∈ g.3

Because of Eq. (5.60) we know that the complete lifting Xc to a the cotangentbundle T∗Q of a vector field X on Q is Hamiltonian with Hamiltonian function PX .Hence the complete lifting ξ c

G of the vector field ξG to T∗G is Hamiltonian withHamiltonian PξQ(αg) = 〈αg, ξG(g)〉, but because ξG is right-invariant, we have thatξG(g) = TRgξ , then 〈αg, ξG(g)〉 = 〈TR∗gαg, ξ〉 = 〈J(αg), ξ 〉 = Jξ (αg), then:

iξ cGωG = dJξ .

Finally, notice that Fξ ◦ J(αg) = 〈J(αg), ξ 〉 = Jξ (α), thus because of Eq. (5.67),checking the result we are looking after, Eq. (5.66), amounts to showing that:

{Jξ , Jζ }T∗G = J[ξ,ζ ] .

However again because of Eq. (5.59) we get [Xc, Yc] = [X, Y ]c, hence [ξ cG, ζ c

G] =[ξG, ζG]c = [ξ, ζ ]cG. Then:

i[ξ cG,ζ c

G]ωG = i[ξ,ζ ]cGωG = dJ[ξ,ζ ] ,

and on the other hand:

i[ξ cG,ζ c

G]ωG = Lξ cG

iζ cGωG = dξ c

G(Jζ cG) = d{Jξ , Jζ }T∗G ,

because Lξ cGωG = 0 (recall that ξ c

G is Hamiltonian) and {Jξ , Jζ }T∗G = ωG(ξ cG, ζ c

G).Hence we get that c(ξ, ζ ) = {Jξ , Jζ }T∗G − J[ξ,ζ ] is a 2-cocycle in g with valuesin R, but remember that this cocycle is obtained from the variation of the 1-form

3 Notice that switch from ‘left action’ to ‘right invariance’ in the previous statement. It happensbecause of TRgξG(h) = d

ds Rg((exp sξ)h) |s=0= dds exp sξ(hg) = ξG(hg).

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θG under the action of G. But remark that θG is invariant because this action is acomplete lifting, and therefore c = 0. Alternatively, the direct computation offeredin Eq. (5.61) gives the desired result. ��Remark 5.1 In the proof of Proposition5.19 we have never used the explicit form ofωG because it actually works the same for complete actions of groups on cotangentbundles. It is illustrative also as an exercise in the geometry of groups, to workthe proof by developing an explicit formula for ωG and the corresponding Poissonbrackets (see for instance [Ab78], Proposition4.4.1. and [Cu77]).

Exercise 5.3 Work out a group-theoretical expression for ωG, i.e., in terms of left(or right) invariant vectors and forms. In other words, write down the formula for ωG

using the global trivialization T∗G ∼= G× g∗.

Hence we conclude these comments by observing that the cotangent bundle ofthe group G provides a symplectic realization of the Lie–Poisson structure. Thisrealization has another relevant aspect: the symplectic manifold is a group itself, buteven more important there is a canonical groupoid structure on T∗G.

A few more remarks are in order concerning dynamical systems and this exampleof a symplectic realization. The analysis of dynamics on groups or cotangent bundlesof groups is extremely rich. The main example is the dynamics of rigid bodies infinite dimensions and of incompressible fluids in infinite dimensions. A lot of efforthas been poured into clarifying the geometry of such systems. We can only cite herethe works by V. Arnold, J.E. Marsden, etc. From our point of view, such dynamicalsystems are systems belonging to a (finite-) dimensional Lie algebra of vector fields.Then integrating the Lie algebra we obtain a group. We will devote a full chapter toexploring the dynamics of systems defined on groups, but instead of following thepath of reduction, we will approach them from the point of view of their integrabilityproperties, that in this case ‘integrability’ means the possibility of obtaining theirsolutions by using (nonlinear in general) superposition rules. Such systems will becalled Lie-Scheffer systems (see Chap.9).

5.4.3.1 Momentum Maps and Group Actions

As we indicated before, Proposition5.19 is not specific of the cotangent bundle of agroup. Actually let G be a Lie group acting on a smooth manifoldQ, and consider thelifted action of G, or cotangent lifting of the action of G, to T∗Q. Then if ξQ denotesthe fundamental vector field associated to ξ ∈ g in Q, then the fundamental vectorfield associated to ξ in T∗Q is ξ c

Q. The action of G on T∗Q preserves the symplecticstructure because (Eq. (5.58)):

LξcQθQ = 0 .

Hence because ξ cQ leaves θQ invariant it will leave ωQ invariant too. Moreover,

we have that the Hamiltonian of ξ cQ will be Pξ (q, p) = 〈p, ξQ〉. But this is nothing

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but the momentum map corresponding to the Lie algebra homomorphism s : g →Ham(T∗Q), s(ξ) = ξ c

Q = XJξ with

Jξ (q, p) = 〈J(q, p), ξ 〉 = 〈p, ξQ(q)〉 = Pξ (q, p) . (5.68)

The same formula shows that

{Jξ , Jζ }T∗Q = J[ξ,ζ ], ∀ξ, ζ ∈ g ,

and the momentum map:J : T∗Q→ g∗

provides another symplectic realization of g∗.In this case, what we discover is that if we have a dynamics � such that it admits a

cotangent bundle structure, and in particular an invariant cotangent bundle structure,hence it will be Hamiltonian with a quadratic Hamiltonian function in the momenta,then the symmetry group G of � consisting of transformations that commute withthe partial Liouville vector field �T∗Q will define a momentum J : T∗Q → g∗ andthe vector field � will be tangent to the level sets of this momentum map.

5.4.4 An Illustrative Example: The Harmonic Oscillator

Wewill start, as inmany instances of this book, by discussing an illustrative example.In Sect. 1.2.7 we introduced complex coordinates to describe the harmonic oscillatorand the canonical action of the U(n) on it (in the isotropic case, other subgroups inthe anisotropic situation). There was also a call for a further analysis of the topologyof its space of orbits. We will proceed now to perform such analysis in the lightprovided by the previous discussions in this chapter and, at the same time, paving theway for further discussions in Chap.6 when the harmonic oscillator will be discussedin the context of Hermitean systems and the geometry of quantum systems.

An important notion in discussing the Hamiltonian structure of the harmonicoscillator, actually for any dynamical system, is the bifurcation set of the system,that is the set where the topology of the level sets of the Hamiltonian function of thesystem changes. The bifurcation set of the system encodes most of the propertiesof the reduced system, i.e., the system that we will get once we have ‘removed’the symmetries of the system or, in a slightly ‘old fashioned’ language, the cyclicvariables [Sm70].

We will start with the isotropic one-dimensional harmonic oscillator whose con-figuration space is R and its phase space is T∗R ∼= R

2 whose points will be denotedas usual by (q, p). The Hamiltonian function is just H = 1

2 (p2 + q2). In complexcoordinates z = q + ip and z∗ = q − ip we have H = 1

2 z∗z and the equations ofmotion turn out to be z = −iz and the phase space is just C. The invariance group is

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now U(1) which is topologically the unit circle S1. The orbits of the action of U(1)in C by complex multiplication are just the orbits of the dynamics of the harmonicoscillator and then the space of orbits, that is the quotient spaceC/U(1) is diffeomor-phic to the half-line R

+, the point 0 corresponding to the critical point (0, 0) (whichhappens to be the bifurcation set of the system), and any other point r = √

q2 + p2

corresponds to the circle of radius r in R2.

5.4.5 The 2-Dimensional Harmonic Oscillator

Much more interesting is the isotropic harmonic oscillator in dimension 2. Now theconfiguration space is R

2 with points (q1, q2) and the phase space is 4-dimensionalT∗R2 ∼= R

4 with points (q1, 12, p1, p2). The Hamiltonian of the isotropic oscillatoris H = 1

2 (p21 + p22 + q21 + q22) which is the sum H1 + H2, with Ha = 1

2 (p2a + q2a),

a = 1, 2, the Hamiltonian of a 1-dimensional oscillator. Notice that Ha are constantsof motion for the dynamics defined by H.

If we denote by T(r, ρ) = S1r × S1

ρ a torus corresponding to the points of T∗R2

such that H1 = h1 = 12 r21 and H2 = h2 = 1

2ρ2, then we have that the level setH = h = 1

2R2 = constant, is the union of all such sets when we vary r, ρ such that0 ≤ r2 + ρ2 ≤ R2. Notice that the level set H = 1

2R2 is the 3-dimensional sphereS3

R of radius R in R4, thus the union of all tori T(r, ρ), 0 ≤ r2 + ρ2 ≤ R2 foliates

the 3-dimensional sphere of radius R (the foliation is singular because there are twoleaves, T(r, 0) and T(0, ρ) with different topologies). Notice that T(0, ρ) = S1

ρ is acircle as well as T(r, 0) = S1

r . The family of tori T(r, ρ) filling the sphere is depictedin Fig. 5.1.

The equations of motion in polar coordinates (ϕ, r) and (θ, ρ) are just:

r = 0, ϕ = −1, ρ = 0, θ = −1 ,

which are integrated trivially. The previous observations allows us to describe thelevel surface H = h as a double cone (see Fig. 5.2) obtained by mapping S3

R = {H =h} into R

3 by (q1, q2, p1, p2) �→ (x, y, z) with x = q1, y = p1 and z = θρ. Noticethat if we consider the circle 2r2 = q21+p21, then ρ2 = R2− r2 and the torus T(r, ρ)

corresponds to the cylinder over the circle of radius√2r2 in the xy plane of height

z varying form −πρ to πρ. Notice that the upper rim (z = πρ) and the lower rimz = −πρ of the cylinder T(r, ρ) are identified.

Thedegenerate toriT(r, 0) corresponds to the horizontal circle of radius sqrt2r2 =√2, and the degenerate tori T(0, ρ) correspond to the vertical segment along the z

axis−πR ≤ z ≤ πR. Trajectories of the harmonic oscillator correspond to helicoidsalong the cylinders with pass ρ. Each trajectory cuts the equatorial disk z = 0 atexactly one point except the unit circle which is a trajectory by itself. Thus the spaceof trajectories can be described as a disk D2 with its boundary identified to a point.This is the topological way of describing the sphere S2. Thus we have obtained a


Fig. 5.1 The family of tori foliating the level surfaces of a 2D harmonic oscillator

Fig. 5.2 The space of orbitsof the 2D harmonic oscillatoris a sphere S2

map πH : S3 → S2 from the sphere S3 describing an energy level set for the isotropicharmonic oscillator, and the sphere S2 describing the space of trajectories of the givenoscillator with fixed energy. This map is called the Hopf fibration and will appear indifferent occasions throughout the book. We will proceed in the next paragraphs toprovide an analytical, rather than topological, description of it.

To finish this part we may also describe the trajectories and the level sets of the2-dimensional isotropic oscillator, but this time using the stereographic projectionof the 3-dimensional sphere. The (North pole) stereographic map S : S3 → R

3 isdefined as (see Fig. 5.3): S(u1, v1, u2, v2) = (ζ1, ζ2, ζ3) = ζ , (u1, v1, u2, v2) ∈ R

4

such that u21 + v21 + u22 + v22 = 1 and ζ = (ζ1, ζ2, ζ3) ∈ R3, with:


Fig. 5.3 The stereographic projection from S3 into R3

ζ1 = u11− v2

, ζ1 = v1

1− v2, ζ3 = u2

1− v2,

or, conversely,

u1 = 2ζ11+ ||ζ ||2 , v1 = 2ζ2

1+ ||ζ ||2 , u2 = 2ζ31+ ||ζ ||2 , u1 = −1+ ||ζ ||

2

1+ ||ζ ||2 .

The South pole of S3 is mapped into the origin (0, 0, 0) and the North pole is mappedinto∞ that should be added to R

3 to complete the picture. Notice that meridians aremapped into straight lines in R

3 starting at the origin and all of them end up at∞.Under this projection the tori T(r, ρ) are mapped into a family of tori foliating R

3

where the tori T(r, 0) is mapped into a circle of radius√2R situated in the xy-plane

and the tori T(0, ρ) becomes the z-axis. The trajectories wind up around the tori andare depicted in Fig. 5.4 for various initial data.

5.4.5.1 The Complex Structure of the 2-Dimensional Harmonic Oscillator

As we did in Sect. 1.2.7, we may introduce complex coordinates to analyze theharmonic oscillator. In doing so we will gain a different perspective that will bevery insightful when dealing with Hermitean systems. Thus we will define complexvariables z1 = q1 + ip1 and z2 = q2 + ip2. The inner product in C

2 is defined as〈z, w〉 = z1w1 + z2w2, with z = (z1, z2) and w = (w1, w2) vectors in C

2. Then theHamiltonian function of the 2-dimensional oscillator is written as: H = 1

2 〈z, z〉 =12 ||z||2.

The complex coordinate vector fields are given by:

∂

∂z= 1

2

(∂

∂q− i

∂

∂p

),

∂

∂ z= 1

2

(∂

∂q+ i

∂

∂p

),

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Fig. 5.4 The foliation ofR3 by tori T(r, φ) under the stereographic projection and some trajectoriesof the 2D harmonic oscillator

and, conversely:∂

∂q= ∂

∂z+ ∂

∂ z,

∂

∂p= i

(∂

∂z− ∂

∂ z

).

It is convenient towrite ∂z instead of ∂/∂z and ∂z for ∂/∂ z (or even ∂ and ∂ respectivelyif there is no risk of confusion).

The canonical symplectic structureω0 = dq1∧dp1+dq2∧dp2 on T∗R2 becomes:

ω0 = i

2(dz1 ∧ dz1 + dz2 ∧ dz2) ,

hence the Poisson bracket of two functions f (z, z), g(z, z) becomes:

{f , g} = 2i∑

a=1,2

∂f

∂za

∂g

∂ za− ∂f

∂ za

∂g

∂za,

and the corresponding Hamiltonian vector fields Xf , become:

Xf = 2i∑

a=1,2

∂f

∂za

∂

∂ za− ∂f

∂ za

∂

∂za.


Thus, for instance:

XH = iz1∂

∂z1− iz1

∂

∂ z1+ iz2

∂

∂z2− iz2

∂

∂ z2.

The group U(2) acts on C2 by matrix multiplication. This action of the group

U(2) is called the spinorial representation of U(2) or the spin 1/2 representation ofU(2). With the standard presentation of U(2) (see Sect. 2.5.4) the Lie algebra u(2)∗of U(2) consists of skew-Hermitean matrices iA with A = A†. We may write anyHermitean matrix A as:

A = 1

2(x0σ0 + x1σ1 + x2σ2 + x3σ3) = 1

2

(x0 + x3 x1 + ix2x1 − ix2 x0 − x3

), (5.69)

where σμ, μ = 0, 1, 2 and 3 are Pauli’s sigma matrices, Eq. (2.138).If we want to emphasize the relation between the matrix A and the vector x =

(x0, x1, x2, x3) ∈ R4 given by Eq. (5.69), we will write A = Ax and, vice versa,

x = xA.Again as we know form Sect. 1.2.7, Eq. (1.68), the real quadratic constants of

motion have the form fA(z, z) = 12 〈z, Az〉 with A Hermitean. Notice that the Hamil-

tonian of the system H is fσ0 . Denoting by Ji = fσi , i = 1, 2, 3, we get that anyconstant of motion fA can be written as:

fA = x0fσ0 + x1fσ1 + x2fσ2 + x2fσ3 = xμJμ ,

which we denote by Jμ = fσμ. If we collect the components J1, J2, J3 as a

3-dimensional vector J = (J1, J2, J3), then J defines a map:

J : C2 → R3, J(z) = 〈z, σ z〉 ,

where σ = (σ1, σ2, σ3) is the 3-dimensional vector whose components are Paulimatrices (Eq. 10.68).

We can think of the previous maps in a group-theoretical way as momentummaps. Recall that the group U(2) can be identified with U(1)× SU(2) mod Z2, viathe map (eiθ , U) �→ eiθU, U ∈ SU(2). Elements U of the special unitary groupSU(2) = {U ∈ M2(C | U†U = I2} are written as:

U =(

z1 −z2z2 z1

), with |z1|2 + |z2|2 = 1 . (5.70)

The Lie algebra of SU(2) is the Lie subalgebra of u∗2 consisting of traceless matrices,that is matrices of the form

A = 1

2(x1σ1 + x2σ2 + x3σ3) = 1

2x · σ .

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Fig. 5.5 Left The dual pair defined by the action ofU(1) and SU(2) onC2.Right TheHopf fibrationas a restriction of the momentum map J

Now the action of U2) on C2 restricts to actions of the subgroups U(1) and SU(2)

and both groups act symplectically. Not only that, they will preserve the canoni-cal symplectic structure ω0. They define momentum maps which are given by theHamiltonian itself and the map J.

Theorem 5.20 The spinorial action of U(2) and its subgroups U(1) and SU(2)described above are Hamiltonian with momentum maps:

J0 : C2 → u∗1 ∼= R, J0 = H

andJ : C2 → su∗2 ∼= R

3, J(z) = 〈z, σ z〉, z ∈ C2 .

Moreover, pair (J0, J) define a dual pair J = J∗0 and it is called the energy-momentummap of the system (see Fig.5.5).

Proof It is a direct computation. In the case of J0 = H, we already know that it isthe momentum map of the action of the center U(1) of U(2) on C

2. Its orbits beingthe trajectories of the harmonic oscillator.

The components of the momentum map J are given by:

J1 = 1

2〈z, σ1z〉 = z1z2 + z2z1 = q1q2 − p1p2 ,

J2 = 1

2〈z, σ2z〉 = −iz1z2 + iz2z1 = q1p2 − q2p1 ,

J3 = 1

2〈z, σ3z〉 = z1z1 − z2z2 = 1

2(q21 + p21 − q22 − p22) .

We compute the Hamiltonian vector fields corresponding to J1 to get:

XJ1(z) = {z, J1} = −iσ1z ,

which is just the linear vector field corresponding to the matrix iσ1, XJ1 = Xiσ1 ,and it is the fundamental vector field associated to the action of the generator iσ1 ofthe Lie algebra of SU(2). After similar computations we get the same results for J2and J3.


Finally notice that J0 is the commutant of J in the Poisson algebra of C2, thus the

maps J0 and J define a dual pair. ��

5.4.5.2 The Momentum Map and the Hopf Fibration

The momentum map J corresponding to the canonical action of SU(2) on C2 maps

C2 into R

3, but when restricted to the 3-dimensional sphere given by a level set ofthe Hamiltonian, we are going to show that its image is the 2-dimensional sphere S2

describing the space of trajectories of the 2-dimensional harmonic oscillator, i.e.,. isthe Hopf fibration (or just Hopf map) described in Sect. 5.4.5. To prove that we needa few results on products of σ matrices whose proofs are simple exercises.

Lemma 5.21 For any z, u, w ∈ C2 we get:

1. 〈u, σ z〉 · σw = 2〈u, w〉z − 〈u, z〉w,2. Re〈z, σu〉 · Re〈z, σw〉· = 〈z, z〉Re〈u, σw〉 − ImRe〈z, σu〉Re〈z, σw〉,3. ||J(z)|| = 1

2 〈z, z〉 = 12 ||z||2 = H(z, z),

4. J(eitz) = J(z).

The next theorem follows immediately.

Theorem 5.22 The restriction of the momentum map J to S3h = {z ∈ C2 | H =

12 ||z||2 = h} is the Hopf fibration πH mapping each point z into the trajectory of theharmonic oscillator passing through it. The preimage J−1(x) is the set {eitz0}, forany z0 such that J(z0) = x (see Fig.5.5).

Finally, we get:

Proposition 5.23 The Hopf map J is a Poisson map, i.e., {J1, J2} = J3 and cyclicpermutations.

There is an alternative, though equivalent, description of the Hopf fibration thatwill be of interest in relation with the geometry of Hermitian systems and Kählermanifolds which is based on the geometry of the group SU(2) itself. In this presen-tation we will obtain the formulas for the invariants defined by the Hopf map as aprincipal fibre bundle over S2 with fibre U(1). We will postpone this discussion untilSects. 6.4.2 and 6.4.4. However we may mention here that the basic idea consists inidentifying the group SU(2) itself with the sphere S3 ⊂ C

2 as we did before, thatis the unitary matrix U given by Eq. (5.70) is identified with the point z = (z1, z2),then we may define the map π : SU(2)→ S2 given by:

π(U) = Uσ3U† = x ,

where the vector x = (x1, x2, x3) is defined as:

x · σ = Uσ3U† .

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A direct computation shows:

x1 = 〈z, σ1z〉, x2 = 〈z, σ2z〉, x3 = 〈z, σ3z〉 ,

and ||x|| = 1, i.e., x ∈ S2 ⊂ R3, in accordance with our previous results.

5.4.5.3 The n-dimensional Harmonic Oscillator

We can consider now the n-dimensional isotropic harmonic oscillator under the lightof the previous discussions. Thus we will denote by (q, p) ∈ T∗Rn ∼= R

2n a point inphase space, The Hamiltonian of the system will be H(q, p) = 1

2

∑na=1(p2a + q2a).

We may as in the discussion of the 2-dimensional harmonic oscillator, introducecomplex coordinates: za = qa + ipa, a = 1, . . . , n, and then H(z, z) = 1

2 〈z, z〉, with〈z, w〉 = ∑n

a=1 zawa. The level set h of the Hamiltonian is the 2n − 1-dimensionalsphere S2n−1

h ⊂ Cn of radius R = √2h.

The equations of motion defined by H are given in complex coordinates by:z = −iz, ˙z = iz, and the dynamical vector field is: � = XH = −iz∂z + iz∂z.

The group U(1) acts on C2 by complex multiplication, i.e., (eit, z) �→ eitz and its

infinitesimal generator defines the fundamental vector field � = XH . Its orbits beingthe trajectories of the harmonic oscillator. The space of orbits of fixed energy h isthen the quotient space S2n−1

h /U(1). Such a space is the n− 1-dimensional complexprojective space CPn− 1.

The description of the complex projective space CPn− 1 is done by introducingcomplex homogeneous coordinates [z] = [z1, . . . , zn] corresponding to a family ofcomplex coordinate charts:

[z] �→

⎧⎪⎪⎨⎪⎪⎩

(z2z1

, . . . , znz1

)∈ C

n−1, if z1 �= 0 ,

· · ·(z1zn

, . . . ,zn−1

zn

)∈ C

n−1, if zn �= 0 .

The projection map πH : S2n−1 → CPn− 1, πH (z) = [z] will be called theHopf map, or Hopf fibration, in dimension n − 1. Notice that there is a canonicalidentification between CP1 and the sphere S2 that equips the sphere with a complexstructure (in such case we call S2 the Riemann sphere). The properties and geometryof this projection as well as of complex projective spaces will be discussed at lengthinChap.6, Sect. 6.4.3 because it provides the natural setting to describe pure quantumstates.

The symmetry group of the system� is the groupU(n)which is defined exactly asthe group of n×n complex matrices, such that 〈Uz, Uz〉 = 〈z, z〉 for all z ∈ C

n. Thenbecause of the factorizationU(n) = (U(1)×SU(n))/Zn, where the subgroupU(1) isthe center of the group, we may restrict the action of U(n) to the two subgroups U(1)and SU(n). The first has momentum map J0 = H and the second has a momentum

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map J : Cn → su(n)∗. The trace bilinear form (A, B) �→ −Tr A†B restricted to

the space of skew-Hermitean matrices allows us to identify su(n)∗ with n× n skew-Hermiteanmatrices iA. Thuswemay consider the basisN (ij) introduced in Sect. 1.2.7,Eq. (1.71), and the corresponding component Jij of the momentum map J given by:

Jij(z, z) = 1

2(zizj − zizj) .

As it was proved in the case of n = 2, we also have that J maps S2n−1h into the

complex projective space CPn−1 and the fibers of this map are the orbits of U(1).It is interesting to understand the natural embedding of CPn−1 inside su(n)∗ as acoadjoint orbit. Recall that the coadjoint orbits of a groupG are the symplectic leavesof the canonical linear Poisson structure on the dual g∗ of its Lie algebra (Sect. 4.3.3).

The complexprojective spaceCPn−1 is the coadjoint orbit ofSU(n) correspondingto a diagonal matrix in su(n)∗ with diagonal (λ, . . . , λ, μ), λ �= μ. The isotropygroup of the action (which is now matrix conjugation A �→ UAU† is the subgroupS(U(n− 1)× U(1)) and C

n−1 = SU(n)/S(U(n− 1)× U(1)).Finally we conclude this section by observing that the pair J0 = H and J define

a dual pair as in the 2-dimensional case. The map (H, J) is called the energy-momentum map and the study of its topological properties play an important role inthe analysis of mechanical systems with symmetry [Sm70]. We will come back tothis subject in the study of Kepler’s problem, Sect. 7.4.3.

5.5 Lagrangian Systems

In our quest for dynamically determined structures we arrive to the discussion of theproblem that in a sense gave birth to all the previous one. This problem is calledthe inverse problem of the calculus of variations and can be stated in our contextby asking when a given dynamical vector field � can be obtained as the Euler–Lagrange equations for some variational principle. In such generality the problem ishopeless and we have to qualify it by describing with precision which ones are thegeometrical structures that arise naturally in the calculus of variations. Once this isdone the inverse problem of calculus of variations for our vector field � amounts toa search for the appropriate geometrical structures.

It is difficult to emphasize the relevance and extraordinary importance that vari-ational principles have played, and still are playing, in the derivation of dynamicallaws for specific physical systems. In many occasions, and more often as we searchfor deeper structures, such variational principles become almost our only guide. Thisis particularly true of field theories such as Yang–Mills and other interaction theories.Thus, apart from historical reasons or the completeness of our exposition, we feelthat devoting the next few sections to discuss this problem could be of relevance forfuture research.

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5.5 Lagrangian Systems 321

5.5.1 Second-Order Vector Fields

The previous chapters dealt with (mostly linear) vector fields defining a dynamics ona Poisson manifold or, as a special case, on a symplectic manifold, actually a Poissonand/or a symplectic linear space in the linear case. In the whole of this section wewill specialize to a particular class of vector fields, namely second-order vector fields(also called, for reasons that will become clear shortly, Second-Order DifferentialEquations, (SODE’s for short) that we will try now to characterize.

Let us start then with a pair (V , �), where V is a vector space, with dim V = k,and � a vector field in V . We shall assume here that V is endowed with a partiallinear structure. Recalling what has been discussed in a previous section, we meanby this that there exists a vector field � ∈ X(V) (a (partial) dilation (or Liouville)vector field) such that the equations

L�g = 0, g ∈ F(V) , (5.71)

andL�f = f , f ∈ F(V) , (5.72)

have respectively, say, n and k−n functionally independent solutions g1, . . . , gn andf 1, . . . , f k−n, i.e., such that

dg1 ∧ · · · ∧ dgn ∧ df 1 ∧ · · · ∧ df k−n �= 0. (5.73)

Notice that the g’s span a linear space, denoted as F0 (dim(F0) = n), and are analgebra undermultiplication aswell, while the f ’s span also a linear space that wewilldenote asF1 (with dimF1 = k−n) and are also anF0-module (i.e., f ∈ F1, g ∈ F0will entail gf ∈ F1 by the Leibnitz rule), but obviously not an algebra.

As it has been discussed elsewhere, the dilation field� can be immediately expo-nentiated in F1 as

exp{t�}f = exp(t)f , ∀f ∈ F1 , (5.74)

whence the name of ‘dilation’ field. It can also be exponentiated in F0 as well, ofcourse, but there the action is trivial. In this way V acquires the structure of a vectorbundle with fibres of dimension k − n spanned by the f i’s. The fibres will be vectorspaces, and that is what is defined as a ‘partial linear structure’. Taking then oneset (f 1, . . . , f k−n) of independent solutions of Eq. (5.72) as basis on the fibres, theLiouville vector field can be written as

� = f i ∂

∂f i, (5.75)

and it is clear that the zeroes of f =: (f 1, . . . , f k−n) coincide with those of�. As thef ’s are functionally independent, the zero is a regular value of themap f : V → R

k−n,


and therefore f−1(0) will be a regular submanifold M of V , the ‘zero section’ of thevector bundle.

Then we can state the following:

Definition 5.24 A vector field � ∈ X(V) will be a second-order vector field withrespect to the partial linear structure defined by the Liouville field � if

L�F0 = F1 . (5.76)

Notice that this implies at once that dimF0 = dimF1 = n, and hence, k = 2n,i.e., V must be an even-dimensional space.

Let now m ∈ M and choose local coordinates g1, , . . . , gn, in a neighbourhoodU ⊂ M, when viewed as functions, g1, . . . , gn ∈ F0, of course (the choice will beglobal if M is also a vector space). Notice that whatever we say here will be only‘local’ (and at most) as far as the ‘base manifold’ M is concerned. Indeed, Eq. (5.74)tells us that we can generate the whole fibre over any point of M via exponentiation.Therefore, our statements here will be valid (at least), in a ‘tubular neighbourhood’in V of any neighbourhood U ⊂ M. By this we mean an open subset of TM thatincludes the whole of the fibres above a neighbourhood U in the base manifold.Then, it is not hard to see that the condition given by Eq. (5.76) implies for � thelocal expression

� = f i ∂

∂gi+ Fi(g, f )

∂

∂f i, (5.77)

with the Fi’s arbitrary (smooth) functions of the gi’s and f i’s. In other words, beingsecond-order will fix unambiguously (only) the ‘first components’ of a vector field.

The equations of motion associated with � are then of the form:

⎧⎪⎨⎪⎩

dgi

dt= f i ,

df i

dt= Fi(g, f ) ,

(5.78)

i.e., altogetherd2gi

dt2= Fi(g,

dg

dt) , (5.79)

i.e., they are second-order Newtonian equations, written in normal form, on the ‘basemanifold’ M. This justifies the name SODE for second-order fields.

Proceeding further, we can define the (1, 1)-type tensor field

S = ∂

∂f i⊗ dgi (5.80)

It is not hard to prove that S and � enjoy the following properties:


(i) � ∈ ker(S),(ii) ker(S) = Im(S) (then S2 = 0),(iii) L�S = −S,(iv) NS = 0 .

where NS denotes the Nijenhuis tensor associated with S (see Appendix E).It was proved in [DeF89] that S and � define a ‘unique’ tangent bundle structure

on V . Therefore, V = TM.

Remark 5.2

(i) It is clear that different choices of the g’s and/or f ’s will (or can) lead in princi-ple to different tangent bundle structures. Under this respect, it is perhaps moredesirable (and more clarifying) to view the pair (V , �) (with no further qualifi-cations) as the ‘standard’ reference structure, and to view any choice of the g’sand f ’s as providing us with a (at least local) diffeomorphism between V and atangent bundle (V ≈ TM then), and to say that � is ‘related’ to a second-ordervector field on TM.

(ii) We can also use a slightly different construction, one that has the advantage ofemphasizing the rôle that the dynamical vector field � plays in the constructionof the tangent bundle structure(s).

Suppose we select, if dim V = 2n, n functionally independent functions g1, . . . ,gn, i.e., such that dg1 ∧ · · · ∧ dgn �= 0, and assume (this is the really crucial assump-tion) that the n functions

f i =: L�gi

are also functionally independent among themselves and also with the g’s, i.e.

dg1 ∧ · · · ∧ dgn ∧ df 1 ∧ · · · ∧ df n �= 0 .

Then the g’s and f ’s can be taken (at least locally) as a new set of coordinates forV . In this new coordinate system the dynamics will be given by

� = f i ∂

∂gi+ Fi ∂

∂f i

where, by construction, Fi = L� f i = L�L�gi. The g’s and f ’s define then (again,

at least locally) a tangent bundle structure, with the g’s providing coordinates for thebase manifold and the f ’s for the fibres, with respect to which � is a second-ordervector field. The Liouville dilation along the fibres field will be given now by

� = f i ∂

∂f i.

Example 5.4 Let us take V = R2 with Cartesian coordinates (x1, x2) and consider

the vector field


� = x2∂

∂x1− x1

∂

∂x2. (5.81)

It is obvious that, with no need of introducing any additional structures in V , �is, in appropriate units, just the standard 1D-harmonic oscillator.

Using the notationq =: x1 and L�q = x2 =: v , (5.82)

� acquires the form

� = v∂

∂q− q

∂

∂v, (5.83)

and V becomes the tangent bundle V = TR, with the base space being R withcoordinate q.

We might as well, however, have taken, e.g.

q = q(x1, x2) = αx1 − βx2 , α2 + β2 = 1 , (5.84)

as the new ‘coordinate’, together with

L� q =: v = αx2 + βx1 (5.85)

as the new ‘velocity’. Notice that dq ∧ dv = dx1 ∧ dx2 �= 0. But then,

� = v∂

∂ q− q

∂

∂v, (5.86)

which is again second-order in a new tangent bundle structure V = TR, with R nowthe q-axis, i.e., an axis rotated counterclockwise by an angle θ (tan θ = β/α) w.r.t.the previous one.

As a less simple example, let us consider, e.g., new coordinates given by Q, Uwith

Q =: sinh x1 , U =: L�Q = x2 cosh x1 , (5.87)

and dQ ∧ dU = (cosh2 x1)dx1 ∧ dx2 �= 0. The transformation inverts to x1 =arcsinh(Q), x2 = U/

√Q2 + 1. As the branch-cuts of the square root are along the

imaginary axis and away from the real axis, the latter is a perfectly smooth functionon the real axis.

Then, a simple algebraic calculus yields

� =(

x2∂Q

∂x1− x1

∂Q

∂x2

)∂

∂Q+

(x2

∂U

∂x1− x1

∂U

∂x2

)∂

∂U, (5.88)

where, of course, x1,2 = x1,2(Q, U).


But ∂Q/∂x2 = 0 and x2∂Q/∂x1 = x2 cosh x1 ≡ U, and hence,

� = U∂

∂Q+ f (Q, U)

∂

∂U, (5.89)

(where the expression for the ‘force’ f (Q, U) can be read off from the previ-ous equation and has the rather unappealing form f (Q, U) = U2Q/(Q2 + 1) −√

Q2 + 1 arcsinh(Q)) will again be a second-order field, though not a simple har-monic oscillator anymore, in the new coordinates and, again V = TM, where thebase M in the R

2 plane will be no more a (1D) vector space but rather the graphof sinh x1, and the linear structure will survive only on the fibres spanned by thecoordinate U.

Despite the nonlinearity of the transformation, it will be seen shortly that the resultcould have been anticipated in a single stroke, the map (q, u)→ (Q, U) being whatwill be defined as a ‘point transformation’ (see below for that), i.e., one that mapsdiffeomorphically fibres into fibres, i.e. it is what is usually called a ‘fibre-preserving’diffeomorphism. It has the only advantage of showing that a second-order field canin general be related only to a ‘partial linear structure’.

To conclude the discussion of this example, let us consider a different coordinatetransformation defined by

Q = x1(1+ (x1)2 + (x2)2

), U = x2

(1+ (x1)2 + (x2)2

).

This is a perfectly invertible transformation, actually a diffeomorphism. Indeed,

dQ ∧ dU =(1+ 4E + 3E2

)dx1 ∧ dx2 ,

where E = (x1)2 + (x2)2. Explicitly, the inversion is given by

x1 = 2Q/(√

1+ Q2 + U2 + 1)

, x2 = 2U/(√

1+ Q2 + U2 + 1)

.

The fact that (x1)2 + (x2)2 is a constant of motion implies that, here too,

U = L�Q .

The harmonic oscillator is then a second-order field in the new coordinate systemas well. However the transformation (x1, x2) ↔ (Q, U) does not maps fibres intofibres, i.e., it is not a fibre-preserving diffeomorhism.


5.5.2 The Geometry of the Tangent Bundle

From now on we shall assume that a given (and fixed) tangent bundle structure hasbeen defined on the ‘evolution space’ V ≈ R

2n , and we will denote henceforththe (local or global) coordinates on the base manifold M as (q1, . . . ., qn) and the(global) coordinates on the fibres as (v1, . . . , vn). In this section we shall discussvarious geometrical objects that can be associated with a tangent bundle structureand that characterize the geometry of a tangent bundle.

5.5.2.1 Vertical and Dilation Fields: Homogeneous Tensors

A tangent bundle TM, with M the base manifold (usually, but not necessarily –seethe example in the previous section– a vector space itself) is endowed with a naturalprojection π : TM → M, π : (q, v) �→ v. We will denote by X(TM) the set ofvector fields on TM, and by �1(TM) the set of 1-forms on TM. The set of 0-forms(smooth functions) will be denoted as F(TM).

Vertical vector fields are those projectable on the zero vector field in M, and theycan be viewed then as vector fields that are ‘along the fibres’, i.e., fields acting onlyon the fibre coordinates. In other words, a vector field X is vertical if LX(π∗f ) = 0,∀f ∈ F(M).

Note that homogenous functions of degree zero, i.e., functions that are pull-backsof functions on the base manifold, are constant along the fibres, and vice-versa.

The local coordinate expression of a vertical field is

X = Xi ∂

∂vi, Xi ∈ F(TM) . (5.90)

and the set Xv(TM) of vertical vector fields is a C∞(TM)-module.An immediate consequence of the definition is that

L[X,Y ](π∗f ) = 0 , ∀X, Y ∈ Xv(TM), ∀f ∈ F(M) , (5.91)

andXv(TM) is therefore an infinite-dimensional Lie algebra generating an involutivedistribution andhence, byFrobenius’ theorem, integrable distribution (Theorem2.16,Sect. 2.5.2; see also [Mm85]).

The dilation along the fibres vector field, also-called the ‘Liouville’ or the ‘starfield’, introduced in the previous section and that characterizes the partial linearstructure has the local expression

� = vi ∂

∂vi, (5.92)

and it can be used to characterize tensor fields in the following manner.

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Definition 5.25 A tensor field τ on TM will be said to be homogeneous of degree p(p ∈ R) in the fibre coordinates if

L�τ = p τ . (5.93)

Example 5.5

1. Homogeneous functions of degree zero are constant along the fibres. This followsfrom the fact that, as far as the dependence on the variables on which it operatesgoes (the fibre coordinates in the present case), a dilation field has no non trivialconstants of motion. Therefore, homogeneous functions of degree zero are pull-backs of functions on the base manifold.

2. Homogeneous functions of negative degree cannot be smooth. Indeed,L�f = pf ,p < 0, implies that f must be singular on the base manifold (the zero section).

3. Let θ = ai dqi + bi dvi be a 1-form, with ai, bi ∈ C∞(TM). Then, L�θ = p θ

is easily seen to imply L�ai = 0 and L�bi = (p − 1)bi. In particular, a 1-formwill be homogeneous of degree zero iff (see ii)) the bi’s are all zero and the ai’sare pull-backs of functions on the base manifold, ai = π∗αi, αi ∈ F(M), i.e., iffθ = π∗α, α = αi dqi ∈ �1(M). The 1-form θ is forced then to be the pull-backof a 1-form on the base manifold.

4. A second-order vector field has the local expression

� = vi ∂

∂qi+ Fi ∂

∂vi, Fi ∈ F(TM) . (5.94)

Observing that the ‘first components’ of � are homogeneous of degree one inthe fibre coordinates, while the ∂/∂vi’s are homogeneous of degree −1, and oneinfers immediately that, in general,

L�� = � + [(L� − 2)Fi] ∂

∂vi. (5.95)

Second-order homogeneous fields are also-called ‘sprays’. Necessarily they areof degree one and correspondingly, the ‘forces’ Fi will have to be homogeneousof degree two in the fibre coordinates.

Second-order fields allow for a simple characterization of zero vector fields onTM: X ∈ X(TM) = 0 iff

LX(π∗f ) = 0 , and LX(L�π∗f ) = 0 , ∀f ∈ F(M), (5.96)

and with � any second-order field in the set X(TM).The proof is an easy exercise in coordinates, so we will omit details.

Remark 5.3 Of course a vector field is entirely characterized by its action on func-tions. What the preceding property tells us is that it is enough to consider functions


that are constant along the fibres and functions that are homogeneous of degree onein the fibre coordinates.

5.5.2.2 Vertical and Tangent Lifts

Let now X = Xi(q) ∂/∂qi ∈ X(M) be a vector field on the base manifold. For anyq ∈ M, X(q) ∈ TqM (the fibre through q), using the identification of TqQ withT(q,v)(TqQ) for any vector (q, v) ∈ TQ we can define the curve in TM

R ! t → (q, v + tX(q)) (5.97)

through (q, v). This defines also a one-parameter of diffeomorphisms, that are justtranslations along the fibres. This one-parameter group defines a new vector field,the vertical lift Xv of X, that is easily seen to be given, in local coordinates, by

Xv =: (π∗Xi)∂

∂vi∈ Xv(TM) . (5.98)

Then one can see that a vertical vector field X ∈ X(TM) is a vertical lift of a vectorfield in the base if and only if ∀f ∈ F(M), there exists a g ∈ F(M) such that

LX(L�π∗f ) = π∗g , (5.99)

where � is any second-order field.We shall denote v : X ∈ X(M)→ Xv(TM) the map associating with any vector

field X ∈ X(M) the corresponding vertical lift i.e.,

v : X(M)→ X(TM) , (5.100)

and correspondingly, we also denote v : TM → T(TM)) the corresponding map.Given now a vector field X ∈ X(M), this will generate a (local) one-parameter

groupof transformations onM thatwewill denote as {φt} :φt : M �→ M,φt=0 = IdMand φt ◦ φτ = φt+τ whenever both φt, φτ and φt+τ are defined. This will be grantedif X is a ‘complete’ field (see e.g. [Ar76]). If this is not the case, we will refer to theflow generated by X as to a ‘pseudo-one-parameter group’.

We can now ‘lift’ φt to the tangent map φct =: Tφt ∈ Diff (TM). We recall that,

given any (smooth) map φ : M → M, the tangent map Tφ is defined, in localcoordinates, as Tφ : (qi, vi) �→ (φi(q), vj∂φi/∂qj), and acts linearly on the fibres.Correspondingly:

Definition 5.26 The infinitesimal generator Xc ∈ X(TM) of the one-parametergroup (or pseudogroup) of diffeomorphisms φc

t will be called the tangent, or com-plete, lift of the vector field X ∈ X(M) to TM.


It is a simple matter to show that, in local coordinates, the tangent lift of X =Xi(q) ∂/∂qi ∈ X(M) is given by

Xc = (π∗Xi)∂

∂qi+ (L�π∗Xi)

∂

∂vi, (5.101)

with � any second-order vector field. Note that this does not depend on the choiceof the SODE �

For reasons that will become clear later, diffeomorphisms of TM that are tangentlifts of diffeomorphisms on the base manifold are called also ‘Newtonian’ diffeo-morphisms.

Again, one can prove easily in coordinates that:

Proposition 5.27 A vector field X ∈ X(TM) is a tangent lift of a vector field Y ∈X(M) iff, ∀f ∈ F(M), there exists g ∈ F(M) such that:

LX(π∗f ) = π∗LY f ,

LX(L�π∗f ) = L�(π∗g .(5.102)

Proof The first condition is a necessary and sufficient condition for X to be pro-jectable onto Y , with g = LY f ). Therefore X must be in local coordinates of theform

X = Yi(q)∂

∂qi+ Zi(q, v)

∂

∂vi.

Taking into account that

LX(L�π∗f ) =(

X = Yi(q)∂

∂qi+ Zi(q, v)

∂

∂vi

) (vk ∂f

∂qk

)

= vkY i(q)∂2f

∂qi∂qk+ Zk ∂f

∂qk,

while

vj ∂

∂qj

(Yl ∂f

∂ql

)= vjY l ∂2f

∂qj∂ql+ vj ∂Yl

∂qj

∂f

∂ql,

and then the second condition says that

Zi = vj ∂Yi

∂qj

which means that X is the complete lift of Y . ��Remark 5.4 ‘Newtonian’ diffeomorphisms.

In local language, a second-order vector field will be of the form � = vi∂/∂qi +Fi∂/∂vi, Fi ∈ F(TM), and the ‘first half’ of the associated equations of motion


will be dqi/dt = vi (and vice-versa, of course), i.e., as already said before, theyare of the (standard) Newtonian form. If now φc

t is the tangent lift of φt , then,φc

t : (qi, vi) �→ (Qi, Ui), with Qi = φit(q), Ui = vj∂φi

t/∂qj. But then, dQi/dt =(∂φi

t/∂qj)(dqj/dt) ≡ vi, and the Newtonian form of the equations of motion ispreserved. This justifies the name ‘Newtonian’ given to diffeomorphisms that aretangent lifts of diffeomorphisms on the base manifold. These are also-called ‘pointtransformations’. One can also show that a diffeomorphism preserving the second-order character of any SODE is forced to be a Newtonian diffeomorphism (see e.g.[Mm85])

It is useful to compute the Lie brackets of both vertical and tangent lifts. They aregiven by the following:

Proposition 5.28 The Lie brackets of vertical and tangent lifts satisfy

[Xv, Yv] = 0 ,

[Xc, Y c] = [X, Y ]c ,

[Xv, Y c] = [X, Y ]v .

Moreover,[Xc, �] ∈ Xv(TM) , (5.103)

as well as[Xv, �] − X ∈ Xv(TM) (5.104)

for any SODE � where, with some abuse of notation, we have used the symbol X forthe vector field (π∗Xi)∂/∂qi.

Remark 5.5 (i) The second of the preceding equations tells us that themapX �→ Xc

is a Lie algebra homomorphism.Thismeans that tangent lifts are aLie subalgebraof the Lie algebra of vector fields on TM and that vertical lifts are an Abelianideal thereof (see e.g., [SW86]).

(ii) Equation (5.103) expresses in a compact way, and at the infinitesimal level,precisely the contents of the previous Remark, namely that tangent lifts sendsecond-order vector fields into second-order ones.

Proof The proof of the first property is immediate using the local coordinate expres-sions of Xv and Yv . Remark also that Xc and Yc are π -related with X and Y , respec-tively, and then [Xc, Y c] is π -projectable onto [X, Y ]. Moreover, using the localcoordinate expressions of Xc and Yc we see that

[Xc, Yc] =[

Xi ∂

∂qi +∂Xi

∂qj vj ∂

∂vi , Yk ∂

∂qk+ ∂Yk

∂qlvl ∂

∂vk

]

=(

Xk ∂Y i

∂qk− Yk ∂Xi

∂qk

)∂

∂qi+

[Xc

(∂Y k

∂qlvl

)− Yc

(∂Xk

∂qlvl

)]∂

∂vk


which can be rewritten as

[Xc, Y c] = [X, Y ]i ∂

∂qi+

(XcL�Yk − YcL�Xk

) ∂

∂vk

for any second-order differential equation vector field �. Now, using the above men-tioned property, this can be written as

[Xc, Yc] = [X, Y ]i ∂

∂qi+ L�(XYk − YXk)

∂

∂vk= [X, Y ]i ∂

∂qi+ L�([X, Y ]k) ∂

∂vk

i.e., [Xc, Yc] = [X, Y ]c.Finally the proof of the last property is trivial, because

[Xi ∂

∂vi, Yk ∂

∂qk+ ∂Yk

∂qlvl ∂

∂vk

]= Xi ∂Yk

∂qi

∂

∂vk− Yj ∂Xi

∂qj

∂

∂vi= [X(Yk)− Y(Xk)] ∂

∂vk,

which shows that [Xv, Y c] = [X, Y ]v . ��It is clear that vertical and tangent lifts of a basis of vector fields in X(M) are

clearly a local basis for X(TM).

5.5.2.3 The Vertical Endomorphism Again

Weshall discuss here inmoredetail the vertical endomorphismS, thatwehave alreadyintroduced in Sect. 5.5.1. On general grounds, a (1, 1) tensor field is a fibrewise linearmap T : TM → TM, and it induces a dual bundle map T∗ : T∗M → T∗M via

〈TX, α〉 = 〈X, T∗α〉 =: T(X, α) , (5.105)

where 〈·, ·〉 denotes the natural (point-wise) pairing between vector fields and1-forms. If there is no risk of confusion we will denote both bundle maps withthe same symbol T .

The vertical endomorphism S is the (1, 1)-tensor field in TM defined in intrinsicterms (see e.g., [Mo91]) from the action on vector fields as S = v ◦ Tπ , where v isthe map associating with any vector field on the base manifold its vertical lift andTπ is the tangent map of the canonical projection π : TM → M.

In local tangent bundle coordinates,

S = ∂

∂vi⊗ dqi . (5.106)


Note that by definition it vanishes on vertical vector field and also

S

(∂

∂qi

)= ∂

∂vi

from where the local expression of S is obtained.Another coordinate-free characterization of the vertical endomorphism is that the

vertical endomorphism S is the unique (1, 1)-tensor field in TM satisfying

S(L�π∗df ) = π∗df , (5.107)

∀f ∈ F(M) and for any SODE �, because in particular for f = qi this conditionmeans

S(L�π∗dqi) = S(dvi) = dqi,

and from

L�π∗df = vj[π∗

(∂2f

∂qi∂qj

)]dqi +

[π∗

(∂f

∂qi

)]dii

the assumed condition implies S(dqi) = 0, from the local-coordinate form follows.We explicitly mention here some of the most important properties of the vertical

endomorphism, namely:

S2 = 0 ,

Im S = ker S = Xv(TM) ,

� ∈ X(TM) is a SODE iff S(�) = �,

(5.108)

where � is the Liouville field.As another remarkable property, both vertical and tangent lifts have the prop-

erties of leaving the vertical endomorphism unchanged, i.e., for any vector fieldZ ∈ X(TM), such that there exists X ∈ X(M) and Z = Xv , we have that LZS = 0and similarly, if Z = Xc also LZS = 0.

Of course, if Z = Xv , then LZ(∂/∂vi) = 0 and LZ(dqi) = d(Zqi) = 0. Similarly,if X = f i(q)∂/∂qi we have that

LZ

(∂

∂vi

)=

[Xc,

∂

∂vi

]= − ∂f j

∂qi

∂

∂vj, LZ (dqi) = df i,

from where

LZS = LZ

(∂

∂vi

)⊗ dqi + ∂

∂vi⊗ df i = 0.


5.5.2.4 Exterior Derivatives Associated with (1, 1) Tensors

Tensors of (1, 1)-type have been defined through their action on vector fields and1-forms. Here we want to discuss the extension of their action to higher-order forms.

As discussed in the literature (see e.g., [Mo91]), there are various possible exten-sions of the action of a tensor T ∈ T 1

1 (TM) mapping forms of a given order ontoforms of the same order. They all share the fact that the resulting action ‘goes through’the coefficients of any form. It is therefore natural, to begin with, to define the actionof T on any function f ∈ F(TM) (a 0-form) as Tf = 0.

Now we can give the following:

Definition 5.29 With any tensor field T ∈ T 11 (TM) we can associate a derivation

δT of degree zero on the graded algebra of forms �(TM) whose action on 0- and1-forms is defined as

δT f = 0 , ∀f ∈ F(TM) ,

δTθ = Tθ, ∀θ ∈ �1(TM) .(5.109)

We recall (see e.g., [ChB82]) that graded derivations on forms are entirely deter-mined by their action on 0- and 1-forms. Given that, one can prove at once that:

Proposition 5.30 If ω ∈ �k(TM), 1 ≤ k ≤ 2n, then:

δT ω(X1, X2, . . . , Xk) =ω(TX1, X2, . . . , Xk)+ ω(X1, TX2, . . . , Xk)

+ + · · · + ω(X1, X2, . . . , TXk) .

This is the extension of the action of T on higher order forms we will concentrateon here. Notice that, if T = I (the identity tensor), then δI will act simply bymultiplying a form by its degree.

The exterior derivative d is an derivation of degree one in �(TM). A knownresult states that the graded commutator of a derivation of degree p and a derivationof degree q is a derivation of degree p+ q [ChB82]. We can then state the following:

Definition 5.31 The operation dT on the algebra of forms defined as

dT =: δT ◦ d − d ◦ δT (5.110)

is a derivation of degree one that will be called the exterior derivative associated withthe (1, 1)-type tensor T .

The action of dT on 0- and 1-forms is easily seen to be given explicitly by

dT f = TdfdTθ(X, Y) = (LTXθ)(Y)− (LTY θ)(X)+ θ(T [X, Y ]) (5.111)


The first one of these equations is an immediate consequence of the definition of dT .The second one can be proved using either one of the identities

dθ(X, Y) = LX(θ(Y))− LY (θ(X))− θ([X, Y ]) ,

dθ(X, Y) = (LXθ)(Y))− (LYθ)(X))+ θ([X, Y ]), (5.112)

so we will omit details. In particular, it is not hard to check that, if T = I, thendI ≡ d.

Example 5.6 It can be useful to have the explicit expression of dT in local coordi-nates. To this effect, let us introduce the collective coordinates ξ i, i = 1, . . . , 2n,with

ξ i = qi, ξ i+n = vi, i = 1, . . . , n . (5.113)

Then the tensor T will be represented as

T = T ij

∂

∂ξ i⊗ dξ j , Ti

j ∈ F(TM) , (5.114)

and

dT f = ∂f

∂ξ kTk

jdξ j , (5.115)

while, with θ = θi dξ i,

dT θ = 1

2

(∂θj

∂ξ kTk

i + θk∂T k

i

∂ξ j− (i↔ j)

)dξ i ∧ dξ j. (5.116)

In particular, for T = S = (∂/∂vi) ⊗ dqi, Sij = δi−n

j for n < i ≤ 2n, and zerootherwise, and we find, going back from the ξ ’s to the q’s and v’s,

dSf = ∂f

∂vidqi ,

dSθ = ∂αj

∂vidqi ∧ dqj + ∂βj

∂vidqi ∧ dvj, (5.117)

if θ = αi dqi + βi dvi.A relevant property of the exterior derivative associated with T , with an easy

proof, is contained in the following:

Proposition 5.32 The degree one derivation dT commutes with the exterior deriv-ative d and the graded commutator of iX and dT satisfies the generalized Cartanidentity,

iX ◦ dT + dT ◦ iX = LTX + [δT ,LX ] , (5.118)


which reduces to the standard Cartan identity when T = I. Moreover,

LX ◦ dT − dT ◦ LX = dLX T . (5.119)

5.5.2.5 Cohomology of dT and Nijenhuis Tensors

Given the exterior derivative dT , we may enquire under which conditions we canassociate a DeRham-type cohomology with dT for a general (1, 1)-type tensor T .In an essentially equivalent way, we may enquire: (i) under which conditions the(coboundary) property d2

T = 0 holds, and (ii) whether or not a Poincaré-type lemmaholds.

First of all, let us remark that it follows at once from the definition (and fromd2 = 0) that

d ◦ dT + dT ◦ d = 0 , (5.120)

and hence thatd2

T ◦ d = d ◦ d2T (5.121)

i.e., that d2T commutes with the external derivative. As dT is a degree-one derivation,

d2T is a (degree-two) derivation commuting with d. Therefore its action on forms isentirely determined by that on functions [ChB82]. A long but otherwise straightfor-ward algebra leads then to

(d2T f )(X, Y) = df (NT (X, Y)) , (5.122)

where the Nijenhuis tensor associated with the tensor field T ∈ T 11 (TM) is the

(1, 2)-type tensor NT given by (see Appendix E):

NT (X, Y) =: [TX, TY ] + T2[X, Y ] − T [TX, Y ] + T [TY , X] . (5.123)

It can be useful to have the explicit expression of the Nijenhuis tensor in localcoordinates. Introducing again collective coordinates it is a long but straightforwardexercise to show that

NT = 1

2(NT )i

km∂

∂ξ i⊗ dξ k ∧ dξm , (5.124)

where

(NT )ikm =

∂Tim

∂ξ jT j

k + ∂Tjk

∂ξmTi

j − (k ↔ m) . (5.125)

So, NT = 0 whenever the components of T are constant. This will be thecase in particular for the vertical endomorphism and, therefore, NS = 0, for


the vertical endomorphism. One could also use Eq. (5.117) to prove at once thatd2

S f = (∂2f /∂vi∂vj)dqi ∧ dqj = 0.Coming back to the problem posed at the beginning of this section, we can state:

Proposition 5.33 The exterior derivative dT associated with a (1, 1)-type tensor Tsatisfies the coboundary property dT ◦ dT = 0 iff T satisfies the Nijenhuis condition:

NT = 0 (5.126)

Definition 5.34 Tensors satisfying the Nijenhuis condition will be said to be of theNijenhuis type.

It follows then that if T is of the Nijenhuis type, then ‘T -exact’ forms, i.e., formsof the type α = dTβ for some β, are also ‘T -closed’, i.e. dT α = 0. The converseproperty, i.e. that dTα = 0 implies that there exists a β such that α = dTβ, at leastlocally), i.e., the analog of the Poincare Lemma, requires however the Nijenhuis typetensor T to be invertible.

Remark 5.6 The invertibility condition is not simply a technical requirement for theconstruction of the proof, as the following example shows. The vertical endomor-phism S is of the Nijenhuis type but, as S2 = 0, it is not invertible. Consider then,e.g., the 2-form

α = 1

2αijdvi ∧ dvj, (5.127)

with αij + αji = 0, αij ∈ R. As dSdvi = 0, then dSα = 0, but α = dSθ with θ

a 1-form fails to be true even locally. For instance, dSθ(∂/∂vi, ∂/∂vj) = 0 for any1-form θ , while α(∂/∂vi, ∂/∂vj) = αij.

5.5.2.6 Horizontal Lifts Associated with Second-Order Vector Fields:Nonlinear Linear Connections on TM

Given a local basis X1, . . . , Xn of vector fields on M, their vertical lifts Xv1 , . . . , Xv

n ,yield a holonomic local basis forXv(TM). We recall that ‘holonomic’ means that thevector fields pairwise commute. Notice that the correspondence X ∈ X(M) �→ Xv ∈Xv(TM) is F(M)-linear, i.e., if X �→ Xv , then f X �→ (f X)v = f Xv , ∀f ∈ F(M).

In the absence of additional structures, there are many ways of finding, at everypoint ofTM, ann-dimensional subspace, transversal to the vertical one, to supplementthe latter. Any such subspace will have the right to be called a ‘horizontal space’, andthe choice of such a subspace in each point of TM is called a ‘horizontal distribution’.

We have already a possible candidate for a supplementary subspace at every point,namely that one provided by the tangent lifts Xc

1, . . . , Xcn . However, as it is easy to

check that (fX)c = fXc + (L�f ) Xv , for any X ∈ X(M) and f ∈ F(M), and then thisassociation is not F(M)-linear.


A different (and intrinsic) way to find an F(M) -linear distribution, is found bysingling out a given (but otherwise arbitrary) second-order field �. A preliminaryresult to this effect is contained in the following:

Proposition 5.35 If X ∈ X(M), � is a SODE and S is the vertical endomorphism,then,

S[Xv, �] = Xv . (5.128)

Indeed, from S(�) = � and LXv S = 0, we obtain S[Xv, �] = S(LXv�) =LXv (S�) = [Xv,�] = Xv , the last passage following from vertical lifts beinghomogeneous of degree −1 in the fibre coordinates. Notice that this result couldhave been inferred also directly from Eq. (5.104).

Now we state (and will justify in a moment) the following:

Definition 5.36 The horizontal lift of a vector field X ∈ X(M) associated with agiven second-order vector field � is the vector field

Xh =: 12{[Xv, �] + Xc} ∈ X(TM) . (5.129)

Proposition 5.37 The horizontal lift Xh satisfies (and is entirely characterized by)

LXh(π∗f ) = π∗LXf , f ∈ F(M) , (5.130)

and

LXh(L�π∗f ) = 1

2LXv (L�L�π∗f ) . (5.131)

It follows from this result and the fact that the association X ∈ X(M) �→ Xv ∈Xv(TM) X �→ Xv is F(M)-linear, i.e., that (fX)h = fXh, that the correspondenceX ∈ X(M) �→ Xh ∈ X(TM) is F(M)-linear as well. Moreover, using the definitionof Xh and (5.128) we see that the horizontal lift satisfies

S(Xh) = Xv , (5.132)

which follows from S(Xc) = Xv .Remark that if the local expression of the given second-order vector field � in

natural coordinates (qi, vi) on the tangent bundle, is

� = vi ∂

∂qi+ Fi(q, v)

∂

∂vi,

then, having in mind that

L�

∂

∂vi= − ∂

∂qi− ∂Fj

∂vi

∂

∂vj,


and L�dqi = dvi, we find that

L�S = − ∂

∂qi⊗ dqi − ∂Fj

∂vi

∂

∂vj⊗ dqi + ∂

∂vi⊗ dvi,

i.e., the matrix representing L�S in the basis {∂/∂qi, ∂/∂vi | i = 1, . . . , n} is( −δi

j 0

− ∂Fi

∂vj δij

),

and then (L�S)2 = I . Therefore it allows us to construct two projection operators,the first one PV = 1

2 (I +L�S), with range in the vertical subbundle, and the secondone PH = I−PV = 1

2 (I−L�S). They give us a splitting of the tangent space at eachpoint, becausePH+PV = I andPHPV = PV PH = 0, providing the abovementionedgeneralized nonlinear connection.

The explicit expression of the projectors are

PV =(

0 0

− 12

∂Fi

∂vj δij

), PH =

(δi

j 012

∂Fi

∂vj 0

),

and then the coefficients of the associated nonlinear connection are given by

�ij = −

1

2

∂Fi

∂vj.

In local coordinates, if X = Xi(q) ∂/∂qi and � = vi ∂/∂qi + Fi ∂/∂vi, then,

Xh = (π∗Xi)Hi , (5.133)

where the ‘horizontal basis’ Hi is given by

Hi =(

∂

∂qi

)h

= ∂

∂qi− �

ji

∂

∂vj= ∂

∂qi+ 1

2

∂Fk

∂vi

∂

∂vk. (5.134)

The local basis for the linear space of vector fields on TQ adapted to this connectionis given by the previous vector fields and

Vi = ∂

∂vi,

the corresponding dual basis for �1(TQ) being

Hi = dqi Vi = dvi + �ij dqj = dvi − 1

2

∂Fi

∂vjdqj .


We will throughout use this basis. Of course, if the second-order vector field � is aspray, the connection so defined is linear.

Let us take now, in natural coordinates, the ∂/∂vi’s and the Hi’s as a basis. Recallthat ∂/∂vi = (∂/∂qi)v and that Hi = (∂/∂qi)h. A corresponding dual basis of1-forms will be given by forms Hi = dqi, Vi =, i = 1, . . . , n such that

⟨V i

∣∣∣ ∂

∂vj

⟩=

⟨Vi

∣∣∣ Hj⟩ = δi

j , i, j = 1, . . . , n , (5.135)

and ⟨Vi|Hj

⟩=

⟨Hi| ∂

∂vj

⟩= 0, i, j = 1, . . . , n . (5.136)

It will be also useful to have at our disposal the explicit form of the Lie bracketsamong the vectors of the basis. A long but straightforward algebra yields then thefollowing:

Proposition 5.38 The vertical and horizontal bases satisfy the commutation rules:

[∂

∂vi,

∂

∂vj

]= 0 ,

[∂

∂vi, Hj

]= 1

2

∂2Fk

∂vi∂vj

∂

∂vk, (5.137)

[Hi, Hj

] = −Rmij

∂

∂vm,

where

Rmij = −

1

2

(∂2Fm

∂qi∂vj− ∂2Fm

∂vi∂qj+ 1

2

[∂2Fm

∂vj∂vl

∂Fl

∂vi− ∂2Fm

∂vi∂vql

∂Fl

∂uj

]). (5.138)

Remark 5.7 (i) It follows from the previous result that, if X = Xi(q)∂/∂qi andY = Y i(q)∂/∂qi, then,

[Xh, Yh] − [X, Y ]h = −XiY jRmij

∂

∂vm, (5.139)

i.e., that the horizontal distribution fails to be involutive (and a Lie algebra homo-morphism) to the extent that the (1, 2)-type tensor

R = 1

2Rm

ij∂

∂vm⊗ dqi ∧ dqj (5.140)

is nonzero.


(ii) It follows also from (5.139) and (5.132) that

S[Xh, Y h] = [X, Y ]v (5.141)

(iii) It is also easy to show that

� = viHi + 1

2(� − [�,�]) , (5.142)

i.e., � will be a horizontal field iff it is a spray (� = [�,�]).(iv) Given a vector field X ∈ X(M), we can define the ‘horizontal lifts’ of its

integral curves as the integral curves of the horizontal lift Xh. Explicitly, if X =Xi∂/∂qi, the horizontal lifts will be solutions of the equations:

⎧⎪⎨⎪⎩

dqi

dt= Xi

dvi

dt= 1

2Xk ∂Fi

∂vk,

(5.143)

with the appropriate initial conditions.

5.5.2.7 Nonlinear Connections and SODE’s

In order to make contact with a somewhat more familiar context, let us assume thata Riemannian (or pseudo-Riemannian) metric

g = gij dqi ⊗ dqj (5.144)

has been given on M, and consider the second-order vector field:

� = vi ∂

∂qi+ Fi ∂

∂vi,

Fi = −�ikmvkvm , (5.145)

�ikm =

1

2gij

(∂gjk

∂qm+ ∂gim

∂qk− ∂gkm

∂qj

),

associated with g. The �ikm’s are of course the familiar Christoffel symbols defin-

ing the Levi-Civita connection, and the second-order (in M) equations of motionassociated with � are

d2qi

dt2+ �i

kmvkvm = 0 (5.146)

i.e., � describes geodesic motion on M. Notice that: (i) � is a spray ([�,�] = �)and (ii) if ∇ (see the Appendix E on connections and covariant derivatives) denotes


the covariant derivative associated with the Levi-Civita connection, then,

d2qi

dt2+ �i

kmvkvm = 0⇐⇒ ∇γ γ = 0 , (5.147)

i.e., the equations for geodesic motion γ express just the condition of vanishing ofsuch covariant derivative. It will be seen in the next section that the equations ofmotion associated with � are precisely the Euler-Lagrange equations that can bederived from the Lagrangian L = 1

2gijvivj ≡ 1

2 (π∗g)(�,�) ∈ F(TM).The horizontal distribution associated with � is easily found to be

Hi = ∂

∂qi− �k

imvm ∂

∂vk. (5.148)

Also,θ i = dvi + �i

jmvmdqj , (5.149)

andRm

ij = Rmkijv

k, (5.150)

where R is the Riemann curvature tensor of the Levi-Civita connection.So far for linear connections associatedwith ametric tensor. Recognizing however

that the basic concept involved in the definition of a connection is that of a liftingprocedure for curves, with the ensuing prescription for parallel transport, then we canconclude that the association of a horizontal distribution with a given second-orderfield � (not a spray in general) will define for us a ‘nonlinear’ connection on M withits associated procedure of parallel transport. We refer to the literature [Cr83, Cr87]for further developments concerning this specific point.

5.5.3 Lagrangian Dynamics

Up to now we have basically studied the geometry of the tangent bundle structurethat can be attributed to a given vector space V (V ∼= TM, then). Now we begin asystematic description of Lagragian dynamics on TM.

5.5.3.1 Lagrangian one- and Two-Forms: Regular and Singular Lagrangians

Let L ∈ F(TM) be a smooth function on TM. L will be called a ‘Lagrangian’ fromnow on. Then we define the Lagrangian, or Cartan, 1-form θL associated with L asthe (semi-basic) 1-form

θL =: dSL . (5.151)


We recall that a 1-form α is ‘semibasic’ iff S∗α = 0, i.e., iff it is of the local formα = αi dqi, with the αi’s functions on TM, in general. Such a 1-form α will be, inparticular, a ‘basic’ form iff α = π∗β, with β ∈ �1(M), i.e., iff the αi’s are functionsof the q’s alone. Semibasic forms are also called ‘horizontal’. The 2-form

ωL = −dθL = −dSdL (5.152)

will be called the Lagrangian 2-form associated with L. The function L will be saidto be a ‘regular Lagrangian’ iff ωL is nondegenerate, hence a symplectic form onTM, and ‘singular’ otherwise.

In local coordinates (qi, vi), we have the explicit expressions

θL = ∂L

∂vi dqi , (5.153)

and

ωL = ∂2L

∂vi∂vjdqi ∧ dvj + 1

2

(∂2L

∂vi∂qj− ∂2L

∂qi∂vj

)dqi ∧ dqj . (5.154)

If X is a vector field on TM having the local expression X = Ai∂/∂qi + Bi∂/∂vi,then,

iXωL = ∂2L

∂vi∂vjAidvj +

[(∂2L

∂vi∂qj− ∂2L

∂qi∂vj

)Ai − ∂2L

∂vi∂vjBi

]dqj . (5.155)

Therefore, ωL will be non-degenerate (i.e., iXωL = 0 implies that X = 0) iff the‘Hessian matrix’

Hij =: ∂2L

∂vi∂vj (5.156)

is of maximum rank (i.e., equals to dim M).

Remark 5.8 Although formulated in local language, the non-degeneracy conditionis clearly an intrinsic one. Note that ω∧n

L �= 0 iff the Hessian matrix is regular.

It is interesting to inquire if the map associating θL with L is injective and samefor the map associating ωL to θL . As to the first map, since the fact that θL vanisheson vertical fields, its kernel is made up clearly of functions that are constant along thefibres, i.e., θL = 0 if and only if L is of the form L = π∗g for some g ∈ F(M). As tothe second, ωL will be zero if the semi-basic 1-forms θL is closed. This can be onlysatisfied if θL is a basic form. Hence, ωL = 0 iff θL = π∗α, with α ∈ �1(M) anddα = 0. But this implies that L be of the form L = i�π∗α with � any second-orderfield. Locally (and also globally if M is a vector space) this means that L has to be a‘total time derivative’, L = df /dt for some f ∈ F(M).


All in all, we have proved that the most general Lagrangian that makes the asso-ciated Lagrangian 2-form to vanish identically has the general expression

L = π∗g + i�π∗α , g ∈ F(M), α ∈ �1(M), dα = 0 , (5.157)

i.e., it is the sum of a ‘pure potential term’ (π∗g) and of what is often called a ‘gaugeterm’, the ‘total time derivative’ term i�π∗α. This can be rewritten in a differentway, because a 1-form α ∈ �1(M) defines a function α : TM → R as followsα(v) = 〈απ(v), v〉, and then i�π∗α = α.

Let Y = Ci∂/∂qi +Di∂/∂vi be a vector field in X(TM). Then, using Eq. (5.155)and the definition of the vertical endomorphism, one proves immediately, in coordi-nates, that ωL(X, SY) = −ωL(SX, Y) = (∂2L/∂vi∂vj)AiCj . Hence,

ωL(X, SY)+ ωL(SX, Y) = 0 , ∀X, Y ,

and, remembering the definition (see subsection2.4) of the zeroth-order derivationassociated with a (1, 1)-type tensor, we have found that any Lagrangian 2-form ωL

satisfies

δSωL = 0 , (5.158)

and, as a consequence of S2 = 0,

ωL(SX, SY) = 0 , ∀X, Y ∈ X(TM) , (5.159)

i.e., ωL will vanish on any pair of vertical fields.Recall that δS is a derivation δS of degree zero such that

(δSωL)(X1, X2) = ω(S(X1), X2)+ ωL(X1, S(X2), . . . , Xk)

Noticing that ωL(SX, Y) = iY iSXωL and ωL(X, SY) = iSY iXωL ≡ iY (SiXωL), thislast result can be rewritten as

iSXωL + SiXωL = 0 , ∀X ∈ X(TM) . (5.160)

5.5.3.2 Conditions Under Which a Closed 2-Form Is a Lagrangian 2-Form

We have seen in the previous Subsection that to every function (let it be regular ornot) L ∈ F(TM) we can associate (a Cartan 1-form and) a Lagrangian 2-form ωLwhich is closed (actually exact) and vanishes on pairs of vertical fields. A relevantquestion is then under which conditions such a 2-form on TM is actually a 2-formfor some L ∈ F(M). This is settled by the following [Ca95]:


Theorem 5.39 Let ω be a closed 2-form,ω ∈ �2(TM). Then,ω will be a Lagrangian2-form, ω = ωL , for some L ∈ F(TM) iff

δSω = 0 , (5.161)

where S is the vertical endomorphism.

The ‘only if’ part of the theorem has already been proved. Let us turn then to the‘if’ part. Let ω and X, Y ∈ X(M) have the local expressions

ω = 1

2Fij dqi ∧ dqj + Gij dqi ∧ dvj + 1

2Hij dvi ∧ dvj ,

X = Ai ∂

∂qi+ Bi ∂

∂vi, (5.162)

Y = Ci ∂

∂qi+ Di ∂

∂vi.

Then, δSω(X, Y) = (Gij −Gji)AiCj +Hij(BiCj + AiDj), and δSω = 0 implies thenHij = 0 and Gij = Gji, and hence,

ω = 1

2Fij dqi ∧ dqj + Gij dqi ∧ dvj . (5.163)

Closure of ω implies, by Poincaré’s Lemma, that there will exist, locally at least,a 1-form α such that ω = −dα. Such an α will be defined, of course, ‘modulo’a closed 1-form. If it has the local expression α = Mi dqi + Ni dvi, then Hij = 0implies ∂Ni/∂vj = ∂Nj/∂vi, which is an integrability conditions along the fibres.There will exist therefore a function f ∈ F(TM) such that Ni = ∂f /∂vi. Takingthen θ = α − df , θ will be semi-basic, θ = Ai dqi, where Ai = Mi − ∂f /∂qi. Then,ω = −dθ and, more explicitly,

ω = ∂Ai

∂vjdqi ∧ dvj + 1

2

(∂Ai

∂qj− ∂Aj

∂qi

)dqi ∧ dqj . (5.164)

Symmetry of Gij will imply then ∂Ai/∂vj = ∂Aj/∂vi, which is again an integrabilitycondition along the fibres, implying the existence of a function L ∈ F(TM) suchthat Ai = −∂L/∂vi. All in all this leads to

ω = −d

(∂L

∂vidqi

)= ωL, (5.165)

and this achieves the proof of the theorem.

Remark 5.9 The way the proof of the theorem has been constructed, the use of thePoincaré Lemma and of the integrability conditions indicate that, in general, theproof is only a local one. This is perhaps a purely academic remark if the tangent


bundle structure is constructed over a (2n-dimensional) vector space V , where thereare no topological obstructions and the proof becomes global. However, we wouldlike to stress that even in more general cases the fibres remain vector spaces withno topological obstructions. Therefore the functions B and L will exist (at least) ina ‘tubular neighbourhood’ of a neighbourhood in the base manifold, and the proofwill be ‘local’ only to this extent.

5.5.3.3 Euler-Lagrange Equations in Hamiltonian and in Standard Form

The Liouville field � is vertical and the vertical endomorphism S is homogeneousof degree minus one in in the fibre coordinates, i.e., L�S = −S. Therefore, asθL = S(dL), we see that

L�θL = (L�S)∨(dL)+ S(dL�L) = θ�(L) − θL = θ�(L)−L.

The functionEL =: (L� − 1)L , (5.166)

is called, for reasons that will become clear shortly, the ‘energy function’ associatedwith the Lagrangian L. Note that, if L is a ‘pure potential’, and hence it is homoge-neous of degree zero in the fibre coordinates, then, EL = −L, while if L is a ‘puregauge’, homogeneous of degree one, EL ≡ 0.

On the other side, i�θL = 0 because θL is semibasic and the vector field � isvertical. Using the homotopy identity, LX = iX ◦d+d ◦ iX , valid for any vector fieldX, for the vector field � we see that

i�ωL = −i�dθL = −L�θL = −dSEL . (5.167)

Let now � be a vector field such that

i�ωL = dEL . (5.168)

Then, using Eq. (5.160), we find at once

Si�ωL = −iS�ωL ≡ dSEL , (5.169)

which implies that, if� exists at all, S�−�will be in the kernel ofωL . If L is regular(and hence ωL is non-degenerate), � will exist, will be unique, and S� = � willforce � to be a ‘second-order vector field. If L is not regular, there is no guaranteethat Eq. (5.168) has any solutions for �. This happens, e.g., if L is a ‘pure potential’(L = π∗g, g ∈ F(M)), which implies ωL = 0 but EL = −L. At the other extreme,if L is a ‘pure gauge’ (loosely speaking, L =df /dt for some f ∈ F(M)), thenωL = EL = 0 and the search becomes pointless, as any vector field would trivially


fit the job. Finally, even if there are solutions, it may well be that among the possiblesolutions no second-order one can be found.

Let us remark also that any � solving Eq. (5.168) will leave ωL invariant, i.e.

L�ωL = 0 . (5.170)

Then we can summarize the discussion:

(i) Any regular Lagrangian L will define a (its own) symplectic structure on thetangent bundle TM.

(ii) The (unique) second-order field � of Eq. (5.168) will be ωL-Hamiltonian w.r.t.the symplectic structure, with the energy function EL playing the rôle of theHamiltonian.

Remark 5.10 (i) If L is of the standard form, also called a Lagrangian ‘of mechan-ical type’, L = T − V , with T a kinetic term quadratic in the velocities andV = V(q) a potential energy term, then, EL = T + V , and this justifies thename ‘energy function’ given to EL.

(ii) At variancewith the canonical (and/or Poisson) formalism on cotangent bundlesstudied in previous chapters, where the symplectic structure is given geomet-rically once and for all independently from the dynamics, here the Lagrangiandefines both the dynamics and the symplectic structure.

(iii) If � has the local form � = vi∂/∂qi +Fi∂/∂vi, it is left as a simple exercise inlocal coordinates to show that Eq. (5.168) yields, explicitly, the equations

∂2L

∂vi∂qj vj + Fj ∂2L

∂vi∂vj −∂L

∂qi = 0, i = 1, . . . , n (5.171)

which, upon usingvi = dqi/dt and Fi = dvi/dt, are just the familiar Euler-Lagrange equations:

d

dt

(∂L

∂vi

)− ∂L

∂qi= 0 , i = 1, . . . , n . (5.172)

Let us elaborate now a bit on these equations. With the familiar identificationof the Lie derivative w.r.t. a vector field with a total time derivative, they can berewritten in the equivalent form L�(∂L/∂vi)−∂L/∂qi = 0. Multiplying then by dqi

(summing over i) and observing that

L�θL = L�

(∂L

∂vi

)dqi +

(∂L

∂vi

)dvi ,

we obtain the equivalent set of equations

L�θL − dL = 0 . (5.173)


This equation as well is written in intrinsic terms, and is what is usually called the‘standard form’ of the Euler-Lagrange equations. It is obviously fully equivalent tothe Hamiltonian Eq. (5.168). Indeed, L�θL = −i�ωL + di�θL , and i�θL = i�dSL =i�SdL = iS�dL = i�dL = L�L. Putting things together, we recover Eq. (5.168).With a completely similar set of manipulations one can easily show that Eq. (5.168)leads indeed to Eq. (5.173).

In this way we have obtained the intrinsic (both Hamiltonian and ‘standard’ or‘Lagrangian’) versions of the Euler-Lagrange equations. These are the forms that wewill use from now on and until the end of this chapter.

In this as well as in the previous subsection the starting point has been the datumof a Lagrangian function. From that we have, so-to-speak, ‘deduced’ the associateddynamical vector field � which, if the Lagrangian is regular, turns out to be uniqueand second-order.

To keep however with the spirit of the present book, where the ‘primary’ objectis always assumed to be a dynamics (i.e., a vector field), and to anticipate to someextent the discussion of the Inverse Problem that will be carried on (hopefully) in fulldetail in Sect. 5.5, we have to invert to some extent the line of reasoning that has ledus to Eqs. (5.168) and (5.173). Let then � be a dynamical vector field (a dynamicalsystem, for short). Then,

Definition 5.40 The dynamical system � on a tangent bundle TM will be said toadmit a Lagrangian description with the admissible Lagrangian L ∈ F(TM) iff theequation:

L�θL − dL = 0 , (5.174)

holds.

We know already that if the Lagrangian is regular, then � has to be second-order,so we will concentrate on second-order fields, and hence on dynamical systems ofthe Newtonian type, from now on.

Not all second-order fields will admit of a Lagrangian description (some specificexamples will be discussed in Sect. 5.5) and the latter, if and when it exists, need notbe unique. As a simple example, let � be the dynamics of an n-dimensional isotropicharmonic oscillator. Setting the frequency and the mass of the oscillator equal to one,� is then given by

� = vi ∂

∂qi− qi ∂

∂vi, (5.175)

and it is a simple exercise to show that any Lagrangian of the form

LB = Bij{vivj − qiqj} , (5.176)

withB = ||Bij|| being a symmetric n×nmatrix (an invertible one, if we insist that LBbe a regular Lagrangian), will be a perfectly admissible Lagrangian for the isotropicharmonic oscillator.


Remark 5.11 If no further restrictions are imposed on the admissible Lagrangians,then the problem posed in the above definition has always (at least) one trivial solu-tion. Indeed, let L be a ‘pure gauge’, i.e., L = i�0π

∗α, α ∈ �1(M), dα = 0, where�0 is a second-order field. As α is basic, �0 can be any second-order field, and there-fore we are free to replace it with � itself. Then, θL = π∗α and dL = d(i�π∗α).This leads to L�θL − dL = i�π∗dα, and the closure of α will ensure that L will beactually an admissible Lagrangian for any second-order field �. The converse is alsotrue, i.e., a Lagrangian that is admissible for any second-order field is necessarily apure gauge, but we will not insist on this point (For details, see [Mo91]).

To close this section, we may inquire under which conditions a 2-form ω is aLagrangian 2-form for a given second-order field � = vi∂/∂qi+Fi∂/∂vi, i.e., whatis the form that Theorem 15 acquires when the dynamics (which is, after all, the‘primary’ object for us) is assigned. If it is so, besides being closed, ω has to beinvariant under the dynamics, L�ω = 0. Let us show briefly that this condition andthe additional condition of vanishing on pairs of vertical fields, ω(SX, SY) = 0,∀X, Y , become then equivalent to the condition δSω = 0 of Theorem 15. It will beuseful to use here the horizontal and vertical basis (Hi, ∂/∂vi) and cobasis (dqi, θ i)

‘adapted’ to �. Then, in local coordinates, a vector field X = ai ∂/∂qi + bi ∂/∂vi

can be rewritten as X = aiHi + bi ∂/∂vi, where

bi = bi − 1

2aj ∂Fi

∂vj.

By taking the Lie derivative w.r.t. � of the condition ω(SX, SY) = 0 one finds atonce that this implies that

ω((L�S)X, SY)+ ω(SX, (L�S)Y) = 0 . (5.177)

In the adapted bases,

L�S = θ i ⊗ ∂

∂vi− dqi ⊗ Hi (5.178)

It is then a long but straightforward exercise in coordinates to show that, if Y =ci∂/∂qi + f i∂/∂vi, then,

ω((L�S)X, SY)+ ω(SX, (L�S)Y) = (aif j − ajf i)ω

(∂

∂vi, Hj

),

and therefore,

ω((L�S)X, SY)+ ω(SX, (L�S)Y) ≡ ω(SX, Y)+ ω(X, SY) (5.179)

and this shows that an equivalent set of assumptions is:

(i) ω is closed and L�ω = 0.


(ii) ω vanishes on pairs of vertical fields, ω(SX, SY) = 0, ∀X, Y ∈ X(TM).

5.5.3.4 Point Transformations

Point-transformations of TM, i.e., transformations that are tangent lifts of transfor-mations on the base manifold, and that are represented, at the infinitesimal level,by tangent lifts of vector fields in X(M), play a privileged rôle in the framework ofLagrangian dynamics. This is due to the fact that, as stated in almost all standardtextbooks onClassical Dynamics, the Euler-Lagrange equations are ‘form-covariant’under point-transformations. By this we mean the following:

Consider the Euler-Lagrange equations in the usual form,

d

dt

(∂L

∂vi

)− ∂L

∂qi= 0 . (5.180)

If we perform a point-transformation (qi, vi)→ (qi, vi), given by

qi = qi(q) , (5.181)

vi = vi(q, u) =: vj ∂ qi

∂qj

(the last one expressing simply vi = dqi/dt, in more conventional language), then,defining

L(q, v) =: L(q(q), v(q, v)) (5.182)

one finds, with some simple algebra

d

dt

(∂L

∂vi

)− ∂L

∂ qi= ∂qj

∂ qi

[d

dt

(∂L

∂vj

)− ∂L

∂qj

], (5.183)

and this expresses the well-known ‘form-covariance’ property of the Euler-Lagrangeequations with respect to point-transformations.

In a more intrinsic way, denoting by, say, φ the map φ : (q, v) �→ (q, v), then thisequation can be rewritten simply as L = φ∗L.

In a more intrinsic language, let Xc ∈ X(TM) be the tangent lift of a vector fieldX ∈ X(M). As discussed in a previous Section, Xc will be the infinitesimal generatorof a one-parameter group {φt}t∈R, φt ∈ Diff (TM) of point transformations of TM.Recall that

LXc S = 0 , (5.184)

andS([Xc, �]) = 0 , (5.185)


when Xc is a tangent lift and � any second-order field, which states that [�, Xc] isvertical. This expresses at the infinitesimal level the fact that tangent lifts of diffeo-morphisms of the base manifold map second-order fields into second-order ones.Given a Lagrangian function, we have that

LXcθL = LXc(SdL) = SLXc dL ,

and hence,LXcθL = θLXc L , (5.186)

andLXcωL = ωLXc L , (5.187)

as well. At the finite level this implies that

φ∗t θL = θφ∗t L , (5.188)

as well asφ∗t ωL = ωφ∗t L , (5.189)

and this is specific of point transformations. Therefore, taking the pull-back of Eq.(5.173), we obtain

Lφ∗t �θφ∗t L − d(φ∗t L) = 0 (5.190)

where the transformed fieldφ∗t � =: (φ−1t )∗� ≡ (φ−t)∗�will be again, aswe alreadyknow, a second-order field, and will admit of a Lagrangian description in terms of thetransformedLagrangianL. This is the intrinsicwayof expressing the form-covarianceproperty of the Euler-Lagrange equations w.r.t. point transformations.

At the infinitesimal level we find, as LXcL�θL = (L[Xc,�] + L�LXc)θL , theinfinitesimal version of the preceding equation

LXc(L�θL − dL) = L�θLXc L − dLXc L + L[Xc,�]θL = 0 , (5.191)

i.e., with L′ =: LXc L,L�θL′ − dL′ = L[�,Xc]θL . (5.192)

Now, using Eq. (5.119) and the Cartan identity, one can prove easily that

i[Xc,�]θL = i[Xc,�](SdL) = iS[Xc,�]dL = 0 , (5.193)

and hence the previous equation can be rewritten as

L�θL′ − dL′ = i[�,Xc]ωL . (5.194)


Again using the Cartan identity and the fact that, by virtue of S(�) = �, i�θL′ =L�L′, and this last equation can be rewritten as

i�ωL′ − dEL′ = i[Xc,�]ωL , (5.195)

which is the Hamiltonian version of Eq. (5.168), where, as before, EL′ = (L�−1)L′.As a consequence of this, we see immediately that the following holds:

Proposition 5.41 Let be � ∈ X(TM) a second-order field and L a regular, admis-sible Lagrangian for �, and let Xc be the infinitesimal generator of a one-parametergroup of point transformations (hence the tangent lift of a vector field X ∈ X(M)).Then,

L′ = LXc L (5.196)

will be an admissible Lagrangian for � iff � and Xc commute, i.e., iff

[Xc, �] = 0 . (5.197)

Remark 5.12 (i) Of course, nothing guarantees that L′ will be regular even if L is.(ii) At the finite level, [Xc, �] = 0 implies that � will be left unaltered by the one-

parameter group φ∗t � = �. The vector field Xc will be then the infinitesimalgenerator of a symmetry for the dynamics represented by �. More on this in thenext Subsection.

5.5.4 Symmetries, Constants of Motion and the Noether Theorem

The concept of symmetries (and of infinitesimal symmetries) for a second-orderfield has been introduced in the previous Subsection in the context, which is the mostrelevant for Lagrangian dynamics, of point-transformations, that are represented, atthe infinitesimal level, by tangent lifts of vector fields on the base manifold. Herewe will enlarge somewhat the discussion by not requiring transformations to benecessarily point-transformations.

We begin with some general and perhaps well-known definitions that we recallhere mainly for the sake of completeness. A ‘constant of motion’ for a vector field(a second-order one for us, but the definition is of course more general) � ∈ X(TM)

is any function F ∈ F(TM) such that

L�F = 0 . (5.198)

This is of course nothing but the familiar notion of vanishing of the ‘time’ derivativeof F along the trajectories of �, dF/dt = 0.

An ‘infinitesimal symmetry’ for � will be instead any vector field X ∈ X(TM)

such that


[X, �] = 0 , (5.199)

and this implies, as already stated, that � will be left invariant by the one-parametergroup φt ∈ Diff (TM) generated by X. Otherwise stated, φt will send trajectoriesinto trajectories (it ‘will permute the trajectories’ among themselves).

Recall the very useful relation among graded derivations valid for any pair ofvector fields X and Y :

i[X,Y ] = LX ◦ iY − iY ◦ LX . (5.200)

If � admits a Lagrangian description with a regular Lagrangian L, then there willbe a unique ωL-Hamiltonian vector field XF associated with F via iXF ωL =: dF.

Using then the previous identity andL�ωL = 0 we obtain i[XF ,�]ωL = −L� iXF ωL =−L�dF = −d(L�F), and hence,

L�F = 0⇒ [XF, �] = 0 , (5.201)

i.e., if F is a constant of motion, then the Hamiltonian vector field XF will be aninfinitesimal symmetry for �.

Vice-versa, let X be an infinitesimal symmetry for �, [X, �] = 0 (not necessarilyan ωL-Hamiltonian vector field). Then, the function

FX =: iX i�ωL = ωL(�, X) (5.202)

is a constant of motion. Indeed, using i�ωL = dEL, one can also write FX = LXEL,and then using [X, �] = 0,

L�FX = L�(LXEL) = LX(LXEL) = 0

In the particular case of X being a Hamiltonian vector field X = XG, then FXG =ωL(�, XG) = {EL, G} = XGEL.

Remark 5.13 This seems to be a general procedure for associating symmetries withconstants of motion, and vice-versa. However, there is no control on how effectivethe procedure may be, and in many relevant cases it may well turn to be empty, asthe following example shows. Let us consider the isotropic harmonic oscillator withthe standard Lagrangian, in the appropriate units,

L = 1

2δij(v

ivj − qiqj) , i, j = 1, . . . , n > 1 .

Then, the vector fields

Xij = vi ∂

∂qj+ vj ∂

∂qi− qi ∂

∂vj− qj ∂

∂vi, i �= j (5.203)


are all symmetries for the dynamics of the isotropic oscillator (see [Mm85]), and areassociated, in the sense specified above, with the components

Qij = vivj + qiqj (5.204)

of the ‘quadrupole tensor’ of the harmonic oscillator, that are all constants of motion,butLXi

jEL = 0, and the procedure we have just outlined for associating ‘backwards’

constants of the motion with symmetries is completely empty.

Let us turn now to Noether’s theorem. We begin with the following:

Definition 5.42 Let � be a second-order vector field admitting a (or at least of one)Lagrangian description with a regular Lagrangian L. A Noether symmetry for � is,at the infinitesimal level, a tangent lift Xc ∈ X(TM) such that

LXc L = i�π∗α , (5.205)

for a closed 1-form α ∈ X∗(M).

Otherwise stated,LXc L is a ‘total time derivative’, i.e., locally at least,LXc L = L�h,

h = π∗g, g ∈ F(M) (loosely speaking, LXc L = dg/dt).Observe that then LXcθL = θLXc L = θα and similarly LXcE = ELXc L = Eα = 0

and consequently

i([Xc, �])ωL = LXc (iγ ωL)− i�(iγωL) = d(LXc EL)

that shows that Xc is a symmetry for �, i.e.

[Xc, �] = 0. (5.206)

Proposition 5.43 (E. Noether) If Xc ∈ X(TM) is a Noether symmetry for � suchthat LXc L = L�h, then,

FXc = iXcθL − h (5.207)


Indeed,

L�FXc = L� iXcθL − L�h = iXcL�θL + i[�,Xc]θL − LXc L = iXc dL − LXc L = 0,

and this achieves the proof.

Remark 5.14

(i) By simply differentiating the equation defining FXc we find that

dFXc = iXcωL + LXcθL − dh.


But LXcθL = θLXc L = θL�h = dh, and hence,

iXcωL = dFXc , (5.208)

i.e., Xc is ωL-Hamiltonian with FXc as Hamiltonian function. This implies, ofcourse,

LXcωL = 0 , (5.209)

as well.(ii) Spelling out explicitly the equation i[Xc,�]ωL = 0, we obtain also easily

LXc EL = 0 . (5.210)

Noether symmetries are therefore a conspicuous example of the case in which Eq.(5.202) yields only a trivial constant of motion.

As an example, let us consider again the n-dimensional harmonic oscillator. Theassociated vector field is, in appropriate units,

� = vi ∂

∂qi− qi ∂

∂vi, (5.211)

which, introducing once again the collective coordinates ξ i, i = 1, . . . , 2n, withξ i = qi, i = 1, . . . , n, and ξ i = vi−n, i = n+ 1, . . . , 2n, can also be written as

� = �ij ξ

j ∂

∂ξ i, (5.212)

where the matrix � = (�ij) is given by

� =(

0 In

−In 0

), (5.213)

with each entry an n× n matrix, 0 the null matrix and In the n× n identity matrix.Consider then the linear vector field

XA = Aij qj ∂

∂qi∈ X(M) , M = R

n , (5.214)

with A = (Aij) any n × n numerical matrix with real entries. As A varies in the set

of n× n real matrices, the XA’s will generate the action (actually the representation)of the general linear group GL(n, R) on M = R

n. The associated tangent lifts willbe given by

XcA = Ai

j ξj ∂

∂ξ i, (5.215)


where

A = (Aij) =

(A 00 A

), (5.216)

and they will generate the tangent lift of the action of GL(n, R) to TM = R2n.

As, given any two linear vector fields defined bymatricesA andB,XA = Aij ξ

j ∂∂ξ i ,

XB = Bij ξ

j ∂∂ξ i , we have [XA, XB] = X[B,A], and in view of the fact that, with the

structure of the above matrices, [A, �] = 0 ∀A, we find at once that the full generallinear group GL(n, R) is a symmetry group for the dynamics of the linear harmonicoscillator.

Let us inquire now what is the rôle of this group in the Lagrangian frameworkwhen the harmonic oscillator is described by a Lagrangian of the type

LB(q, v) = 1

2Bij

(vivj − qiqj

),

as discussed in a previous Subsection. In terms of collective coordinates,

LB(q, v) = 1

2Bij ξ

i ξ j

with

B =(−B 0

0 B

). (5.217)

Recalling that, by construction, B has to be a symmetric matrix, some long butstraightforward algebra leads to

LXcALB = 1

2Cij ξ

i ξ j , (5.218)

where

C =(−C 0

0 C

)(5.219)

andC = BA+ AtB. (5.220)

It is pretty obvious that, unless C = 0, LXcALB will be a ‘quadratic’ form in the

velocities, and hence it cannot be a total time derivative. Therefore:

Proposition 5.44 The vector field XcA will be a (an infinitesimal) Noether symmetry

for the isotropic harmonic oscillator with the Lagrangian LB iff

C = BA+ AtB = 0 (5.221)


in which case, LXcALB = 0, and the associated Noether constant of motion will be

given byFA = iXc

AθLB = vhBhkAk

jqj (5.222)

Remark 5.15

(i) If we consider the orthogonal subgroup O(n) of the general linear group, whichis generated, at the infinitesimal level, by vector fields XA associated with skew-symmetric matrices (A+ At = 0), then the orthogonal transformations that arealso Noether symmetries for LB will be those (and only those) that are generatedby matrices A that are in the ‘commutant’ of B,

C(B) = {A | [A, B] = 0} .

If B = I (the ‘standard’ Lagrangian for the isotropic harmonic oscillator), thenthe whole of the orthogonal group will be also a group of Noether symmetries,and the associated constants of motion will be given byFA = vh δhk Ak

j qj . Thiswill not be the case when B �= I .

(ii) Of course, B being symmetric it can always be diagonalized via an orthogonaltransformation followed, if necessary, by a scale transformation. So, withoutloss of generality, we can always consider B to be of the form

B = diag{1, 1, . . . , 1,−1,−1, . . . ,−1} (5.223)n−m times m times

in which case the invariance group of the corresponding Lagrangian, and henceof Noether symmetries, is easily seen to be O(n−m, m). The consequences onthe association of symmetries with constants of motion will be investigated inthe following Subsection.

(iii) Notice that Eq. (5.221) expresses in general the (necessary and sufficient) con-dition under which the linear transformations (a one-parameter group obtainedby exponentiation) generated by A leave invariant, say, the quadratic form:

B = 1

2Bijq

iqj. (5.224)

As the standard orthogonal matrices are those leaving invariant the quadratic formcorresponding toB = I , it is natural to generalize the notion of a linear transformationbeing orthogonal to that of being ‘B-orthogonal’ whenever it satisfies (5.221) for ageneral symmetric matrix B. We can then rephrase the previous theorem by sayingthat the vector field Xc

A will be an infinitesimal Noether symmetry for the fharmonicoscillator with the Lagrangian LB iff A is B-orthogonal.

If B is positive, the B-orthogonal matrices will provide us with a realization of theorthogonal groupO(n) different from the standard one, a realization ofO(n−m, m) ifB has the signature as in Eq. (5.223). These realizations will be of course isomorphic


to the standard one, actually they will provide us with a different ‘placement’ ofO(n)

(orO(n−m, m))within the full general linear group, the two realizations being relatedby the conjugation defined by the orthogonal transformation that diagonalizes B.

5.5.4.1 Consequences on Noether’s Theorem of the Existence of AlternativeLagrangians

Noether’s theorem is commonly assumed to give a unique prescription for associatingconstants of motion with symmetries. This is not so if the same dynamical systemadmits of different and genuinely alternative Lagrangian description, as the followingexample shows.

Let us consider a three-dimensional isotropic harmonic oscillatorwith the standardLagrangian

L = 1

2[(v1)2 + (v2)2 + (v3)2 − (q1)2 − (q2)2 − (q3)2] , (5.225)

which is O(3)-invariant. The Noether theorem associates then the three conservedcomponents of the angular momentum with the infinitesimal generators of the rota-tion group O(3), and this is all standard material.

Consider instead the alternative, O(2, 1)-invariant, Lagrangian function:

L′ = 1

2[(v1)2 + (v2)2 − (v3)2 − (q1)2 − (q2)2 + (q3)2] . (5.226)

The (2+ 1) Lorentz group O(2, 1) has the infinitesimal generators:

K1 = q3∂

∂q1+ q1

∂

∂q3, K2 = q3

∂

∂q2+ q2

∂

∂q3, (5.227)

and

J = q1∂

∂q2− q2

∂

∂q2, (5.228)

corresponding respectively to the two ‘boosts’ in the q1 and q2 directions and tothe rotations in the (q1, q2) plane. They close on the Lie algebra o(2, 1) of O(2, 1),namely,

[K1, J] = K2, [K2, J] = −K1 , (5.229)

and[K1, K2] = J . (5.230)

The tangent lifts Kc1, Kc

2 and Jc will close on the same Lie algebra, of course.As already stated, the Lagrangian L′ is invariant under the action of Kc

1, Kc2 and

Jc. The associated Noether constants of motion turn then out to be


F1 = iKc1θL′ = q3v1 − q1v3

F2 = iKc2θL′ = q3v2 − q2v3 (5.231)

and, as in the previous case, iJcθL′ = q1v2 − q2v1. We see therefore that the (three)components of the angular momentum become the Noether constants of motionassociated with the generators of the Lorentz group if the Lagrangian of the harmonicoscillator is taken to be the O(2, 1)-invariant Lagrangian L′.

Remark 5.16 If the standard Lagrangian L is assumed in place of L′, two out of thethree generators of the (2+ 1) Lorentz group (the generators of the Lorentz boosts)are of course still symmetries for the dynamics, but no more Noether symmetries.We can use then Eq. (5.202) to evaluate the constants of motion associated with theLorentz boosts. They turn out to be

G1 =: LKc1EL = Q13, G2 =: LKc

2EL = Q23 , (5.232)

whereQ13 = v1v3 + q1q3, Q23 = v2v3 + q2q3 , (5.233)

i.e., the constants of motion turn out to be now two of the components of the quadru-pole tensor of the harmonic oscillator.

What this admittedly simple example shows is that an essential ingredient in theassociation of constants of motion with symmetries via Noether’s theorem is thechoice of a Lagrangian and that the association can become ambiguous wheneverthe latter is not unique.

5.5.5 A Relativistic Description for Massless Particles

Following Wigner’s program [Wi39], we take advantage of the fact that upon quan-tization our system should provide us with a unitary irreducible representation of thePoincaré group P . This suggest that this time we look for a Lagrangian descriptionwritten on the group manifold of P .

Now the carrier space will be the connected component of the Poincaré groupP↑+ ∼= R

4�L↑+ whereL↑+ is the proper orthochronous Lorentz group (see Sect. 4.6),

Eq. (4.108). We make the following identification of variables:

1. x = (t, x) ∈ R4 represents the physical coordinate in space-time.

2. p is the 4-momentum of the massless particle. It is defined in such a way thatwe take into account that the 3-momentum cannot be transformed to zero by anyLorentz transformation.

We introduce a basis (e0, e1, e2, e3) on R4 such that the Minkowski metric is

defined as:

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e20 = −1 ; ei · ej = δij ; e0 · ei = 0 .

We set:p = ω(�e0 +�e3) ,

where � ∈ L↑+ and ω is positive. With the help of the above coordinates theLagrangian is given by:

L = pμxμ + λTr (T21�−1�),

where T12, T23, T31, are the generators of the subgroup of the space rotations. Thetrace operation is actually a scalar product in L↑+. By writing the Lagrangian in moredetail:

L = xμ(�(e0 + e3))μ + λTr (T21�−1�)

we see that L is invariant under the left action of P , once we notice that x transformslike the dual of�(e0+e3) = p. To find the equations of motion we proceed as usual,that is consider the Euler-Lagrange equations:

d

dtiXt θL = LXt L .

The 1–form θL associated with the Lagrangian is

θL = pμdxμ + λTr (T21�−1d�).

We use X = aμ∂/∂xμ, aμ ∈ R, and we find for any a ∈ R4,

d

dt(pμaμ) = 0 ,

which implies that dpμ/dt = 0. By using vector fields Y ’s, infinitesimal generatorsof the Lorentz group, and defining matrices TY , in the Lie algebra of the Lorentzgroup, as

Tμν = iYμν d� ·�−1 ,

we find

LY � = iY d� = (iY d��−1)� = TY�,

and also

iY�−1d� = �−1(iY d��−1)� = �−1TY�.


These right invariant generators define the left action. Therefore LYt L = 0 and weget

d

dtiYθL = 0.

We notice that LY (pμxμ) = 0 because it preserves L. Therefore (LY pμ)xμ =−pμLY xμ = −pμiY dxμ and we can compute now iYθL to be

iY θL = iYλTr (T21�−1d�)− (LY p) · x

= λTr (T21�−1iY d��−1�)− x · TY�(e0 + e3)

= λTr (T21�−1TY�)− x · TY (e0 + e3) ,

and we find also

iYμν θL = λTr (T21�−1Tμν�)− x · Tμν�(e0 + e3),

i.e.,d

dt(λ(�μ1�ν2 −�μ2�ν1))+ zμpν − zνpμ) = 0.

As aμ and Tμν are a basis for the algebra of the Poincaré group, the equationsof motion are provided by the conservation of pμ, and the conservation of Jμν =λ(�μ1�ν2 −�μ2�ν1))+ zμpν − zνpμ. By looking at θL and dθL it is immediatelyclear that the dynamics actually takes place on the coadjoint orbit of the Poincarégroup passing through (e0 + e3, T21). The stability group of this element underthe coadjoint representation is provided by the translation group generated by thefollowing elements

π1 = T10 + T13; π2 = T20 + T23

and the rotation associated with T21. We then get the Euclidean group E(2). Thuswe obtain

L↑+/E(2) = R× S2.

5.6 Feynman’s Problem and the Inverse Problemfor Lagrangian Systems

5.6.1 Feynman’s Problem Revisited

We examine now Feynman’s problem in the framework of tangent bundles. As wasdiscussed in the previous section, this is the natural setting for Lagrangian mechan-ics. Thus it makes sense to try to describe all dynamical vector fields which are

5.6 Feynman’s Problem and the Inverse Problem for Lagrangian Systems 361

Hamiltonian with respect to some Poisson structure subject to certain fundamentalconditions. Then, we can ask whether all of them are associated with an Euler–Lagrange SODE defined by a Lagrangian. Finally, we consider the local structure ofsuch Lagrangian.

5.6.1.1 Localizability

We will be discussing systems whose classical phase space is the tangent bundle ofa configuration space Q. The first condition that we will impose on Poisson tensors� defined on TQ is that they will describe localizable systems, i.e.,

{ qi, qj } = 0 . (5.234)

This condition was used in the Feynman procedure as presented in Sect. 4.5.2, andlater on was a crucial ingredient for the introduction of interactions Eq. (4.73). Froma physical point of view, this condition reflects that the classical limit of the quantumtheory defines a configuration space whose points can be used to label the quantumstates of the system. This notion is thus borrowed from quantum theory and anotherway of stating it is simply that position operators commute. Geometrically, thismeans that the algebra of observables on Q is an Abelian subalgebra of the fullalgebra F(TQ).

Definition 5.45 Wewill say that a Poisson tensor� onTQ is localizable if τ∗(F(Q))

is an Abelian Poisson subalgebra of F(TQ).

Notice that if � is localizable, then the Hamiltonian vector fields correspondingto coordinates on the configuration space Q commute. On the other hand it is clearthat if a Poisson dynamical system� has a Lagrangian realization, the Poisson tensor� is localizable. Because of this, this condition is called ‘variational admissibility’in [No82] and was used to show the existence of a (local) Lagrangian description forcertain Poisson dynamical systems.

In other words, if the Poisson tensor � is given by

� = aij ∂

∂qi∧ ∂

∂qj+ bij ∂

∂qi∧ ∂

∂vj+ cij ∂

∂vi∧ ∂

∂vj, (5.235)

localizability amounts to aij = 0. But these conditions are equivalent to imposing therequirement that the Hamiltonian vector fields Xqi corresponding to the coordinateson Q be vertical. In fact,

Xqi = bij ∂

∂vj+ 2aij ∂

∂qk, (5.236)

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which means that Xqi are vertical if and only if aij = 0. Moreover, the vector fieldsXqi commute among themselves, then

[Xqi , Xqi ] = 0 ⇐⇒ bij ∂bkl

∂vj − bkj ∂bil

∂vj = 0 .

5.6.1.2 Hamiltonian Second-Order Equations

Another assumption which is often implicitly assumed is that there are dynamicalsystems which are SODE’s on TQ and which are Hamiltonian with respect to a givenPoisson tensor �. For instance, in the derivation of Feynman’s results in Sect. 4.5.2and later on Sect. 4.5.3, the existence of a SODEwhichwas Hamiltonian with respectto � was assumed. In general, this need not to be the case. In this regard, considerthe example of TR

2, with coordinates (q1, q2; v1, v2) and the Poisson tensor � =∂/∂q1 ∧ ∂/∂v1. For this case, there is no SODE � such that, � = −�(dH).

Definition 5.46 We will say that a Poisson tensor � in TQ is Newtonian if thereexists a SODE � such that � = XH for some Hamiltonian function H on TQ, inother words,

� = −�(dH) . (5.237)

Notice that there is a weaker assumption on the Newtonian character of �, whichis as follows: Suppose that there is a SODE � such that L�� = 0. Then we will saythat � is canonical. If � is Newtonian then the latter condition automatically holds,but the converse is not necessarily true. In the previous example on R

2 it is easy tocheck that the SODE � = v1∂/∂q1 + v2∂/∂q2 is canonical.

From Eq. (4.33), � being canonical implies that

�{F, G } = {�(F), G } + {F, �(G) } , (5.238)

i.e., the Poisson bracket is compatible with the dynamics defined by �, but thisdoes not guarantee the existence of a Hamiltonian function for �. However if � isnondegenerate, it is sufficient that � be canonical to insure the existence of a (local)Hamiltonian function. Notice that if � has a regular Lagrangian description, thenautomatically is a SODE and � is Newtonian.

Then we can state the result as follows:

Lemma 5.47 If � is a canonical SODE in TQ for a localizable Poisson tensor �,then the bracket { qi, vj } = bij is symmetric. In particular if � is Newtonian andlocalizable the same result holds.

Proof If� is localizable we have that { qi, qj } = 0. However because of Eq. (5.238),we get

0 = �{ qi, qj } = {�(qi), qj } + { qi, �(qj) } = −bji + bij. ��

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Another important observation concerning Hamiltonian SODE’s is the following:

Lemma 5.48 If � is a Hamiltonian SODE and α is a non-null basic 1-form, then�(α) �= 0. In particular, there are no nonconstant basic Casimir functions.

Proof Let α = αi dqi a basic form. Then,

i�(α)dH = �(α, dH) = i�α = αivi = α(q, v),

and consequently, if �(α) = 0 then α = 0.In particular, if there is a basic Casimir function φ, then �(dφ) = 0 and therefore

dφ = 0. ��Conversely, it can be easily seen that if � is a canonical SODE for � and it has

basic Casimir functions, then � must be nonregular. Otherwise, a canonical SODE� would be Hamiltonian and no basic Casimir functions would exist.

5.6.1.3 Regular Poisson Tensors

Let us recall the important rôle played by the condition iSω = 0 in the geometryof closed 2-forms on TQ. As it was indicated in Theorem5.39, this condition isnecessary and sufficient to insure the existence of a (local) Lagrangian L such thatω = ωL . Has this condition similar implications for Poisson tensors?

To answer this question we first make the following observation:

Lemma 5.49 If � is localizable and there is a SODE � which is canonical (inparticular if it is Newtonian), then iS� = 0.

Proof Consider � as in Eq. (5.235). Then, a simple computation shows that,

iS� = 2ail ∂

∂vi∧ ∂

∂ql+ bil ∂

∂vi∧ ∂

∂vl.

Thus, if � is localizable, ail = 0, and if � is canonical, (Lemma 5.47) bil is sym-metric. ��

Notice that the converse is not true. (To prove this result one can use the aboveexample.)

Thus, the condition iS� = 0 is too weak to insure the existence of a localLagrangian since it does not first insure the regularity of �. On the other hand, wehave seen that the condition that it is localizable plus Newtonian implies iS� = 0.In fact, we can show that these conditions are sufficient to insure that � is regular.

Theorem 5.50 If � is a localizable and Newtonian Poisson tensor on TQ, then �

is regular, and hence it defines a symplectic structure on TQ.


Proof Notice that �∧n = � ∧ · · · ∧� is proportional to

det(bij)∂

∂q1∧ · · · ∧ ∂

∂qn∧ ∂

∂v1∧ · · · ∧ ∂

∂vn.

Thus, � will be regular iff det(bij) �= 0.We shall show it by proving that � defines a monomorphism when restricted

to closed basic 1-forms. In fact, let F ∈ F(U), where U is an open set in Q, andf = τ ∗F ∈ F(τ−1(U)). Computing �(df ) we find

�(df ) = bij ∂f

∂qi

∂

∂vj, (5.239)

where we have used that � is localizable, and hence aij = 0. On the other hand, if �

were not a monomorphim on closed basic 1-forms, this would imply the existenceof a function φ on an open set U in Q such that �(dφ) = 0, i.e., that φ is a (locallydefined) Casimir basic function. However, because of Lemma 5.48, this cannot occurif � is Newtonian. Thus det bij �= 0, because of Eq. (5.239). ��

It is interesting to point out that if � is nondegenerate, it does not necessarilyfollow that � is localizable and/or Newtonian. If fact, the Poisson tensor � =∂/∂q1 ∧ ∂/∂q2 + ∂/∂v1 ∧ ∂/∂v2 is neither one nor the other.

Therefore, if there exists such a SODE vector field � the Poisson tensor is invert-ible. When writing it in block matrix form

(�) =(

0 b−b c

), (5.240)

we find the inverse matrix given by

(ω) =(

b−1cb−1 −b−1b−1 0

), (5.241)

and we have a closed non-degenerate 2-form ω by inverting the Poisson structure �

as described in Sect. 5.2.1, Eq. (5.1). We can write ω as

ω = Gijdqi ∧ dvj + Fijdqi ∧ dqj , (5.242)

whereGij = bikcklblj; Fij = −bij ,

and bikbkj = δji .

On the other hand, if iS� = 0 and ω is the inverse of �, then iSω = 0 andconversely. Consequently, by Them 1, if there is a second order differential equationvector field � which is Hamiltonian with respect to a localizable Poisson structure


�, then there will be a (locally defined ) Lagrangian function L in TQ such that� is the dynamical vector field defined by L. In this sense, as indicated by Dyson,Feynman’s approach does not go beyond the Lagrangian formalism and internaldegrees of freedom should be considered for such a purpose. Notice that we havemade no use of further conditions, such as { qi, vj } = δij [Hu92]. These results alsoprovide the general setting for the remarks in [No82]. Thus the impossibility of goingbeyond the Lagrangian formalism using Feynman’s procedure on tangent bundlesis a fundamental restriction. Nevertheless, the Lagrangian functions arising in thismanner can be quite different from the standard ‘mechanical’ type (in contrast withthe result in [Mo93]). We will see however that this is not the case, if we impose anadditional physical condition on �.

5.6.1.4 Dynamical Independence of the Poisson Brackets from the Stateof Motion of Test Particles

From a physical point of view, it is reasonable to assume that the fields acting on aparticle are external. This means that if we are using a test particle to probe the field,then the field itself will not depend on the state of motion of the test particle, i.e., itwill not depend on the velocity of the particle. In this sense, we can assume that thePoisson tensor � will not depend on the velocities. In other words, translation alongthe fibres of TQ will be Poisson maps for �. Yet another way of stating this is: Thecoefficients �ij of the Poisson tensor � in natural coordinates qi, vi are defined onthe configuration space.

Definition 5.51 We will say that a Poisson tensor is velocity independent if L∂/∂vi

� = 0, for all i = 1, . . . , n, i.e., if translations along the velocities are Poisson maps.

We have seen that the condition of localizability and that � is Newtonian are notequivalent to that of the regularity of �. In spite of this, we can show the following:

Theorem 5.52 If � is a non-degenerate Poisson tensor such that iS� = 0 and it isvelocity independent, then � is Newtonian.

Proof The matrix bij is non-degenerate because � is regular. We will denote theinverse of bij by bij. Because iS� = 0, then bij is symmetric, and because � isvelocity independent bij = bij(q) is a basic function. Then, H = 1

2bij(q)vivj is suchthat

XH = −�(dH) = vi ∂

∂qi+

(1

2vivjblm ∂bij

∂ql+ 2clmbilv

i)

∂

∂vm,

which is a Hamiltonian SODE. ��Notice that if the Poisson tensor� is velocity independent, then the transformation

from one reference frame to another, having uniform velocity with respect to theoriginal frame, is canonical. If the Poisson tensor � is velocity independent, thenthe functions Gij and Fij depend only on the coordinates qi. Using the results in


the previous section (Theorem5.39), we know that there exists a semibasic 1-formθ = Ai(q, v) dqi such that dθ = ω. But then

Gij = ∂Ai

∂vj,

and,

Fij = ∂Ai

∂qj− ∂Aj

∂qi.

Because Gij depends on just the q’s, then Ai(q, v) = Gij(q)vj + ai(q). On the otherhand (again from Theorem5.39), we get that Ai = ∂L/∂vi. Then integrating oncemore, we get

L = 1

2Gij(q)vivj + ai(q)vi − V(q) ,

which is a mechanical type Lagrangian.We thus conclude:

Theorem 5.53 If � is a localizable, Newtonian and velocity independent Poissontensor on TQ, then � defines a family of Hamiltonian SODE’s whose Lagrangiansare of mechanical type differing on the potential term, and whose Cartan 2-form isthe inverse of �.

5.6.2 Poisson Dynamics on Bundles and the Inclusion of InternalVariables

So far we have proved that the generalization of Feynman’s attempt at constructinglocalizable second-order Hamiltonian dynamics on tangent bundles always leads(locally) to Lagrangian systems with a well defined structure. Nevertheless, inSects. 4.5.3 and 4.5.4 we have constructed examples of dynamics which are notof the Lagrangian type (at least at first sight). This was done by introducing internaldegrees of freedom.Wewill discuss such a situation inwhat follows. For that purposewe will first need to understand some geometrical aspects of Poisson structures onbundles, which now follow.

5.6.2.1 Poisson Structures on Bundles: Localizability

We shall consider a bundle Eπ→ B over the base manifold B. We can assume for

simplicity that E is a vector bundle, but this is not strictly necessary for what follows.If we are given a Poisson tensor�B on the base space B, we can ask for the existenceof a Poisson tensor �E on the total space E such that π∗(ζ )�E(ζ ) = �B(π(ζ )),ζ ∈ E. We will then say that �E is π -related to �B. This is also equivalent to

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saying that the map π is a Poisson map from E to B. In fact, upon introducing localcoordinates bi for B and ζa along the fibres of E, we can then write

�E = �ijB(b)

∂

∂bi∧ ∂

∂bj+ Ai

a(b, ζ )∂

∂bi∧ ∂

∂ζa+ Cab(b, ζ )

∂

∂ζa∧ ∂

∂ζb, (5.243)

where the first term on the r.h.s. of Eq. (5.243) is simply the local expression for �B.Equivalently, the fact that�E isπ -related to�B implies that themapπ∗ : F(B)→

F(E) induced by π is a Lie algebra homomorphism.A particular case arises when �B = 0, i.e., the base manifold carries a triv-

ial Poisson bracket. In such a case, the possible Poisson tensors on E which areπ -related to it have the general form

�E = Aia(b, ζ )

∂

∂bi∧ ∂

∂ζa+ Cab(b, ζ )

∂

∂ζa∧ ∂

∂ζb. (5.244)

For instance, if B is the configuration space Q for a physical system, and E is thecorresponding tangent bundle, E = TQ, then Poisson tensors �TQ π -related to�Q = 0 are those such that �TQ is localizable.

Another example that will concern us here is the introduction of internal variablesfor a system defined on a configuration space Q. Internal variables are modeled bydefining a bundle ρ : F → Q. Attached to each base point m ∈ Q there is a fibreρ−1(m) that can be a representation space M for a group of gauge symmetries. Againthe Poisson bracket on Q is trivial, and a Poisson tensor �I which is ρ-related to itwill have the form

�I = Aia(q, I)

∂

∂qi∧ ∂

∂Ia+ Cab(q, I)

∂

∂Ia∧ ∂

∂Ib, (5.245)

where Ia denote local coordinates along the fibres of F.If the Ia’s are to be considered as internal variables, they should be localized at

points in configuration space. This means that they must be ‘measurable’ simultane-ously with position operators, i.e.,

Aia = { qi, Ia }I = 0 . (5.246)

We will call such a localizability property for Poisson tensors on F ‘localizability onthe bundle’, or just ‘localizability’, which we trust will not cause any confusion. Inthis case, Eq. (5.245) reduces to

�I = Cab(q, I)∂

∂Ia∧ ∂

∂Ib. (5.247)

Notice that �I is a Poisson tensor iff [�I ,�I ] = 0. This condition is true for allpoints q ∈ Q. This then implies that on each fibre ρ−1(q), the Poisson tensor �I


induces a Poisson tensor structure which is obtained from �I by fixing q. We canthen conclude that a localizable Poisson structure on a bundle F → Q is equivalentto a bundle of Poisson manifolds, but the Poisson structure can change from fibre tofibre.

Definition 5.54 A localizable Poisson tensor on a fibre bundle ρ : F → Q is abundle Poisson tensor, i.e., a smooth assignment to each fibre of a Poisson tensor�I .

Notice that Hamiltonian vector fields on such a bundle are vertical vector fields,with their restriction to each fibre being Hamiltonian in the corresponding Poissonstructure. A special case of such localizable Poisson tensors on F occurs when thetransition functions of the bundle F are Poisson maps. Then we could say that thebundle F → Q is Poisson and the standard fibre is a Poisson manifold modeling allthe others.

Examples of this situation occur when we consider a principal fibre bundle Pover Q with structure group G. If Q is simply a point, then P is the group itself, andthe Poisson structure �P is nothing but a Poisson structure on G. This case will bediscussed in detail in Sect. 5.6.4.

Let us instead consider the coadjoint representation of G on the dual g∗ of theLie algebra g of G. Then we can construct the associated bundle ad ∗P → Q, withstandard fibre g∗. Recall that this bundle is obtained as the quotient of P× g∗ by thenatural action of G on it. On g∗ we consider the linear Poisson tensor,

�g = cijkxk

∂

∂xi∧ ∂

∂xj, (5.248)

where xi denote linear coordinates on g∗ and cijk are the structure constants of g on

a basis Ei which is dual to the basis defined by xi on g∗. We then define a Poissontensor � on P × g∗ as the direct sum of the zero tensor in P and �g. This Poissontensor is G-invariant. Then it will induce a localizable Poisson tensor on ad ∗P. If thebundle P → Q were already carrying a G-invariant localizable Poisson tensor �P,we could form �P +�g to obtain a new localizable Poisson tensor on ad ∗P.

Alternatively, we can proceed as follows: Let �P be a G-invariant localizablePoisson tensor on P and T : G→ V a linear representation of G on V . Then in theassociated vector bundleET = P×V/G, there is a localizable Poisson tensor inducedfrom �P. Some specific examples of this situation will be discussed in Sect. 5.6.4.

5.6.2.2 Extension of Poisson Structures and Internal Variables

We note that if E → B is a bundle, then it could happen that there are no smoothsections. For instance, if E → B is a principal bundle, then there are no smooth, norcontinuous, sections unless the bundle is trivial. In physical terms, this means thatit is not possible to globally fix the state of the inner variables in a continuous way.In this sense, there is no way to ‘switch off’ the inner variables and to restrict our


attention to a system without them. On the other hand, if E → B is a vector bundle,i.e., if the inner variables are defined in a vector space, then there are always smoothsections. In fact, there is the zero section which assigns to each inner variable thezero value, thereby switching off the internal variables.

Choosing a section σ : B→ E of the projection map π provides a different linkbetween a Poisson tensor �E on E and a Poisson tensor �B on B. We will say that�E is an extension of �B if the map σ is Poisson, or equivalently if the induced mapσ ∗ : F(E) → F(B) is a Lie algebra homomorphism. This will mean in particularthat if E is a vector bundle and σ is the zero section, then in bundle coordinates ξ i

and ζ a, the Poisson brackets will have the form

{ ξ i, ξ j }(ξ, 0) = �B(ξ), { ξ i, ζ a }(ξ, 0) = 0, { ζ a, ζ b }(ξ, 0) = 0.

Whenever this accretive condition on inner variables is required from physical prin-ciples, we will impose in addition that the Poisson tensor �E will be an extension of�B with respect to the zero section.

5.6.2.3 Feynman’s Problem in Tangent Bundles with InnerDegrees of Freedom

We are now prepared to try to determine all Poisson structures on a tangent bundlewith internal degrees of freedom and satisfying some fundamental physical restric-tions. We will follow a similar chain of reasoning as that in Sect. 5.6.1.

Our setting will be that of a system with configuration space Q and possessinginternal degrees of freedom modeled on a bundle ρ : F → Q. We can pull back thebundle τQ : TQ → Q along the map ρ, to obtain a new bundle E = ρ∗(TQ) overF. The pull-back bundle ρ∗(TQ) is defined as the set of pairs (I, v) where I ∈ F,v ∈ TQ, and their base points agree, i.e., ρ(I) = q = τQ(v). There is a natural mapp1 : ρ∗(TQ) → F, sending each pair (I, v) into its first component p1(v, I) = v.Then the fibre of ρ∗(TQ) at the point I ∈ F is simply the vector space TqQ withq = ρ(I). Thus p1 : ρ∗(TQ) → F is a vector bundle with the fibres of TQ → Q,p−11 (I) = Tρ(I)Q (see Fig 5.6.). Notice that if we take the pull-back of the bundleF → Q along the map τQ : TQ→ Q, then we obtain the bundle τ ∗Q(F) made up ofpairs (v, I) with the same property as before, and projection map p2 : τ ∗Q(F)→ TQgiven by p2(v, I) = v. The total spaces ρ∗(TQ) and τ ∗Q(F) are the same (this will bedenoted by TQ &' F), but the fibre structures are different. The fibre of TQ &' F asa bundle over TQ with projection map p2 is given by Fq = ρ−1(q).

We will be searching for Poisson tensors � on TQ &' F = ρ∗(TQ) that will bep1-related to a localizable Poisson tensor �I on F → Q given by Eq. (5.247). SuchPoisson tensors will be called localizable on TQ &' F. This requirement is equivalentto � to having the form


Fig. 5.6 Internal variablesand configuration bundles ρ ∗ (TQ )

p2

p1

TQ

τQ

F ρ Q

τ ∗Q(F )

r2=p1

r1=p2

E

ρ

TQ τqQ

� = bij(q, v, I)∂

∂qi∧ ∂

∂vj+ cij(q, v, I)

∂

∂vi∧ ∂

∂vj

+ Aia(q, v, I)

∂

∂vi∧ ∂

∂Ia+ Cab(q, I)

∂

∂Ia∧ ∂

∂Ib. (5.249)

Notice that the fundamental Poisson brackets of the variables qi, vi and Ia can alldepend on q, v and I , except for Cab which depends only on the q’s and I’s.

We have discovered in the previous section that the existence of a HamiltonianSODE for a Poisson tensor has important consequences. Something similar happensnow. Even though the bundle p1 : TQ &' F → F is not a tangent bundle, there is anatural tensor S on it that plays the same rôle as the tangent structure on TQ discussedin Sect. 5.5.3. Locally, this tensor S has the same form as the tangent structure onTQ, i.e.,

S = ∂

∂vi⊗ dqi . (5.250)

It is easy to show that S is well defined, either by direct computation or by noticingthat there is a natural projection � from TF to E = ρ∗(TQ), given by

� : (qi, Ia; vi, Ia) �→ (qi, Ia; vi), (5.251)

and that the natural tangent structure SF on TF is projectable along this map (seeFig. 5.7). The projected tensor results from S given by Eq. (5.250).

Fig. 5.7 The almost tangentstructure on TQ &' F

F Q

TQ �� F TQ

TF

ρ

p2

p1 τQ

Π

τF

Tρ


Wewill be considering dynamical systemswhich are second-order in the variablesqi and vi and first-order on the variables Ia. The corresponding vector fieldswill againbe called second-order differential equations (SODE’s) on E, or also driven SODE’s[Ib91]. These vector fields � are characterized by S(�) = �, where � is the dilationfield along the fibres of ρ∗(TQ), i.e., � has the form vi∂/∂vi. The most general formof a driven SODE is given by

� = vi ∂

∂qi+ Fi(q, v, I)

∂

∂vi+ fa(q, v, I)

∂

∂Ia, (5.252)

and its integral curves satisfy the coupled system of differential equations:

qi = Fi(q, v; I) , (5.253)

Ia = fa(q, v; I) . (5.254)

Compare these equations with Eq. (4.72).We will denote by FTQ&'F(Q) the functions on E = TQ &' F obtained by pull-

back of functions on Q, FTQ&'F(Q) = (ρ ◦p1)∗F(Q), i.e., functions depending onlyon q’s. We will denote by FTQ&'F(F) the functions on E obtained by pull-back offunctions on F by the map p1, i.e., functions on E depending on both q’s and I’s.

Proposition 5.55 Let � be a localizable Poisson tensor on TQ &' F (Eq.5.249). If� is a Hamiltonian SODE, then there are no nonconstant locally defined Casimirfunctions on FTQ&'F(Q). Moreover, if the Poisson tensor �I is regular along the fibresof ρ, then there are no nonconstant locally defined Casimir function s on FTQ&'F(F)

either.

Proof Let φ be a locally defined Casimir function on FTQ&'F(Q). Then �(φ) = 0if � is a Hamiltonian SODE on E. But clearly this implies that ∂φ/∂qi = 0, as inLemma 5.48.

On the other hand, if φ is a locally defined Casimir function on FTQ&'F(F), then�(φ) = 0 again because � is Hamiltonian. Because of Eq. (5.252), this implies that

0 = vi ∂φ

∂qi+ fa(q, v, I)

∂φ

∂Ia.

Since φ does not depend on v, fa(q, v, I) = fia(q, I)vi and

∂φ

∂qi+ fia(q, I)

∂φ

∂Ia= 0 . (5.255)

On the other hand, if φ is a locally defined Casimir function on FTQ&'F(F), then�(dφ) = 0, and this implies that

(bij ∂φ

∂qj− Ai

a∂φ

∂Ia

)∂

∂vi+ Cab

∂φ

∂Ia

∂

∂Ib= 0 .

http://dx.doi.org/10.1007/978-94-017-9220-2_4


Then Cab(q, I)∂φ/∂Ia = 0. But if �I is non-degenerate along the fibres of ρ, therecannot be nonconstant locally defined Casimir functions along the fibres of ρ, i.e.,∂φ/∂Ia = 0. Hence because of Eq. (5.255), ∂φ/∂qi = 0. ��

As in the previous section we will say that a Poisson tensor on TQ &' F isNewtonian if it possesses a Hamiltonian SODE.

What about the regularity of � on TQ &' F? To answer this question, we canrepeat the discussion leading to Theorem 5.50, but first we notice that:

Lemma 5.56 A localizable Poisson tensor � on TQ &' F of the form given by Eq.(5.249) will be regular iff bij and Cab are invertible.

Proof Thematrix defined by� on the basis constructed from ∂/∂qi, ∂/∂Ia and ∂/∂vi

is ⎛⎝ 0 0 b

0 C A−bt −At c

⎞⎠ ,

which is invertible iff b and C are. ��Theorem 5.57 If � is a localizable Poisson tensor on TQ &' F such that the Poissontensor �I on the configuration bundle F → Q is regular along the fibres, andpossessing a Hamiltonian SODE, then � is regular.

Proof It is easy to show that bij is invertible using locally defined Casimirs onFTQ&'F(Q). In fact, � must be a monomorphism acting on FTQ&'F(Q). (If not therewould be locally defined Casimir functions onFTQ&'(Q). This is impossible becauseof Proposition5.55.) On the other hand, it is easy to check that �(dφ) is againbij∂φ/∂qi∂/∂vi, implying the invertibility of bij.

Finally, � is also a monomorphism on FE(F), because if �I is regular alongthe fibres there are no locally defined Casimir functions on F(F) (Proposition5.55again). Then, computing �(dφ) we find

�(dφ) = Cab∂φ

∂Ia

∂

∂Ib+

(bij ∂φ

∂qj− Ai

a∂φ

∂Ia

)∂

∂vi,

then, Cab must be invertible. ��From this result we cannot conclude the existence of a Lagrangian function on

TQ &' F, because it is not a tangent bundle. However we could try to constructa singular Lagrangian L on TF, such that after some reduction the Cartan form ωLinduces a formω onE that is the inverse of�. The necessary and sufficient conditionsfor this to happen were established in [Ib91] in a situation slightly different from theone we are dealing with. We will then state it, adapting it to the present context, asfollows:


Theorem 5.58 Let p1 : TQ &' F → TQ be the vector bundle obtained by pull-backfrom a bundle F over Q, and let p2 : TQ &' F → F be the natural projection on thesecond factor. Let S be the natural (1, 1)-tensor field on TQ &' F given by Eq. (5.250)that projects onto the natural tangent structure SQ on TQ along the map � definedby Eq. (5.251). If ω is a symplectic form on TQ &' F, then there will exist a (locallydefined) function L on TF such that the Cartan 2-form ωL will project onto ω along�, if and only if the following two conditions are satisfied:

i. iSω = 0.ii. ω(V1, V2) = 0 for all vector fields vertical with respect to p2, i.e., vectors tangent

to the tangent spaces TqQ ⊂ TQ &' F.

Proof The necessity of conditions (i) and (ii) is almost obvious. LetL be aLagrangianfunction on TF such that ωL projects onto ω along�. Then, ker ωL = ker T�. Fromthe fact thatωL vanishes on vertical bivectors on TF, we get condition (ii). In general,because of Eq. (5.158), ωL is such that iSF ωL = 0, where SF is the tangent structureon TF. We get S by projecting SF along the map �, and consequently condition (i)is satisfied.

Conversely, let us assume that there is a symplectic form ω on TQ &' F satisfying(i) and (ii). Let � = �∗ω be the pull-back of ω to TF. Then we will show thatiSF � = 0. In fact,

iSF �(X, Y) = �(SF(X), Y)+�(X, SF(Y)) ,

for any two vector fields X and Y on TF. Taking X = ∂/∂qi and Y = ∂/∂qj, we get

iSF �

(∂

∂qi,

∂

∂qj

)= �

(∂

∂vi,

∂

∂qj

)+�

(∂

∂qi,

∂

∂vj

)

= ω

(∂

∂vi,

∂

∂qj

)+ ω

(∂

∂qi,

∂

∂vj

)= iSω

(∂

∂qi,

∂

∂qj

)= 0 .

If we take now X = ∂/∂qi and Y = ∂/∂Ia, we get

iSF �

(∂

∂qi,

∂

∂Ia

)= �

(∂

∂vi,

∂

∂Ia

)+�

(∂

∂qi,

∂

∂ Ia

)

= ω

(∂

∂vi,

∂

∂Ia

)= iSω

(∂

∂qi,

∂

∂Ia

)= 0 .

Finally consider X = ∂/∂Ia and Y = ∂/∂Ib. Then,

iSF �

(∂

∂Ia,

∂

∂Ib

)= �

(∂

∂Ia,

∂

∂ Ib

)+�

(∂

∂ Ia,

∂

∂Ib

)= 0 .

The evaluation of iSF � on all the remaining pairs of basis vectors vanishes becausethey always contain a vertical vector field. Then because of Theorem5.39, there will


exist a (locally) defined Lagrangian function L on TF such that � = ωL , completingthe proof. ��

Thus, we can finally state:

Theorem 5.59 If � is a Newtonian localizable Poisson tensor on TQ &' F suchthat the Poisson tensor �I on the configuration bundle F → Q is regular along thefibres, then there is a Lagrangian realization of it on TF, in the sense that there existsa locally defined Lagrangian function L on F → Q, such that � is obtained by directreduction from ωL.

In thiswaywehave obtained another no-go theoremanalogous to that of Feynman.Thus even with the inclusion of inner degrees of freedom we do not escape theLagrangian formalismwhenever the Poisson brackets on the inner degrees of freedomare regular. The latter need not be the case as was seen in Sects. 2 and 3. In fact, forparticles in a Yang-Mills field or for spinning particles, the inner degrees of freedom(the isospin and spin, respectively) are defined in a space with degenerate Poissonbrackets. However, in both situations, we can restrict the classical inner degreesof freedom to a subbundle of F made up of symplectic leaves on each fibre; forexample, spheres for the spinning particle and coadjoint orbits for the Yang-Millsparticle, with the restriction that �I is regular. Then we will apply the previoustheorem to this bundle obtaining a Lagrangian representation for the system. Noticethat the Lagrangian will be singular and only locally defined if the topology of thefibres of the symplectic bundle is complicated enough.

5.6.3 The Inverse Problem for Lagrangian Dynamics

5.6.3.1 The Inverse Problem of the Calculus of Variations

In previous Sections we have given examples of second-order vector fields thatadmit more than one Lagrangian description with genuinely nonequivalent, i.e., notdifferingmerely by a ‘total time derivative’ (a gauge term), Lagrangians, andwe havealso mentioned that there are second-order fields for which no non trivial Lagrangiandescription is available.

The problem of whether or not a given second-order field � admits of Lagrangiandescriptions and, in the affirmative case, whether or not the description is uniqueconstitutes what is commonly known as the ‘Inverse Problem’. The full name thatone finds in the literature is actually ‘Inverse Problem in the Calculus of Variations’,and that for the historical reason that the problem was stated for the first time byHelmoltz back in 1887 within the setting of Hamilton’s principle.

It has been established long ago [CS55] that the Inverse Problem for one-dimensional systems has actually infinitely many solutions. So, we will considerhere second-order dynamical systems in two or higher space dimensions.


We will discuss briefly here only two relatively modern approaches to the InverseProblem. The first one has to do with the uniqueness of the Lagrangian descriptiononce a (regular) Lagrangian is available. The second one deals instead also with thevery existence of Lagrangian descriptions (if any).

To illustrate the first approach, let us assume that a given second-order field admitsof a Lagrangian description with a regular Lagrangian L, and let us denote by ωL

the associated Lagrangian 2-form and by H = (Hij) the Hessian matrix of L (hence:det H �= 0 by assumption).

If a second Lagrangian exists, say L′ with associated Lagrangian 2-form ωL′ , wecan define a (1, 1) tensor R, called also the recursion operator associated with thetwo Lagrangians via:

ωL(RX, Y) =: ωL′(X, Y) (5.256)

for any pair X and Y of vector fields on the tangent bundle.The main properties of the recursion operator are reviewed in Appendix F, and

we will only summarize them here, referring to the Appendix (and to the literaturecited therein) for more details:

(i) If ωL is nondegenerate, R is uniquely defined, and will be invertible iff ωL′ isnon-degenerate as well. In other words: ker(R) ≡ ker(ωL′).

(ii) In local coordinates, R can be written as

R = Mji(

∂

∂qi⊗ dqj + ∂

∂vi⊗ dvj

)+ (L�jM) i ∂

∂vi⊗ dqj , (5.257)

whereM = H ′·H−1 , (5.258)

H ′ being theHessianmatrix of L′. This implies thatM·H be a symmetricmatrix.(iii) The matrix M satisfies

L�M = [A, M] , (5.259)

where, if � = vi∂/∂qi + Fi∂/∂vi, the matrix A = (Aij) is given by

Aij =: −1

2

∂Fj

∂vi. (5.260)

(iv) Finally, M satisfies also the constraint

[M,L�A+ B] = A[A, M] + [A, M]A , (5.261)

where

B = (Bij) , Bi

j =: ∂Fj

∂qi(5.262)


With the usual identification of � with a total time derivative with respect to anevolution parameter, we can view Eq.5.4 as a ‘Heisenberg-type’ evolution equationforM (which is now assumed to be unknown), supplemented by the constraints givenby Eq.5.6 and by the request that M be a symmetric matrices, i.e. by the additionalconstraint:

M = Mt . (5.263)

Of course both the evolution equation and the constraints are trivially solvedbyM ∝ I(the identity matrix), and the Lagrangian description will be unique (up to equiva-lence) if this is the case. Both constraints have to be always satisfied in order that the‘dynamical’ evolution generates admissible M’s, and this is somehow reminiscentof the situation one encounters in Dirac’s theory of constraints. Taking successiveLie derivatives with respect to � of the constraints will (or may) generate thereforenew constraints that have to be satisfied, thereby further restricting the admissibleform of M. If it happens that the full set of constraints generated in this way forcesM to be the trivial solution of the problem, i.e., a multiple of the identity, then, up toequivalence, the Lagrangian description will be unique.

For example, for velocity-independent forces, Aij ≡ 0, and the relevant set of

equations reduces to:L�M

[M, B] = 0M = Mt .

(5.264)

This will be the case, e.g., in two space dimensions, for simple Lagrangians ‘of themechanical type’, i.e., of the form:

L = 1

2(u2x + u2y)− V(x, y) (5.265)

Then, it has been shown that the Lagrangian description for two-dimensional systemsof the mechanical type is indeed unique unless V is a separable potential, and this isthe case only for the free particle and the harmonic oscillator.

Remark 5.17 This conclusion seems to be at variance with a result of Henneauxand Shepley [HS84], according to which a family of admissible (and alternative)Lagrangians for the Kepler problem is given by:

Lγ = 1

2δijv

ivj + c

q+ γ

J

q2(5.266)

where: q =:√

δijqiqj, c is a fixed real constant,

J =:√

q2δijvivj − (δijqivj)2 (5.267)


is the absolute value of the angular momentum and the family is parametrized by thereal parameter γ . There is however no contradiction, as, in two space dimensions(this is nothing but Kepler’s second law, of course)

J

q2= i�dθ (5.268)

with θ the polar angle, and the additional term reduces to a ‘gauge term’, which isnot the case, as Henneaux and Shepley have shown, in higher space dimensions.

We turn now to the second method, that is due to M. Henneaux. First of all, let usintroduce some preliminary notations and conventions. Denoting then by X ∧ Y thebivector (i.e., the skewsymmetrized tensor product) associated with a pair of vectorfields X and Y ,

X ∧ Y =: X ⊗ Y − Y ⊗ X, (5.269)

it is not hard to check that, if � is a 2-form, then:

�(X, Y) = 1

2〈�|X ∧ Y〉 (5.270)

where 〈·|·〉 stands for the fully-contracted multiplication of � and the bivector.We can now introduce an equivalence relation (denoted by the symbol)) between

pairs of bivectors by declaring two of them to be equivalent iff a F(TM)-linearcombination of them (their difference in particular) vanishes upon contraction witha given, and fixed, 2-form.

If now a given second-order field � admits of a Lagrangian description with anadmissible Lagrangian L, then the associated Lagrangian 2-form ωL , besides beinginvariant under the dynamics and closed, will have to vanish on pairs of verticalfields. This amounts to the constraint

ωL(SX ∧ SY) = 0 , ∀X, Y ∈ X(TM) , (5.271)

i.e., SX ∧ SY ) 0 in the notation established above. This constraint too has tobe preserved by the dynamical evolution. Taking then successive derivatives withrespect to. � will generate the set of constraints

ωL((L�)k(SX ∧ SY)) = 0, k = 0, 1, 2, . . . (5.272)

that have to be obeyed by the (a priori unknown) Lagrangian 2-form ωL . If n =dim(M), the dimension of the space of bivectors is the same as that of 2-forms, i.e.,n(2n−1), and this will be also the maximum number of independent constraints thatcan be generated in the above way.

An important observation is that if for a certain value of k the (k + 1)-th bivectordepends on the previous ones, then the same will be true for all the subsequent ones.The generation of independent bivectors will stop then, and k will be the maximum


number of independent constrains. If k = n(2n− 1), the bivectors generated in thisway will form a basis, implying ωL = 0, and the Inverse Problem will have nosolution at all.

Let us prove then the statement we have just made. Let us denote by simplicity asv0, v1, . . . , vk, . . . the set of bivectors generated by taking successive Lie derivatives,i.e.

vi+1 = L�vi , i ≥ 0 . (5.273)

Independence of the first k bivectors and dependence of the (k + 1)-th from theprevious ones are expressed by:

v0 ∧ v1 ∧ · · · ∧ vk �= 0

v0 ∧ v1 ∧ · · · ∧ vk ∧ vk+1 = 0 . (5.274)

Then,

0 = L�(v0 ∧ v1 ∧ · · · ∧ vk ∧ vk+1) =k+1∑i=0

v0 ∧ · · · ∧L�vi ∧ · · · ∧ vk+1 . (5.275)

But all the terms but the last vanish on account of the definition of the vi’s. We arethus left with:

0 = v0 ∧ v1 ∧ · · · ∧ vk ∧ L�vk+1 ≡ v0 ∧ v1 ∧ · · · ∧ vk ∧ vk+2 , (5.276)

and this completes the proof.Returning then to the set of constraints that have been generated with this proce-

dure, we see that the number of independent 2-forms that satisfy all the constraintsis given by the ‘codimension’ of the set of bivectors that has been generated. If thecodimension is zero, there will be no possible Lagrangian description. If it is greaterthan zero those forms in the set that are also closed and that are invariant under thedynamics (and this will have to be checked separately) will be admissible Lagrangian2-forms for � and will correspond to genuinely alternative Lagrangians. The codi-mension of the set of bivectors will give then the ‘maximum’ number of possiblealternative Lagrangians.

To illustrate the method, let us consider TR2, (n = 2), with coordinates

(x, y, ux, uy) and the second-order field:

� = ux∂

∂x+ uy

∂

∂y+ (x + y)

∂

∂ux+ xy

∂

∂uy(5.277)

Let us denote by Xi, i = 0, . . . , 5, the standard basis of bi vector fields on TR2,


X(0) = ∂

∂ux∧ ∂

∂uy, X(1) = ∂

∂x∧ ∂

∂y, X(2) = ∂

∂x∧ ∂

∂ux

X(3) = ∂

∂x∧ ∂

∂uy, X(4) = ∂

∂y∧ ∂

∂ux, X(5) = ∂

∂y∧ ∂

∂uy.

Taking thenLie derivativeswith respect to� starting from v0 =: X(0) will generatethe set of n(2n− 1) = 6 bivectors:

vi = aijX

(j) , i, j = 0, . . . , 5, (5.278)

where

a = (aij) =

⎛⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 00 0 1 −1 0 00 1 0 0 0 00 0 −1 −x 1 y0 0 0 −ux 0 uy

0 0 0 −(x + y) 0 xy

⎞⎟⎟⎟⎟⎟⎟⎠

(5.279)

Now: v0 ∧ . . . ∧ v5 = DX(0) ∧ . . . ∧ X(5), where : D = det a, and will van-ish iff D does. But D = xyux − (x + t)uy �= 0. Therefore the vi’s are a basis,codim span {(v0, . . . , v5)} = 0, and the present criterion shows that � does notadmit non trivialLagrangian descriptions.

At the opposite extreme, let us consider the n-dimensional isotropic harmonicoscillator. Then,

� = vi ∂

∂qi− qi ∂

∂vi. (5.280)

Starting from the bivectors

�(0)ij =

∂

∂vi∧ ∂

∂vj, i, j = 1, . . . , n, (5.281)

and taking Lie derivatives, by making use of L�∂/∂qi = ∂/∂vi and L�∂/∂vi =−∂/∂qi, one obtains

�(1)ij =: L��

(0)ij =

∂

∂qj∧ ∂

∂vi− ∂

∂qi∧ ∂

∂vj. (5.282)

But then,�

(2)ij =: L��

(1)ij ≡ 0 (5.283)

and the procedure terminates. As there are n(n − 1)/2 independent bivectors in theset of the �

(0)ij ’s and as many in that of the �

(1)ij ’s (that are by construction skew-

symmetric), the codimension of the space of bivectors that has been generated turnsout to be n2.


Consider then the family of 2-forms with constant coefficients:

ωB,C = Bij dqi ∧ dvj + 1

2Cij dqi ∧ dqj. (5.284)

They vanish on vertical bivectors and are closed (actually exact). If they have to beLagrangian 2-forms, then B = (Bij) must be the Hessian matrix of the Lagrangian,hence a symmetric matrix, while C = (Cij) is skew-symmetric by construction. Thetotal number of independent ωB,C’s is then precisely n2 (recall that there are n(n +1)/2 independent symmetric n×n matrices and n(n− 1)/2 skew-symmetric ones) .However, imposing invarince under the dynamics we obtain L�ωB,C = Cijdvi ∧ dqj

(all the other terms vanishing by symmetry) and, in the next order, L�(L�ωB,C) =Cij

(dvi ∧ dvj − dqi ∧ dqj

), which forces the matrix C to be zero. We are then

left with 2-forms that are parametrized only by symmetric matrices and that aregenerated, of course, by the family {LB} of Lagrangians that were introduced anddiscussed in a previous Section.

These simple examples suggest that generically, i.e., unless the dynamical vectorfield admits of a fairly large symmetry group (as is the case for the harmonic oscil-lator), the set of constraints that is generated in both methods will be enough eitherto prove the the nonexistence of a Lagrangian description or it will force the latterto be unique. We conclude that generic second-order dynamical systems either willnot admit of a Lagrangian description at all, or the Lagrangian description will begenerically unique.

These conclusions seem to suggest that, in the case of non-uniqueness of theLagrangian description, one might try to add a small perturbation to the dynamics,find out the (hopefully unique) Lagrangian description for the perturbed dynamicsand then switch off the perturbation to recover the ‘right’ Lagrangian for the originaldynamics. However, as the space of second-order vector fields is not endowed withany natural topology, what ‘small’ means in this context is a highly ambiguousconcept. Moreover, a generic perturbation may well lead to a new vector field thatsimply does not admit of any Lagrangian description, and we are not aware of anysensible procedure to select perturbations of the appropriate form (whatwemight call‘Lagrangian perturbations’). Last but not least, in physical problems one is almostalways confronted not with ‘generic’ but with very ‘specific’ dynamical systems,and if there are ambiguities in their Lagrangian description they will remain there.

Remark 5.18 As a final remark, let us stress what may seem an obvious fact, butneed not be such, namely that all the methods for tackling the Inverse Problem thathave been described here assume implicitly that the tangent bundle structure has beengivenonce and for all. So, all the conclusions are true relative to a given tangent bundlestructure, butmightwell not hold true under alternative tangent bundle structures like,e.g., the one that was used in the last example. In other words, and although we donot have at the moment any specific examples, we cannot rule out the possibility fora vector field � on a vector space V to be second-order with respect to two differenttangent bundle structures on V and to admit, say, of Lagrangian descriptions with


one or more alternative Lagrangians with respect to one of the structures and of feweror none at all with respect to the second one.

Before addressing the central discussion on this paper, we will review its classicalcounterpart from a variety of different perspectives.

5.6.3.2 The Hamiltonian Inverse Problem for Classical Systems

Given a vector field � defined on a state space modelled by the manifold with localcoordinates xk , the Hamiltonian inverse problem for � consists in finding a Poissonbracket { ·, · } and a smooth function H such that

{H, xk } = �k(x), (5.285)

and � = �k(x)∂/∂xk . In other words, the Hamiltonian inverse problem for � isequivalent to determine whether or not � is a Hamiltonian vector field with respectto some Poisson brackets. It is remarkable that in this formulation, neither the Poissonbrackets nor the Hamiltonian are determined a priori, this is, if they carry physicalinformation it has to be determined by further information not contained in the vectorfield � alone.

The inverse problem can be considered locally or globally, i.e., we can discussthe existence of a Poisson structure in the neighborhood of an ordinary point for thevector field � such thatL�{ ·, · } = 0, and in a second step the existence of a globallydefined Poisson structure. We will not pursue this aspect of the problem here (see[Ib90] for a discussion of the related issue of global aspects of the inverse problemin the Lagrangian formalism).

5.6.3.3 The Symplectic Inverse Problem for Classical Systems

The inverse Hamiltonian problem can be restricted requiring that the Poisson tensor� defined by the fundamental commutation relations

{ xi, xj } = �ij(x), (5.286)

will be nondegenerate, i.e., det�ij �= 0. In such case, inverting � we will get a2-form ω, defined by

ω = ωij dxi ∧ dxj,

whith �ikωkj = δij , such that ω is closed and,

i�ω = dH, (5.287)


that is,� is the Hamiltonian vector field defined by the functionH and the symplecticform ω. Notice that in particular solving the inverse problem for �, i.e., finding asymplectic form ω and a function H such that Eq. (5.287) holds, requires that thephase space of the system will be even dimensional.

5.6.3.4 The Lagrangian Inverse Problem

If the state space manifold has the form (locally at least) of the tangent bundle ofa configuration space Q with local coordinates xk , vk , then the symplectic inverseproblem for � would be solved if we were able to find a regular Lagrangian for it,that is, a Lagrangian L such that its Euler–Lagrange equations will be equivalent tothe vector field� and det ∂2L/∂vk∂vj �= 0. It is well-known that the Euler–Lagrangevector field obtained from a regular Lagrangian defines a second-order differentialequation (sode). Thus, if � is a SODE,

� = vk ∂

∂xk+ f j(q, v)

∂

∂vj,

i.e., the equations deining its integral curves are

dxk

dt= vk,

dvj

dt= f j(q, v),

then the Lagrangian inverse problem for � will be to find a function L such that itsatisfies the linear partial differential equation,

∂2L

∂vkvjf j − ∂2L

∂vkxjvj − ∂L

∂xk= 0,

and the regularity condition on L,

det∂2L

∂vkvj�= 0,

which spoils the linearity of the previous equation.There is also a Poisson version of the previous discussion, as reflected by the

following equations,

{H, xi } = vi, {H, {H, xi } } = f i(x, {H, x }).

Adding the localizability condition,

{ xi, xj } = 0,


we recover Feynman’s problem as reported by Dyson [Dy90]. It is possible to gobeyond the Lagrangian description while preserving the commutator description?The answer to this question is contained in the following theorem (see [Ca95] for anupdated revision of Feynman’s problem and references therein).

Theorem 5.60 Let � = vi∂/∂xi+f i∂/∂vi be a SODE, then it has a Poisson descrip-tion in terms of a function H(x, v) and a localizable Poisson bracket iff it can begiven a (local) Lagrangian description.

A simple proof of the previous theorem is obtained if we make the additionalassumption that,

∂

∂vk{ ·, · } = 0.

Then, if we start with vk = {H, xk }, taking the derivative with respect to vj we find,

δkj = {

∂H

∂vj, xk } = ∂2H

∂vj∂vl{ vl, xk }.

Thuswe can define theLegendre transform and therewill be a Lagrangian descrip-tion.

5.6.4 Feynman’s Problem and Lie Groups

5.6.4.1 Standard Lagrangian Realizations for External Variables

We know that Feynman’s procedure gives a negative answer in many instances;negative in the sense that there is always a Lagrangian description for dynamicscompatible with the requirements described in Theorem 5.57. In this section we willdiscuss Feynman’s procedure from a group theoretical setting, showing its relationto the construction of Lagrangian realizations of systems defined on groupmanifoldsor on associated bundles to principal fibre bundles. This construction again includesthe cases of Yang-Mills particles and spinning particles.

We will start by considering Feynman’s procedure for the case where the config-uration space is a Lie group G.

5.6.4.2 Feynman’s Procedure on Group Manifolds

The classical velocity phase space of the system we will examine here is TG withthe Lie group G as the configuration space of the system. We will be looking forall dynamics and Poisson brackets on TG which are localizable and Newtonian. Weknow already because of Theorem5.50 that the relevant Poisson tensors must beregular and that there will exist Lagrangian descriptions for such systems. If the


Poisson tensor is velocity independent, we know that the only possible non trivialdynamics are due to interactions with a background gravitational field arising fromsome non constant metric tensor η on G, and with an electromagnetic field, definedon the group by a vector and scalar potentials A and φ. If we impose G-invariance onthe dynamical system and on the Poisson tensor, then all fields must be G-invariant.We will show how Feynman’s procedure can be applied to this situation.

We first need to introduce some notation and facts about the geometry of Liegroups, their tangent bundles and SODE’s on them.

Let L (R) denote the action of G on itself by left (right) translations, i.e., L : G×G→ G, L(g, h) = Lg(h) = gh, (R(g, h) = Rg(h) = hg) for all g, h ∈ G. A vectorfield X on G is called left (right) invariant if TLgX(h) = X(gh) (TRgX(h) = X(hg)).IfX is left (right) invariant thenX(g) = TLgX(e) (X(g) = TRgX(e)). The Lie algebrag ofG is the Lie subalgebra of left (right) invariant vector fields onG. These invariantvector fields can be identified with their values at the identity element of the group.Then fixing a basis Ei of TeG we define the corresponding basis of invariant vectorfields XL

i (g) = TLgEi, XRi (g) = TRgEi. From now on we will identify g with TeG

and we fix a basis Ei of g.We next define left (right) invariant forms on G. We shall denote by θ i

L (θ iR) a

basis of left (right) invariant 1-forms on G dual to XLi (XR

i ). They obviously satisfy

〈XLi , θ

jL〉 = δ

ji , (〈XR

i , θjR〉 = δ

ji ). The canonical Maurer-Cartan g-valued 1-forms �L

and �R on G can be written as

�L = Ei ⊗ θ iL; �R = Ei ⊗ θR

i . (5.288)

The structure constants cijk of the Lie algebra g are defined from

[XLi , XL

j ] = cijkXL

k ,

or from the analogous relations using right invariant vector fields. The vector bundleTG is trivial and can be identifiedwithG×g, using left or right translations as follows:l : TG → G × g, l(g, g) = (g, TL−1g g), (r : TG → G × g, r(g, g) = (g, TR−1g g)),for any g ∈ TgG. We will denote by vL = TL−1g (g) (vR = TR−1g (g)) and vL =vi

LEi. Coordinates on TG obtained using the map l will be called body or convectivecoordinates, and coordinates defined bymeans of r will be called spatial coordinates.It is clear that body and spatial coordinates are related by (g, vL) = (g,Ad gvR). Fromnow on, we will be using only left-invariant or body coordinates, as it is equivalent tothe description in terms of right-invariant ones.Wewill then suppress the indexes ‘L’.

It is clear that the canonical 1-tensor SG on TG has the following form whenwritten in body coordinates:

SG = ∂

∂vi⊗ θ i ,

and a SODE D will be written as


D = viXi + f i(g, v)∂

∂vi.

IfL is aLagrangian function onTGwhich is left-invariant, i.e.,L = l∗Lg, orL(g, g) =Lg(TL−1g g), for a function Lg defined on g, then the Cartan 1-form is

θL = ∂L

∂viθ i ,

and the Cartan 2-form,

ωL = ∂2L

∂vivjθ i ∧ dvj − 1

2cjk

i ∂L

∂viθ j ∧ θ k .

If it is regular, the Lagrangian SODE defined by L is given by

� = viXi +Wlkvjcjki ∂L

∂vi

∂

∂vl,

where as usual Wij are the elements of the inverse matrix of W with Wij = ∂2L∂vi∂vj .

5.6.4.3 Free Motion

By free motion we mean geodetic motion with respect to a metric η on G. Thenη = ηL

ijθiL ⊗ θ

jL = ηR

ijθiR⊗ θ

jR, where ηL

ij and ηRij are functions on G. If η is left (right)

invariant, then the functions ηLij (η

Rij ) are constant. If the metric η is biinvariant then

it satisfies:ηircjk

r + ηkrcjir = 0 .

Let Lη be the kinetic energy defined by the left-invariant metric η,

Lη = 1

2ηij(g)g

igj = 1

2ηijv

ivj . (5.289)

The Cartan 1-form will beθη = ηijv

iθ j ,

while the Cartan 2-form is

ωη = ηijθi ∧ dvj − 1

2ηijv

ickljθ k ∧ θ l .

The energy of the system is the Lagrangian itself, ELη = Lη, and the Euler–LagrangeSODE �η is obtained from


�η = viXi + vkvjckjl ∂

∂vl. (5.290)

Thus the equations of motion look like:

dg

dt= TLgv; dv

dt

i

= cjkivjvk , (5.291)

which defines the geodesic flow on G associated with η. If the metric is biinvariant,then the equations of motion reduce to

dv

dt= 0 . (5.292)

The Poisson tensor defined by ωLη is given by

�η = ηij ∂

∂vi∧ Xj − cjk

i vi ∂

∂vj∧ ∂

∂vk,

and cjki = ηilη

jmηkncmnl. Because the Hamiltonian vector field with Hamiltonian vi

is given by ηijXj, the fundamental Poisson brackets defined by �η are

{ gi, gj } = 0, { gi, vj } = ηikXk(gi), { vi, vj } = cij

kvk , (5.293)

where gi represent a parametrization of the elements g ∈ G.We will now discuss Feynman’s procedure in this setting. For this we examine

the particular case of G = SU(2).

5.6.4.4 The Isotropic Rotator and 3He Superfluid

It is well known that SU(2) is diffeomorphic to S3. Thus we will be consideringFeynman’s problem on the sphere S3. We will first examine free motion on S3, i.e.,geodetic motion with respect to the biinvariant metric on SU(2). Physically, thiscorresponds to a noninteracting isotropic rotator. After discussing the free system,we introduce interactions bymodifying the Poisson structure and equations ofmotionin a similar manner to what was done in Sect. 4.5.3.

Let g denote an SU(2)matrix, g−1 = g†, where † denotes hermitean conjugation.Matrices g in SU(2) will be parametrized as

g =(

x0 + ix3 −x2 + ix1x2 + ix1 x0 − ix2

)= x0I + ixkσk, (5.294)

with x20 + x21 + x22 + x23 = 1 and σk are the Pauli matrices. In spatial coordinates, freemotion means that, Eq. (5.292),

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d

dt(gg†) = 0 , (5.295)

This is the equation of motion for a noninteracting isotropic rotator. These equationsare invariant under left and right multiplication by SU(2)

g→ ug, g→ gv , u, v ∈ SU(2) . (5.296)

The associated velocity phase space of the system is TSU(2) which we identify withSU(2) × su(2) by means of right translations. We will denote the spatial (angular)velocities (formerly vR) by ωi, i = 1, 2, 3, this is gg† = w = i

2wkσk . From Eq.(5.293), the Poisson brackets between SU(2) matrix elements are zero, with theremaining brackets given by

{ g, ωi } = − i

2σig , (5.297)

{ωi, ωj } = εijk ωk , (5.298)

For this choice ωi generates right translations on SU(2). The equation of motion(5.295) is recovered using the Hamiltonian function

H0 = −1

2

3∑i=1

ω2i = −

1

2ω2 . (5.299)

which is the standard free Hamiltonian for an isotropic rotator. (We have set themoment of inertia equal to one.)

The chiral transformations (5.296) can be canonically implemented for the abovePoisson structure. For this we must specify the following action of u and v on theangular velocities:

ωiσi → uωiσiv . (5.300)

Thus ωi is invariant under the left action of SU(2) parametrized by u, while rightSU(2) parametrized by v induces rotations. The Hamiltonian (5.299) is invariantwith respect to both left and right SU(2), and the transformations are associated withcanonical symmetries.

5.6.4.5 Interactions on S3

We next apply the analogue of Feynman’s assumptions in order to introduce inter-actions on S3. As on R

3, we only obtain the coupling to electromagnetism.We consider equations of motion of the form


gg† = i

2σiωi , (5.301)

ωi = fi(g, ω) . (5.302)

They are the analogues of Eqs. (4.39) and (4.40). As always we demand that theequations of motion are Hamiltonian. For the Poisson brackets we again assume(5.297) and that the Poisson brackets betweenmatrix elements of g are zero, implyinglocalizability on S3. As with the case on R

3, we generalize the Poisson brackets ofthe velocities (or here, angular velocities) with themselves. We take

{ωi, ωj } = εijk(ωk + Bk) . (5.303)

Particle interactions are due to the functions fi and Bk which were zero for the freesystem. We next determine fi and Bk , assuming that they depend on the coordinatesg and the velocities ωi. (For simplicity we shall assume that they have no explicitdependence on the time.) The procedure is identical to that carried out on R

3.It is not hard to see that Bk is a function of only the spatial coordinates, as was

the case for R3. For this we can expand the Jacobi identity

{ g, {ωi, ωj } } + {ωi, {ωj, g } } + {ωj, { g, ωi } } = 0,

to find that { g, Bk } = 0, and hence Bk is independent of ωi. As on R3, Bk is diver-

genceless. This follows from the Jacobi identity involving three angular velocitiesand can be expressed by

{ωi, Bi } = 0 .

Next we take the time derivatives of Poisson brackets (5.297) and (5.303). From(5.297) we get

{ g, ωi } + { g, ωi } = − i

2σig .

Upon substituting the equations of motion (5.301) and (5.302), we find that

{ g, fi } = i

2εijkσjBk(g)g ,

and as a result, the force can be at most linear in the angular velocity ωi,

fi = εijkBj(g)ωk + Ei(g) . (5.304)

Equation (5.304) is the Lorentz force equation on S3 and it defines the electric fieldEi(g).

Conditions on Ei(g) result from taking the time derivative of the Poisson bracket(5.303). We obtain

εijk{ ωi, ωj } = ωk + Bk(g) . (5.305)

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Next we substitute (5.304) and use the condition that the magnetic fields have noexplicit time dependence. Implicit time dependence through g means that Bi doesnot vanish, but instead

Bi + {Bi, ωj }ωj = 0 .

From (5.305) we are then left with

εijk{Ei, ωj } − Ek = 0 , (5.306)

which is the statement that the electric field has zero curl in the presence of staticmagnetic fields.

The Hamilton function for this system is

H = −1

2ω2 + φ(g) . (5.307)

In order to recover the equations of motion (5.301), (5.302) and (5.304) using thisHamiltonian we need to define the electric field Ei in terms of the scalar potentialφ(g) as follows:

Ei(g) = {ωi, φ(g) } . (5.308)

Furthermore, using this definition we can obtain the condition (5.306) after applyingthe Jacobi identity

{ωi, {ωj, φ } } + {ωj, {φ,ωi } } + {φ, {ωi, ωj } } = 0 .

It is easy to generalize the above treatment of a particle moving on SU(2) to thecase of a particle on an arbitrary Lie groupmanifold. For this we need only replace thePauli matrices by the appropriate group generators and εijk by the structure constants.For instance we can apply the previous procedure to a closely related group SO(3).Then, we will obtain as Poisson brackets the following:

{Rij, Rkl } = 0, {wi, Rjk } = εijlRlk, {wi, wj } = εijk(wk + Bk), (5.309)

where we denote by R = (Rij) the elements of SO(3) with.

(RRt)ij = εijkwk,

and Rij = cos θδij + (1 − cos θ)ninj + sin θεijknk , n2 = 1. The dynamics has theform of Eq. (5.133), with

H(R, w) = −1

2w2 + V(R),


and V(R) = V(cos θ) = C( 12 +2 cos θ)2. We obtain in this way the Legget equationfor the B-phase of the 3He superfluid [Br78]. Thus we can construct a Lagrangianfor such system on TSO(3).

5.6.4.6 Standard Formulation for Internal Variables

If the external configuration space is trivial, i.e., the system has only internal degreesof freedom, we can call it a pure isospin system. Such systems are generally associ-ated with first-order equations of motion as was the case in Sect. 4.5.3. The isospinvariables Ia parametrizing the system will be defined on the Lie algebra g of a com-pact semisimple Lie group G, typically SU(n). We can identify g with its dual spaceg∗ using a biinvariant metric on G, the Killing-Cartan metric for instance. In anycase, we can assume that the inner space is the dual g∗ of the Lie algebra g of theLie group G. The standard Poisson structure for a pure isospin system will be thelinear one defined by Eq. (5.248). In order to apply to it the results on Sect. 5.6.2 wecan think of it as a trivial bundle F with fibre g∗ over a single point Q = { point }.The Poisson tensor �I on it will again be chosen as the natural linear Poisson tensor�g, Eq. (5.248), induced by the Lie group G. Notice that because TQ is a singlepoint, the bundle TQ &' F is simply g∗, and the Poisson tensor � will be �g.But this tensor satisfies all conditions from Theorem5.57 except that is in generalnonregular. If we select a coadjoint orbit O ⊂ g∗, then �O = �g|O becomesregular and the associated symplectic form is the well-known Kirillov–Kostant–Souriau form ωO on O. Then, Theorem5.57 applied to �O in O, shows that theremust be a Lagrangian (possibly local) realization on TO. This is evident becauselocally ωO must be exact. Then we can choose any locally defined symplectic poten-tial A such that dA = ωO and define the Lagrangian L(ξ, ξ ) = Ai(ξ)ξ i, where ξ i

denote local coordinates on O. This Lagrangian is locally defined and singular. Forinstance, for the standard formalism for spin variables Si discussed in §3 and §4with the standard bracket defined by Eq. (67), the coadjoint orbits spanned by Si arespheres S2r of radius r. The Kirillov–Kostant–Souriau 2-form is a multiple of the areaform on S2 which is locally exact. This allows us to define local Lagrangians for thespin dynamics Eq. (24). In this particular example, a global (singular) Lagrangianrealization can be constructed using the Hopf fibration S3 → S2 (see for instance[Ba82]). In general then, the Lagrangians constructed in this way will be singular. Itwould be desirable to have a Lagrangian realization by a regular Lagrangian. SuchLagrangian regularization exists and will be described in the following paragraph.

5.6.4.7 A Lagrangian Realization of the Natural Poisson Bracket on g∗

The space g∗ can be considered as the Poisson reduction ofT∗G, the cotangent bundleof the groupG. We will identify T∗GwithG×g∗ using left translations on the group,i.e., l∗ : T∗G→ G× g∗, l∗(αg) = (g, TL∗gαg), for all αg ∈ T∗g G, and we will denote

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by x = TL∗gαg . With respect to a basis θ i of left-invariant 1-forms we will have

x = xiθi. In these coordinates the Poisson bracket on g∗ is again, { xi, xj } = cij

kxk .On T∗G ∼= G × g∗ we consider the canonical 1-form θ0 = xiθ

i. The canonicalsymplectic 2-form ω0 = −dθ0, given by

ω0 = θ i ∧ dxi − 1

2xicjk

iθ j ∧ θ k ,

on the cotangent bundleT∗G is biinvariant with respect to the lifting of the action ofGby left or right invariant translations. For instance considering a basis of left invariantvector fields X1, . . . , Xn on G, we can consider their cotangent lifts X∗1 , . . . , X∗n toT∗G by setting

LX∗j (π∗f ) = π∗(LXj f ), ∀f ∈ F(G),

with π : T∗G→ G the cotangent bundle projection, and

LX∗j θ0 = 0.

We notice that LX∗j θ0 = 0 implies that iX∗j dθ0 = −d〈X∗j , θ0〉 and 〈X∗j , θ0〉 dependsonly on Xj. Then the nondegeneracy of ω0 uniquely defines X∗j . The vector fieldsX∗i are thus Hamiltonian with Hamiltonian Ji = 〈X∗j , θ0〉. Similar results occur forright-invariant vector fields. Because the infinitesimal generators of the left actionof G on itself are right-invariant vector fields, and vice versa, we have that themap JR = Ji θ

iL : T∗G → g∗, represents the momentum map of the Hamiltonian

action defined by right-translations on the group. Then it is easy to see that JR, themomentum map corresponding to right translations, is defined by JR(αg) = TL∗gαg,where αg ∈ T∗g G. Notice that JR corresponds to the projection πL of T∗G onto g∗ bythe left action of G on T∗G.

Elements of g define linear functions on g∗ and the elements Xi define the linearfunctions xi. Thus the projection JR : T∗G → g∗ gives J∗R(xi) = Ji. The naturalsymplectic structureω0 onT∗Gdefines aPoisson structure�0 which canbewritten as

�0 = XLi ∧

∂

∂xi+ cjk

ixi∂

∂xj∧ ∂

∂xk. (5.310)

Themost important fact about JR is that it is a Poissonmapwith respect to the Poissonstructure �0 defined on T∗G by ω0 and the Poisson structure �g. This is easily seenfrom the particular expression of the Poisson Brackets on Eqs. (5.310) and (5.248).

Thus, T∗G provides a symplectic realization for a pure isospin system, however,because T∗G is not a tangent bundle, it will not provide a Lagrangian realization. Forthat, we just need to choose a right invariant Riemannian metric η on G and definethe kinetic energy Lagrangian Equation (5.289) on TG. The Cartan 2-form ωLη willthen be the pull-back of ω0 by the identification between TG and T∗G provided bythe metric η. Then, the action of G on (TG, ωLη )will define a Lagrangian momentum


map J : TG→ g∗, given by J(g, v) = ηijviθ j, that will be Poisson. More generally,

we can consider any Lagrangian function L ∈ F(TG) which is nonsingular, i.e., ωLis symplectic. We consider the left or right action on G and define a lift to TG whicha priori does not coincide with the tangent lift. We define

LXLi(τ∗Gf ) = τ ∗G(LXL

if ); LXL

iθL = 0,

where τG : TG → G is the tangent bundle projection, and similarly, for the rightaction.

It is clear that Pi = 〈XLi , θL〉 allows us to define a projection P : TG→ g∗ which

is a Lagrangian realization of the Poisson manifold (g∗,�g).At this point we may wonder if the Lagrangian vector field associated with L will

be P-projectable onto g∗. Of course a necessary and sufficient condition for this isthat {EL, Pi }L must be P-projectable onto g∗ for any Pi. This will be the case if theLagrangian function is invariant under the flow lift of right invariant translations.When this is the case the flow lift coincides with the Liouville lift. (We call Liouvillelift the one defined by using θL).

What is the relation between the regular Lagrangian realization of g∗ on TGdescribed above and the (local) singular Lagrangian realization on TO? The answeris easily obtained if we extend the coadjoint orbit to the full group. That is, weconsider O as an homogeneous space of G, O = G/GO where GO is the isotropygroup of an arbitrary element μ ∈ O. Then μ will define a left-invariant 1-form αμ

on G by the formulaαμ(g) = TL∗

g−1μ, ∀g ∈ G ;

Namely, αμ = μiθi. A simple computation shows that dαμ = μicjk

iθ j ∧ θ k is aclosed presymplectic 2-form on G which is Gμ-invariant, i.e., the 2-form dαμ isprojectable to O. Moreover, the characteristic distribution of dαμ is spanned by theelements on gμ, the Lie algebra of Gμ. Then, the projected 2-form on O will besymplectic and it coincides with the Kirillov–Kostant–Souriau symplectic form ωO.

Consider now the linear Lagrangian on TG defined by the 1-form αμ, i.e.,

Lμ(g, v) = μivi. (5.311)

It is clear that θLμ= τ ∗Gαμ and the Cartan 2-form is given by,

ωLμ = τ∗Gdαμ.

The Lagrangian Lμ is of course degenerate and the kernel K of its Cartan 2-form isgiven byK = gμ×g, the factor g corresponding to all vertical vectors on TG ∼= G×gand gμ corresponding to the kernel of dαμ on G. It is then clear that TG/K = O, andin this sense, the singular Lagrangian Lμ, Eq. (5.311), provides a singular Lagrangianrealization for the coadjoint orbit O.


5.6.4.8 Lagrangian Realization of Hydrodynamical Systems ofKirchhoff Type

We can apply the previous ideas to a system quite different to the spin dynamicsconsidered above. The equations ofmotion for a rigid body in an ideal incompressiblefluid at rest at infinity are given by Kirchhoff equations [Mi60] assuming the motionof the fluid to be potential,

p = p× w; M =M× w + p× u, (5.312)

M andp are the total angular and linearmomentum and ui = ∂H/∂pi,wi = ∂H/∂Mi,where H is the energy of the system and is assumed to be positive and quadratic inboth variables M and p. This equations can be put in Hamiltonian form consideringthe variables M, p lying in the dual of the Lie algebra of the Euclidean group E(3).The Poisson tensor has the form

�E(3) = εijkpk∂

∂Mi∧ ∂

∂pj+ εijkMk

∂

∂Mi∧ ∂

∂Mj. (5.313)

Notice that if we think of the dual of the Lie algebra of E(3) as a trivial bundleR3×R

3→ R3, the tensor�E(3) is not localizable on the bundle because of the term

corresponding to the bracket { pi, Mj } = εijkpk (see Eqs. 5.104 and 5.107). Anothersystem that can be reduced to a Poisson dynamics using the tensor (5.313) is theLegget equation of spin dynamics in the A-phase of the superfluid 3He (comparewith the B-phase discussed in section before). The construction discussed abovepermits to lift this dynamics to a Lagrangian setting on TE(3). For this we shall usea metric η on E(3) and the momentum map JR : TE(3) → R

6 and the Lagrangianwill be simply the kinetic energy Lη.

The ideas developed in the previous paragraphs can be extended to the followingsituation: Let P(G, Q) be a principal fibre bundle over the base manifold Q with fibrethe groupG.We shall consider thePoisson bundle ad∗P→ Q described inSect. 5.6.2.This bundle can be obtained by symplectic reduction of the universal phase space forparticles in a Yang-Mills field [We78, St77]. The details of the computations havebeen exhaustively described elsewhere (see for instance [Gu84]), which we summa-rize below. Consider the cotangent bundle T∗P. In addition, consider the cotangentlift of the action of G on P to T∗P. The action is Hamiltonian with momentum mapJ : T∗P→ g∗. The quotient of T∗P by the action of G is diffeomorphic to the bundlep : E = π∗(ad∗P)→ T∗Q. The bundle E carries a natural Poisson structure whichis localizable, Newtonian and velocity independent. A Lagrangian realization can beeasily obtained using a regular Lagrangian function L on TP and lifting the action ofG to TP using θL as indicated in the previous subsection. (See [Al94] for details onthe tangent bundle setting.)


5.6.4.9 Nonstandard Formulations

In this section we will analyze some solutions to the inverse problem of Poissondynamics on group manifolds. We will rediscover Lie-Poisson structures on groupmanifolds that will be used to provide an alternative Hamiltonian description of theisotropic rotator. The equations of motion (5.295) are SODE’s and consequently areassociated with external variables. In general, solutions to the inverse problem willprovide different models for variables, beside the linear Poisson structure used in thetwo previous section.

5.6.4.10 The Inverse Problem of Poisson Dynamics on Lower DimensionalSpaces

To deal with Poisson structures we consider a general approach which is very usefulfor lower dimensional problems. The strategy has been developed in general terms in[Gr93, Ko85] and [Ca94], and consists of using the exterior differential calculus onforms, instead of the graded Lie bracket on multivector fields. This can be done byassociating (n− k)-forms with k-vector fields V on R

n (or on an arbitrary manifold)by setting

�V = iV� , (5.314)

� being a volume formonRn. Themap� provides an isomorphismbetween k-vector

fields and (n− k)-forms. On monomial multivector fields we have,

(iV�)(Y1, . . . , Yn−k) = �(X1, . . . , Xk, Y1, . . . , Yn−k) ,

for V = X1 ∧ · · · ∧ Xk . For a bivector field � = �ij∂/∂xi ∧ ∂/∂xj and � =dx1 ∧ · · · ∧ dxn, we have

�� = 2∑i<j

(−1)i+j�ijdx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxj ∧ · · · ∧ dxn ,

where the symbol stands for omission. We should notice that this isomorphism isnot natural and depends on �.

By using the Schouten bracket (see Sect. 4.3.1) on multivector fields V1 and V2,we find

i[V1,V2] = −iV1 iV2d − diV2∧V1 + iV1diV2 + iV2diV1 ,

which generalizes the analogous relation,

i[X,Y ] = iX iY d − diX∧Y + iXdiY − iY diX ,

for vector fields X and Y .A relevant proposition for our purposes is the following:

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Proposition 5.61 [Gr93] A bivector field � is a Poisson structure if and only if2i�d�� = d��∧�. Two Poisson structures �1 and �2 are compatible if and onlyif d��1∧�2 = i�1d��2 + i�2d��1 .

In low dimensions we have some immediate results:OnR

2, with coordinates (x, y), any bivector field� = f (x, y)∂/∂x∧∂/∂y definesa Poisson structure because there [�,�] is a skew-symmetric three tensor on a two-dimensional manifold, which therefore vanishes identically.

OnR3, with coordinates (x1, x2, x3), we have� = �ij∂/∂xi∧∂/∂xj andwith� =

dx1∧dx2∧dx3, we get�� = εijk �ij dxk . The Jacobi identity gives��∧d�� = 0,i.e., the 1-form �� must admit an integrating factor. Locally this implies that anyPoisson structure is parametrized by two functions, f0, f1, such that �� = f0 df1 andthe associated Poisson bracket has the form

{ xi, xj } = f0 εijk ∂f1∂xk

.

This result was used in Sect. 4.5.3 to write Eq. (4.49). If we have a dynamical vectorfield � and we look for a Hamiltonian description, we need to solve the equation�(dH) = −�. For this problem it is better to start with a volume in contravariantform, say � = g ∂/∂x1∧ . . .∧∂/∂xn. Then instead of i�� = �, we have i�� = �.If it is possible to select � such that L�� = 0, we have �(dH) = i�∧dH� = � ori�� = �∧dH. Therefore if� admits an invariant volume, a necessary and sufficientcondition for � to admit a Hamiltonian description is that i�� is the product of two1-forms α ∧ β with dβ = 0 and α ∧ dα = 0. Thus, if α = f0 df1, the function f1must be a constant of motion for �.

Because of d��∧� = 0, we find that

i�1d(f2 df3)+ i�2d(f0 df2) = 0 ,

reduces tod(f0 df1 + f2 df3) ∧ (f2 df3 + f0 df1) = 0 ,

i.e.,(f0 df2 − f2 df0) ∧ df1 ∧ df3 = 0 .

where we have used our formula on compatible Poisson structures.If the bracket is an admissible bracket for �, then f1 and f3 are two constants of

motion and we get a family of admissible brackets by taking f0 and f2 to also beconstants of motion.

In general,(f0 df2 − f2 df0)(�) = 0

gives d(log |f2| − log |f0|)(�) = 0, i.e., the ratio of f0 and f2 is a constant of motionfor �. Thus, if f1 and f3 are two independent constants of motion, we get

http://dx.doi.org/10.1007/978-94-017-9220-2_4

http://dx.doi.org/10.1007/978-94-017-9220-2_4


� = f0 df1 + f0 F(f1, f3) df3 ,

i.e.,

{ xi, xj } = f0 εijk(

∂f1∂xk+ F(f1, f3)

∂f3∂xk

)

is the most general expression for Poisson brackets compatible with �.

5.6.4.11 Poisson Structures on Three-Dimensional Lie Groups

We now consider a matrix Lie group G. In this way if g = (gij), we have dg = (dgij)

and the left and rightMaurer–Cartan 1-forms Eq. (5.288) valued 1-form are given by,

�L = g−1 · dg; �R = dg · g−1,

respectively. If we consider a basis Ei for the Lie algebra g of G, the associated leftinvariant vector fields XL

1 , . . . , XLn , and right invariant ones XR

1 , . . . , XRn , satisfy,

iXLj(g−1 · dg) = Ej, or iXR

j(dg · g−1) = Ej.

Any bivector field on G can be written as � = �ijL XL

i ∧ XLj = �

ijR XR

i ∧ XRj ,

where �ijL, �

ijR ∈ F(G). By using a basis of 1-forms we can also define a volume

�L = θ1L ∧ θ2L ∧ · · · ∧ θnL or �R = θ1R ∧ θ2R · · · ∧ θn

R which coincide for unimodulargroups. Then we can associate a (n−2)-form �� with any bivector field � by usingEq. (5.314).

We shall now restrict our considerations to 3-dimensional Lie groups.We consider�L = θ1L ∧ θ2L ∧ θ3L and � = �

ijL XL

i ∧ XLj to find the 1-form,

�� = εijk �ijL θk

L = fkθkL = α ,

with fk = εijk�ijL. The Jacobi identity once more gives α ∧ dα = 0, i.e.,

fj dfk ∧ θjL ∧ θ k

L =1

2fk fm cij

k θmL ∧ θ i

L ∧ θjL .

On the other hand, it is possible to write the differential of any function f in the formdf = (LXL

if )θ i

L . Then by using this expression we get

fj LXLm

fk θmL ∧ θ

jL ∧ θk

L +1

2fk fm cij

k θmL ∧ θ i

L ∧ θjL = 0 .

Therefore the equation for (f1, f2, f3) reads as:


fi LXLm

fj − fj LXLm

fi = cijk fk fm , (5.315)

and gives all Poisson structures on the 3-dimensional Lie group G.To be more specific, we give some examples:

Example 5.7 The group SU(2). The group SU(2) is unimodular, and thus�L = �R.If we look for a left-invariant 1-form α = fj θ

jL , fj ∈ R, then Eq. (5.315) gives, fj = 0

for all j = 1, 2, 3. Thus there are no Poisson brackets on SU(2) which are left-invariant. Similarly, one shows that there are no right-invariant Poisson brackets.

The simplest non trivial case would be to consider

� = �ijL XL

i ∧ XLj +�

ijR XR

i ∧ XRj .

Because of the unimodularity of SU(2) and carrying on the computation of α ∧ dα,we find a solution for �

ijL and �

ijR as

�ijL = −�

ijR ∈ R .

Therefore,� = XL

i ∧ XLj − XR

i ∧ XRj (5.316)

is a Poisson structure on SU(2). This bracket defines a Lie-Poisson bracket on SU(2)[Sk82, Ya92, Ahl93] (see [Ta89, Ma90, Tj92] for a review).

It may be useful to find the precise Poisson structure associated with � when werealize SU(2) as a group of 2× 2 complex matrices, Eq. (5.294)

g =(

x0 + ix3 −x2 + ix1x2 + ix1 x0 − ix3

), (5.317)

with the condition x20 + x21 + x22 + x23 = 1. If we consider the left and right invariant

vector fields XL1 , XR

1 , XL2 , XR

2 , associated with E1 =(0 −11 0

)and E2 =

(0 ii 0

),

we find

�SU(2) = (x22 + x21)∂

∂x3∧ ∂

∂x0+ x3x1

∂

∂x0∧ ∂

∂x1+ x0x2

∂

∂x0∧ ∂

∂x2

+x0x1∂

∂x1∧ ∂

∂x3+ x0x2

∂

∂x2∧ ∂

∂x3(5.318)

=(

x1∂

∂x1+ x2

∂

∂x2

)∧

(x3

∂

∂x0− x0

∂

∂x3

)+ (x22 + x21)

∂

∂x3∧ ∂

∂x0,

(5.319)

along with


��SU(2) = (x22 + x21)x2 dx1 − x1 dx2

x22 + x21∧ d

(1

2(x20 + x21 + x22 + x23)

).

Example 5.8 The group SL(2, R). It is possible repeat the above procedure forSL(2, R). In this case we let our group matrices g have the form,

g =(

x1 x2x3 x4

), (5.320)

with the condition x1x4 − x2x3 = 1. We find,

�SL(2,R) =x2x1∂

∂x2∧ ∂

∂x1+ 2x2x3

∂

∂x4∧ ∂

∂x1+ x1x3

∂

∂x3∧ ∂

∂x1

+ x2x4∂

∂x4∧ ∂

∂x2+ x3x4

∂

∂x4∧ ∂

∂x3,

with

��SL(2,R)= (x22 + x23)

x3 dx2 − x2 dx3x22 + x23

∧ d(x1x4 − x2x3) .

Let us now consider the inverse problem for � on a three dimensional group. Weconsider �L = θ1L ∧ θ2L ∧ θ3L , i��L = α and look for solutions of L�� = 0. Bysetting

α = fi θiL, � = gi XL

i ,

we get:

i. figi = 0, i.e., i�α = 0,ii. fk div � = −gi(LXL

kfi − LXL

ifk + fjcik

j) along with the conditions on functionsfj’s given by Eq. (5.315).

When div� = 0 we obtain some particular solutions by using constants of motionfor �, say F with L�F = 0. Then we set α = dF and find i�I �L = dF. Fromdiv� = 0, we get L��I = 0, with � in the image of �I because � is tangent to thefoliation induced by F.

5.6.4.12 Alternative Hamiltonian Description of the Isotropic Rotator

We have seen that the solution of the inverse problem for Poisson dynamics on Liegroups leads to families of Poisson brackets, the simplest ones being Lie-Poissonbrackets on the group. Thus we can relax the localizability condition in the classicaltreatment of Feynman’s problem to assume that the Poisson tensor in the total spaceis π -related to a Lie-Poisson tensor on the configuration group manifold. We shallexplore such assumption in the simple situation of the group SU(2), but most ofthe ideas can be extended to arbitrary Lie–Poisson structures. The main result we


will obtain is that the Hamiltonian description of a particle moving on S3 ∼= SU(2)is not unique [Mr93, St94]. We end up with a Hamiltonian formulation which iscanonically inequivalent to the free system given in the Sect. 5.6.4.1, yet it yields thesame equations of motion, i.e., (5.295).

We begin by examining the spatial coordinates. As our spatial manifold is S3

we can once again utilize SU(2) matrices g as coordinates Eq. (5.317). Then weintroduce the weak assumption of nonlocalizability which means that the Poissonbrackets between matrix elements take the quadratic relations Eq. (5.319). It will beconvenient in what follows to change the notation and to use tensor product notationfor the Poisson brackets on SU(2). Then the Poisson brackets defined by the Poissontensor Eq. (5.319), reads as [Sk82, Ya92, Ahl93]

{ g1, g2 } = [r12, g1g2] , (5.321)

The indices 1 and 2 refer to two separate vector spaces on which the SU(2) matricesact. The matrix r12, known as the classical r-matrix, acts non trivial ly on both vectorspaces 1 and 2, while g1 and g2 are defined by the tensor products:

g1 = g ⊗ 1 g2 = 1⊗ g .

As usual the Poisson bracket should be antisymmetric and satisfy the Jacobiidentity. Antisymmetry of (5.321) requires that

[r12 + r21, g1g2] = 0 , (5.322)

or r12+r21 is an adjoint invariant. The Jacobi identity involving three SU(2) elementsg1, g2 and g3 leads to certain quadratic relations for the classical r matrix known asthe modified classical Yang-Baxter relations. They are:

[r23, r31] + [r31, r12] + [r12, r23] = adjoint invariant . (5.323)

Solutions to (5.322) and (5.323) are well known. If we express g in terms of itsdefining matrix representation then r can be written in terms of 4 × 4 matrices. Aone parameter family of such matrices satisfying (5.322) and (5.323) is

r = iλ

4

⎛⎜⎜⎝1−1

4 −11

⎞⎟⎟⎠ , (5.324)

where we assume the parameter λ to be real. Only in the limit λ→ 0 do we recoverthe ordinary condition of localizability.

As the dimension of the phase space should be six, we must introduce three morevariables in addition to the spatial coordinates g. We denote the new variables byS3 and S−, where S3 is real and S− is complex. From them we shall later define the


analogue of the angular velocity ωi of section Sect. 5.6.4.1. For the Poisson bracketsof Si with themselves we take

{ S±, S3 } = ± iλ

2S±S3 , { S+, S− } = −iλ

((S3)

2 − (S3)−2

), (5.325)

where S+ is the complex conjugate of S−. These brackets were utilized previouslyin an alternative description of spin, Eq. (4.64). They were seen to satisfy the Jacobiidentity, and to have the classical Casimir function

f1 = S+S− + (S3)2 + (S3)

−2 . (5.326)

Thus { Si, f1 } = 0. Moreover the Poisson brackets (5.325) were shown to correspondto the classical limit of the Uq(sl(2)) Hopf algebra.

The Poisson brackets (5.325) can be re-expressed so that they resemble the brack-ets (5.321) for g. For this define the 2× 2 lower triangular matrices �(−),

�(−) =(

S3 0S− S−13

). (5.327)

The set of such matrices forms the SB(2, C) group. Furthermore, if we denote their

conjugate inverses by �(+) = �(−)†−1, then (5.325) can be written according to

{�(±)1 , �

(±)2 } = [ r21 , �

(±)1 �

(±)2 ] , (5.328)

{�(+)1 , �

(−)2 } = [ r21 , �

(+)1 �

(−)2 ] . (5.329)

Eqs. (5.328) and (5.329) satisfy the requirement of antisymmetry and the Jacobiidentity for the same reasons as does (5.321). In terms of the matrix �(−) the classical

Casimir function can be written f1 = Tr �(−)�(−)†.It remains to specify the Poisson bracket relations for Si with the spatial coordi-

nates g. This choice must of course be consistent with the Jacobi identity and thePoisson brackets (5.321) and (5.325). One such choice is:

{ S3, g } = − iλ

4S3σ3g ,

{ S±, g } = −iλ

(1

4S±σ3 + S−13 σ∓

)g , (5.330)

where σ± = 12 (σ1±iσ2). These relations can be re-expressed in terms of thematrices

�(±) as follows:

{�(−)1 , g2} = −�

(−)1 r12 g2 ,

{�(+)1 , g2} = �

(+)1 r21 g2 . (5.331)

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Next we introduce the Hamilton function which we denote byH(λ). The equationsof motion for the free system (5.295) state that the angular velocities are constants ofmotion, so they should have zero Poisson brackets with H(λ). We will construct theangular velocities from Si alone. Then H(λ) should have zero Poisson bracket withSi, and the Hamiltonian can therefore depend only on the Casimir function (5.326).We take

H (λ) = − 1

2λ2(f1 − 2) . (5.332)

We will justify this choice later.It then remains to compute the equation of motion for g. From (5.330) and (5.332)

we get

− 2igg† = −2i{ g, H(λ) } g† = �3σ3 +�−σ+ +�+σ− . (5.333)

�i are angular velocities, analogous to ωi of section Sect. 5.6.4.1. Here they aredefined according to

�3 = − 1

2λ

((S3)

2 − (S3)−2 + S−S+

),

�± = �1 ± i�2 = −1

λS∓S−13 , (5.334)

or

�i = − 1

2λTr �(−)σi�

(−)† .

Since we also have that�i = {�i, H(λ) } = 0 ,

we have recovered the equation of motion (5.295) of the isotropic rigid rotator.The above formulation of the isotropic rotator is a one parameter deformation

of the standard formulation given in Sect. 5.6.4.1. Thus, it is possible to recover thestandard formulation as a limiting case. The relevant parameter here is λ and thelimiting value for the parameter is λ = 0. In this regard, we have already seen thatthe Poisson bracket (5.321) of matrix elements of g with themselves vanishes whenλ → 0. The remaining Poisson brackets reduce to the canonical ones (i.e., (5.297)and (5.298)) when λ → 0 provided that we make the variables Si functions of theangular velocity ωi of section Sect. 5.6.4.1, as well as the parameter λ,

Si = S(λ)i (ω) .

For this, we need to assume the following behavior for small λ


S3 = 1− λ

2ω3 + λ2

8(ω3)

2 + O(λ3) ,

S± = −λω∓ + O(λ3) . (5.335)

Then (5.297) and (5.298) result from (5.325) and (5.330), respectively, upon takingthe limit λ→ 0. Because the variables Si are functions of angular velocities ωi andthe parameter λ, then so are angular velocities �i functions of ωi and λ:

�i = �(λ)i (ω) .

From (5.334) and (5.335) it follows that the two angular velocities coincide in thelimit λ→ 0,

�(λ→0)i (ω) = ωi .

Finally we need to show that the Hamiltonians (5.299) and (5.332) of the twodifferent formulations of the rotator agree in the λ → 0 limit. For this we see thatthe classical Casimir function f1 has the following small λ behaviour

f1 → 2+ λ2ωiωi + O(λ3) ,

from which it follows thatH(λ→0) = H0 ,

H0 given by (5.299). This of course was the reason for choosing H(λ) of the formgiven in (5.332).

For λ �= 0, the treatment given here is canonically inequivalent to that given insection Sect. 5.6.4.1. Here, for instance, we see that the Hamiltonian (5.332) is notproportional to the square of the angular velocity (unlike (5.299)), nor does the Pois-son bracket algebra of the angular velocity�i correspond to an SU(2) algebra (unlike(5.298)). Perhaps the most distinguishing feature of the formulation presented hereconcerns the symmetries. In the standard formulation we saw that chiral transforma-tions were canonical symmetries. This is not the case in the alternative formulation.In fact, chiral transformations can not be canonically implemented for the Poissonstructure given by (5.321), (5.325) and (5.330). For example, under the right actionof SU(2),

g→ gv , �(±) → �(±) , v ∈ SU(2) , (5.336)

the brackets (5.321) undergo the following transformation:

{ g1, g2 } → { g1v1, g2v2 } �= [r12, g1v1g2v2] . (5.337)

Instead of being canonical the right action (5.336) is Lie–Poisson. Poisson Liegroup transformations are classical analogues of quantumgroup transformations. Theformer are defined so that group multiplication is a Poisson map. Thus with regard


to transformation (5.336) we want that { v1, v2 } �= 0, v1 = v⊗ 1, v2 = 1⊗ v. If onechooses the following Poisson bracket for matrix elements of v with themselves

{v1, v2} = [ r12 , v1v2 ] , (5.338)

(and zero with g and �(±)) then the inequality in (5.337) can be replaced by anequality. In fact, from (5.338) it follows that all Poisson brackets (5.321), (5.325)and (5.330) are unchanged by this transformation. Then right multiplication is aPoisson map and (5.336) is a Poisson Lie group transformation. Further, since �(±)

are unchanged by this transformation, and the Hamiltonian (5.332) depends only onthese variables, then this transformation leaves the dynamics invariant. We then saythat (5.336) is a Poisson Lie group symmetry.

There exists another Poisson Lie group symmetry and it is the analogue of theleft SU(2) canonical symmetry of the standard formalism. Unlike right SU(2) givenby (5.336) it has a non trivial action on �(±) (as well as g). In order to specify thisaction it is convenient to introduce another set of variables d(±), known as classicaldouble variables [Dr83, Dr86]. These variables are defined by

d(±) = �(±)g . (5.339)

It follows that d(±) is an element of SL(2, C), (5.339) being an Iwasawa decomposi-tion. In terms of these variables the right action of SU(2) is given by d(±) → d(±)v.We now define the left action of SU(2) by

d(±) → ud(±) , u ∈ SU(2) . (5.340)

The Poisson brackets for d(±) can be constructed from those of g and �(±). One finds

{d(±)1 , d(±)

2 } = −d(±)1 d(±)

2 r12 + r21 d(±)1 d(±)

2 ,

{d(−)1 , d(+)

2 } = −d(−)1 d(+)

2 r12 − r12 d(−)1 d(+)

2 . (5.341)

These relations are invariant under (5.340) upon insisting that u ∈ SU(2) has thefollowing Poisson bracket with itself

{u1, u2} = [ r12, u1u2 ] ,

and zero Poisson bracket with d(±). Then SU(2) left multiplication is a Poissonmap and (5.340) is a Poisson Lie group transformation. Further, since the classicalHamiltonian can be written

H(λ) = − 1

2λ2

(Tr d(−)d(+)−1 − 2

),


we can use the cyclic property of trace to show that it is unchanged under (5.340),and hence SU(2) left multiplication is a Lie Poisson symmetry.

The Hamiltonian formalism presented here of free motion on SU(2) can be quan-tized and the result is free motion on a noncommuting space, namely the quantumgroup SUq(2) [St94]. Furthermore, upon quantization, the two Lie Poisson symme-tries turn into quantum group symmetries.

An intriguing question concerns whether or not there exists an analogous alter-native Hamiltonian description for the interacting system given in Sect. 5.6.4.1,Eqs. (5.301–5.308). It would involve introducing electromagnetic fields, which whenquantized would be defined on a non-commutative space, namely SUq(2).

References

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structure. Ann. Inst. H. Poincaré 50, 205–218 (1989)[Mm85] Marmo, G., Saletan, E.J., Simoni, A., Vitale, B.: Dynamical Systems: A Differential

Geometric Approach to Symmetry and Reduction. Wiley, Chichester (1985)[Ar76] Arnol’d, V.I.: Méthodes mathématiques de la mécanique classique. Ed. Mir, 1976. Math-

ematical Methods of Classical Mechanics. Springer, Berlin (1989)[SW86] Sattinger, D.H., Weaver, O.L.: Lie Groups and Algebras with Applications to Physics,

Geometry and Mechanics. Springer, New York (1986)[Mo91] Morandi, G., Ferrario, C., Lo, G.: Vecchio, G.Marmo and C. Rubano. The inverse problem

in the calculus of variations and the geometry of the tangent bundle. Phys. Rep. 188, 147–284 (1991)

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[Cr87] Crampin, M., Ibort, L.A.: Graded Lie algebras of derivations and Ehresmann connections.J. Math. Pures et Appl. 66, 113–125 (1987)

[Ca95] Cariñena, J.F., Ibort, A., Marmo, G., Stern, A.: The Feynman problem and the inverseproblem for Poisson dynamics. Phys. Rep. 263, 153–212 (1995)

[Wi39] Wigner, E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann.Math.40, 149–204 (1939)

[No82] Novikov, S.P.: The Hamiltonian formalism and a many valued analogue of Morse theory.Russ. Math. Surveys. 37, 1–56 (1982)

[Hu92] Hughes, R.J.: On Feynman’s proof of the Maxwell equations. Amer. J. Phys. 60, 301–306(1992)

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[CS55] Currie, D.I., Saletan, E.J.: q-equivalent particle Hamiltonians. I. The classical one-dimensional case. J. Math. Phys. 7, 967–974 (1966)

[HS84] Henneaux, M., Shepley, L.C.: Lagrangians for spherically symmetric potentials. J. Math.Phys. 23, 2101–2107 (1984)

[Ib90] Ibort, A., López-Lacasta, C.: On the existence of local andGlobal’s for ordinary differentialequations. J. Phys. A: Math. Gen. 23, 4779–4792 (1990)

[Dy90] Dyson, F.J.: Feynman’s proof of theMaxwell equations. Amer. J. Phys. 58, 209–211 (1990)[Br78] Brinkman, W.F., Cross, M.C.: Spin and orbital dynamics of superfluid 3He. In: Brewer,

D.F. (ed.) Progress in low temperature physics. North-Holland, Amsterdam (1978)[Ba82] Balachandran, A.P., Marmo, G., Stern, A. : A Lagrangian approach to the non-interaction

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in the presence of a Yang-Mills field. Proc. Nat. Acad. Sci. 74, 5253–5254 (1977)[Gu84] Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University

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http://arxiv.org/abs/hep-th/9310068

Chapter 6The Geometry of Hermitean Spaces:Quantum Evolution

A measurement on one property can produce unavoidablechanges in the value previously assigned to another property,and it is without meaning to speak of a microscopic systempossessing precise values for all its attributes. This contradictsthe classical representation of all physical quantities bynumbers. The laws of atomic physics must be expressed,therefore, in a nonclassical mathematical language thatconstitutes a symbolic expression of the properties ofmicroscopic measurement.

Julian Schwinger, Quantum Kinematics and Dynamics 1970.

6.1 Summary

The basic geometrical structures arising in Quantum Mechanics are analyzed as inprevious chapters, that is, we ask when a given dynamics possesses simultaneouslyinvariant symplectic and metric stuctures. After solving this inverse problem in thelinear case, we will discuss some of its implications. All the fundamental mathemat-ical structures needed to discuss quantum mechanical systems will unfold in frontof our eyes. However we should warn the reader that the systems obtained so farare treated as classical ones, i.e., the physical epistemology of them is not inferredfrom the mathematical structures. Some questions will be raised in this sense and adiscussion of the physical meaning of the emerging mathematical structures will beconsidered at the end of the chapter.

6.2 Introduction

In Chap.5 we concentrated our attention on studying contravariant geometricalstructures compatible with a given dynamical system: vector fields and contravari-ant 2-tensors (Poisson tensors) were our primary concern, even though at thesame time zeroth order tensors (constants of motion) and covariant 2-tensors


407

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408 6 The Geometry of Hermitean Spaces: Quantum Evolution

(symplectic forms) appear naturally into the picture as well as the geometricalstructures needed to develop an intrinsic approach to the calculus of variations.

We have pointed out at various times that even if skew-symmetric structuresplayed a significant role, symmetric ones will also play their part in the descriptionof physical world. This chapter is devoted to them. In it we will discover that theexistence of compatible symmetric structures for a given dynamical system imposesextremely tight conditions on it, to the point of almost determining its structureuniquely. Not only that, the geometry of systems admitting a compatible Hermiteanstructure is exactly the geometry needed to describe quantum evolution in a veryspecific form that will be explained later on.

We must point out here that the so-called ‘geometry of quantum systems’ can beshared by many systems, quantum or not; that is, no attempt will be made here todescribe the epistemological and phenomenological intricacies of quantum systems,least of all to try to present the ‘theory of measurements’ founding the physicaltheory of quantum systems. However it is remarkable that the extremely elegantand comprehensive view offered by the geometrical picture of quantum systemsoffers a crisp picture of many aspects of quantum physics that on many occasionsare hindered by obscure reasonings and that, from this point of view will simplyconstitute straightforward consequences of the geometry of the system and do notdepend on its classical or quantum physical character.

A one-parameter group of transformations hasmany featureswhich do not dependon the specific space on which it acts. When the space is the space of states or thespace of observables, the evolution should be compatible with additional structuresdictated by the physical setting we are considering, i.e., classical or quantum.

To be more precise, in the quantum case we are going to study the properties ofdynamical systems on Hilbert spaces exploring the consequences of the existenceof invariant structures that this time is going to be an Hermitean structure. We shallstress here thatwe do not pretend to describe the geometrical foundations ofQuantumMechanics, which is something very different. Neither we are going to approach theproblem of the axiomatic foundation of quantum mechanics, i.e., we are not goingto describe the phenomenological and physical arguments leading to construction ofthe Hilbert space of states of a quantum system and their unitary evolution, togetherwith some extremely subtle aspects of the physical interpretation of observables andthe measuring process. In this sense this chapter deals more with what can be calledquantum-like systems, i.e., systems whose dynamics takes place in Hilbert spacesand preserves certain Hermitean structures. But we do not imply that the systemis quantum, that is, such a system could very well be a classical system, hence thestates of the system will simply be phase space like classical states and the space offunctions will be the space of classical observables (this idea has been used often,see for instance [St66, Du90, Ma96]).

However we must point out that a big deal of the results we are going to describecan be applied directly to actual quantum systems, hence this chapter can also be readas an exposition on the geometrical properties of the dynamics of quantum systems.The structures we are considering do have a physical interpretation and are coherentwith the experimental findings and therefore may be falsified.

6.2 Introduction 409

To avoid technical difficulties we will deal mostly with finite dimensional carrierspaces V , i.e., only ‘finite level quantum systems’ will be considered, an assumptionwhich is not far from experimental constraints.

6.3 Invariant Hermitean Structures

6.3.1 Positive-Factorizable Dynamics

We will assume as customary that � is a linear vector field on a finite-dimensionallinear space E . While studying conditions for a linear vector field � = X A to beHamiltonian with constant symplectic form �, see Sect. 5.2.6, we found, Eq. (5.27):

AT �+�A = 0; �T = −�, (6.1)

with H = �A a symmetricmatrix, whereas, the condition for the same vector field tobe compatible with a constant symmetric (2, 0)-tensor, see Sect. 4.2.2, it was foundto be, Eq. (4.2):

AT G + G A = 0; GT = G, (6.2)

with the matrix B = G A being skew-symmetric. If we had A = �H , with �

skew-symmetric, it is clear that det A �= 0 would imply det� �= 0 and det H �= 0.Recall that the previous conditions are distilled from the factorization problem for

a linear dynamical system, i.e., the existence of a skew symmetric contravariant, i.e.,(2, 0), tensor� on E and a covariant symmetric tensor G on E such that A = �◦G.Let us also recall that � can be thought as a linear map � : E∗ → E such that〈β,�(α)〉 = −〈�(β), α〉 for all α, β ∈ E∗, and the symmetric covariant tensor Gcan be thought as a linear map G : E → E∗, such that 〈G(u), v〉 = 〈u, G(v)〉 for allu, v ∈ E . Then the composition�◦G is an endomorphism A : E → E . Clearly if Ais invertible both maps� and G must be invertible too, thus the linear map� = �−1will define a non-degenerate 2-form on E , i.e., a linear symplectic structure on Eand the dimension of E must be even. Moreover

� ◦ A = G. (6.3)

Replacing the linear maps A, � and G by the corresponding matrices once a lin-ear basis has been chosen (and keeping the same notation for the correspondingmatrices) we obtain that equation (6.3) becomes the matrix equation �A = Gand then (�A)T = GT = G = �A, hence Eq. (6.1). Moreover, from the fun-damental factorization condition A = �G, we obtain that AT = −G� and thenAT G = −G�G = −G A, that is Eq. (6.2).

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Therefore we conclude that any such a linear factorizable system� with invertiblerepresentative matrix A will preserve both, a symplectic structure � and a non-degenerate symmetric tensor G.

If we assume now that the tensor G is definite positive we can obtain moreinformation about the structure of� and G satisfying Eq. (6.3). Certainly, consider aLagrangian subspace L ⊂ E for�. The symmetric tensor G defines a scalar producton E , hence on L . Then consider now the subspace L⊥G orthogonal to L with respectto the metric G. Clearly

AL⊥G = L (6.4)

because if u ∈ L⊥G and v ∈ L , we have �(Au, v) = G(u, v) = 0. Thus Au ∈L⊥� = L because L is Lagrangian, then AL⊥G ⊂ L . Moreover because A isinvertible we have that both subspaces must be the same. Before addressing thegeneral structure theorem for such situation we can recapitulate some of the previousdiscussion as:

Proposition 6.1 Suppose that on a real vector space E of dimension m we haveA = � G with � skew-symmetric, ker A = 0 and G positive definite symmetric.Then E is even dimensional, m = 2n and:

1. If we use G as a scalar product on E, then the adjoint A† of the operator A withrespect to G satisfies A† = −A, in other words, A is G-skew-symmetric.

2. When considering the polar decomposition of the matrix A in the form A = |A| Jwith respect to the scalar product G, then the matrix J is a complex structurecommuting with A and |A|.

Proof We have shown before that E must be even dimensional. Item 1 is just anotherway of expressing the compatibility condition Eq. (6.2). Notice that if we define ascalar product 〈u, v〉G = G(u, v) on E and A is a linear operator A : E → E , then (Efinite-dimensional) 〈A†u, v〉G = 〈u, Av〉G . Then by choosing a basis and denotingthe corresponding matrices with the same symbols, we get A† = G−1AT G. ThenA† = −A is equivalent to AT G = −G A.

Finally to prove 2 let us consider the polar decomposition of the matrix A asA = |A| J where J is an orthogonal matrix with respect to the scalar product G, i.e.,J T G J = G, and |A| is a positive G-symmetric operator (uniquely defined becauseker A = 0).

Again, because ker A = 0 such a factorization is unique with |A| = √A† A > 0.As A† = −A, A†A = −A2 > 0. Therefore J = A/|A| is a function of A and then Jcommutes with A. Finally, a left multiplication by J in both members of the relationcorresponding to A† = −A, leads to |A| = −J 2 |A|, and as |A| is invertible we getJ 2 = −1. �

In the proof of 2 in the previous theorem, we derived two additional facts that areconvenient to highlight.

6.3 Invariant Hermitean Structures 411

Corollary 6.2 Under the conditions of the previous theorem, Theorem 6.1, we getthat the complex structure J obtained from the dynamics is orthogonal with respectto the scalar product G, i.e.

G(Ju, Jv) = G(u, v), ∀u, v ∈ E, (6.5)

and�(Ju, u) > 0, ∀u �= 0. (6.6)

Proof The G-orthogonality of J , Eq. (6.5), follows from its definition J = A |A|−1;thus:

G(Ju, Jv) = G(A|A|−1u, A|A|−1v) = G(u, A†A|A|−2v) = G(u, v),

because |A|2 = A†A = −A2 and A commutes with |A|.The positivity condition, Eq. (6.6), follows from the positivity of |A|, that is A†A

is G-positive because:

G(A†Au, u) = 〈A†Au, u〉G = 〈Au, Au〉G > 0, u �= 0.

The operator A†A is self-adjoint. We can extend A†A to the complexificationEC = E ⊗ C of the real linear space E , then because of the spectral theoremA†A =∑n

k=1 λk Pk where λk > 0 are the eigenvalues of A†A and Pk the orthogonal

projectors on the corresponding eigenspaces. Then |A| = √A†A = ∑k=1

√λk Pk

and

G(|A|u, u) = 〈|A|u, u〉G =∑k=1

√λk ||Pku||2 > 0,

for all u �= 0. � Definition 6.3 Let � be a linear symplectic structure on E . We will say that acomplex structure J on E is tame with respect to �, or �-tame, if:

�(Ju, u) > 0, ∀u �= 0.

Thus because of the previous corollary, the complex structures obtained from fac-torizable invertible dynamics are tame with respect to the corresponding symplecticstructure.

We notice that it is also possible to consider a degenerate situationwith ker A �= 0.

Proposition 6.4 Suppose that on a real vector space E of dimension m we haveA = � H, with ker A �= 0, � skew-symmetric and H positive definite symmetric;then H defines a scalar product on E and:

1. E = ker A⊕

(ker A)⊥ is a decomposition of E that is invariant under the actionof A.


2. On (ker A)⊥ there exists an orthogonal operator J1 such that J 21 = −I and

commutes with A (when restricted to (ker A)⊥); as a by-product, this implies thatthe dimension of (ker A)⊥ is even.

3. On ker A, A vanishes and we can define J0 with J 20 = −1 only when dim(ker A)

is even.

Proof To prove 1, notice that as in Proposition 6.1, we have A† = −A, with theadjoint operator A† defined with respect to the scalar product determined by H(this property is just another way of writing the compatibility conditions, Eq. (6.2),derived from the factorization property). Now if v ∈ (ker A)⊥wehave that 〈Av, u〉 =〈v, A†u〉 = −〈v, Au〉 = 0 for all u ∈ ker A, thus Av ∈ (ker A)⊥.

To prove 2 and 3, notice that because of the invariance of (ker A)⊥ with respectto A we can apply Proposition 6.1 to the restriction of A to (ker A)⊥ and obtainan orthogonal operator J1 such that J2

1 = −I which commutes with A (hence thedimension of (ker A)⊥ must be even). �

Collecting these results, when dim E = 2 n, we can define J = J1⊕

J0 whichcommutes with A, is orthogonal, and J 2 = −1. Moreover, from � = A H−1 itfollows that � also decomposes according to the orthogonal decomposition E =ker A

⊕(ker A)⊥.

Before starting to unfold the consequences of these observations, we would liketo end this section by stating a structure theorem for linear dynamics on linear spacessatisfying the previous conditions.

Notice that selecting an orthonormal basis {ui | i = 1, . . . n} on an arbitraryLagrangian subspace L ⊂ E with dim E = 2n, then {Jui } is an orthonormal basisof J L , because J is G-orthogonal. If L were a real subspace, i.e., J L ∩ L = 0 andJ L were Lagrangian, then the collection {ui , Juk} would be an orthonormal basis,and rescaling the vectors suitably, a symplectic basis. In general, it is not true thatany Lagrangian subspace is real; neither is J L a Lagrangian subspace (in Sect. 6.3.2we will establish under what conditions that is true). However we can prove that sucha Lagrangian subspace always exists.

Theorem 6.5 Let E be a even-dimensional real linear space, � a linear symplecticstructure and G a scalar product on it. Then there exists a G-orthogonal symplecticbasis.

Proof Consider the invertible linear operator A defined by � and G, i.e., �A = G.We are in the situation of Proposition 6.1 and Corollary 6.2. If we consider as in theproof of Corollary 6.2 the complexfied space EC = E ⊗ C, then the extended lineroperator i A is Hermitian because A† = −A. Because of the spectral theorem, therewill exists an orthonormal basis of eigenvectors Wk , k = 1, . . . , n, with eigenvalues±√λk where λk > 0 are the egivenvalues of A†A = −A2 as in the proof of Corollary6.2. The eigenvectors Wk ∈ EC will have the form:

Wk = uk + ivk, k = 1, . . . , n,


and uk, vk ∈ E . Then it is easy to show that

G(ui , u j ) = G(vi , v j ) = 1

2δi j ; G(ui , v j ) = 0, ∀i, j = 1, . . . , n. (6.7)

Notice that because

δ jk = 〈W j , Wk〉G = 〈u j + iv j , uk + ivk〉G= 〈u j , uk〉G + 〈v j , vk〉G + i

(〈u j , vk〉G − 〈v j , uk〉G),

then:

〈u j , uk〉G + 〈v j , vk〉G = δi j (6.8)

and

〈u j , vk〉G = 〈v j , uk〉G (6.9)

for all u j , vk . We also have that (AWk)∗ = A∗W ∗

k but A∗ = A because A is areal matrix and ∗ denotes complex conjugation. Then AW ∗

k = −i√

λk W ∗k , if the

eigenvalue of A with eigenvector Wk were i√

λk . The simple computation

〈AW ∗j , Wk〉G = 〈W j , A†Wk〉G = −〈W j , AWk〉G,

shows:

〈−i√

λ j (u j − iv j ), uk + ivk〉G = −〈u j + iv j , i√

λk(uk + ivk)〉G .

that is,

√λ j

(〈u j , uk〉G − 〈v j , vk〉G) = −√

λk(〈u j , uk〉G − 〈v j , vk〉G

),

and,

√λ j

(〈u j , vk〉G + 〈v j , uk〉G) = −√

λk(〈u j , vk〉G + 〈v j , uk〉G

).

Then because√

λk > 0 for all k, we get:

〈u j , uk〉G = 〈v j , vk〉G (6.10)

and

〈u j , vk〉G = −〈v j , uk〉G (6.11)


for all j, k. Equation (6.10) combined with Eq. (6.8) gives the first expression inEq. (6.7). finally Eq. (6.11) together with Eq. (6.9) implies that 〈u j , vk〉G = 0for all i, j , concluding the proof of relations Eq. (6.7). Then the basis {u j , vk} isG-orthogonal.

Now recall that AWk = i√

λk Wk and |A|Wk = √λk Wk . Then J Wk = iWk , andconsequently, Juk = −vk and Jvk = uk , k = 1, . . . , n. Then:

�(u j , uk) = �(−√λ j−1

Av j , uk) = −√

λ j−1

G(v j , uk) = 0 (6.12)

and

�(v j , vk) = �(√

λ j−1

Au j , vk) =√

λ j−1

G(u j , vk) = 0 (6.13)

and finally,

�(u j , vk) = �(−√λ j−1

Av j , vk) = −√

λ j−1

G(v j , vk) = − 2√λ j

thus scaling the vectors vk (for instance) with the factor −√

λ j

2 we get the desiredsymplectic basis. �

Because of the previous theorem, linear dynamics � with operators determinedby linear operators A possessing the previous factorization properties, i.e., A =� ◦G, with � a non-degenerate constant Poisson structure and G a definite positivesymmetric (2, 0) tensor, i.e., a scalar product, have a very definite structure. We willobtain a more detailed description of the dynamics defined by such operators in theforthcoming sections and because of the central role that they play in the analysis ofHermitean dynamics we will call them positive factorizable dynamics.

Definition 6.6 A linear dynamical system � on a linear space E with associatedlinear operator A, will be called positive-factorizable if it is factorizable, that isA = �G with � a skew symmetric (0, 2) tensor and G a symmetric (2, 0) tensor,invertible, i.e., both � and G are nondegenerate, and G defines a scalar product inE , i.e., it defines a positive definite, symmetric bilinear form on E .

We conclude this analysis by stating the following theorem about the structure ofA, � and G, that recollects most of the previous results.

Theorem 6.7 Let A be positive factorizable linear operator on the vector space E,A = �G, then there exists a complex structure J on E which is orthogonal withrespect to the scalar product defined by G and �-tame. Moreover, there exists a realLagrangian subspace L, that is J L ∩ L = 0, with respect to the symplectic structure� = �−1 such that L⊥G is Lagrangian too and AL = J L = L⊥G . Hence the linearspace E decomposes as an orthogonal sum as E = L ⊕ J L and the operator A isanti-diagonal with respect to this decomposition.


Proof Because of Proposition 6.1 there exists a complex structure J on E whichis orthogonal with respect to the scalar product defined by G. We also showed, Eq.(6.4) that AL⊥G = L for any Lagrangian subspace L . Because of Theorem 6.5 thereexists a G-orthogonal symplectic basis {u j , vk} on E adapted to J that satisfies Eqs.(6.7). Then consider the subspace L = span{u1, . . . , un}. Notice that the vectorsuk are non zero and orthogonal, then dim L = n. Because J is G-orthogonal thenthe space J L has dimension n and the vectors vk provide an orthogonal basis for it.Again the orthogonality conditions (6.7) imply that J L = L⊥G because the vectorsu j and vk are orthogonal to each other. Thus J L ∩ L = 0 and L is a real subspace.

Moreover Auk = J |A|uk = √λk Juk = √

λkvk . Thus AL = J L and A isanti-diagonal with respect to the orthogonal decomposition E = L ⊕ L⊥G . Noticethat we always have that AL⊥G = L; actually the previous computation shows thatAvk = √λk

−1A2uk = −√λkuk for all k.

Finally because of Eqs. (6.12), (6.13) both subspaces L and J L areLagrangian. �

When the symmetric bilinear form associated with � is not positive definite, Gbecomes a pseudo-scalar product and the inner product will be pseudo-Hermitean. Itis now clear that a given linear dynamical system admits more than one Hamiltoniandescription and we may associate with it Hermitean or pseudo-Hermitean structures.The flow associated with the dynamical evolution will preserve all of them.

The complex structure J associated to a positive factorizable dynamics is also asymplectic map for the symplectic structure � as the following computation shows.

�(Ju, Jv) = �(AA−1 Ju, Jv) = G(J A−1u, Jv) = G(A−1u, v) = �(u, v);∀u, v ∈ E,

where we have used that J and A commute. Complex structures satisfying the pre-vious condition are called compatible with the symplectic structure.

Definition 6.8 Given a symplectic linear space (E,�), a complex structure Jsuch that:

�(Ju, Jv) = �(u, v), ∀u, v ∈ E, (6.14)

and

�(Ju, u) > 0, ∀u �= 0, (6.15)

is said to be a complex structure compatible with the symplectic structure �, or�-compatible for short. In other words J is �-compatible if it is a symplectic mapand is �-tame.

If J is �-compatible we will denote by gJ (or simply by g if there is no risk ofconfusion) the associated scalar product, i.e., g(u, v) = �(Ju, v).


If we concentrate our attention now on the structures � and J alone, we will getthe following result.

Proposition 6.9 Let (E,�) be a symplectic linear space and J a complex structurecompatible with �, then a subspace L ⊂ E is Lagrangian iff J L is Lagrangian.Moreover J L is the orthogonal complement of L with respect to the scalar productg associated to J .

Proof It is easy to prove that J L is the orthogonal complement of L . Let u, v ∈ L ,then

g(u, Jv) = �(Ju, Jv) = �(u, v) = 0.

Moreover because J is symplectic J L is Lagrangian iff L is Lagrangian. � If J is�-compatible, the combination of�, J and g defines anHermitean structure

on E .

Definition 6.10 Let E be a complex linear space, then aHermitean structure (also aninner product) is a positive definite sesquilinearmap on E , that is, amap h : E×E →C such that it is complex linear in the second argument, satisfies:

h(u, v) = h(v, u), ∀u, v ∈ E ,

and u, u > 0 for all u �= 0.

Notice that if h is an Hermitean structure h(iu, v) = −ih(u, v). It is a simplecomputation to check that if the complex structure J is �-compatible, then

h(u, v) = g(u, v)+ i�(u, v) = �(Ju, v)+ i�(u, v) ,

defines a Hermitean structure on E . Conversely, if h defines a Hermitean structurethen the structures:

g(u, v) = Re h(u, v), �(u, v) = Im h(u, v) (6.16)

define a compatible pair.Then we may summarize the previous discussion by stating that when the con-

ditions in the preceding propositions hold true, we have a scalar product g definedby H , a constant Poisson structure defined by � along with a complex structure J ,orthogonal with respect to g, which commutes with A. This allows us to complexifythe real vector space E and we may also define an Hermitean scalar product h onit, and i A will be self-adjoint on the resulting n-dimensional Hilbert space. Thuswe have again all the ingredients to associate with A a unitary flow on the complexvector space we have constructed out of E . Notice that any positive factorizablelinear dynamics gives raise to such structure. We will analyze their properties in thefollowing section.


Before embarking on a detailed analysis of such dynamics we would like to stressthat most of the previous results are also valid in infinite dimensions, in the followingsense. Let E be a real linear space, not necessarily finite-dimensional. Let A be aninvertible linear operator on E and we will assume as before that A is factorizable,i.e., A = � ◦ G. If A is invertible � must be invertible too, then � = �−1 is calleda strong symplectic form on E . We will say now that A is positive factorizable if Gdefines a scalar product on E . If we assume now that E is complete with respect to thenorm defined by G, or in other words that E is a (real) Hilbert space with the scalarproduct G and that A is a bounded linear operator such that A−1 is bounded too,then the results stated in Proposition 6.1, Corollary 6.2, Theorem 6.5, and Theorem6.7 are true with some obvious changes in the statements. Proposition 6.9 is alsotrue if � is a strong symplectic structure and J is a bounded symplectic map. Wewill not pay too much attention right now to the possibility of extending to infinitedimensions the previous results, but we will come back to them at the end of thischapter when dealing with quantum mechanical systems.

6.3.2 Invariant Hermitean Metrics

We will summarize from a slightly different perspective the results obtained at theend of the Sect. 6.3.1. Let E be a real linear space. We may use the tensorializationprinciple to promote the linear tensorial objects, �, J , G studied in the previoussection, to smooth tensors on E . Thus we introduce the geometrical tensors ω, g andJ corresponding to positive factorizable linear dynamical system � = X A:

ω = ωi j dxi ∧ dx j , g = gi j dxi ⊗ dx j , J = J ji dxi ⊗ ∂

∂x j. (6.17)

At this stage we will not be worried about further integrability conditions imposedwith the tensorialization principle as our structures are constant and only their alge-braic properties are relevant, that is: J : E → E is a complex structure and ω aconstant symplectic structure.

Definition 6.11 We will say that ω is J -invariant if

ω(J X, JY ) = ω(X, Y ),

for all vector fields X, Y in E .In this case

ω(X, JY ) = ω(J X, J 2Y ) = −ω(J X, Y ),

i.e.

ω(X, JY )+ ω(J X, Y ) = 0; (6.18)


for all vector fields X, Y in E .Notice that conversely if ω and J satisfy the preceding relation, Eq. (6.18), then,

ω(J X, JY ) = −ω(X, J 2Y ) = ω(X, Y ),

i.e., ω is J -invariant. Moreover, using J -invariance we see that the bilinear form gdefined as:

g(X, Y ) = ω(J X, Y ) (6.19)

defines a bilinear symmetric form, because

g(Y, X) = ω(JY, X) = −ω(Y, J X) = ω(J X, Y ) = g(X, Y ).

Note that

g(J X, JY ) = ω(J 2X, JY ) = −ω(X, JY ) = ω(JY, X) = g(Y, X) = g(X, Y ),

i.e., J is g-orthogonal:

g(J X, Y ) = g(J X, Y ). (6.20)

We say that (ω, J ) is a compatible pair (as after Definition 6.10, Eq. (6.16), in thelinear case) ifω is J -invariant and g(X, Y ) = ω(J X, Y ) defines a symmetric positivedefinite form. In such case J isω-compatible using the terminology in the Sect. 6.3.1.Then we can define a Hermitean product in E , considered now as a complex spacebecause of the complex structure J , by means of

h(X, Y ) = g(X, Y )+ i ω(X, Y ) = ω(J X, Y )+ i ω(X, Y ).

Recall also that the real linear space E can be complexified and the complexextension of J to EC, JC, admits two eigenvalues, ±i . Moreover,

EC = E+ ⊕ E−

where

E± = {X ∈ EC | JCX = ±i X}.

A tensorial triple (g, J, ω), with g a metric, J a complex structure and ω a sym-plectic structure satisfying the conditions (6.18), (6.19) and (6.20) will be called anadmissible triple. Because admissible triples are in one-to-one correspondence withHermitean products we will use both notions interchangeably. For reasons that willbe clear later on, wemay also refer to an admissible triple as a linear Kähler structure.


Definition 6.12 A linear vector field � will be called Hermitean if it preserves acompatible pair (ω, J ), i.e.

L�ω = 0, L� J = 0, (6.21)

or equivalently, if L�h = 0

Then, the inverse problem forHermitean dynamics can be stated as follows: Givena linear dynamics � find a Hermitean product h such that L�h = 0.

Notice that in using the Kähler geometry on E defined by the Hermitean product h(see later on Sect. 6.4.3 and [Er10], [Ib96a] for more details on the Kähler geometryof Hermitean systems), we are using a classical picture to describe the dynamics ofa system that later on we may be able to identify with a quantum system �. This wasprecisely the point of view adopted in [St66].

Thus, if we represent � in a real basis for E we get, � = Alkξ

k ∂/∂ξ l , and thefirst invariance condition in Eq. (6.16), L�ω = 0, is equivalent to the factorizationproperty for A. This forces us to lookfirst for the inverse symplectic problem for linearvector fields [Gi93]. Because of the discussion on Sect. 5.2.6, we obtain Eq. (5.29),i.e., all odd powers of A are traceless matrices, Tr A2k+1 = 0 for all k = 0, 1, . . ..The second invariance condition in Eq. (6.21) becomes

[A, J ] = 0, (6.22)

or, in other words, A is a complex linear map with respect to the complex structuredefined in E by J .

How can we determine all compatible pairs ω, J , that exist for a given linearfield �? Let us consider any invertible linear map T : E → E and denote by AT thetransformed linear operator, that is, AT = T−1AT . If we apply T to the previousrelations, Eq. (6.21) or in matrix form, to the factorization property of A, and Eq.(6.22). Then we get,

AT = �T · gT , [AT , JT ] = 0 (6.23)

with

�T = T−1�(T−1)t ; gT = T tgT , (6.24)

and

JT = T−1 J T . (6.25)

Therefore, if T commutes with A, that is AT = A, then we obtain a newHamiltoniandescription provided that �T �= � or gT �= g. Notice that because of the generalfactorization theorem, Theorem 4.2, even powers of A provide such transformations

http://dx.doi.org/10.1007/978-94-017-9220-2_5

http://dx.doi.org/10.1007/978-94-017-9220-2_5

http://dx.doi.org/10.1007/978-94-017-9220-2_4


in a natural way: Tt = exp(t A2k), k = 1, 2, . . .. In general we have the followingresult:

Proposition 6.13 Let� a Hermitean linear dynamics with associated linear operatorA. Any invertible nonunitary transformation T which is a symmetry for A takes aHermitean description of � onto an alternative one and any alternative Hermiteanstructure compatible with � is obtained by such a transformation.

Proof Let (ω, J, g) or h = g+ iω, be a Hermitean structure invariant under �. Thegroup of unitary transformations for the Hermitean structure h denoted by U (h) isthe intersection of the group of symplectic transformations Sp(ω) and the group ofisometries of g, O(g), then T is non unitary, it is either non-symplectic, non-isometricor both. In any case �T �= � or gT �= g.

Conversely, by using a Gram-Schmidt procedure, each Hermitean structure isreduced to the standard one in C

n . Therefore composing one map with the inverse ofthe other, we get the required T . In the symplectic case this amounts to the so-calledDarboux theorem and, in the Riemannian case it amounts to the diagonalizationprocedure. �

This shows that the space of alternative compatible pairs is a homogeneous spaceCA = G A/(U (h) ∩ G A), where G A is the isotropy group at A with respect to theadjoint action of GL(E) on the space of linear operators on E , i.e., the group oflinear symmetries of A, and U (h) is the unitary group defined by the Hermiteanstructure h.

How can we describe the space CA? or, more especifically, which are the sym-metries of A which are not unitary?

Proposition 6.14 For a generic matrix A, the symmetry group G A of A is Abelianand is generated by the flows corresponding to all powers of A, A j , j = 0, 1, . . . ,dim V while U (h) ∩ G A is generated by the flows corresponding to the odd powersof A. The homogeneous space is the subgroup of G A generated by the flows corre-sponding to the even powers of A.

Thus even powers of A generate symmetries which are not unitary. Then, anyfinite level quantum system admits alternative quantum descriptions parametrizedby the Abelian subalgebra generated by A2.

6.3.3 Hermitean Dynamics and Its Stability Properties

Among the linear Hamiltonian vector fields preserving an invertible Poisson struc-ture, we concentrate, in this section, on the subset of dynamical systems which arestable linear.


In particular, we show that:

Proposition 6.15 A stable linear Hamiltonian vector field � has at least one Hamil-tonian description with positive definite Hamiltonian function.

Proof Let, as usual, A be the representative matrix of�, with the property A = � H .We decompose (the complex extension of) A as a sum, A = S+ N , of a semisimplematrix S plus a nilpotent one N , with N S = S N . Then, for some integer k,

et A = et S et N = et S

(k∑

m=0

tm

m! N m

),

since N k+1 = 0. The evolution of the initial condition x0 is given by x(t) = et Ax0.To have a finite norm for x(t) when t → ∞ the nilpotent part must be absent, i.e.,k = 0. Moreover, the boundness of the orbit implies that the semisimple part S musthave only imaginary eigenvalues.

With the help of the associated eigenvectors we can construct an invariantHermitean structure. The imaginary part is a symplectic structure and the con-traction with the dynamical vector field � provides the positive definite Hermiteanfunction. �

Thus in the Hamiltonian decomposition we have:

Corollary 6.16 A linear Hamiltonian vector field which is stable, must have a rep-resentative matrix with purely imaginary eigenvalues and, therefore, will admit apositive definite Hamiltonian function.

Summarizing: stable linear Hamiltonian systems may be associated with unitaryflows on complex Hilbert spaces of finite dimension. We may call them ‘quantum-like’ systems. It is possible to formulate an inverse problem for these systems directlyon complex vector spaces.

6.3.4 Bihamiltonian Descriptions

In Sect. 6.3.1 (Definition 6.10) we have seen that on a complex linear space E theHermitean structure is associated with a triple (g, ω, J ) on the real vector spacecorresponding to E . There we have seen that g(u, v) = ω(Ju, v). At the grouplevel, this association is translated into the statement that transformations preservingthe Hermitean structure (unitary transformations) coincide with the intersection oftransformations that preserve the metric tensor g (orthogonal transformations) withthe transformations which preserve ω (symplectic transformations) or, also, withthose preserving the complex structure of J . Thus the intersection of the groupsof transformations preserving any two of the involved structures, gives the unitarygroup of transformations.


In this section, starting with two triples (g1, ω1, J1) and (g2, ω2, J2) on the realvector space E , dimR E = 2n, we would like to define when two triples are compat-ible. This definition arises from our desire to establish a situation similar to the onewe have for compatible Poisson brackets in the canonical setting when we considerdynamical systems which are biHamiltonian.

As the space of physical states is not the vector space itself but rather the complexprojective space, or the ray space, the analogy with the classical situation should beestablished not on the vector space but on the ray space. We notice that on the vectorspace E we have a linear structure represented by �. The reduction to the ray spacerequires also the generator of the shift in the phase, i.e., J (�), as we have two tripleswe are going to have a quotienting procedure associated with � and J1(�) for thefirst triple, and � and J2(�) for the second triple. In conclusion, if we denote byP1(E) and P2(E) the two quotient ray spaces, we could like to find conditions to beable to project tensor fields associated with (g2, ω2, J2), expressed into contravariantform, onto P1(V E (see more details in Sects. 6.5.4, 6.5.5).

To be able to follow the subsequent arguments we consider first a simple examplein two (real) dimensions. Here we are not considering the associated ray space, itwould be just a point, we examine instead the meaning of the definition which wenow give.

Definition 6.17 TwoHermitean structures h1 and h2, or their associated triples, willbe said to be compatible if

LJ1(�)h2 = LJ2(�)h1 = 0 . (6.26)

6.3.4.1 A Simple Example in Two Dimensions

Starting from the observation that two quadratic forms, one of which is assumedto be positive, can always be diagonalized simultaneously (at the price of using anon-orthogonal transformation, if necessary) we can assume from start g1 and g2 tobe of the form

g1=(

1 00 2

), g2=

(σ1 00 σ2

). (6.27)

The more general J such that J 2 = −1 will be of the form

J =(

a b−(1+ a2)/b −a

). (6.28)

Compatibility with g1 requires that J be anti-Hermitean (with respect to g1), and thisleads to

J = J1± =⎛⎝ 0 ±

√ 2 1

∓√

1 2

0

⎞⎠ (6.29)


and similarly

J = J2± =⎛⎝ 0 ±

√σ2σ1

∓√

σ1σ2

0

⎞⎠ (6.30)

from the requirement of admissibility with g2.As a consequence:

ω = ω1± =(

0 ±√ 2 1∓√ 2 1 0

), ω = ω2± =

(0 ±√σ2σ1

∓√σ2σ1 0

).

(6.31)

Nowwehave all the admissible structures, i.e., (g1, ω1±, J1±) and (g2, ω2±, J2±).Let us compute the invariance group for the first triple, having made a definite

choice for the possible signs (say: J = J+). The group is easily seen to be

O1(t) = cos(t)I+ sin(t)J1 =⎛⎝ cos(t)

√ 2 1

sin(t)

−√

1 2

sin(t) cos(t)

⎞⎠ , (6.32)

while for the second triple we obtain

O2(t) = cos(t)I+ sin(t)J2 =⎛⎝ cos(t)

√σ2σ1

sin(t)

−√

σ1σ2

sin(t) cos(t)

⎞⎠ , (6.33)

and in general we obtain two different realizations of SO(2).The two realizations will commute when ρ2/ρ1 = σ2/σ1. The latter condition

is easily seen (by imposing e.g., [J1,2, T ] = 0) to be precisely the condition ofcompatibility of the two triples.

To conclude the discussion of the example, let us see what happens in the com-plexified version of the previous discussion.

To begin with we have to define multiplication by complex numbers on R2, thus

making it a complex vector space, and this can be done in two ways, namely as:

(x + iy)

(ab

)= (xI+ J1y)

(ab

), (6.34)

or as

(x + iy)

(ab

)= (xI+ J2y)

(ab

). (6.35)


Correspondingly, we can introduce two different Hermitean structures on R2 as

h1 = g1 + iω1, or as h2 = g2 + iω2 . (6.36)

They are antilinear in the first factor and in each case the correspondingmultiplication by complex numbers must be used. The O1(t) and O2(t) actionsboth coincide with the multiplication of points of R

2 by the complex numbers eit

(i.e., with different realizations ofU (1)), but the definition of multiplication by com-plex numbers is different in the two cases.

6.3.5 The Structure of Compatible Hermitean Forms

Having clarified the setting by means of a previous example, let us proceed now tothe general situation and the various consequences emerging from the compatibilitycondition.

Consider two different Hermitean structures on E , h1 = g1 + iω1 and h2 =g2 + iω2. We define the associated quadratic functions g1 = g1(�,�) and g2 =g2(�,�). Their associated Hamiltonian vector fields will be exactly �1 = J1(�)

and �2 = J2(�), if we use 12g1 and 1

2g2 as Hamiltonians. This follows easily fromω(�,�) = g(�,�) and i�� = d H , and,

L�H = i�i�� = d H(�) = 2H = g(�,�) .

From the definition of compatible Hermitean structures we get:

L�1,2�2,1 = L�1,2g2,1 = 0 .

Remark 6.1 Notice that, if ω = ωi j dxi ∧ dx j is a constant symplectic structureand X = Ai

j x j∂/∂xi is a linear vector field, then the condition: LXω = 0 can bewritten in terms of the representative matrices as the requirement that the matrix ωAbe symmetric, i.e., (recall Eq. (6.1)):

ωA − (ωA)T = ωA + AT ω = 0 , (6.37)

while the condition LXg = 0 is equivalent to the matrix gA being skew-symmetric,i.e., (recall now Eq. (6.2)):

gA + (gA)T = gA + AT g = 0 . (6.38)

Now because L�1g2 = 0 we get that the matrix g2A1 will be skew-symmetric(Eq. (6.38)), but then

L�1g2 = xT (g2A1 + AT1 g2)x = 0 , (6.39)


and we obtain:0 = L�1(i�2ω2) = i�2L�1ω2 + i[�1,�2]ω2 ,

and as L�1ω2 = 0, we conclude:

i[�1,�2]ω2 = 0,

and similarly for ω1. As neither ω1 nor ω2 are degenerate, this implies that �1 and�2 commute, i.e. that

[�1, �2] = 0.

Moreover, remembering that the Poisson bracket of any two functions f and g, { f, g}is defined as { f, g} = ω(Xg, X f ), where X f and Xg are theHamiltonian vector fieldsassociated with f and g respectively, we find 0 = L�1g2 = dg2(�1) = 2ω2(�2, �1).Hence we find:

{g1, g2}2 = 0 ,

where {·, ·}2 is the Poisson bracket associated with ω2 and similarly for ω1.

Proposition 6.18 Two Hermitean structures h1 and h2 with associated quadraticforms g1 = g1(�,�), g2 = g2(�,�), and Hamiltonian vector fields �1 and �2 arecompatible iff {g1, g2}1,2 = 0 and L�1ω2 = L�2ω1 = 0.

Proof It remains to be proved that L�1g2 = 0. Notice that this is equivalent tothe matrix g2A1 being skew-symmetric. Then the condition {g1, g2}2 = 0 impliesexactly that because of Eq. (6.39). �

Remembering that the weak form of the tensorialization principle states that state-ments concerning linear vector fields translate into equivalent statements for the(1, 1)-type tensor fields having the same representative matrices, and recalling thatthe defining matrices of �1 and �2 are precisely those of the corresponding complexstructures, we see at once that:

[�1, �2] = 0⇐⇒ [J1, J2] = 0 ,

i.e., that the two complex structures will commute as well.In general, given any two (0, 2) (or (2, 0)) tensor fields g1 and g2 one (at least) of

which, say g1, is invertible, the composite tensor g−11 ◦g2 will be a (1, 1)-type tensor.Then, out of the two compatible structures we can build up the two (1, 1)-type tensorfields

G = g−11 ◦ g2 ,


and

T = ω−11 ◦ ω2.

Actually one can prove at once that the two are related, and a direct calculationproves that:

G = J1 ◦ T ◦ J−12 = −J1 ◦ T ◦ J2

or, in other words,

T = −J1 ◦ G ◦ J2 . (6.40)

It turns out that T (and hence G) commutes with both complex structures i.e.,

[G, Ja] = [T, Ja] = 0, a = 1, 2 . (6.41)

This follows from the fact that both G and T are �-invariant, i.e.,

L�1,2G = L�1,2T = 0 ,

and from Eq. (6.40).It follows also from Eqs. (6.40) and (6.41) that G and T commute, i.e.:

[G, T ] = 0 .

Moreover, G enjoys the property that

ga(Gu, v) = ga(u, Gv), a = 1, 2 . (6.42)

Indeed one can prove by direct calculation that

g1(Gu, v) = g1(u, Gv) = g2(u, v) ,

while

g2(Gu, v) = g2(u, Gv) = g−11 (g2(u, ·), g2(v, ·)) ,

and this completes the proof.Notice that Eq. (6.42) can be read asG being self-adjointwith respect to both metrics.

Notice that the derivation of this result does not require the compatibility conditionto hold. If the latter is assumed, however, one can prove also that T is self-adjointwith respect to both metrics, and that both J1and J2 are instead skew-adjoint withrespect to both structures, i.e., that


ga(T u, v) = ga(u, T v), a = 1, 2 ,

and thatg1(u, J2v)+ g1(J2u, v) = 0, ∀u, v ∈ E ,

with a similar equation with the indices interchanged.Indeed, from, e.g.: L�1ω2 = 0 we obtain, in terms of the representative matrices

and using: J1 = g−11 ω1:

ω2g−11 ω1 = ω1g

−11 ω2 ⇔ ω2ω

−11 g1 = g1ω

−11 ω2.

Remembering the definition of T , that is equivalent to: g1T = (g1T )T , and thisleads to

T = g−11 T T g1 = (T †)1

where (T †)1 is the adjoint of T with respect to g1. Interchanging indices, one canprove that: (T †)2 = T as well.

Concerning the J ’s (that have already been proved to be skew-adjoint with respectto the respective metric tensors), consider, e.g.

(J †1 )2 =: g−12 J T

1 g2 = −g−12 g1 J1 ,T g−11 g2 = −G−1 J1G = −J1

as G and the J ’s commute. A similar result holds of course for J2.Summarizing what has been proved up to now, we have found that G, T, J1 and

J2 are a set of mutually commuting linear operators. G and T are self-adjoint, whileJ1 and J2 are skew-adjoint, with respect to both metric tensors.

If we now diagonalize G, the 2n-dimensional vector space E ∼= R2n will split

into a direct sum of eigenspaces: E = ⊕k Vλk , where the λk’s (k = 1, . . . , r ≤ 2n)are the distinct eigenvalues of G. According to what has just been proved, the sumwill be an orthogonal sum with respect to both metrics, and, in Vλk , G = λkIk , withIk the identity matrix in Vλk . Assuming compatibility, T will commute with G andwill be self-adjoint. Therefore we will get a further orthogonal decomposition ofeach Vλk of the form

Vλk =⊕

r

Wλk ,μk,r

where the μk,r ’s are the (distinct) eigenvalues of T in Vλk .The complex structurescommute in turn with both G and T . Therefore they will leave each one of theWλk ,μk,r ’s invariant.

Now we can reconstruct, using the g’s and the J ’s, the two symplectic structures.They will be block-diagonal in the decomposition of E , and on each one of theWλk ,μk,r ’s they will be of the form

g1 = λkg2, ω1 = μk,rω2 ,


where, with some abuse of notation, we have written g1, g2, ω1, ω2 instead of theirrestrictions to the subspaces Wλk ,μk,r . Therefore, in the same subspaces:

J1 = g−11 ω1 = μk,r

λkJ2 .

It follows from: J 21 = J 2

2 = −1 that: (μk,rλk

)2 = 1, whence: μk,r = ±λk (andλk > 0 ). The index r can then assume only two values, corresponding to ±λk andat most Vλk will have the decomposition of Vλk into the orthogonal sum: Vλk =Wλk ,λk ⊕Wλk ,−λk . All in all, what we have proved is the following:

Lemma 6.19 If the two Hermitean structures h1 = (g1, ω1) and h2 = (g2, ω2),coming from admissible triples (g1, ω1, J1) and (g2, ω2, J2), are compatible, thenthe vector space E ∼= R

2n will decompose into the (double) orthogonal sum:

E ∼=⊕

k=1,...,r ,α=±1Wλk ,αλk , (6.43)

where the index k = 1, . . . , r ≤ 2n labels the eigenspaces of the (1, 1)-type tensorG = g−11 ◦ g2 corresponding to its distinct eigenvalues λk > 0, while T = ω−11 ◦ω2will be diagonal (with eigenvalues ±λk) on the Wλk ,±λk ’s, on each one of which:

g1 = λkg2, ω1 = ±λkω2, J1 = ±J2 . (6.44)

As neither symplectic form is degenerate, the dimension of each one of theWλk ,±λk ’s will be necessarily even.

Now we can further qualify and strengthen the compatibility condition by statingthe following:

Definition 6.20 Two (compatible) Hermitean structures will be said to be in ageneric position iff the eigenvalues of G and T have minimum (i.e., double)degeneracy.

In general, two appropriate geometrical objects like two (0, 2)or (2, 0)-type tensorfields are said to be in a generic position if they can be ‘composed’ to yield a 1-1tensor whose eigenvalues have minimum degeneracy. For instance g1 and g2 are ina generic position if the eigenvalues of G = g−11 ◦ g2 have minimum degeneracy,which possibly depends on further conditions: when compatibility is required, thisdegeneracy is double. The results that we have just proved will imply that each oneof the Wλk ,λk , Wλk ,−λk will have the minimum possible dimension, that is two.

Denoting them by Ek ( these now two-dimensional subspaces k = 1, . . . , n), allthat has been said up to now can be summarized in the following:

Theorem 6.21 (Structure theorem for generic compatible Hermitean structures) Ifthe Hermitean structures h1 and h2 are compatible and in a generic position, then the


2n-dimensional carrier linear space E splits into a sum of n mutually bi-orthogonal(i.e., orthogonal with respect to both metrics g1 and g2 ) two-dimensional vectorsubspaces E = E1 ⊕ E2 ⊕ . . . ⊕ En. All the structures ga, Ja, ωa decomposeaccordingly into a direct sum of structures on these two-dimensional subspaces, andon each one of the Ek’s they can be written as

g1|Ek = λk(e∗1 ⊗ e∗1 + e∗2 ⊗ e∗2); λk > 0 g2|Ek = k g1|Ek ; k > 0J1|Ek = (e2e∗1 − e1e∗2) J2|Ek = ±J1|Ek

ω1|Ek = λk(e∗1 ∧ e∗2) ω2|Ek = ± k ω1|Ek

where e2 = J1e1, e1 is any given vector in Ek and the e∗’s are the dual basis of thee′’s. In other words on each subspace g1 and g2 are proportional, while J1 = ±J2and accordingly ω2 = ± ω1.

Every linear vector field preserving both h1 = (g1, ω1) and h2 = (g2, ω2) willhave a representative matrix commuting with those of T and G , and it will be block-diagonal in the same eigenspaces Ek . Therefore, in the generic case, the analysis canbe restricted to each 2-dimensional subspace Ek in which the vector field willpreserve both a symplectic structure and a positive-definite metric. Therefore it willbe in Sp(2)∩ SO(2) = U (1) and, on each Ek , it will represent a harmonic oscillatorwith frequencies depending in general on the Vk ’s .

Going back to the general case, we can make contact with the theory of completeintegrability of a biHamiltonian system by observing that T plays here the role ofa recursion operator. Indeed, we show now that it generates a basis of vector fieldspreserving both the Hermitean structures ha given by:

�1, T �1, . . . , T n−1�1 . (6.45)

To begin with, these fields preserve all the geometrical structures, commute pair-wise and are linearly independent. In fact these properties follow from the observa-tion that T , being a constant 1-1 tensor, satisfies the Nijenhuis condition [T, T ] = 0.Therefore, for any vector field X :

LT X T = TLX T (6.46)

that, T being invertible, amounts to

LT X = TLX (6.47)

So, ∀k ∈ N:LT k�1

= TLT k−1�1= . . . = T kL

�1(6.48)

and

T kL�1ωa = 0 = T kL�1

ga ; (6.49)


Moreover, ∀s ∈ N:

[T k+s�1, T k�1] = LT k+s�1T k�1 = T sLT k�1

T k�1 = T s[T k�1, T k�1] = 0.(6.50)

Besides, the assumption of minimal degeneracy of T implies that the minimalpolynomial of T be of degree n. Indeed, we have shown that the diagonal form ofT is

T =⊕

k=1,...,n{±ρkIk} (6.51)

where Ik is the identity on Vk . Any linear combination

m∑r=0

αr T r = 0 , m ≤ n − 1 , (6.52)

yields a linear system for the αr ’s of n equations in m + 1 unknowns whose matrixof coefficients is of maximal rank and that, for m = n − 1, coincides with the fullVandermonde matrix of the ρk’s .

Then, we can conclude that the n vector fields T r�1, r = 0, 1, . . . , n − 1 forma basis.

6.4 Complex Structures and Complex Exterior Calculus

In Sect. 6.3.5 we discussed the inverse problem for Hermitean dynamics. As we dis-cussed there, Hermitean structures combine metric and symplectic structures anddefine in a natural way a complex linear structure. Before continuing our analysisand the discussion of an extremely interesting family of systems fitting into this geo-metrical description, quantum systems, we will digress on the geometry of complexlinear spaces. In the same way as in Chap. 3 we were provided with a purely tensorialcharacterization of real linear structures, we would like to proceed for complex struc-tures, that is to a geometrization of complex calculus. This task will be accomplishedin the next few sections.

6.4.1 The Ring of Functions of a Complex Space

Throughout this book the dual approach to dynamical systems, that is evolutionconsidered as derivations on an algebra of observables, has been stressed. When weencounter complex spaces new interesting phenomena occur.

Thus suppose that E is a complex linear space, i.e., a real space equipped with alinear complex structure J , that is a (1, 1)-tensor such that J 2 = −I . The ringF(E)

of smooth real functions defined on E , as was discussed in Chap.3, is generated by

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6.4 Complex Structures and Complex Exterior Calculus 431

its linear part, the eigenvectors with eigenvalue 1 of the Liouville vector field �E .The linear part of F(E) is the real dual space E∗ of E . Now, it happens that E isequipped with a complex structure, so it is a complex space, thus we shall considerC-valued functions instead. However remember that in the real case we selectedsmooth functions on E as the natural space capturing the differential calculus onE . If we have a complex linear space, what shall we consider instead of smoothfunctions?

Let us first study the linear part of what this algebra would be. The space of reallinear maps E∗ can be complexified to obtain E∗ ⊗ C = (E∗)C, that is the spaceof real linear maps from E to CR (i.e., C is equipped with its natural real spacestructure). The natural notion of transformation between complex linear spaces isa set of complex linear maps. We can show immediately that any real linear mapα : E → C decomposes as the sumof a complex linear and a complex antilinearmap:

α = α(1,0) + α(0,1). (6.53)

In fact, let

α(1,0)(v) = 1

2(α(v)− iα(iv)) , α(0,1) = 1

2(α(v)+ iα(iv)).

It is clear that

α(1,0)(iv) = iα(1,0)(v) , and α(0,1)(iv) = −iα(0,1)(v) .

We denote the complex linear maps α(1,0) as �(1,0)(E∗) and by �(0,1)(E∗) thecomplex antilinear maps α(0,1). Hence,

(E∗)C = �(1,0)(E∗)⊕�(0,1)(E∗) . (6.54)

If we are given a smooth map f : E → C, its differential at each point x ∈ Edefines a linear map d f (x) : Tx E = E → C. But, according to Eq. (6.53), d f (x) =d f (x)(1,0) + d f (x)(0,1). Notice that the complex structure J , being a (1, 1) tensor,acts on d f in a natural way, (J ∗d f )(X) = d f (J (X)) for any vector field X . Thusthe complex linear and antilinear parts of d f are eigenvectors of J with eigenvalues±i respectively.

We will say that a function f is holomorphic at x if d f (x)(0,1) = 0 (respectively,we will say that f is anti-holomorphic at x if d f (1,0)(x) = 0). We will say that f isholomorphic if it is holomorphic at x for all x ∈ E (respect. antiholomorphic). Noticethat f being holomorphic is equivalent to the expression d f (x)◦ J (x) = J0 ◦d f (x),for all x ∈ E , or simply

d f ◦ J = J0 ◦ d f. (6.55)


where J0 denotes the complex structure on C, i.e., multiplication by i .If we introduce complex coordinates zk in E , with real structure zk = xk + iyk ,

k = 1, . . . , n, dimR E = 2n, and we write f = u+ iv, then the previous expressiond f (0,1) = 0 takes the familiar form of Cauchy-Riemann equations for f :

∂u

∂xk= ∂v

∂yk; ∂u

∂yk= − ∂v

∂xk, k = 1, . . . , n . (6.56)

Now notice that the space of complex linear maps αx : Tx E → C is just thecomplexification of the cotangent space T ∗x E , that is (T ∗x E)C = T ∗x E ⊗ C. But(T ∗x E)C can be decomposed into its holomorphic and antiholomorphic parts, i.e.,(T ∗x E)C = �(1,0)(T ∗x E)⊕�(0,1)(T ∗x E) following Eq. (6.54). Often we will simplywrite (T ∗x E)C = T ∗x E (1,0) ⊕ T ∗x E (0,1) for every x ∈ E . Globally we get:

(T ∗E)C = T ∗E (1,0) ⊕ T ∗E (0,1) .

The previous decomposition introduces a similar decomposition on the space ofcomplex-valued 1-forms on E , i.e., if we denote by �1(E, C) the space of complex-valued 1-forms on E , that is, �1(E, C) = �1(E) ⊗ C, they may be identifiedwith sections of the bundle (T ∗E)C. Then we may decompose a 1-form α intoits holomorphic and antiholomorphic part: α = α(1,0) + α(0,1) (pointwise as inEq. (6.53)), then

�1(E, C) = �(1,0)(E)⊕�(0,1)(E).

The exterior differential d maps a smooth function f into a 1-form d f , thenbecause of the decomposition of d f into its holomorphic and antiholomorphic parts,we can write:

d f = d f (1,0) + d f (0,1) ,

Thus we can define the Cauchy–Riemann first-order differential operators ∂ and ∂ asthe holomorphic and antiholomorphic parts of the exterior differential respectively,that is:

∂ f = d f (1,0), ∂ f = d f (0,1) . (6.57)

Hence a function f is holomorphic iff d f = ∂ f or equivalently, ∂ f = 0 (respectively,anti-holomorphic if ∂ f = 0).

Using complex coordinates zk , the Cauchy–Riemann operators take the simpleform:

∂ f = ∂ f

∂zkdzk, ∂ f = ∂ f

∂ zkd zk


with zk denoting the complex conjugate of zk .We may consider now arbitrary complex differential forms on the complex linear

space E . The differentials dz1, . . . , dzn, dz1, . . . , dzn constitute a basis for themod-ule of (complex) differential forms of degree 1. If follows that a differential form ofdegree k can be expressed as a linear combination with complex-valued coefficientsof exterior products of order k of the 1-forms dzi and dz j , i, j = 1, . . . , n. Anhomogeneous term containing p factors dzi and q = k − p factors dz j is said to beof bidegree (p, q). A differential form of bidigree (p, q) is a sum of homogeneousterms of bidegree (p, q).

The notion of bi-degree is well defined in the realm of complex linear spaces andholomorphic changes of coordinates, i.e., a smooth change of coordinatesφ : E → Eis holomorphic if the differential map dφ(x) : Tx E = E → Tφ(x)E = E is linearcomplex for all x ∈ E . In other words the map φ : E → F is a holomorphic map if:

dφ ◦ J = J ◦ dφ .

It is clear now that if we introduce complex coordinates zi as before andwi = zi ◦ φ are the transformed coordinates, then wi are holomorphic functionsiff φ is holomorphic and dwi = dzi ◦ φ, dwi = dzi ◦ φ. We will denote the moduleof differential forms of bidegree (p, q) as �(p,q)(E). Clearly:

�k(E, C) =⊕

p+q=k, p,q≥0�(p,q)(E) .

Notice that, in particular, �2(E, C) = �(2,0)(E)⊕ �(1,1)(E)⊕ �(0,2)(E). A sim-ilar bidegree structure can be introduced in the tensor algebra over E and in thecorresponding algebra of differentiable tensors over E .

6.4.2 Complex Linear Systems

From the discussion before we conclude that a linear complex structure is definedby the constant (1,1)-tensor

TJ = Jik ∂

∂xk⊗ dxi ,

that we will keep denoting by J . Notice that in complex coordinates

J = dzi ∂

∂ zi− dzi ∂

∂zi.

Thus if a given dynamical system � on E is possess an invariant linear complexstructure J , then


L� J = 0,

which, is equivalent to:

[A, J ] = 0 , (6.58)

if � is linear with � = X A. But Eq. (6.58) means that the linear map A is linearcomplex, i.e., the matrix A can be identified with an n × n complex matrix Z andthe vector field � defines a complex linear vector field on C

n ,

� = Z ji z

i ∂

∂z j

whose flow is given by the one-parameter family of complex isomorphisms,

φt = et Z .

In the real space E , if we consider a basis u1, . . . , un, v1, . . . , vn , adapted to J ,i.e., such that J (uk) = vk and J (vk) = −uk , and denoting the corresponding set oflinear coordinates by xk, yk , we will have:

� = X ji

(xi ∂

∂x j + yi ∂

∂ y j

)+ Y j

i

(xi ∂

∂ y j − yi ∂

∂x j

),

with Z = X + iY .In this sense the simplest dynamical system possessing an invariant complex

structure is the vector field defined by the complex structure itself, i.e., if J is a givencomplex structure � = X J has TJ = J as a invariant complex structure. The vectorfield X J is just the isotropic harmonic oscillator.

Notice that the Liouville vector field �E characterizing the linear space E iscomplex with respect to any complex structure on E because�E = X I and [I, J ] =0, hence a linear complex structure is characterized by the two objects, a vector field�E and the (1, 1)-tensor J . However it can be shown (see later) that the complexlinear vector space structure E can be described by two commuting vector fields �,� with the properties of �E = X I and � = X J = J (�E ).

We denote by Jl(E) the space of linear complex structures on E . It is immediateto see that the group GL(E) of real linear automorphisms of E acts transitively onthis space. In fact, if J1, J2 are two complex linear structures, choose adapted basesto both, thus, if { uk, vk } and { u′k, v′k } denote such basis for J1, J2 respectively,define the linear isomorphism P : E → E , by P(uk) = u′k , P(vk) = v′k . Then,J2 ◦ P = P ◦ J1 and the action of GL(E) on Jl(E) by conjugation is transitive. Interms of the model R

2n , the group GL(2n, R) acts on Jl(R2n) by J0 �→ P−1 J0P ,

and the stabilizer group at the linear complex structure J0 is the set of matrices suchthat [P, J0] = 0, i.e., the set of complex matrices GL(n, C).


Proposition 6.22 The space Jl(E) of linear complex structures on E isdiffeomorphic to the homogeneous space GL(2n, R)/GL(n, C) and has dimen-sion 2n2. This space has two connected components corresponding to matrices withpositive or negative determinant. The component J +l (E) is diffeomorphic to thehomogeneous space GL+(2n, R)/GL(n, C) and contains the complex structure J0.

Proof Notice that GL(2n, R) has two components depending on the sign of thedeterminant. We shall denote by GL+(2n, R) those matrices with positive determi-nant. Notice that det J0 = 1 and thus J0 belongs to the component GL+(2n, R)/

GL(n, C). � Exercise 6.1 Show that if dim E = 2, then Jl(E) ∼= S2.

Consider the space of compatible linear complex structures with a given lineardynamics. We shall denote such space by Jl(E, �). Thus, if Jl(E, �) �= ∅, then� = X A where A is a complex linear map with respect to a given complex structure.Using the model space (R2n, J0) for this complex structure, then [A, J0] = 0, andthe space Jl(E, �) is diffeomorphic to the space of matrices such that J 2 = −I and[A, J ] = 0.

For a complex generic A, Jl(R2n, A) = ±J0. On the opposite side, we find

however Jl(E,�E ) = Jl(E).

6.4.3 Complex Differential Calculus and Kähler Manifolds

We can now consider the geometrical notion of complex structure on E given by anon-necessarily constant, (1,1)-tensor J = Ji

k (x)∂/∂xk ⊗ dxi that at each pointdefines a complex structure on the tangent space Tx E ∼= E of E .

Definition 6.23 A (1,1)-tensor J on E such that for each x ∈ E , J (x)2 = −I willbe called an almost complex structure.

An almost complex manifold is an even-dimensional real manifold M endowedwith a (1, 1)-type tensor field J , called an almost complex structure, satisfying

J 2 = −I . (6.59)

We will discuss later the strong holonomy principle for linear complex structures,i.e., under what conditions an almost complex structure has a local normal formgiven by a constant linear complex structure. We will start introducing the notion ofcomplex and almost complex manifold.

A complex manifold [Ch67, Sc86] is a smooth manifold M that can be locallymodeled on C

n for some n, and for which the chart-compatibility conditions arerequired to be holomorphic maps.


A complex manifold as a real smooth manifold M has dimension 2n, hence it canbe embedded inR

2N with N large enough. The local charts define holomorphic tran-sition functions. It is not necessarily true that M can be embedded as a holomorphicsubmanifold of C

N .Alternatively, the algebra of holomorphic functions on C

N is not a differen-tiable algebra, hence we cannot extend the theory developed in Chap.3 to complexmanifolds.

Complex manifolds that can be embedded holomorphically in CN are called

Steinmanifolds. A compact complexmanifold cannot be embedded holomorphicallyin C

N (notice that compact complex manifolds have only constant holomorphicfunctions).

Nirenberg’s theorem [Ni57] shows that an almost complex manifold is a complexmanifold iff the almost complex structure J satisfies the integrability condition NJ =0, where NJ is the Nijenhuis torsion associated with J (see Appendix E).

Finally, let M be a real, even-dimensional, smooth manifold with a complexstructure J and closed 2-form ω which is J -invariant, that is:

ω (X, JY )+ ω (J X, Y ) = 0; X, Y ∈ T M . (6.60)

Notice that as we discussed in the case of linear Hermitean structures, this impliesthat the bilinear form:

g (X, Y ) = ω (J X, Y ) ;

is symmetric and nondegenerate. When g is positive, that is g(X, X) > 0 for allX �= 0, then (M, g, J, ω) is a said to be Kähler manifold [Ch67, Sc86, We58].1

Also, J 2 = −I implies

ω (J X, JY ) = ω (X, Y ) ; g (J X, JY ) = g (X, Y ) , (6.61)

for all X, Y ∈ T M . Notice that Eq. (6.61) implies the analog of Eq. (6.60) for g,namely

g (X, JY )+ g (J X, Y ) = 0 . (6.62)

A tensorial triple (g, J, ω), with g ametric, J a complex structure andω a symplec-tic structure satisfying conditions (6.60)-(6.61) will be called an admissible triple.

Moreover a Kähler manifold satisfies ∇ J = 0 with ∇ the Levi-Civita connectionof the metric g.

1 If g is not positive definite, then M is also-called a pseudo-Kähler manifold.


6.4.4 Algebras Associated with Hermitean Structures

We will conclude the analysis of the geometry of invariant Hermitean structuresby discussing various algebraic structures which are naturally associated with suchstructures and that are going to play a relevant role in the structure of quantumsystems.

We start again with a given Hermitean tensor h = g + iω. Associated with it wehave a Poisson bracket {·, ·} defined by the symplectic structureω = Im h. Moreoverthe symmetric part of the tensor h allows us to define also a Riemannian bracket

( f, g) = G(d f, dg) = Gkj ∂ f

∂ξ k

∂g

∂ξ j, (6.63)

with G the contravariant symmetric (2, 0) tensor obtained inverting the real part gof h. The symmetric product (·, ·) is obviously commutative but not associative.

We can define a Kähler bracket combining both products as:

[[ f, g]] = ( f, g)+ i{ f, g }. (6.64)

However in the spirit of deformations is natural to consider the products,

f ◦ν g = ν

2( f, g)+ f g

and

f �ν g = f g + ν

2[[ f, g]]. (6.65)

The product ◦ν defines a nonassociative commutative real deformation of thecommutative associative algebra F(E)C, whereas �ν determines a complex non-commutative deformed algebra that reduces to f g when ν = 0. By using the naturalcomplex conjugation they become involutive algebras. In addition we have the rela-tions,

{ f, g } = 1

iν( f �ν g − g �ν f ),

f ◦ν g = 1

2( f �ν g + g �ν f ).

The previous definitions can be extended without pain to Kähler manifolds. Adistinguished class of functions on a Kähler manifold are those real functions fsuch that

LX f G = 0 ,


where X f is the Hamiltonian vector field defined by f . This definition implies thatthe natural derivation corresponding to the Hamiltonian vector field X f defined byf is also a Killing vector and its flow are isometries of the metric structure.

Definition 6.24 Let (M, h) be a Kähler manifold. If f is a complex-valued functionon M , we say that f is Kählerian if the real and the imaginary parts u, v of f satisfythat LXu G = 0,LXu G = 0.

Kählerian functions capture in a geometrical way the class of Hermitean (or self-adjoint) operators on a space with inner product. Thus if H is a finite-dimensionalcomplex space with inner product2 〈·, ·〉, that is H carries a Hermitean structure hgiven by h(u, v) = 〈u, v〉, u, v ∈ H, a Hermitean (or self-adjoint) operator is acomplex linear map A : H → H such that A = A† or, equivalently, 〈Au, v〉 =〈u, Av〉 for all u, v ∈ H. Clearly, if A is a Hermitean operator then f A(u) = 〈u|Au〉is a Kählerian function. Notice that the function fA is real, i.e., f A = f A. Moreoverthe Hamiltonian vector field X f A is the linear vector field i Au, then because A isHermitean, its flow�t = exp i t A is a one-parameter group of unitary operators, thatis 〈�t u,�tv〉 = 〈u, v〉, for all u, v ∈ H. As a consequence, �t preserves both thereal and the imaginary part of h, that is they are orthogonal transformations for themetric g and isometries of the tensor G, then LX f A

G = 0.Notice that if T is any complex linear map on H, then T = A + i B with A =

(T + T †)/2, B = (T − T †)/2i are Hermitean operators. Then the complex functionfT (u) = 〈u, T u〉 = f A(u) + i fB(u) is Kählerian, in other words, the quadraticfunctions fT with T a complex linear operator on any finite-dimensional Hilbertspace H are Kählerian.

However on a generic Kähler manifold, Kählerian functions are not abundant.A set of real Kähler functions is said to be complete if they separate derivations.We have:

Proposition 6.25 Let (M, h) be a Kähler manifold. The following statements areequivalent:

1. The function f is Kählerian.2. The Hamiltonian vector field X f is a derivation of the Riemannian product (·, ·).3. The Hamiltonian vector field X f is a derivation of the Kähler product [[·, ·]].4. The Hamiltonian vector field X f is a derivation of the ◦ν-product.5. The Hamiltonian vector field X f is a derivation of the �ν-product.

The structure of the space of Kählerian functions is described in the followingtheorem [Ci90, Ci91, Ci94].

Theorem 6.26 The set of Kähler functions is complete and closed under the �ν

product if the holomorphic sectional curvature,3 is constant and equal to 2/ν and,moreover for any triple of Kähler functions, then:

2 In other words, H is a finite-dimensional instance of a complex Hilbert space.3 See for instance [Go82], pp. 200–201 Theorems 6.1.2 and 6.1.3.


( f �ν g) �ν h = f �ν (g �ν h) , (6.66)

The statement is also true if ν →∞.

6.5 The Geometry of Quantum Dynamical Evolution

So far we have discussed dynamical systems admitting a compatible Hermiteanstructure and explored some of their properties, but we havemade no effort to connectthem to any particular physical model. We have already indicated though, that suchsystems are related to Quantum Physics. In this section we will make the connectionexplicit and we will show how a geometrical picture of Quantum Mechanics nicelyemerges from it.

Of course it will not be possible to offer in such a limited space a self-containeddescription of the foundations of Quantum Mechanics, so that references to theappropriate literature will be offered when needed. The emphasis here will be, asin the rest of this book, to offer the reader a new perspective where, starting alwaysfrom the dynamics, a geometrical picture emerges.

This geometrical picture in the case of quantum systems has a number of aspectsthat have both a mathematical and also a physical interest. We will try to exhibitsome of their most important features in what follows.

6.5.1 On the Meaning of Quantum Dynamical Evolution

We will refine first the notion of dynamics for a physical system with respect to the‘naive’ approach taken in the beginning of Chap. 2.

From a minimalist point of view, a description of any physical system, be it ‘clas-sical’ or ‘quantum’, requires the introduction of a family of observable quantities,say A, a family of ‘states’ representing the maximal information we can gather onour system, say S, and a pairing between them:

μ : A× S → P ,

with P the space of (Borelian) probability measures on the real line R.The interpretation of the pairing μ being that given a state ρ ∈ S, an observable

A ∈ A, that is, the number μ(A, ρ)(�) for any Borel set � ⊂ R, is the probabilitythat a measurement of A while the system is in the state ρ gives a result in �.

The evolution of the physical system characterized by (A,S, μ)will be describedeither by a one-parameter group of automorphisms �t of the state space S, or of thespace of observables A, or of the space probability measures. That is, either wemay consider that the state of the system actually changes with respect to a certaintime parameter t while the observables are detached from the dynamics, the so-called

http://dx.doi.org/10.1007/978-94-017-9220-2_2


‘Schrödinger picture’, or alternatively, wemay consider that the observables attachedto the system are actually evolving in time, the ‘Heisenberg picture’. We may eventhink that the probability measures μ(A, ρ) change in time instead.

We will accept first as a fundamental fact that the dynamical description of aquantum system will be given by a ‘vector field’ � on a carrier space that could bethe space of states, observables or probability measures. The realization of the vectorfield � and of the carrier space will depend on the picture we will choose to describethe system under investigation.

At the dawn of Quantum Mechanics, the founding fathers elaborated two mainpictures: one, which may be associated with the names of Schrödinger and Dirac,starts by associating a Hilbert space with any physical system and such Hilbert spaceis related to the space of states S. The second one, associated with the names ofHeisenberg, Born, von Neumann and Jordan, starts by associating with any physicalsystem a ∗-algebra related to the space of observablesA. Later on, also probabilisticapproaches have been developed. We may quote the names of Weyl, Wigner andEhrenfest in this sense.

In what follows we shall concentrate on the Hilbert space approach but beforethat we would like to outline how the various pictures are related.

Hilbert spaces were introduced and used in a systematic way in the descriptionof quantum systems first by Dirac [Di45] as a consequence of the fact that oneneeds a superposition rule (and hence a linear structure) in order to accommodatea consistent description of the interference phenomena that are fundamental forQuantum Mechanics, i.e., the double-slit experiment. Parenthetically, we shouldnote that a complex Hilbert space carries with it in a natural way a complex structure(multiplication of vectors by the imaginary unit). The rôle of the latter was discussedin the early 1940s by Reichenbach [Re44]. Later on Stückelberg [St60] emphasizedthe rôle of the complex structure in deducing in a consistent way the uncertaintyrelations of Quantum Mechanics (see also the discussion in [Es04] and [Ma05b]).

In a naive way however the Hilbert space approach started with Schrödinger whointroduced the space of square integrable functions on R

3 depending on a time para-meter, say �( x, t). These functions were called wave functions and were connectedto the evolution of the system by solving Schrödinger’s wave equation:

i�d

dt� = H�. (6.67)

Here the Hamiltonian operator H is required to be Hermitean with respect to the L2

inner scalar product on R3:

〈�,�〉 =∫R3

�( x)�( x)d3 x .

Born suggested that |�( x)|2 = �( x)�( x) should be interpreted as a probabilitydensity and the wave function � as an amplitude of probability. The linearity ofthe equation of motion allowed one to easily accommodate the description of inter-

6.5 The Geometry of Quantum Dynamical Evolution 441

ference phenomena which were observed in diffraction experiments with electrons.Nowadays interference phenomena are interpreted as paradigmatic examples of theessence of Quantum Mechanics.

Remark 6.2 Physical interpretation of the wave function.Let us consider briefly the physical meaning that can be attributed to �( x, t).

Born’s proposal was to regard ρ(x, t) = �(x, t)�(x, t) as a probability measure. Inotherwords, the quantity

∫�

ρ(x, t)d3x should represent the probability of observinga particle at time t within the volume element �.

This interpretation is possible if� is square-integrable and normalized. Born wasled to his postulate by comparing how the scattering of a particle by a potential isdescribed in classical andQuantumMechanics. For instance, if an electron is allowedto interactwith a short-range potential V , that is V has support on a compact set, and ifa screen is placed at a distance large enough from the interaction region, the electronwill be detected at a fixed point on the screen. On repeating the experiment, theelectronwill be detected at a different point. After several experiments, the fraction ofthe number of times the electron is detected at x at time t is proportional to |�(x, t)|2,which is therefore the probability density of such an event. Remarkably, the electronexhibits both a corpuscular and a wavelike nature by virtue of the probabilisticinterpretation of its wave function. This means that the ‘quantum electron’ doesnot coincide with the notion of ‘classical electron’ we may have developed in aclassical framework.

Ultimately, it is necessary to give up the attempt to describe the particle motionas if we could use what we are familiar with from everyday’s experience. We haveinstead to limit ourselves to an abstract mathematical description, which makes itpossible to extract information from the experiments that we are able to perform. Thiscrucial point (which is still receiving careful consideration in the current literature)is well emphasized by the interference from a double slit.

6.5.1.1 Dirac’s Insight and Heisenberg’s Equation

Out of the description in terms of wave functions Dirac suggested using an abstractHilbert space to be associated with any physical system, that is, a possible infinite-dimensional, complex separable linear spaceH carrying an Hermitean structure (orinner product) 〈·, ·〉 which is complete with respect to the topology defined by thenorm || · || associated to the inner product, i.e., ||�||2 = 〈�,�〉.

Dirac introduced the braket notation for vectors (’ket’) | · 〉 inH and forms (’bra’)〈 · | inH′. Because of Riesz theorem there is a natural isomorphism between a Hilbertspace and its dual, thus any vector |�〉 inH defines a (bounded) complex linear maponH, denoted by 〈�| such that 〈�|(|�〉) = 〈�,�〉 for any |�〉 ∈ H, and conversely,hence the notation for the inner product: 〈�|�〉 := 〈�,�〉.

The wave function was replaced by a vector |�(t)〉 ∈ H in the Hilbert spaceH and we will call them (pure) quantum states. Observables were introduced as aderived concept, namely as Hermitean operators on H. Mean values or expectation


values of an observable A in a given state |�〉 are given by:

eA(�) = 〈�|A|�〉〈�|�〉 . (6.68)

Given a state |�〉 a probability distribution is defined with the help of a resolutionof the identity operator inH, say:

I =∑

k

Pk =∑

k

|�k〉〈�k |〈�k |�k〉〈�|�〉 ,

for a countable family of rank-one orthogonal projectos Pk . Each operator Pk hasthe form Pk = |�k 〉〈�k |〈�k |�k 〉 for some vector |�k〉 ∈ H. Then the probability distributionfor the vector |�〉 is given by the non-negative numbers:

Pk(�) = 〈�|�k〉〈�k |�〉〈�k |�k〉 .

For a continuous index a labeling the projectors Pa , say:

I =∫

da|a〉〈a|〈a|a〉 ,

with da an appropriate measure in the index space, we obtain a probability density:

Pa(�) = 〈�|a〉〈a|�〉〈a|a〉〈�|�〉 . (6.69)

When the rank-one projectors |a〉〈a|/〈a|a〉 are associated with the spectral decom-position of the operator A, say:

A =∫

a|a〉〈a|〈a|a〉 da ,

we have the relation between the expectation value function and the probabilitydistribution Eq. (6.69):

eA ==∫

a Pa(�)da =∫

a〈�|a〉〈a|�〉〈a|a〉〈�|�〉 da .

Thus a description in terms of expectation value functions may be consideredto be a probabilistic description. In a search for an implementation of the so-called‘correspondence principle’ or the ‘quantum-to-classical’ transition, Ehrenfest elabo-rated a description of the evolution equations in terms of expectation value functionsknown today as the ‘Ehrenfest picture’.


From what we have said, it is clear that the probabilistic description depends noton the Hilbert space vector � but in the ‘ray’ defined by it, i.e., the one-dimensionalcomplex subspace

[�] = {λ|�〉 | λ ∈ C}

in H. Even more, it is well known that a “complete" measurement in QuantumMechanics (a simultaneous measurement of a complete set of commuting observ-ables4 [Di58, Es04, Me58]) does not provide us with a uniquely defined vector insome Hilbert space, but rather with a “ray", i.e., an equivalence class of vectorsdiffering by multiplication through a non zero complex number. Even fixing thenormalization, an overall phase will still remain unobservable.

In explicit terms, states are not elements of the Hilbert spaceH, but rather pointsof the complex projective space P(H) = H0/C0, with H0 = H\{0}. The space ofrays P(H) will be also referred to as the ray space of H.

Of course this transition from a linear Hilbert space to a Hilbert manifold raisesthe question of the description of superposition and interference phenomena (seelater the discussion in Sect. 6.5.2). Moreover the linear equations of motion on alinear space like Schrödinger’s equation (6.67), should be replaced by equations ofmotion on a manifold.

It is not difficult to show that out of Schrödinger’s evolution equation Eq. (6.67)for an Hermitean operator H we obtain the evolution equation:

i�d

dtρ� = [H, ρ� ] ,

for the rank-one operator ρ� = |�〉〈�|/〈�|�〉 associated to the ray [�] definedby |�〉. This immersion of the space of rays into the space of Hermitean operatorsacting on H, i.e.,

[�] �→ ρ� = |�〉〈�|〈�|�〉 , (6.70)

allows us to extend the previous equation to any convex combination of rank-oneprojectors ρk = Pk = |�k〉〈�k |/〈�k |�k〉 (not necessarily orthogonal) by meansof probability vectors p = (p1, p2, . . . , pn, . . .), pk ≥ 0,

∑k pk = 1. Then the

evolution equation for the Hermitean operator ρ =∑k pkρk is given by:

i�d

dtρ = [H, ρ]. (6.71)

This equation is usually called vonNeumann’s equation and the convex combinationsρ are usually called ‘density states’ or density operators. They were introduced in

4 We will not worry at this stage about the technical complications that can arise, in the infinite-dimensional case, when the spectrum of an observable has a continuum part.


the description of quantum physical systems by L. Landau and J. von Neumannindependently [La27, Ne27].

Clearly, if we replace the convex combinations of rank-one orthogonal projectorswith probability vectors {pk}, by any sequence of complex numbers {c1, c2, . . . ,cn, . . .} and arbitrary rank-one Hermitean operators ρk , we get the class of Hermitean(generically unbounded) operators C =∑

k ckρk with equation of motion given by:

i�d

dtC = [H, C]. (6.72)

This equation of motion on the space of operators is quite often called the Heisenbergform of the equations of motion.

Then, in the case of quantum systems, it is the previous equation, Eq. (6.72), theone that we will consider as defining the dynamics of the system, even though as ithas been discussed already, there are other ‘pictures’ equivalent to it.

In the subsequent parts of this chapter we are going to spell out some details ofwhat was already said. To avoid the complications, and why not, embarrassments,due to the subtle and in many cases difficult mathematical problems connected withthe topology of functional spaces, we should restrict our considerations for the mostpart to finite-dimensional quantum systems, even if in many instances the derivedresults extend to the infinite-dimensional situation.

6.5.2 The Basic Geometry of the Space of Quantum States

Following Dirac we will consider that associated to a quantum system there is acomplex Hilbert spaceH and that a class of states of the system are given by rays inH. Such states will be called in what follows pure states [Ha70] and they are pointsof the projective Hilbert space P(H).

IfH has finite complex dimension, say n + 1, the projective Hilbert space P(H)

is just the n-dimensional complex projective space CPn = {[z] | z ∈ Cn+1, z′ ∈

[z] ⇔ z′ = λz, 0 �= λ ∈ C}.Quotienting with respect to multiplication by non zero complex numbers C0 ∼=

R+ ×U (1), gives raise to the following double fibration:

R+ −→ H0

↓U (1) −→ S

↓P (H)

(6.73)

where the first row indicates the action of the radial part R+ of C, while the second

row is the action of the angular part U (1) of C on the sphere S of unitary vectors


(which in the finite-dimensional case is the sphere S2n+1 ⊂ Cn+1), and whose final

result is the projective Hilbert space P(H).

Remark 6.3 Notice that theHilbert spaceH acquires the structure of a principal fiberbundle [Hus75, Ch09, St51], with base P(H) and typical fiber C0, both in finite andinfinite dimensions.

Even if we are dealing with finite-dimensional Hilbert spaces, in this section wewill keep the abstract notation |�〉 for vectors instead of z, just showing the most ofthe computations work in infinite dimensions too. The natural identification ofH andits dual, allows for the (unique) association of every equivalence class [�] with therank-one orthogonal projector ρ� defined by Eq. (6.70) with the known properties:

ρ†� = ρ�, ρ2

� = ρ�, Tr ρ� = 1 .

It is clear by construction that the association depends on the Hermitean structurewe consider (see Sect. 6.5.3 for further comments on this point).

The space of rank-one projectors is usually denoted as D11 (H) (see Sect. 6.6.1

[Gr05] for additional information). It is then clear that in this way we can identifyD1

1 (H) with the projective Hilbert space P(H). Hence, what the best of measure-ments will yield will be always, no more and no less, than a rank-one projector.

Also, transition probabilities that, together with the expectation values of self-adjoint linear operators representing dynamical variables, are among the only observ-able quantities one can think of, will be insensitive to overall phases, i.e., they willdepend only on the (rank-one) projectors associated with the states. If A = A† is anysuch observable, then its expectation value 〈A〉� in the state |�〉 will be given by

〈A〉� = 〈�|A|�〉〈�|�〉 ≡ Tr (ρ� A) . (6.74)

Transition probabilities are in turn expressed via a binary product that can bedefined on pure states. Again, if |�〉 and |�〉 are any two states, then the (normalized)transition probability from |�〉 to |�〉 will be given by:

|〈�|�〉|2〈�|�〉〈�|�〉 = Tr (ρ�ρ�) (6.75)

and the trace on the right-hand side of Eq. (6.75) will define the binary productamong pure states (but more on this shortly below). Therefore the most naturalsetting for Quantum Mechanics is not primarily the Hilbert space itselfH but ratherthe projective Hilbert space P(H) or, equivalently, the space of rank-one projectorsD1

1 (H), whose convex hull will provide us with the set of all density states [Ne27,Neu2, Fa57].

On the other hand, the superposition rule, which leads to interference phenomena,remains one of the fundamental building blocks of Quantum Mechanics, one that,among other things, lies at the very heart of the modern formulation of Quantum


Mechanics in terms of path integrals [Br05], [Gl81], an approach that goes actuallyback to earlier suggestions by Dirac [Di58, Di33].

To begin with, if we consider, for simplicity, two orthonormal states

|�1〉, |�2〉 ∈ H, 〈�i |� j 〉 = δi j , i, j = 1, 2 (6.76)

with the associated projection operators

ρ1 = |�1〉〈�1|, ρ2 = |�2〉〈�2| . (6.77)

A linear superposition with (complex) coefficients c1 and c2 with: |c1|2 + |c2|2 = 1will yield the normalized vector

|�〉 = c1|�1〉 + c2|�2〉 (6.78)

and the associated projector

ρ� = |�〉〈�| = |c1|2 ρ1 + |c2|2 ρ2 +(c1c∗2ρ12 + h.c.

)(6.79)

where: ρ12 = |�1〉〈�2|, which cannot however be expressed directly in terms of theinitial projectors.

A procedure to overcome this difficulty by retaining at the same time the informa-tion concerning the relative phase of the coefficients can be summarized as follows[Cl07, Ma05, Ma05b].

Consider a third, fiducial vector |�0〉 with the only requirement that it is notorthogonal5 neither to |�1〉 nor to |�2〉. It is possible to associate normalized vectors|�i 〉 with the projectors ρi (i = 1, 2) by setting

|�i 〉 = ρi |�0〉√Tr (ρiρ0)

, i = 1, 2 . (6.80)

Remark 6.4 Note that, as all the ρ’s involved are rank-one projectors6:

Tr (ρiρ0)Tr(ρ jρ0

) = Tr(ρiρ0ρ jρ0

) ; ∀i, j = 1, 2 , (6.81)

and:

|�i 〉〈�i | = ρiρ0ρi√Tr (ρiρ0ρiρ0)

≡ ρi , i = 1, 2 . (6.82)

Forming now the linear superposition: |�〉 = c1|�1〉+ c2|�2〉 and the associatedprojector: ρ = |�〉〈�|, one finds easily, using also Eqs. (6.81) and (6.82), that:

5 In terms of the associated rank-one projections: Tr (ρi ρ0) �= 0, i = 1, 2, with: ρ0 = |�0〉〈�0|.6 The proof of Eqs. (6.81) and (6.82) is elementary and will not be given here.


ρ = |c1|2 ρ1 + |c2|2 ρ2 + c1c∗2ρ1ρ0ρ2 + h.c.√Tr (ρ1ρ0ρ2ρ0)

, (6.83)

which can be written in a compact form as

ρ =2∑

i, j=1ci c

∗j

ρiρ0ρ j√Tr

(ρiρ0ρ jρ0

) . (6.84)

The results (6.83) and (6.84) are now written entirely in terms of rank-one pro-jectors. Thus, a superposition of rank-one projectors which yields another rank-oneprojector is possible, but requires the arbitrary choice of the fiducial projector ρ0.

Remark 6.5 This procedure is equivalent to the introduction of a connection on thebundle, usually called the Pancharatnam connection [Ma95].

Remark 6.6 If the (normalized) probabilities |c1|2 and |c2|2 are given, Eq. (6.78)describes a one-parameter family of linear superposition of states, and the same willbe true in the case of Eq. (6.83). Both families will be parametrized by the relativephase of the coefficients.

Remark 6.7 Comparison of Eqs. (6.79) and (6.83) shows that, while the first twoterms on the right-hand side of both are identical, the last terms of the two differ byan extra (fixed) phase, namely:

ρ1ρ0ρ2√Tr (ρ1ρ0ρ2ρ0)

= ρ12 exp{i[arg (〈ψ1|ψ0〉 − arg (〈ψ2|ψ0〉))

]}. (6.85)

Remark 6.8 The result of Eq. (6.84) can be generalized in an obvious way to the caseof an arbitrary number, say n, of orthonormal states none ofwhich is orthogonal to thefiducial state. The corresponding family of rank-one projectors will be parametrizedin this case by the (n − 1) relative phases.

If now we are given two (rank-one) projectors, or more with an obviousgeneralization, and only the relative probabilities are given, we are led to concludethat the system is described by the convex combination (a rank-two density matrix):ρ = |c1|2 ρ1 + |c2|2 ρ2, which is again Hermitean and of trace one, but now:ρ − ρ2 > 0 (strictly). The procedure leading from this “impure"state to one of thepure states given by, say, Eq. (6.84), i.e., the procedure that associates a pure statewith a pair of pure states, is a composition law for pure states that has been termedin the literature as a “purification" of “impure"states [Man00].

In the Hilbert space formulation of Quantum Mechanics one needs also to findthe spectral family associated with any observable, represented by a self-adjointoperator on the Hilbert space. Limiting ourselves for simplicity to observables withpure point-spectrum, these notions can be made easily to “descend" to the projectiveHilbert space PH by noticing that, if A = A† is an observable, and consideringfrom now on only normalized vectors, the expectation value (6.74) associates with


the observable A a (real) functional on PH. We will develop this approach fully inthe coming sections.

Unitary (and, as a matter of fact, also anti-unitary) operators play also a rele-vant rôle in Quantum Mechanics [Me58]. In particular, self-adjoint operators can bethought as infinitesimal generators of one-parameter groups of unitaries. Both unitaryand anti-unitary operators share the property of leaving all transition probabilitiesinvariant. At the level of the projective Hilbert space they represent isometries of thebinary product (6.75).

The converse is also true. Indeed, it was proved long ago byWigner [Wi32, Wi59]that bijectivemaps on PH that preserve transition probabilities (i.e., isometries of theprojective Hilbert space) are associated with unitary or anti-unitary transformationson the original Hilbert space.7

6.5.3 The Hermitean Structure on the Space of Rays

We have mentioned already that the complex projective Hilbert space P(H)we havedefined as the space of pure states does not depend on the particular Hermiteanstructure we have in the Hilbert space H, that is, it just depends on the vector spacestructure of H and the action of C0 = C\{0}. This action defines H0 = H\{0},the Hilbert space without the zero vector as a principal bundle with base space thecomplex projective space P(H).

The identification, or immersion, of P(H) with the space of rank-one orthogonalprojectors depends, on the contrary, on the specific Hermitean product we use todefine the scalar product in the Hilbert space.

As an instance of this let us consider the simplest possible situationwhereH = C2.

We could define a family of inner products by setting:

〈z | w〉λ = (z1, z2)

(λ21 00 λ22

)(w1w2

)= λ21 z1w1 + λ22 z2w2 . (6.86)

where λ1, λ2 are now real numbers. With this 2-parameter family of scalar products,we would have a 2-parameter family of realizations of the abstract unitary groupU (2). Its elements would be given by 2× 2 matrices

U =(

u11 u12u21 u22

)

with complex entries satisfying:

7 The association being up to a phase, this may lead to the appearance of “ray"(or “projective")representations [Ba54, Ha92, Mac68, Mac04, Me58, Sa97] of unitary groups on the Hilbert spaceinstead of ordinary ones, a problem that we will not discuss here though.


(u11, u21)

(λ21 00 λ22

)(u11u21

)= λ21|u11|2 + λ22|u21|2 = 1 .

This means that the bra-vector associated with the ket-vector

(10

)should be

(0, 1/λ22). We would write now:

[�] �→ ρ�,λ = |�〉λ λ〈�|〈�|�〉λ ,

tomake sure this rank-one projector is orthogonal with respect to the λ-scalar product〈·|·〉λ given by Eq. (6.86).

As itwill be discussed at length later on, thiswill become relevant in the frameworkof the GNS construction. Actually the family of products 〈·|·〉λ are nothing but theinner products constructed in a family ofHilbert spaces associatedwith a 2-parameterfamily of states using the GNS construction (see Sect. 6.6.2).

To summarize the content of this section, we have argued that all the relevantbuilding blocks of Quantum Mechanics have to be formulated in terms of objectsthat “live"in the projective Hilbert space P(H). The latter, however, is no longer alinear vector space. As will be discussed in the following sections, it carries insteada rich manifold structure. In this context, the very notion of linear transformationsloosesmeaning, andwe are led in a natural way to consider it as a nonlinearmanifold.

This given, only objects that have a tensorial character will be allowed. We willdo that by following the steps laid in Sect. 6.4 in the first part of this chapter. In thelast part of the chapter, having achieved this goal, we will turn back to discuss theproblem of alternative Hermitean structures in the context of Quantum Mechanics.

6.5.4 Canonical Tensors on a Hilbert Space

Given a linear space H carrying a Hermitean structure 〈·, ·〉, we have seen how wecan construct the tensor fields g, J and ω on HR, the realification of H (see Sect.6.3.2).

The (0, 2)-tensors g and ω define maps from THR to T ∗HR. The two of thembeing non-degenerate,we can also consider their inverses, i.e., the (2, 0) contravarianttensors S = g−1 (a metric tensor) and � = ω−1 (a Poisson tensor) mapping T ∗HR

to THR. The contravariant (2, 0) tensors S and� can be combined together to definean Hermitean product for any two forms α, β in the dual H∗

R:

〈α, β〉H∗R= S(α, β)+ i�(α, β). (6.87)

This expression induces the two (non-associative) real brackets on the space ofsmooth, real-valued functions on HR that were already introduced in Sect. 6.4.3(here ν = 1), that is:


1. The (symmetric) Jordan bracket ( f, h)g = S(d f, dh) (recall Eq. (6.63)) and,2. The (antisymmetric) Poisson bracket { f, h}ω = �(d f, dh).

By extending both these brackets to complex functions by linearity we obtaineventually a complex bracket [[·, ·]] defined as (recall Eq. (6.64)):

[[ f, h]] = 〈d f, dh〉H∗R= ( f, h)g + i{ f, h}ω. (6.88)

To obtain explicit expressions for these structures in finite dimensions, we mayintroduce complex coordinates zk and the corresponding orthogonal-symplecticcoordinates (qk, pk), that is we select an orthonormal basis {ek}k=1,...,n in H andthe global linear coordinates (qk, pk) for k = 1, . . . , n on HR defined as

〈ek, u〉 = zk = qk + i pk, ∀u ∈ H .

Now we will change our notation and to make it closer to the notation used whendealing with calculus on manifolds, we denote points in the Hilbert space H byx, u, v, etc. Then after simple computations8 we get the coordinate expressions forthe tensors J, g and ω:

J = dpk ⊗ ∂

∂qk− dqk ⊗ ∂

∂pk, g = dqk ⊗ dqk + dpk ⊗ dpk, ω = dqk ∧ dpk ,

as well as the corresponding contravariant expressions for G and �:

G = ∂

∂qk⊗ ∂

∂qk+ ∂

∂pk⊗ ∂

∂pk, � = ∂

∂pk∧ ∂

∂qk. (6.89)

Hence,

( f, h)g = ∂ f

∂qk

∂h

∂qk+ ∂ f

∂pk

∂h

∂pk, { f, h}ω = ∂ f

∂pk

∂h

∂qk− ∂ f

∂qk

∂h

∂pk.

Using complex coordinates: zk = qk + i pk , zk = qk − i pk , we can also write:

G + i� = 4∂

∂zk⊗ ∂

∂ zk, (6.90)

where

∂

∂zk= 1

2

(∂

∂qk− i

∂

∂pk

),

∂

∂ zk= 1

2

(∂

∂qk+ i

∂

∂pk

). (6.91)

Then we have :

8 Summation over repeated indices being understood here and in the rest of the Section.


[[ f, h]] = 4∂ f

∂zk

∂h

∂ zk,

or, in more detail:

( f, h)g = 2

(∂ f

∂zk

∂h

∂ zk +∂h

∂zk

∂ f

∂ zk

), { f, h}ω = −2i

(∂ f

∂zk

∂h

∂ zk −∂h

∂zk

∂ f

∂ zk

).

(6.92)Notice also that:

J = −i

(dzk ⊗ ∂

∂zk − dzk ⊗ ∂

∂ zk

). (6.93)

In particular, for any linear operator A inH we can define the quadratic function:

fA(u) = 1

2〈u, Au〉 = 1

2z†Az (6.94)

where z is the column vector (z1, . . . , zn) with the complex coordinates of u. Itfollows immediately from Eq. (6.92) that, for any linear operators A, B we have:

( fA, fB)g = f AB+B A, { fA, fB}ω = −i f AB−B A . (6.95)

The Jordan (or symmetric) bracket of any two quadratic functions fA and fB isrelated to the (commutative) Jordan product of A and B, [A, B]+, defined9 as:

[A, B]+ = AB + B A , (6.96)

while their Poisson bracket is related to the commutator product [A, B]− defined as:

[A, B]− = −i (AB − B A) . (6.97)

In particular, if A and B are Hermitean, their Jordan product (6.96) and their Liebracket will be Hermitean as well. Hence, the set of Hermitean operators on HR,equipped with the binary operations (6.96) and (6.97), becomes a Lie-Jordan algebra[Jo34, Jo34b, Fa12], and the binary product:

A · B = 1

2

([A, B]+ + i [A, B]−

)(6.98)

is just the associative composition of linear operators. We remark parentheticallythat all this extends without modifications to the infinite-dimensional case when weconsider bounded operators.

Returning our attention to quadratic functions, it is not hard to check that:

9 That is actually twice the Jordan Bracket A ◦ B as it is usually defined in the literature [Em84],but we find here more convenient to use here this slightly different definition.


[[ f A, fB ]] = 2 f AB , (6.99)

whichproves the associativity of theKähler bracket (6.88) onquadratic functions, i.e.,

[[[[ f A, fB]], fC ]] = [[ fA, [[ fB, fC ]]]] = 4 f ABC , ∀A, B, C ∈ End(H). (6.100)

We will study now smooth real-valued functions on HR. First of all, it is clearthat f A will be a real function iff A is Hermitean. The Jordan and Poisson bracketswill define then a Lie-Jordan algebra structure on the set of real, quadratic functions,and, according to Eq. (6.100), the bracket [[·, ·]] will be an associative bracket.

For any such f ∈ F (HR) we may define two vector fields, the gradient ∇ f of fand the Hamiltonian vector field X f associated with f , given respectively by:

g(∇ f, ·) = d fω(X f , ·) = d f

orG(d f, ·) = ∇ f,�(d f, ·) = X f

,

which allow to write down the Jordan and the Poisson brackets as:

( f, h)g = g(∇ f,∇h), { f, h}ω = ω(X f , Xh) . (6.101)

Explicitly, in the coordinates (qk, pk) before:

∇ f = ∂ f

∂qk

∂

∂qk+ ∂ f

∂pk

∂

∂pk= 2

(∂ f

∂zk

∂

∂ zk+ ∂ f

∂ zk

∂

∂zk

)(6.102)

and,

X f = ∂ f

∂pk

∂

∂qk− ∂ f

∂qk

∂

∂pk= 2i

(∂ f

∂zk

∂

∂ zk− ∂ f

∂ zk

∂

∂zk

), (6.103)

which shows that

J (∇ f ) = X f , J (X f ) = −∇ f .

Let us recall that to any (bounded) linear operator A : H→ H we can associate:

1. A quadratic function f A as in Eq. (6.94) ,2. A vector field X A : H→ TH, x �−→ (x, Ax) as in Eq. (2.112),3. A (1, 1) tensor field: TA : TxH " (x, y) �−→ (x, Ay) ∈ TxH as in (2.107).

Clearly, as remarked already, f A is real if and only if A is Hermitean. In this casethe vector field X A becomes the gradient vector field associated to fA:

∇ f A = X A , (6.104)

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and

X f A = J (X A) . (6.105)

Indeed, using also the notation 〈·, ·〉 for the standard pairing between vectors andcovectors, Eq. (6.104) holds because

g (y, X A (x)) = g (y, Ax) = 1

2(〈y, Ax〉H + 〈Ax, y〉H) = 〈d f A (x) , y〉 ,

while Eq. (6.105) follows from the second expression in Eq. (6.15), i.e., from:g (y, Ax) = ω (y, (J X A)(x)) = ω (y, i Ax). Thus, we will also write, with a slightabuse of notation,

∇ f A = A, and X f A = i A (6.106)

In particular, for the identity operator I, we obtain the Liouville vector field� = XI:

∇ fI = �, X fI = J�,

with the last vector in the previous equation called the phase vector field and denoted:

� = J (�) = pk∂

∂qk− qk ∂

∂pk. (6.107)

6.5.5 The Kähler Geometry of the Space of Pure Quantum States

In Sect. 6.5.4 we have described the canonical tensors associated to an Hermiteanstructure, however as it was discussed before, the space of pure states of a quantumsystem are the elements on a projective Hilbert space. In this section we would liketo discuss in some detail the geometrical structure of the complex projective Hilbertspace PH. We will restrict ourselves to the finite-dimensional case even if all of ourresults extend naturally to the infinite-dimensional situation.

Notice first that we may describe the quotient π : H0 → P(H) by consideringthe distribution generated by the Liouville vector field � and the phase vector field�. Clearly both vector fields commute [�,�] = 0 (notice that � is homogeneous ofdegree 0), then we can integrate trivially the distribution generated by � and � byusing their respective flows: ϕ�

t (u) = et u and ϕ�s (u) = eisu. Thus if we consider

the action of C0 on H as λu = reisu with r = |λ| and s = argλ, it corresponds tothe element ϕ�

t ◦ϕ�s with log r = t . In other words, the fields � and � are the vector

fields defined by the natural action of the multiplicative group C0 onH0.The orbits of such action correspond to the orbits of the 2-parameter group of

transformations ϕ�t ◦ϕ�

s and they are just the fibres of the projection map π onto the


projective Hilbert space P(H). They are also the leaves of the foliation described bythe integrable distribution defined by � and �.

Contravariant tensorial objects on H will pass to the quotient (i.e., will be pro-jectable with respect to the projection map π ) if and only if they are left invariant byboth � and �, namely if they are homogeneous of degree zero (which means invari-ance with respect to �) and invariant under multiplication of vectors by a phase(invariance with respect to �).

Typical quadratic functions that pass to the quotientwill be normalized expectationvalues of the form (recall Eq. (6.68)):

eA(u) = Tr(ρu A) = 〈u|A|u〉〈u|u〉 , (6.108)

with A any linear operator and any Hermitean structure on H.Concerning projectability of tensors, the complex structure J , being (cfr., Eq.

(6.93)) homogeneous of degree zero and phase-invariant, will be a projectable tensor,while it is clear that the Jordan and Poisson tensors S and � defined respectivelyin Eq. (6.89) or, for that matter, the complex-valued tensor of Eq. (6.90) will notbe projectable (as they are phase-invariant but homogeneous of degree −2). To turnthem into projectable objects we will have to multiply them by the a phase invariantquadratic factor, e.g., θ (z) = z†z, thus defining new tensors

� (z) = θ (z)� (z) , S (z) = θ (z) S (z) . (6.109)

The induced tensors in the quotient space P(H) will be denoted with the samesymbols, �, S and J respectively, and they will define a Hermitian structure onP(H). Such Hermitean structure is called the Fubini-Study Hermitean structure andwill be denoted by hF S .

Accordingly the inverses of the contravariant projected tensors will be denoted byωF S and gF S respectively, that is ωF S = �−1, gF S = S−1. Notice that the tensor �

onH is not a Poisson tensor anymore because of the multiplicative factor θ , howeverthe projected tensor is Poisson. We will prove this, along with the fact that P(H)

inherits a Kähler structure, by a direct computation.

6.5.5.1 Local Expressions in the Projective Space

Thus, let us examine the structures induced on P(H) providing explicit expres-sions for them in local coordinates. Recall that, in the finite-dimensional case,P(H) = CP

n and it is therefore made up of equivalence classes of vectorsz = (z0, z1, · · · , zn) ∈ C

n+1 with respect to the equivalence relation z′ ∼= λz,λ ∈ C0.

The complex chart defined by the local coordinates zk will be denoted by z. Coor-dinates on CP

n induced from the complex coordinates zk are called homogeneouscoordinates and are denoted by [z] = [z0 : z1 : . . . : zn]. The pull-back under the


map π of the Fubini-Study metric gFS to Cn+1 is given by [Be06]:

π∗gFS = (z · z)dz⊗S d z− (dz · z)⊗S (z · d z)(z · z)2 ,

where z · z =∑k zk zk , dz · z =∑

k dzk zk , dz⊗S d z =∑k dzkdzk + dzkdzk , and

so on. Similarly the pull-back of the symplectic form gives:

π∗ωFS = i(z · z) dz ∧ d z− (dz · z) ∧ (z · d z)

(z · z)2 = −dθFS ,

where:

θFS = 1

2i

z · dz− z · dzz·z .

The isometries of the Fubini-Study structure are just the usual unitary transfor-mations which, in infinitesimal form, are written as:

zk = i Akj z

j

where A = (Akj ) is a Hermitean matrix. These are the equations for the flow of a

generic Killing vector field, which therefore has the form 10:

X A = zk∂zk − ˙zk∂zk = i Akj (z

j∂zk − z j∂zk ) .

A straightforward calculation shows that

ωFS(X A, ·) = d(iX AθFS

),

i.e., X A is the Hamiltonian vector field associated with the expectation value func-tion (6.68):

eA(z, z) = z · Azz · z = iX AθFS (6.110)

for the Hermitean matrix A, that is:

ωFS(·, XeA ) = deA .

Again a simple algebraic computation shows that, given any two expectationfunctions eA, eB (A, B being Hermitean matrices), their corresponding Hamiltonian

10 Notice that these are exactly the Killing vector fields of the unit sphere S2n+1. In particular, forA = I we obtain Xk = � which is a vertical vector field with respect to the Hopf projection [St51]πH : S2n+1 → CP

n .


vector fields satisfy:

ωFS(XeA , XeB ) = XeA (deB) = −ieAB−B A .

Therefore, the Poisson brackets associated with the symplectic form ωFS:

{ f, g}ωFS = ω(X f , Xg) ,

are such that:

{eA, eB}ωFS = −ieAB−B A .

In a similar way, one can prove that the gradient vector field ∇eA has the form:

∇eA = Akj (z

j∂zk + z j∂zk )

so that

gFS(∇eA ,∇eB ) = ∇eA(deB) = eAB+B A − eA · eB .

Remember that a real function f on PH is Kählerian iff its Hamiltonian vectorfield X f is also Killing (Definition 6.24). Such functions represent quantum observ-ables. The above calculations show that the space {eA | A = A†} of expectationvalues of Hermitean operators on P(H) consists exactly of all real Kählerian func-tions. To extend this concept to the complex case, one says that a complex-valuedfunction on P(H) is Kählerian iff its real and imaginary parts are Kählerian. Clearly,any such f is a function of the form (6.110) with A a (bounded) linear operator onH. Also, on the space, KC(P(H)), of complex Kählerian functions one can definean associative bilinear product (also-called a star-product) via (recall Eq. (6.65)):

f � g = f · g + 1

2hFS(d f, dg) = f · g + 1

2

[( f, g)gFS + i{ f, g}ωFS

], (6.111)

under which the space KC(PH) is closed since eA � eB = eAB . Thus we haveobtained another realization of the C

∗-algebra of bounded linear operators B(H).11

6.5.6 The Momentum Map and the Jordan–Scwhinger Map

We will study now the action of the unitary group U (H) on H. The unitary groupU (H) is the group of linear isomorphisms that preserve the admissible tensorial triple(g, ω, J ). In the following, we will denote as u(H) the Lie algebra of U (H) which

11 Actually both algebras are isomorphic as ∗-algebras and as C∗-algebras where the norm inKC(PH) is the supremum norm.


is just the space of anti-Hermitean operators onH.12 Thus, in finite dimensions, wewill identify the real linear space of all Hermitean operators with the dual u(H)∗ ofu(H) via the non-degenerate pairing:

〈A, T 〉 = i

2Tr(AT ), (6.112)

with A a Hermitean operator and T ∈ u(H). Having identified u(H)∗ with the reallinear space of Hermitean operators, we can define on it a Lie bracket (cfr. Sect.6.5.4):

[A, B]− = 1

i(AB − B A), (6.113)

with respect to which it becomes a Lie algebra, and also a Jordan bracket:

[A, B]+ = AB + B A , (6.114)

and, as we have discussed before, both structures together equip u(H)∗ with thestructure of a Lie-Jordan algebra.

In addition, u(H)∗ is equipped with the scalar product

〈A, B〉u∗ = 1

2Tr(AB) (6.115)

which satisfies

〈[A, ξ ]−, B〉u∗ = 1

2Tr([A, ξ ]−B) = 1

2Tr(A[ξ, B]−)

= 1

2Tr(ξ [B, A]−) = 〈A, [ξ, B]−〉u∗ , (6.116)

and,

〈[A, ξ ]+, B〉u∗ = 1

2Tr([A, ξ ]+B) = 1

2Tr(A, [ξ, B]+)

= 1

2Tr(ξ [A, B]+) = 〈A, [ξ, B]+〉u∗ . (6.117)

With any A ∈ u(H)∗, we can associate the fundamental vector field X A on theHilbert space corresponding to the element 1

i A ∈ u(H) defined by the formula13:

12 At this point the finite and infinite-dimensional situations separate as in the infinite-dimensionalcase, because of Stone’s theorem, the Lie algebra of U (H) will consists of, in general unbounded,anti-Hermitean operators.13 In what follows we will denote by x the vectors in the real Hilbert space HR.


d

dteit A(x)|t=0 = i Ax, ∀x ∈ HR .

In other words, X A = i A. We already know from Sect. 6.5.4 that i A is a Hamiltonianvector field with Hamiltonian function f A: ω(·, X A) = d f A. Thus, for any x ∈ HR

we obtain an element μ(x) ∈ u(H)∗ such that:

〈μ(x), i A〉 = fA(x) = 1

2〈x, Ax〉H (6.118)

We obtain a mapping:

μ : HR → u(H)∗ , (6.119)

which is just the momentum map of the action of the group U (H) on H [Ch09].More explicitly, it follows from Eq.(6.112) that:

μ(x) = |x〉〈x | . (6.120)

Wemay therefore conclude that the unit sphere inH can be mapped into u(H)∗ inan equivariant way with respect to the coadjoint action of U (H). Actually we have:

μ(U x) = U†|x〉〈x |U = Ad∗U†μ(x) .

In finite dimensions, the unit sphere is odd dimensional and the orbit in u(H)∗ issymplectic, hence even dimensional.

With every A ∈ u(H)∗ we can associate, with the familiar identification of thetangent space at every point of u(H)∗ with u(H)∗ itself, the linear function (hencea one-form) A : u(H)∗ → R defined as: A = 〈A, ·〉u∗ . Then, we can define twocontravariant tensors, a symmetric (Jordan) tensor:

R( A, B) (ξ) = 〈ξ, [A, B]+〉u∗ (6.121)

and a Poisson tensor, which is just the Kostant-Kirillov-Souriau [Ki76, Ko70]) tensor�u∗ (see Sect. 4.3.4):

�u∗( A, B) (ξ) = 〈ξ, [A, B]−〉u∗ (6.122)

A, B, ξ ∈ u(H)∗. We notice that the exterior derivative of the quadratic function fAis the pull-back of A via the momentum map since, for all x ∈ H:

μ∗( A)(x) = A ◦ μ(x) = 〈A, μ(x)〉u∗ = 1

2〈x, Ax〉H = f A(x)

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i.e.,

μ∗ A = d f A . (6.123)

This means that if ξ = μ(x):

(μ∗G)( A, B) (ξ) = G(d fA, d fB) (x) = { f A, fB}g(x) = f[A,B]+(x) = R( A, B) (ξ)

i.e.,

μ∗G = R .

Similarly, using now Eq. (6.95), we find:

(μ∗�)( A, B) (ξ) = �(d f A, d fB) (x) = { f A, fB}ω(x) = f[A,B]−(x) = �u∗( A, B) (ξ)

i.e.,

μ∗� = �u∗ .

Thus, themomentummapμ relates the contravariant metric tensor G and the Poissontensor�with the corresponding contravariant tensors R and�u∗ . Together they formthe complex tensor:

(R + i�u∗)( A, B) (ξ) = 2〈ξ, AB〉u∗ (6.124)

which provides the contravariant form of the Hermitean product on u(H)∗.

6.5.7 A Simple Example: The Geometry of a Two-Level System

Let H = C2 be the Hilbert space appropriate to describe a two-level system (also

referred sometimes in the quantum information literature as a q-bit). We can writeany A ∈ u∗(H) = U (2) as

A = y0AI+ yA · σ (6.125)

where I is the 2×2 identity, yA ·σ = y1Aσ1+ y2Aσ2+ y3Aσ3 and σ = (σ1, σ2, σ3) arethe well-known Pauli matrices (see Sect. 10, Eq. (10.68)), that satisfy the well-knownidentities [Me58]:

σhσk = δhkI+ iεhklσl (6.126)

(h, k, l = 1, 2, 3) and

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σ jσkσl = iε jklI+ σ j δkl − σkδ jl + σlδ jk . (6.127)

Hence, explicitly

A =(

y0A + y3A y1A − iy2Ay1A + iy2A y0A − y3A

)(6.128)

Every A ∈ u(2)∗ is then represented by the (real) ‘four-vector’ (yμA) = (

y0A, yA):

yμA = 〈A|σμ〉, μ = 0, 1, 2, 3 .

If we represent vectors in C2 as two-components complex spinors, i.e.,

|x〉 =(

z1z2

), z1, z2 ∈ C (6.129)

we find, using Eq. (6.128):

f A(z1, z2) = 1

2

{(y0A + y3A

) |z1|2 + (y0A − y3A

) |z2|2 + (y1A − iy2A

)z1z2 +

(y1A + iy2A

)z2z1

}.

Exercise 6.2 Show that rank-one projectors ρ= |x〉〈x |, (ρ†= ρ,Tr ρ= 1, ρ2= ρ),

can be parametrized as [Mo01]:

ρ = ρ (θ, φ) =(

sin2 θ2

12eiφ sin θ

12e−iφ sin θ cos2 θ

2

); 0 ≤ θ < π, 0 ≤ φ < 2π .

(6.130)

Then, because of Exercise 6.2, rank-one projectors correspond to vectors yμρ of

the form:

y0 = 1

2, y1 = 1

2sin θ cosφ, y2 = −1

2sin θ sin φ, y3 = −1

2cos θ , (6.131)

thus y2 = (y0)2 = 1/4 for all rank-one projectors.Using Eqs. (6.128) and (6.129), the momentum map μ = |x〉〈x | can be written

explicitly in the form (6.125).

Exercise 6.3 Show that the components of the vector yμρ of the momentum map μ

are given by:

y0 = |z1|2 + |z2|22

, y1 = z1z2 + z2z12

y2 = iz1z2 − z2z1

2, y3 = |z1|2 − |z2|2

2(6.132)


We can associate with every vector (yμA) = (

y0A, yA)the vector field

X A = y0A∂0 + y1A∂1 + y2A∂2 + y3A∂3 ,

with ∂μ = ∂/∂yu , μ = 0, 1, 2, 3. Also (see the discussion immediately beforeEq. (6.121)), A = 〈A, ·〉u∗ will be represented by the one-form:

A = y0Ady0 + y1Ady1 + y2Ady2 + y3Ady3 . (6.133)

Exercise 6.4 Show, using Eq. (6.125)m that:

AB =(

y0A y0B + yA · yB

)I+

(y0AyB + y0ByA + iyA×yB

)· σ

where “×” denotes the standard cross-product of three-vectors.

Hence

〈AB〉u∗ = 1

2Tr (AB) = y0A y0B + yA · yB .

In particular: 〈ρ (θ, φ) ρ(θ ′, φ′

)〉u∗ = {1+ sin θ sin θ ′ cos

(φ − φ′

)+ cos θ cos θ ′}

/4 for rank-one projectors. Moreover

[A, B]+ = 2(

y0A y0B + yA·yB

)I+

(y0AyB + y0ByA

)· σ ,

while

[A, B]− = 2(yA×yB) · σ .

Also, we check, using Eqs. (6.132) and (6.133) that μ∗ A = d f A, in agreement withEq. (6.123).

Proceeding further, we find the explicit expressions for the tensors R and �u∗ :

R( A, B) (ξ) = 〈ξ, [A, B]+〉u∗ = 〈[ξ, A]+ , B〉 == 2ξ 0

(y0A y0B + yA · yB

)+ 2

(y0AyB + y0ByA

)· ξ =

= 2(

y0Aξ 0 + yA · ξ)

y0B + 2(

y0Aξ + ξ0yA

)· yB (6.134)

and hence, explicitly [Gr05]:

R (ξ) = 2∂0 ⊗(ξ 1∂1 + ξ2∂2 + ξ 3∂3

)+ 2

(ξ1∂1 + ξ 2∂2 + ξ3∂3

)⊗ ∂0

+ 2ξ0 (∂0 ⊗ ∂0 + ∂1 ⊗ ∂1 + ∂2 ⊗ ∂2 + ∂3 ⊗ ∂3) (6.135)


Quite similarly, one finds:

�u∗( A, B) (ξ) = 2(ξ × yA) · yB = 2(yA × yB) · ξ (6.136)

and:

�u∗ (ξ) = 2(ξ 1∂2 ∧ ∂3 + ξ2∂3 ∧ ∂1 + ξ3∂1 ∧ ∂2

). (6.137)

We thus find the tensor:

R+i�u∗ = 2[∂0 ⊗ yk∂k + yk∂k ⊗ ∂0 + y0(∂0 ⊗ ∂0 + ∂k ⊗ ∂k)+ iεhkl yh∂k ⊗ ∂l

].

Remark 6.9 To conclude this section, we notice that we can define two additional(1, 1) tensors, R and J , that will appear below in Sect. 6.6.1, via

Rξ (A) = [ξ, A]+ = R( A, ·)(ξ) ,

and

Jξ (A) = [ξ, A]− = �u∗( A, ·)(ξ) ,

for any A ∈ Tξu∗ (H) ∼= u∗ (H), the last passage in both equations following from

Eqs. (6.116) and (6.117).

Exercise 6.5 Show that in coordinates yμ, we have:

Rξ = 2(ξ0dy0 + ξ · dy

)⊗ ∂0 + 2

(ξ i dy0 + ξ0dyi

)⊗ ∂i

and

Jξ = 2εi jk ξ i dy j ⊗ ∂k .

6.6 The Geometry of Quantum Mechanicsand the GNS Construction

In the previous sections of this chapter, we haveworked out the geometrical structuresthat naturally arise in the standard approach toQuantumMechanics,which starts froma complex separable Hilbert space and identifies the space of physical states with itsassociated complex projective space. In this framework, algebraic notions, such asthat of the C

∗-algebra whose real elements are the observables of the system, arisesonly as a derived concept.

6.6 The Geometry of Quantum Mechanics and the GNS Construction 463

In this section, we would like to show how geometrical structures emerge also in amore algebraic setting, where one starts from the very beginning with an abstractC∗-algebra to obtain theHilbert space of pure states as a derived concept via the so-calledGelfand-Naimark-Segal (GNS) construction [Br87]. A more detailed discussion canbe found in [Ch09].

6.6.1 The Space of Density States

We have seen in Sect. 6.5.5 that it is possible to obtain P(H) as a quotient of H0with respect to the involutive distribution associated with � and �. Equation (6.120)shows that the image ofH0 under themomentummap associated to the natural actionof U (H) consists of the set of all non-negative Hermitean operators of rank one, thatwill be denoted as P1(H), i.e.,14

P1(H) = {|x〉〈x | | x ∈ H, x �= 0} .

On the other hand, the coadjoint action of U (H) on P1(H):

(U, ρ) �→ Uρ U † , ρ ∈ P1(H), U ∈ U(H) ,

foliates P1(H) into the spaces D1r (H) = {|x〉〈x | | 〈x, x〉H = r2}. In particular we

will denote by D11(H) the space of rank-one orthogonal projection operators, which

is the image via the momentum map of the sphere S = {x ∈ H | 〈x, x〉H = 1} andcan be identified with the complex projective space P(H) via:

[x] ∈ P(H)↔ |x〉〈x |〈x, x〉 ∈ D1

1(H) .

We have also discussed the geometry of P(H) as a Kähler manifold. In the fol-lowing we will examine this fact in more detail, by showing explicitly that D1

1(H)

carries a natural Kähler manifold structure.Let ξ ∈ u(H)∗ be the image through the momentummap of a unit vector x ∈ SH,

i.e., ξ = |x〉〈x | with 〈x |x〉 = 1, so that ξ 2 = ξ . The tangent space of the coadjointU (H)-orbit at ξ is generated by vectors of the form [A, ξ ]−, for any Hermitean A.From Eq. (6.116), it follows that the Poisson tensor � defined in (6.122) satisfies:

�( A, B) (ξ) = 〈ξ, [A, B]−〉u∗ = 〈[ξ, A]−, B〉u∗ (6.138)

This defines an invertible map � that associates to any 1-form A the tangent vector atξ : �( A) = �( A, ·) = [ξ, A]−. We will denote with ηξ its inverse: ηξ ([ξ, A]−) = A.This allows us to define, on each U (H) orbit in u(H)∗, a canonical 2-form η whichis given by:

14 Note that here the vectors are not necessarily normalized.


ηξ ([A, ξ ]−, [B, ξ ]−) =: (ηξ ([ξ, A]−), [B, ξ ]−) = ( A, [B, ξ ]−) (6.139)

for all [A, ξ ]−, [B, ξ ]− ∈ Tξu(H)∗.It is also easy to check that η satisfies the equalities:

ηξ ([A, ξ ]−, [B, ξ ]−) = −( A, [B, ξ ]−) = −〈A, [B, ξ ]−〉u∗ = −〈ξ, [A, B]−〉u∗= 〈[A, ξ ]−, B〉u∗ ,

for any A, B ∈ u(H)∗.

Exercise 6.6 Compute ker ηξ .

We can summarize these results in the following:

Theorem 6.27 The restriction of the 2-form (6.139) to the U (H)-orbit D11(H)

defines a canonical symplectic form η characterized by the property

ηξ ([A, ξ ]−, [B, ξ ]−) = 〈[A, ξ ]− , B〉u∗ = −〈ξ, [A, B]−〉u∗ (6.140)

In a very similar way, starting from the symmetric Jordan tensor R given in(6.121) , one can construct a (1, 1) tensor R( A) = R( A, ·) = [ξ, A]+ and its inverse:σ ([ξ, A]+) = A. Thus we obtain a covariant tensor σ such that:

σξ ([A, ξ ]+, [B, ξ ]+) = 〈[A, ξ ]+, B〉u∗ = 〈ξ, [A, B]+〉u∗ . (6.141)

Notice that, at this stage, σξ is only a partial tensor, being defined on vectors of theform [A, ξ ]+, which belong to the image of the map R. However, on TξD1

1(H), wehave [A, ξ ]− = [A, ξ 2]− = [[A, ξ ], ξ ]+, so that, after some algebra, one can alsoprove that:

σξ ([A, ξ ]−, [B, ξ ]−) = σξ ([[A, ξ ]−, ξ ]+, [[B, ξ ]−, ξ ]+) = 〈ξ, [[A, ξ ]−, [B, ξ ]−]+〉u∗= 1

2Tr(ξ [[A, ξ ]−, [B, ξ ]−]+) = 1

2Tr(ξ [A, ξ ]−[B, ξ ]−)

= 〈[A, ξ ]−, [B, ξ ]−〉u∗ .

Therefore we have also the following:

Corollary 6.28 On the U (H)-orbit D11(H) we can define a symmetric covariant

tensor σ such that:

σξ ([A, ξ ]−, [B, ξ ]−) = 〈[A, ξ ]−, [B, ξ ]−〉u∗ . (6.142)

Moreover, turning back to the the (1, 1) tensor � given above, one has the fol-lowing result [Gr05]:


Theorem 6.29 When restricted to D11(H), the (1, 1) tensor � , which satisfies:

�3 = −� (6.143)

will become invertible. Hence: �2 = −I and therefore it will define a complexstructure I such that

ηξ ([A, ξ ]−, Iξ ([B, ξ ]−)) = σξ ([A, ξ ]−, [B, ξ ]−) , (6.144)

and

ηξ (Iξ ([A, ξ ]−), Iξ ([B, ξ ]−)) = ηξ ([A, ξ ]−, [B, ξ ]−) .

Proof Eq. (6.143) follows from a direct calculation by taking into account thatξ2 = ξ . The last two expressions follow by combining Eqs. (6.140) and (6.142).

To prove that I is a complex structure one has first to show that it defines an almostcomplex structure (which follows easily from the fact that [[[A, ξ ]−, ξ ]−, ξ ]− =−[A, ξ ]−) and then that its Nijenhuis torsion vanishes. Detailed calculations of thiscan be found in [Gr05]. �

Putting everything together, we can now conclude that, as expected:

Theorem 6.30 (D11(H), I, σ, η) is a Kähler manifold.

At last, we notice that there is an identification of the orthogonal complementof any unit vector x ∈ H with the tangent space of the U (H)-orbit in u(H)∗ atξ = |x〉〈x |. Indeed, for any y perpendicular to x (‖x‖2 = 1) the operators:

Pxy =: (μ∗)x (y) = |y〉〈x | + |x〉〈y| (6.145)

can be written as Pxy = [Ay, ξ ], where Ay is a Hermitean operator such that

Ay x = iy, Ay y = −i‖y‖2x and Ayz = 0 for any z perpendicular to both xand y, as it can be directly checked by applying both expressions to a generic vec-tor in H which can be written as ax + by + cz with a, b, c ∈ C. Then, fromEqs. (6.140) and (6.142), it follows immediately that, for any y, y ′ orthogonal to x

ηξ (Pxy , Px

y′) = −1

2Tr (ξ [Ay, Ay′ ]−) = − 1

2i(〈y, y′〉 − 〈y′, y〉) = −ω(y, y′)

(6.146)

σξ (Pxy , Px

y′) =1

2Tr (ξ [Ay, Ay′ ]−) = −1

2(〈y, y′〉 + 〈y′, y〉) = g(y, y′) (6.147)

In conclusion, we have the following:

Theorem 6.31 For any y, y′ ∈ H, the vectors (μ∗)x (y), (μ∗)x (y) are tangent tothe U (H)-orbit in u(H)∗ at ξ = μ(x) and:


σξ ((μ∗)x (y), (μ∗)x (y)) = g(y, y′) (6.148)

ηξ ((μ∗)x (y), (μ∗)x (y)) = −ω(y, y′) (6.149)

Iξ (μ∗)x (y)) = (μ∗)x (J y) (6.150)

where the last formula follows from Eq. (6.144).

More generally, with minor changes, we can reconstruct similar structures for anyD1

r (H), obtaining Kähler manifolds (D1r (H), I r , σ r , ηr ). An analog of the above

theorem shows then that the latter can be obtained from a sort of ‘Kähler reduction’starting from the original linear Kähler manifold (HR, J, g, ω).

Example 6.7 Let us go back to the previous example of rank-one projectors onH = C

2. According to (6.131), the latter are described by 3-dimensional vectorsξ = (y1, y2, y3) such that ξ2 = 1/4 (y0 = 1/2 always), which form a 2-dimensionalsphere of radius 1/2. A generic tangent vector X A and a generic one form A at ξ are ofthe form X A = y0A∂0+y1A∂1+y2A∂2+y3A∂3 and A = y0Ady0+y1Ady1+y2Ady2+y3Ady3

with y0A = 0 and yA · ξ = 0.It is clear from (6.136) that the map � that associates to any 1-form A the tangent

vector at ξ : �( A) = �( A, ·) = [A, ξ ]− ismanifestly invariant and given by: �( A) =2(ξ × yA) · ∂ , where we have set ∂ = (∂1, ∂2, ∂3). It follows that the 2-form ηξ issuch that:

ηξ ([A, ξ ]−, [B, ξ ]−) = 2ξ · (yA × yB) (6.151)

so that

ηξ = 2εi jk yi dy j ∧ dyk (6.152)

which is proportional by a factor(y21 + y22 + y23

)− 32 to the symplectic 2-form on a

2-dimensional sphere,15 when pulled back to the sphere.In a similar way, from (6.134), one can prove that R( A) = R( A, ·) = [ξ, A]+ =

2(y0A y0 + yA · ξ)∂0 + 2(y0Aξ + y0yA) · ∂ . Thus, because of (6.142), we have:

σξ ([A, ξ ]−, [B, ξ ]−) = 4(ξ × yA) · (ξ × yB) = yA · yB (6.153)

where the last equality follows from the fact that ξ2 = 1/4 and ξ is orthogonal toboth yA and yB .

Finally, starting for example from Eq. (6.144), it is not difficult to check that

Iξ ([B, ξ ]−) = y′B · ∂ wi th y′B = ξ × yB . (6.154)

A direct calculation shows that I 3ξ = −Iξ .

15 that is also the volume element of a 2-dimensional sphere of radius r = 1/2, as it should be.


6.6.2 The GNS Construction

The algebraic approach to quantum physics started, among others, with the work ofHaag and Kastler [Ha64] which is also at the basis of the mathematical approach toquantum field theory [Ha92], relies on the so-called GNS construction which goesback to the work of Gelfand and Naimark on one side and Segal on the other.

The starting point of this construction is an abstract C∗-algebra A [Br87, Em84]with unity, the latter being denoted as I. The elements a ∈ A such that a = a∗constitute the set Are (a vector space over the reals) of the real elements16 of thealgebra. In particular: I ∈Are. The obvious decomposition: a = a1 + ia2, with

a1 = 1

2(a + a∗); a2 = 1

2i(a − a∗), (6.155)

means that, as a vector space, A is the direct sum of Are and of the set Aim (alsoa vector space over the reals) of the imaginary elements, i.e., of the elements of theform ia, a ∈ Are. The subspace of real elementsAre can be given the structure of aLie-Jordan algebra [Ch09, Em84, Fa12], where, using here the conventions of Sect.6.5.4, the Lie product is defined as

[a, b] = 1

2i(ab − ba) , (6.156)

while the Jordan product is given by

a ◦ b = 1

2(ab + ba) , (6.157)

for all a, b ∈ Are. The product in the algebra is then recovered as

ab = a ◦ b + i [a, b] . (6.158)

Remark 6.10 The main example of a C∗-algebra is the algebraB (H) of all boundedoperators on a Hilbert space H. In this case [Em84]: Are is the set of all boundedself-adjoint operators on H.

Definition 6.32 Astate on aC∗-algebraAwith unit is a linear functionalω : A→ C

satisfying:

1. Real: ω(a∗) = ω(a), ∀a ∈ A,2. Positive: ω (a∗a) ≥ 0, ∀a ∈ A and3. Normalized: ω (I) = 1.

The space of all states of the C∗-algebra A will be denoted by D(A).

16 Also-called the observables.


The space of states D(A) is convex, that is μ1ω1 + μ2ω2 is a state for anyωa ∈ D(A), μa ≥ 0, a = 1, 2, and μ1 + μ2 = 1.

Each stateω defines a non-negative sesquilinear 〈 · | · 〉ω between pairs of elementsa, b ∈ A via:

〈a|b〉ω := ω(a∗b) . (6.159)

Reality and positivity of the state guarantee that the pairing (6.159) satisfies theSchwartz inequality, i.e.,

|〈a|b〉ω| ≤√〈a|a〉ω√〈b|b〉ω (6.160)

Exercise 6.8 Prove Schwartz’ inequality Eq. (6.160).

The form 〈 ·, · 〉ω however might be degenerate. We are thus led to consider theGelfand ideal [Em84, Ha92]Jω consisting of all elements j ∈ A such thatω( j∗ j) =0 and to define the set A/Jω of equivalence classes:

�a = [a + Jω] (6.161)

Exercise 6.9 Prove that Jω is a bilateral ideal of the C∗-algebra A.

Now it is immediate to see that A/Jω is a pre-Hilbert space with respect to thescalar product:

〈�a, �b〉 = ω(a∗b) , (6.162)

because the Schwartz inequality (6.160) implies: 〈 j |a〉ω = 〈a| j〉ω = 0 for all a ∈ A,j ∈ Jω, and hence the scalar product Eq. (6.162) does indeed depend only on theequivalence classes�a ,�b, of a and b, and not on the specific representatives chosen.

Completing this space with respect to the topology defined by the scalar product,one obtains a Hilbert space Hω on which the original C∗-algebra A acts via thenatural representation:

πω(a)�b = �ab , ∀a, b ∈ A . (6.163)

Notice that if such a representation is faithful, i.e., the map πω : a �→ πω(a) is anisomorphism, the operator norm of πω(a) equals the C

∗-norm of a [Br87].Clearly the equivalence class of the unit element in A, denoted by � = �I,

satisfies: ‖�I‖ := √〈�I|�I〉 = 1 and provides a cyclic vector for the representationπω. We recall [Ha92] that a vector � ∈ Hω is called cyclic if πω (A)� is dense inHω. Moreover:

〈�|πω(a)|�〉 = ω(a) . (6.164)


This tells us that, if we consider that A acts by duality on D(A), the Hilbert spaceHω corresponding to a given state ω is the orbit of A through ω itself. Notice thatany other element b ∈ A such that the vector � = πω(b)� is of unit norm, definesa new state ω� by:

ω�(a) = 〈�|πω(a)|�〉 = ω(b∗ab) . (6.165)

These states are called vector states of the representation πω, and are particularexamples of more general states of the form:

ωρ(a) = Tr (ρ πω(a)) , (6.166)

where ρ ∈ B(Hω) is a density operator [Em84, Ha92]. States of the form (6.166)are called a “folium" of the representation πω. Also, one says that a state ω is pureif it cannot be written as a convex combination of other states in D(A), so that theset of pure states defines a set of extremal points in D(A).

The universality and uniqueness of the GNS construction is guaranteed [Br87] bythe following:

Theorem 6.33

1. If πω is a cyclic representation of A on a Hilbert space H, any vector represen-tation ω� for a normalized �, see Eq. (6.166), is equivalent to πω.

2. A GNS represenation πω of A is irreducible iff ω is a pure state.

Example 6.10 The GNS construction can be very simple for finite-dimensionalC∗ -algebras. Consider, e.g., the algebraA = End(Cn) of linear operators onC

n , i.e.,of the n×n matrices with complex entries. Any non-negative operatorω ∈ End(Cn)

defines a state by:

ω(A) = Tr (ωA) , ∀A ∈ A , (6.167)

while we can define the scalar product in Hω as:

〈A|B〉 = ω(A∗B) = Tr (BωA∗) . (6.168)

If we write ω as ω = R R∗, we find:

〈B | B〉ω = Tr (B R)(B R)∗ ,

showing that the Gelfand ideal consists of the elements that annihilate R.If ω is a rank-one projector and {ek} is an orthonormal basis for which ω =

|e1〉〈e1|, writing Akm for the matrix elements of A in such a basis, the scalar productassumes the form:


〈A|B〉 =n∑

k=1Ak1Bk1 (6.169)

while the Gelfand ideal Jω is given by:

Jω = {X ∈ A | Xk1 = 0 , k = 1, · · · , n} (6.170)

Thus Hω = A/Jω is nothing but Cn itself and πω is the defining representation.

If ω is a rank-m density operator: ω = p1|e1〉〈e1| + · · · + pm |em〉〈em | withp1, · · · , pm > 0 and p1 + · · · pm = 1, the scalar product is given by:

〈A|B〉 =n∑

k=1

m∑j=1

pm Ak j Bk j (6.171)

and the Gelfand ideal is given by:

Jω = {X ∈ A | Xkj = 0 , k = 1, · · · , n; j = 1, · · · , m} (6.172)

showing that Hω is the direct sum of m copies of Cn . Now the representation πω is

no longer irreducible, decomposing into the direct sum of m copies of the definingrepresentation:

πω(A) = Im ⊗ A (6.173)

where Im is the m × m identity matrix.

Let us go back now to the GNS construction and consider first a pure state ω overA, which gives raise to the irreducible representation πω in the Hilbert space Hω.We have already seen that self-adjoint operators, that correspond to the real elementsofA, may be identified with the dual u∗(Hω) of the Lie algebra u(Hω) of the unitarygroup U (Hω) and how the momentum map

μω : Hω → u∗(Hω) , μω(ψ) = |ψ〉〈ψ | (6.174)

relates the Poisson tensor on u∗(Hω) with that on Hω, via the pull-back.We have also seen that the unit sphere inHω\ {0} can be projected onto u∗(Hω) in

an equivariant way, in such a way that the Poisson and the Riemann tensor inP(Hω)

are both related to the same tensors defined on u∗(Hω) by using the Lie and theJordan product that are defined on it. Thus the momentummap provides a symplecticrealization of u∗(Hω), which we call a Kählerian (or Hermitean) realization on thecomplex projective space. If G is a group represented unitarily onU (Hω), then thereis a natural projection ν : u∗(Hω) → g∗ and the composition ν ◦ μ : P(Hω) → g∗is called a classical Jordan-Schwinger map.

6.7 Alternative Hermitean Structures for Quantum Systems 471

6.7 Alternative Hermitean Structures for Quantum Systems

6.7.1 Equations of Motion on Density Statesand Hermitean Operators

As we showed before, the equations of motion on density states are given by:

i�d

dtρ = [H, ρ],

out of

i�d

dt|�〉 = H |�〉, and − i�

d

dt〈�| = H〈�|,

i.e., assuming that H is Hermitean. When we consider, for some Hermitean operatorA, the expectation values functional:

eA(�) = 〈�|A|�〉〈�|�〉 = Tr(Aρ�) = Tr

(A|�〉〈�|〈�|�〉

)

and consider

0 = d

dt

〈�|A|�〉〈�|�〉 = d

dtTr(Aρ�) = Tr(

d

dtAρ�)+ Tr(A

d

dtρ�)

We will consider for simplicity finite-dimensional carrier spaces V , i.e., ‘finitelevel systems’, an assumption which is not far from experimental constraints,where only systems on finite volumes and finite energy are considered, the infinite-dimensional systems being idealizations of ‘actual’ systems.

6.7.2 The Inverse Problem in Various Formalisms

6.7.2.1 The Schrödinger Picture

Wewill reviewbriefly someof the relevant aspects of the geometry of the Schrödingerpicture [Du90, St66]. Given a Schrödinger-type equation (recall the examples inSect. 6.6.2):

idψ

dt= H ψ , ψ(x) ∈ H = C

n , (6.175)


with H being a linear operator on the complex vector spaceH, the inverse problemconsists of finding invariant scalar products under the associated time evolution, ifthey exist.

Given the vector field −i H , previous results show that the following statementsare equivalent:

1. There exists an (invariant) scalar product such that H = H†.2. H is diagonalizable and has a real spectrum.3. All the orbits ei Ht�, for any initial condition ψ are bounded sets.

Remark 6.11 It is clear that if H fulfills any of the previous conditions, the samewill be true for the operator H = T−1 H T , where T is any invertible operator; theoperator H will have the same spectrum of H and be self-adjoint with respect to thenew scalar product hT (φ,ψ) = h(φ, T ∗T ψ). In particular, if T commutes with Hwe get alternative ‘quantum descriptions’ for the same equations of motion. Indeed,we have:

If H is diagonalizable and has a real spectrum, the same properties are satisfiedfor all operators in the similarity class of H , i.e., T−1 H T ) for T ∈ GL(n, C).

The family of alternative Hamiltonian structures for H is parametrized by thecommutant of H in GL(n, C) quotiented by the commutant of H in U (n, C).

6.7.2.2 The Ehrenfest Picture

The Ehrenfest picture of Quantum Mechanics describes the dynamical evolution ofexpectation values of linear operators A on V , that is in this picture the dynamicsis defined as a derivation in the Lie algebra Q of real quadratic functions on H , i.e.,functions of the form fB(ψ) = 1

2 〈ψ |B|ψ〉 with B a symmetric operator. The Liealgebra structure is defined by the Poisson bracket { ·, · } associated to the symplecticstructure ω. It is easily noticed that Q is a Lie subalgebra with respect to { ·, · }. It iseasy to show that { f A, fB } = f[A,B].

Then the dynamical behaviour of the system with Hamiltonian H is defined bythe inner derivation DH = { fH , · }, i.e.,

id

dtfB = { fH , fB }. (6.176)

Thus, the inverse problem in the Ehrenfest picture can be put up as the search fora Poisson bracket on quadratic functions such that there exists fH such that,

L� f A = { fH , fA }.

The combined use of the Poisson bracket structure on the space of quadratic realfunctionals together with the complex structure in H to describe the dynamics ofquantum systems were already discussed in [St66].


The geometrical setting for the previous discussion is the category of Kählermanifolds, and the dynamical sytems we are describing preserve Kähler structures.

Consider now the Hermitean tensor h in contravariant form,

h = hkj ∂

∂ξ k⊗ ∂

∂ξ j= 1

2

(sk j ∂

∂ξ k

s⊗ ∂

∂ξ j

)+ i

2

(�k j ∂

∂ξ k∧ ∂

∂ξ j

). (6.177)

As indicated previously, the fundamental Poisson brackets defined by �, define aLie bracket on quadratic functions.

Defining

(A f ) jk = ∂2 f

∂ξ j∂ξ k,

we get,

[A f , Ag] jk = ∂2{ f, g }∂ξ j∂ξ k

,

and

[A f , Ag] = A f ·� · Ag − Ag ·� · A f .

From here we get immediately Heisenberg’s picture. In fact, if � leaves invariant h,i.e., L�h = 0, then necessarily,

d

dtfB = { fH , fB } , (6.178)

as in Eq. (6.176). Then acting on both sides of the equation above, Eq. (6.178), we get

d

dtB = [H, B] ,

because obviously, A fB = B.

6.7.2.3 The Heisenberg Picture

A different approach to Quantum Mechanics is given by what is known as theHeisenberg picture. Here dynamics is encoded in the algebra of observables, consid-ered as the real part of an abstract C

∗-algebra.First, we have to consider observables as associated with Hermitean operators

(finite dimensional matrices if the system is finite-dimensional). These matrices donot define an associative algebra because the product of two Hermitean matrices isnot Hermitean. However we may complexify this space by writing a generic matrix


as the sum of a real part A and an imaginary part i B, A and B being Hermitean. Inthis way we find that:

Proposition 6.34 The complexification of the algebra of observables allows us towrite an associative product of operators A = A1 + i A2, where A1 and A2 are realHermitean. We shall denote by A the corresponding associative algebra.

Finally we can proceed to define the equations of motion on this complexifiedalgebra of observables. It is introduced by means of the Heisenberg equation:

i�d

dtA = [A, H ] , A ∈ A, (6.179)

where H is called the Hamiltonian of the system we are describing. To take intoaccount an explicit time-dependence of the observablewemay alsowrite the equationof motion in the form

d

dtA = − i

h[A, H ] + ∂ A

∂t, A ∈ A. (6.180)

From a formal point of view, this expression is similar to the expression ofHamilton equation written on the Poisson algebra of classical observables (i.e., onthe algebra of functions representing the classical quantities with the structure pro-vided by the Poisson bracket we assume our classical manifold is endowed with).This similarity is not casual and turns out to be very useful in the study of thequantum-classical transition. We shall come back to this point later on.

Remark 6.12 The equations ofmotionwritten in this formare necessarily derivationsof the associative product and can therefore be considered as ‘intrinsically Hamil-tonian’. In the Schrödinger picture, however, if the vector field is not anti-Hermitean,the equation still makes sense, but the dynamics need not be Kählerian. To recovera similar treatement, one has to give up the requirement that the evolution preservesthe product structure on the space of observables.

Let us recall that the inverse problem as it was posed in the introduction intended tofind a Lie product [·, ·] in the space of observables of a quantum system an a functionH such that the equations of motion will have the Hamiltonian form B = [B, H ].We should mention first that the search for Lie products [·, ·] ends up very fast if weconsider the associative product on the space of operators on a Hilbert space definedby their composition because of the following theorem by Dirac [Fo76]:

Theorem 6.35 Any Lie algebra bracket on the space of operators of a complexHilbert space satisfying the derivation property,

[A, B · C] = [A, B] · C + B · [A, C],

is necessarily of the form [A, B] = μ(A · B − B · A), with μ a complex number.


Thus, if we use the associative product on the space of operators defined by thecomposition of linear operators acting on a Hilbert space, the Lie product must bethe ordinary commutator up to a constant factor, leaving open only the problem offinding the Hamiltonian function H . However, it is not necessary to use the definingrepresentation of the operators to equip them with an algebra structure. In fact,we can proceed in an alternative way, thinking of the space of observables as aprimary notion and representing them afterwards. This point of view is very close toSchwinger’s approach in [Sc70]. Thus we are led to consider as a fundamental entitythe measurement algebra of a physical system, a linear space generated by abstractoperators X (α) representing selectivemeasurements defining the states of the system.An associative algebra structure X (α)X (β) = ∑

γ (αβγ )X (γ ) is introduced onthem, that will depend in principle on the physical realization we choose for theselective measurements X (α). Then we can explore alternative associative productsdefined by the structure constants (αβγ ) and such that the dynamics � will beHamiltonian with respect to the Lie product defined by it.

A natural way of obtaining associative products in the space of operators is sug-gested by the following argument. If f is a quadratic function on V , we can define

(A f )kl = ∂2 f

∂ξ k∂ξ l,

then we get a Lie product by the formula,

[A f , Ag]kl = ∂2

∂ξ k∂ξ l{ f, g } .

If the fundamental commutation relations on our quantum space V are given by{ ξ j , ξ k } = � jk , we obtain, [A, B]kl = Aki�

i j B jl − Bki�i j A jl . Then, inspired

by this product we define a new Lie product on the space of linear operators on Vintroducing the associative product,

A ◦K B = AeλK B,

with λ a real number. Hence a new Lie product is defined as

[A, B]K = A ◦K B − B ◦K A.

We find also, [A, B ◦K C]K = [A, B]K ◦K C + B ◦K [A, C]K . Now if we define,

φK (A) = e12λK Ae

12λK = AK ,

we get

[φK (A), φK (B)] = φK ([A, B]K ). (6.181)


More generally, we can define

FK1,K2(A) = eλK1 AeλK2 ,

gives

[FK1,K2(A), FK1,K2(B)] = FK1,K2([A, B]K )

with exp λK = exp λK1 exp λK2.

Proposition 6.36 Any linear vector field that is a derivation of the associative prod-uct ◦K for λ = 0, is also a derivation for λ �= 0 if K is a constant of motion for thelinear vector field.

Then, also in the Heisenberg picture we have many alternative quantum descrip-tions. In fact, the Lie products [·, ·]K define alternative structures for our dynamics.However this creates a problem because the operators are composed with respectto the new associative product ◦K , which is not the natural product induced by thetautological representation of the operators on V . But we have already observed,Eq. (6.181), that the map φK intertwines the defining representation of the algebraof operators with a representation defining the Lie product [·, ·]K (even though themap is not induced by a linear isomorphism of the underlying Hilbert space). Thissuggests that we should define a new Hilbert space structure by,

〈ψ1|ψ2〉K = 〈ψ1|e 12λK e

12λK |ψ2〉.

Notice that the unity for the deformed associative product is given by IK = e−λK .Thus, 〈ψ1|IK |ψ2〉K = 〈ψ1|ψ2〉.

6.7.2.4 The Dirac Picture

What we call here the Dirac picture has its origin in what in Quantum Mechanicsgoes under the name of Dirac’s interaction picture. It starts from the considerationthat the propagator U (t, t0) maps by construction the initial wave function or state,ψ(t0), into ψ(t):

ψ(t) = U (t, t0)ψ(t0),

and hence satisfies the equation

i�d

dtU (t, t0) = H(t) U (t, t0),

with the initial condition U (t0, t0) = 1.


In QuantumMechanics one starts from here to write U = U0 W , with W anotherunitary operator which encodes all the effects resulting from the interaction and itallows us to write U in terms of the comparison ‘free evolution’ described by U0.

One proceeds to write an equation for W , which is usually solved by perturbationprocedures. For us it is simply a way to complete our description of linear systems:

1. We have then on the vector space, Schrödinger picture,

i�d

dtψ = H ψ,

2. On the algebra of observables or operators, Heisenberg picture,

i�d

dtB = [H, B],

3. On quadratic functions, Ehrenfest picture,

i�d

dtfB = { fH , fB}

4. On the group of unitary transformations, Dirac picture,

i�d

dtU = H U .

All these pictures are equivalent and in finite dimensions it is only a matter of con-venience to use one or the other, even though some aspects might bemore transparentin a picture with respect to another.

In infinite dimensions, that we rarely consider in this book, their potentiality areradically different. When domains problems are relevant, Heisenberg picture andEhrenfest picture may be quite different when one deals with unbounded operators.

The Dirac picture, which is the equation for the propagator, when evaluated inthe position representation becomes an equation for the Green function and becomeshighly relevantwhen the Schrödinger picture is dealingwith operators H with contin-uous spectrum and ‘eigenfunctions’ which would be square integrable only locally.

For the aimof this book this picture becomes verymuch relevantwhenwe considerLie–Scheffers systems. Moreover, the equation on the group might be the ‘abstractversion’ of a nonlinear equation as it happens for Riccati-type equations. In otherterms, the equation on the group would be the same even if the group would act non-linearly on a manifold M and therefore give raise to a nonlinear equation. This wouldbe usually the case when the Hamiltonian matrix H is invariant under some closedsubgroup K and we would consider the associated equation on the homogeneousspace G/K .

From our point of view the most relevant aspect of these equations is that theyonly rely on the properties of the group G, algebraical or geometrical, and do not


depend on the specific action or representation. For instance, on a given vector space,say three-dimensional, we would use different metric tensors, say

a21 dx1 ⊗ dx1 + a2

2 dx2 ⊗ dx2 + a23 dx3 ⊗ dx3,

with a1 a2 a3 �= 0.Each choice of these coefficients would give raise to different realizations of the

rotation group, say,

R3 = a1a2

x1∂

∂x2− a2

a1x2

∂

∂x1, R2 = a3

a1x3

∂

∂x1− a1

a3x1

∂

∂x3,

R1 = a2a3

x2∂

∂x3− a3

a2x3

∂

∂x3.

The corresponding linear equation associated to H = b1 R1 + b2 R2 + b3 R3would be

x1 = −b1a2a1

x2 + b2a3a1

x3 ,

and similarly for other coordinates.The Poisson bracket for the Ehrenfest picture would be

{x j , xk} = ε jkl a2l xl .

At the group level we would have

d S

dtS−1 = b1 R1 + b2 R2 + b3 R3 ,

and then

S(t) = et (b1 R1+b2 R2+b3 R3) ,

independently of the particular realization of the operators R j .Similarly at the abstract Lie algebra level, the Heisenberg picture, not on the

operators but on the level of the C∗-algebra, we would have

d

dt(Y j R j ) = [Y k Rk, Y l Rl ] ,

independently of the realization.


6.7.2.5 The Fundamental Example Again: The Harmonic Oscillator

We will revisit the inverse problem for the harmonic oscillator (see [Ib97] fora detailed description). Thus, following the ideas in the last paragraphs of theSect. 6.7.2.4, we can obtain alternative quantum descriptions by looking for solu-tions of the functional equation,

[H , a]K = −iωa,

with K = K (a†a). Thus, any decomposition of H = ω(a†a+ 1/2) as H = HeλK ,gives us alternative commutation relations and alternative Hamiltonians because[H, a] = [H , a]K .

By setting

φk(a) = e12λK (a†a)ae

12λK (a†a) = A, and φk(a

†) = e12 λK (a†a)a†e

12λK (a†a) = A†,

we can construct a new Fock space,

|N 〉 = 1√n! (A†)n|0〉,

with a|0〉 = 0 and with scalar product defined by 〈M |N 〉 = δmn .We shall consider as an example of the previous discussion the one dimensional

quantum harmonic oscillator. The equations of motion are given by

id

dt

(aa†

)= ω

(1 00 −1

)(aa†

),

where the operators a, a† verify the canonical commutation relations

[a, a†] = 1 ,

and the Hamiltonian H of the system is given by

H = ω

(a†a + 1

2

),

and the equations of motion are written as

d

dta = i[H, a] .

Let K be now an arbitrary function of a†a,

K = K (a†a) = K †,


which is a constant of motion [H, K ] = 0. Then, we define the following family ofalternative associative products,

a ◦K a† = aeλK a† = aeλK/2eλK †/2a† = AK A†K ,

where AK = a eλK/2.The new ‘creation’ and ‘anihilation’ operators AK and A†

K verify the commutationrelations

[a, a†]K = aeλK a† − a†eλK = AK A†K − A†

K AK = 1 .

The search for alternative Hamiltonians HK for the dynamics implies the equations

[HK , a]K = −a, [HK , a†]K = a† ,

namely,

HK eλK a − aeλK HK = −a ,

that has a family of solutions,

HK = ω

2(a†a + 1)e−λK .

We can represent this family in the Hilbert space constructed from the vacuum state|0〉 defined by,

A|0〉 = 0 = a|0〉 ,

and the N -particle states defined by

|N 〉 = 1√n! (A†)n |0〉 .

The inner product is defined as

〈M |N 〉 = δmn .

This new product induces a new Poisson structure on the Hilbert space of quadraticfunctions.


6.7.3 Alternative Hermitean Structures for Quantum Systems:The Infinite-Dimensional Case

Wenow analyze the same kind of problems in the framework ofQuantumMechanics,taking advantage of the experience and results we have obtained in the previoussections where we dealt with a real 2n-dimensional vector space.

In Quantum Mechanics the Hilbert space H is given as a vector space over thefield of complex numbers. Now we assume that two Hermitean structures are givenon it, which we will denote as 〈·, ·〉1and 〈·, ·〉2 (both linear, for instance, in the secondfactor). As in the real case, we look for the group that leaves invariant both structure,that is the group of unitary transformations with respect to both Hermitean structures.We call them bi-unitary operators.

In order to assure that 〈·, ·〉1 and 〈·, ·〉2 do not define different topologies onH itis necessary that there exists A, B ∈ R , 0 < A, B such that

A ‖x‖2 ≤ ‖x‖1 ≤ B ‖x‖2 , ∀x ∈ H .

The use of Riesz’s theorem on bounded linear functionals immediately impliesthat there exists an operator F defined implicitly by the equation

〈x, y〉2 = 〈Fx, y〉1 , ∀x, y ∈ H ,

and F replaces the previous G and T tensors of the real vector space situation, i.e.,now it contains both the real and imaginary parts of the Hermitean structure and,in fact

F = (g1 + iω1)−1 ◦ (g2 + iω2) (6.182)

It is trivial to show that F is bounded, positive, and self-adjoint with respect to bothHermitean structures and that

1

B2 ≤ ‖F‖1 ≤ 1

A2 ;1

B2 ≤ ‖F‖2 ≤ 1

A2 . (6.183)

If H is finite-dimensional, F can be diagonalized, the two Hermitean structuresdecompose in each eigenspace of F , where they are proportional and we get imme-diately that the group of bi-unitary transformations is indeed,

U (n1)×U (n2)× . . .×U (nk) , n1 + n2 + . . .+ nk = n = dimH

where ni denotes the degeneracy of the i th eigenvalue of F .In the infinite-dimensional case F may have a point part of the spectrum and a

continuum part. From the point part of the spectrum one gets U (n1)×U (n2)× . . .

where now ni can be also infinite. The continuum part is more delicate to discuss. It


will contain for sure the commutative group UF of bi-unitary operators of the formei f (F) where f is any measurable real-valued function.

The concept of genericity in the infinite-dimensional case can not be given aseasily as in the finite-dimensional case. One can say that the eigenvalues should benon-degenerate but what do we say for the continuous spectrum? We give here analternative definition that works for the finite and infinite case as well.

Note first that any bi-unitary operator must commute with F . Indeed: 〈x, U†

FU y〉2 = 〈U x, FU y〉2 = 〈FU x, U y〉2 = 〈U x, U y〉1 = 〈x, y〉1 = 〈Fx, y〉2 =〈x, Fy〉2, from this:

U†FU = F , [F, U ] = 0 .

The group of bi-unitary operators therefore belongs to the commutant F ′ of theoperator F. The genericity condition can be restated in a purely algebraric form asfollows:

Definition 6.37 Two Hermitean forms are in a generic position iff F′′ = F ′, i.e.,

the bicommutant of F coincides with the commutant of F.

In other words this means that F generates a complete set of observables.This definition reduces, for the case of a pure point spectrum, to the condition of

nondegeneracy of the eigenvalues of F and, in the real case, to the minimum possibledegeneracy of the eigenvalues of T and G, which is two.

To grasp how the definition works, we will give some simple examples. Consider:(Fψ)(x) = x2ψ(x) on the space L2([−b,−a] ∪ [a, b]) with 0 < a < b: then theoperator x , its powers xn and the parity operator P belong to F ′ while F

′′does

not contain x (and any odd power of x) because they do not commute with P. So ifF = x2 the two Hermitean structure are not in a generic position because F

′′ ⊂ F ′.On the contrary, on the space L2([a, b]), F

′′ = F ′ because a parity operator P doesnot exist in this case, so the two Hermitean structure are now in a generic position.

In this case the group of bi-unitary operators is{

ei f (x2)t}for the appropriate class of

functions f . In some sense, when a continuous part of the spectrum is considered,there appears a continous family of U (1)’s as a counterpart of the discrete family ofU (1)’s corresponding to the discrete part of the spectrum.

Remark 6.13 i) Suppose that complex Hilbert spaces with two Hermitean structureshave been constructed from a given real vector space V using two compatible andadmissible triples (g1, ω1, J1) and (g2, ω2, J2). Then, by complexification, we gettwo different Hilbert space, each one with its proper multiplication by complex num-bers and with its proper Hermitean structure. The previous case we have just studiedis obtained if we assume J1 = J2. It is easy to show that this is a sufficient condi-tion for compatibility. That is the reason why in the quantum-mechanical case thegroup of bi-unitary transformations is never empty, and the compatibility conditionis encoded already in the assumptions.

ii) If J1 �= J2 but the compatibility condition still holds, we know that V splits intoV+ ⊕ V−, where J1 = ±J2 on V± respectively. On V+ we have the previous case,


while on V− we get two Hermitean structures, one C-linear and one anti-C-linear inthe second factor (which one is which depending on the complexification we havedecided to use). From the point of view of the group of unitary transformations,this circumstance is irrelevant, because the set of unitary transformations does notchange from being defined with respect to an Hermitean structure or with respectto its complex conjugate. We conclude from this that our analysis goes through ingeneral, provided the compatibility condition holds.

We will try now to summarize our main result, by restating it at the same timein a more concise group-theoretical language. What we have shown is, to beginwith, that once two admissible triples: (g1, ω1, J1) and (g2, ω2, J2) are given ona real, even-dimensional vector space E ∼= R

2n , they define two 2n-dimensionalreal representations, Ur (2n; g1, ω1) and Ur (2n; g2, ω2) of the unitary group U (n),Ur (2n; ga, ωa) (a = 1, 2) being the group of transformations that leave simultane-ously ga and ωa (and hence Ja) invariant. Their intersection

Wr = {Ur (2n; g1, ω1) ∩Ur (2n; g2, ω2)} (6.184)

will be their common subgroup that is an invariance group for both triples. Theassumption of compatibility17 implies that Wr should not reduce to the identityalone.

If the two triples are in a generic position, then

Wr = SO(2)× SO(2)× . . .× SO(2)︸︷︷︸n factors

(6.185)

where, here: SO(2) ∼= U (1) or, more generally if the genericity assumption isdropped

Wr = Ur (2r1; g, ω)×Ur (2r2; g, ω)× . . .×Ur (2rk; g, ω) (6.186)

where: r1 + r2 + . . . + rk = n and (g, ω) is any one of the two pairs (g1, ω1) or(g2, ω2).

The real vector space V ≈ R2n will decompose then into a direct sum of even-

dimensional subspaces that are mutually orthogonal with respect to bothmetrics, andon each subspace the corresponding (realization of the) special orthogonal groupwillact irreducibly.

Alternatively, we can complexify E ∼= R2n in two different ways, using the two

complex structures that are at our disposal. The equivalent statement in the complexframework will be then:

Given two Hermitean structures ha, a = 1, 2 on a complex n-dimensional vectorspace C

n , they define two representations U (n; ha), a = 1, 2 of the group U (n) on

17 As the previous two-dimensional example shows explicitly, but it should be clear by now also ingeneral.


the same Cn . Then U (h1, n) (resp. U (h2, n)) will be the group of transformations

that are unitary with respect to h1 (resp. h2). The group W of simultaneous invariancefor both Hermitean structures :

W ≡ {U (h1, n) ∩U (h2, n)} . (6.187)

will be a subgroup of both U (h1, n) and U (h2, n), and our assumption of compati-bility of the ha ’s implies that the component of W connected to the identity shouldnot reduce to the identity alone.

The assumption of genericity implies that

W = U (1)×U (1)× . . .×U (1)︸︷︷︸n factors

(6.188)

If the assumption of genericity is dropped, one can easily show, along the same linesas in the generic case, that W will be of the form:

W = U (r1)×U (r2) . . .×U (rk) , (6.189)

with r1+ r2 + . . .+ rk = n. The space Cn will decompose accordingly into a direct

sum of subspaces that will be mutually orthogonal with respect to both ha ’s, and oneach subspace the appropriate U (r) will act irreducibly.

We have also shown that these results generalize to the infinite-dimensional caseas well. Some extra assumptions must be added on the Hermitean structures in orderthat they define the same topology in H and an appropriate definition of genericitymust also be given. Then, a decomposition like in Eqs.(6.188) and (6.189) is obtained,possibly with denumberable discrete terms and a continuum part as well. We notethat, in the spirit of this work where two Hermitean structures are given from thevery beginning, it is natural to supplement the compatibility condition, in the infinite-dimensional case, with a topological equivalence condition. However from the pointof view of the study of bi-hamiltonian systems, where a fixed dynamics is given, itwould be more natural to assume some weaker regularity condition, for instance thatthe given dynamics should be continuous with respect to both structures.

Bi-Hamiltonian systems ‘generated’ out of a pencil of compatible Poisson struc-tures have been considered in connection with the separability problem [Ib00]. Itshould be noticed that our compatible structures would give raise to a pencil ofcompatible triples defined by:

gγ = g1 + γ g2 , ωγ = ω1 + γω2 , Jγ (6.190)

We will conclude this section by stressing again the fact that the auxiliary mathe-matical structures used in the description of physical systems, classical or quantum,usually are not uniquely determined by the physical evidence collected from theobservation of the system. This situation is relevant because it reflects an intrinsic


ambiguity, for instance in the correspondence between symmetries and constants ofmotion. At the classical level, some of these ambiguities have been already discussedin relation with the quantization problem but at the quantum level they have receivedlittle attention.

Even more fundamental than all that is the possibility of using alternative linearstructures to describe quantum systems. A different path pointing to test the linearityof Quantum Mechanics was taken for instance by S. Weinberg [We89], where thelinearity of the equations of motion was questioned. Alternative linear structuresdescribe a more fundamental aspect, the nonexistence of an absolute linear structureon the space of quantum states.

Finally, we should point it out that many of the previous considerations can beextended to systems admitting higher order geometrical structures, such as NambuMechanical systems [Na73, Ma81, Ta94, Ma95], generalized Poisson structures[Az96, Az97, Iba98] etc.

References

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[Du90] Dubrovin, B.A., Marmo, G., Simoni, A.: Alternative Hamiltonian descriptions for quan-tum systems. Mod. Phys. Lett. A 5, 1229–1234 (1990)

[Ma96] Marmo, G., Vilasi, G.: Symplectic structures and quantum mechanics. Mod. Phys. Lett.B 10, 545–553 (1996)

[Er10] Ercolessi, E., Marmo, G., Morandi, G.: From the equations of motion to the canonicalcommutation relations. Riv. Nuovo Cim. 33, 401–590 (2010)

[Ib96a] Ibort, A., Marmo, G.: From the equations of motion to quantum commutation relations.Proceedings of the V Workshop on Geometry and Physics, Jaca (1996)

[Gi93] Giordano, M., Marmo, G., Rubano, C.: The inverse problem in the Hamiltonian formal-ism: integrability of linear Hamiltonian fields. Inv. Probl. 9, 443–467 (1993)

[Ch67] Chern, S.S.: Complex Manifolds without Potential Theory. Van Nostrand, Princeton(1967)

[Sc86] Schwartz, L.: Lectures on Complex Analytic Manifolds. Narosa, New Dehli (1986)[Ni57] Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex mani-

folds. Ann. Math. 65, 391–404 (1957)[We58] Weil, A.: Introduction a’ l’Étude des Varietés Kähleriennes. Hermann, Paris (1958)[Ci90] Cirelli, R., Maniá, A., Pizzocchero, L.: Quantum mechanics as an infinite-dimensional

Hamiltonian system with uncertainty structure I. J. Math. 31, 2891–2897 (1990) (ibid.Quantum mechanics as an infinite-dimensional hamiltonian system with uncertaintystructure II. J. Math. Phys. 31, 2898–2903 (1990))

[Ci91] Cirelli, R.,Maniá, A., Pizzocchero, L.: Quantumphase-space formulation of Schrödingermechanics. Int. J. Mod. Phys. A 6, 2133–2146 (1991)

[Ci94] Cirelli, R., Maniá, A., Pizzocchero, L.: A functional representation for non-commutativeC∗-algebras. Rev. Math. Phys. 6, 675–697 (1994)

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Phys. 17, 195–199 (1945)[Re44] Reichenbach, H.: Philosophical Foundations of QuantumMechanics. University of Cal-

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[St60] Stueckelberg, E.C.G.: Quantum Theory in Real Hilbert Space I, II. Helv. Phys. Acta 33,727–752 (1960) (ibid. 34, 621–28 (1961))

[Es04] Esposito, G., Marmo, G., Sudarshan, G.: From Classical to Quantum Mechanics: anintroduction to the formalism. Cambridge Univ,Press(2004)

[Ma05b] Marmo, G., Scolarici, G., Simoni, A., Ventriglia, F.: The quantum-classical transition:the fate of the complex structure. Int. J. Geom. Meth. Mod. Phys. 2, 127–145 (2005)

[Di58] Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press, Oxford (1958)[Me58] Messiah, A.: Mecanique Quantique, vol. I . Dunod, Paris (1958)[La27] Landau, L.: DasDampfungsproblem inWellen-mechanik (The damping problem inwave

mechanics). Z. Phys. 45, 430–441 (1927)[Ne27] von Neumann, J.: Wahrscheinlichkeitstheoretischer aufban der Quantenmechanik. Goet-

tingenische Nachrichten 11, 245–272 (1927)[Ha70] Haefliger, A.: Homotopy and integrability. Lect. Note Math. 197, 133–164, Springer

(1970)[Hus75] Husemoller, D.: Fibre Bundles, 2nd ed. Springer, New York (1975) (3d edn, Springer,

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quantum states. Open Syst. Inf. Dyn. 16, 157–167 (2009)[St51] Steenrod, N.: The Topology of Fibre Bundles. Princeton University Press, Princeton

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Point of View)[Di33] Dirac, P.A.M.: The Lagrangian in Quantum Mehanics.[Cl07] Clemente-Gallardo, J., Marmo, G.: The space of density states in geometrical quantum

mechanics. In: Cantrijn, F., Crampin, M., Langerock, B. (eds.) Differential GeometricMethods in Mechanics and Field Theory, pp. 35–56. Ghent University Press, Ghent(ISBN 9789038211282) (2007)

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[Ma95] Marmo, G., Morandi, G.: Some geometry and topology. In: Lundqvist, S., Morandi, G.,Lu, Y. (eds.) Low-Dimensional Quantum Field Theories for Condensed-Matter Physi-cists. World Scientific, Singapore (1995)

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[Wi59] Wigner, E.P.: Group Theory and its Applications to the Quantum Mechanics of AtomicSpectra. Academic Press, New York (1959)

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(1963) (Dover 2004)

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(1976)[Sc70] Schwinger, J.: QuantumKinematics andDynamics. Frontiers in Physics, Benjamin, New

York (1970) (Proc. Nat. Acad. Sci. 46, 261–265 (1960))[Ib97] Ibort, A., Man’ko, V., Marmo, G., Zaccaria, F.: The inverse problem for quantum and

classical systems: from the classical equations of motion to the quantum commutationrelations. Group 21. In: Doebner, H.-D., Nattermann, P., Scherer, W. (eds.) PhysicalApplications and Mathematical Aspects of Geometry, Groups and Algebras, vol. I, pp.265–274. World Scientific, Singapore (1997)

[Ib00] Ibort, A., Magri, F., Marmo, G.: Bihamiltonian structures and Stäckel separability. J.Geom. Phys. 33, 210–228 (2000)

[We89] Weinberg, S.: Testing quantum mechanics. Ann. Phys. 194, 336–386 (1989)[Na73] Nambu, Y.: Generalized Hamiltonian mechanics. Phys. Rev. D 7, 499–510 (1973)[Ma81] Marmo, G., Saletan, E.J., Simoni, A., Zaccaria, F.: Liouville dynamics and Poisson

brackets. J. Math. Phys. 22, 835–842 (1981)[Ta94] Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math.

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[Iba98] Ibáñez, R., de León,M., Marrero, J.C., Padrón, E.: Nambu-Jacobi and generalized Jacobimanifolds. J. Phys. A: Math. Gen. 31, 1267–1286 (1998)

Chapter 7Folding and Unfolding Classical and QuantumSystems

The unmanifest world is simple and linear, it is the manifestworld which is ‘folded’ and nonlinear.Giuseppe Marmo, XXI International Workshop on DifferentialGeometric Methods in Theoretical Mechanics, 2006

7.1 Introduction

Reduction procedures, the way we understand them today (i.e. in terms of Poissonreduction) can be traced back to Sophus Lie in terms of function groups, reciprocalfunction groups and indicial functions [Ei61, Fo59, Lie93, Mm85]. Function groupsprovide an algebraic description of the cotangent bundle of aLie group but are slightlymore general because they can arise from Lie group actions which do not admit amomentummap [Ma83]. Function groups are also know today as ‘dual pairs’ [Ho85]as discussed already in Sect. 4.3.4. The present chapter exhibits many instances ofreduction procedures appearing in a variety of physical situations, both classical andquantum. This choice may give the impression of an episodic description, howeverit contains an illustration of the essential aspects of any reduction procedure, both inthe algebraic and geometrical setting, pointing out analogies and differences betweenthe classical and the quantum situation.

7.2 Relationships Between Linear and Nonlinear Dynamics

The aim of this chapter is to analyze the relationships between linear and nonlineardynamics. We show that some nonlinear dynamics enjoy interesting properties, likesome sort of integrability or separability, arising as reduction from linear dynam-ics. Conversely, starting from linear ‘free’ dynamics, we can obtain by appropriate


489

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490 7 Folding and Unfolding Classical and Quantum Systems

reduction procedures well known dynamics that appear quite often in the physicsliterature.

7.2.1 Separable Dynamics

Physicists have used reduction procedures as an aid in trying to integrate the dynamics‘by quadratures’. Dealing, as usual, with a coordinate formulation, reduction andcoordinate separability have overlapped a good deal. A first approach to the notionof separability was offered at the linear level in Sect. 3.6.2. From the point of view ofthe integration of the equations ofmotion, separability or the so called decompositioninto independent motions, may be formalized as follows.

Definition 7.1 Consider a dynamical vector field � on a carrier manifold M and adecomposition

� =r∑

i=1

�i ,

with the requirement that:

1. [�i , � j ] = 0 for all i, j = 1, . . . , r , and2. span {�i (x) | i = 1, . . . , r} = Tx M , ∀x ∈ N ⊂ M , where N is an open and

dense submanifold in M .

When such a decomposition exists, the evolution is given by the composition of theone-parameter groups associated with each � j and the system � will be said to beseparable.

Looking for superposition rules which would generalize the usual linear super-position rule of linear systems, Lie [Lie93] introduced dynamical systems admittinga decomposition of the form:

� =∑

i

a j (t)� j ,

with [�i , � j ] = ∑kci j

k�k and ci jk ∈ R (i.e., the vector fields �k span a finite-

dimensional real Lie algebra) and still {span�i (x)} = Tx M , ∀x ∈ N ⊂ M , whereN is open and dense in M . The integration of these systems may be achieved byfinding a fundamental set of solutions: they admit a superposition rule even if thesystem is nonlinear. These systems have been called Lie–Scheffers systems andwill be extensively studied in Chap.9. An important representative of Lie-Schefferssystems is given by the Riccati equation. It is worth illustrating this example becauseit is an instance of a nonlinear equation obtained as a reduction from a linear one.

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7.2 Relationships Between Linear and Nonlinear Dynamics 491

7.2.2 The Riccati Equation

In 1720, when he proposed his today famous equation, Riccati was interested in thefollowing problem: Given a linear system of differential equations describing thedynamical evolution of a point (x1, x2) of the Euclidean plane,

d

dt

(x1x2

)=

(b11 b12b21 b22

) (x1x2

)(7.1)

where the bi j are t-dependent functions, find the dynamical evolution of x = x2/x1.He found that such evolution is given by:

x = c0(t) + c1(t) x + c2(t) x2 , (7.2)

where

c0 = b21 c1 = b22 − b11, c2 = −b12 .

Equation (7.2) is known as the Riccati equation and it is a first-order equation thatonly involves one variable, while the linear system Eq. (7.1) involves two unknownvariables. In this sense we have reduced the number of degrees of freedom. Note alsothat the final equation does not depend on b11 and b22 but only on their difference.Therefore we can restrict ourselves to the case in which the matrix

A =(

b ca −b

)(7.3)

defining the linear system is traceless and then belongs to the Lie algebra sl(2, R).That is, rewriting the coordinates in R

2 by (x, y) and considering the linear systemdefined by the matrix A:

⎧⎪⎨⎪⎩

dx

dt= b x + c y

dy

dt= a x − b y

,

the vector field for such dynamics is given now by:

�A = ax∂

∂y+ b

(x

∂

∂x− y

∂

∂y

)+ cy

∂

∂x(7.4)

with a, b, c ∈ R time-dependent functions.We remark now that because the system is linear, �A commutes with the Euler

vector field � = x∂/∂x + y∂/∂y, the dilation vector field on R2. Therefore, the

algebra of functions f such that �( f ) = 0 on R2\{0, 0} will be invariant under �A,


i.e., �( f ) = 0 implies that �(�A( f )) = 0. Such functions f , being homogeneousof degree zero, depend only on the variable x/y or y/x . If we consider first thedynamics in terms of ξ = x/y, y �= 0, we find

dξ

dt= x

y− x y

y2= cy + bx

y− x

y

(ax

y− by

y

)= c + 2bξ 2 − aξ 2 (7.5)

i.e.,

dξ

dt= c + 2bξ − aξ2 (7.6)

The transformation: (x, y) �→ x/y is often called the Riccati transformation. There-fore we have obtained the Riccati equation, which is nonlinear, by reducing our lineardynamics with an invariant equivalence relation, namely

(x1, y1) ∼ (x2, y2) i f f x1y2 − y1x2 = 0 ,

which is the equivalence relation defined by the integral curves of the vector field�. Notice that the equivalence relation defined by such set of curves is invariantbecause, as indicated above, �A is a linear vector field and then [�,�A] = 0.

We should notice here that the integral curves of � are open half-lines starting at(0, 0), and the origin itself, therefore the set of such integral curves of � is singularat the origin. This implies that in the quotient space the integral curve (0, 0) is notseparated from all the others and the quotient space is not a Hausdorff space. Remov-ing it, the quotient space is simply the circle S1. The two choices for parametrizingits points ξ = x/y and ζ = y/x define two charts covering it. In the language ofdifferentiable spaces, we have obtained that the subalgebra of �-invariant functionson F(R2\{0, 0}) is isomorphic to the algebra of smooth functions on the circle S1.

Equation (7.6) provides in fact the local expression of the vector field � on S1

induced by �A in the chart ξ . Because the range of this chart is R we can say thatthe induced vector field reduces to the Riccati equation on the line. Notice that in thesecond chart ζ the projected vector field has the expression

dζ

dt= a − 2bζ − cζ 2 , (7.7)

which again is a Riccati equation obtained from the previous one (7.6) by the changeof variable ζ = 1/ξ .

Wemight comment a littlemore on this last example. It is clear that the vector fieldsx∂/∂y, y∂/∂x and x∂/∂x − y∂/∂y close on the Lie algebra of the group SL(2, R)

(see Sect. 2.6.1). In fact, they are the linear vector fields Xe, X f , Xh , associated withthe traceless matrices

e =(0 01 0

), f =

(0 10 0

), h =

(1 00 −1

), (7.8)

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which are the usual generators of the Lie algebra sl(2, R) with the commutationrelations

[e, h] = 2e , [ f, h] = −2 f , [e, f ] = h . (7.9)

The vector field� is a linear combination of these three vector fieldswith functionsof time as coefficients and constitutes the prototype of a Lie-Scheffers system. Still,using this representation we can say that Eq. (7.2) is associated with a ‘free’ motionon the group SL(2, R):

g g−1 = −∑

j

c j (t)A j ,

where A0 = e, A1 = f , A2 = h is an appropriate basis of the Lie algebra ofSL(2, R). Associated to it we find a nonlinear superposition rule for the solutions:if x(1), x(2), x(3) are independent solutions, every other solution x is obtained from aconstant value k of the following double ratio:

(x − x(1)

) (x(2) − x(3)

)(x − x(2)

) (x(1) − x(3)

) = k . (7.10)

Exercise 7.1 Derive the double ratio relation from invariants (constants of motion)of the linear system �A.

Notice the geometric meaning of this construction. Starting from a linear equationin R

2, we consider a map π : R2 → R, given by π(x1, x2) = x2/x1, and then the

t-dependent vector field describing the given linear system of first-order differentialequations projects on a corresponding one which describes a Riccati equation, adifferential equation involving only one degree of freedom and simpler in this sensethan the original system. The knowledge of the general solution of the differentialequation allows us to find the general solution of the given system.

But we can also use this example in the opposite direction, as a suggestion thatspecific dynamics, for instance exhibiting superposition rules, can be deduced fromlinear ones under an appropriate reduction process.

Riccati type equations arise also in the reduction of the Schrödinger equation tothe space of pure states (see also [Ch07]).

Another example, but for partial differential equations, is provided by the follow-ing variant of the Burgers equation.

7.2.3 Burgers Equation

To illustrate the procedure for partial differential equations in one space x and onetime t variables, we consider the following variant of the Burgers equation


∂w

∂t+ 1

2

(∂w

∂x

)2

− k

2

∂2w

∂x2 = 0 , (7.11)

where k is an arbitrary but fixed real parameter.This equation admits a (at least partial) superposition rule of the following kind:

for any two solutions, w1(t, x) and w2(t, x), the function:

w = −k log

(exp

(−w1 + �1

k

)+ exp

(−w2 + �2

k

)),

is again a solution of Eq. (7.11) with �1 and �2 arbitrary real constants.The existence of a superposition rule might suggest that the equation may be

related to a linear one. That is indeed the case and we find that the heat equation

∂u

∂t= k

2

∂2u

∂x2,

is indeed related to the nonlinear equation by the replacement of the dependentvariable w by the new variable u = exp(−w/k).

Out of this experience onemay consider the possibility of integratingmore generalevolution systems of differential equations by looking for a simpler system (“simpler”here meaning that it is an explicitly integrable system) whose reduction provides thesystem that we would like to integrate. In some sense, in a short sentence, we couldsay that the reduction procedure provides us with interacting systems out of free (orHarmonic) ones.

The great interest for new completely integrable systems boosted the research inthis direction in the past 25years andmany interesting physical systems, both in finiteand infinite dimensions were shown to arise in this way (see for instance [OlPe81]and references therein).

In the same ideology one may also put the attempts for the unification of all thefundamental interactions in Nature by means of Kaluza-Klein theories. In addition,the attempt to quantize theories described by degenerate Lagrangians called for adetailed analysis of reduction procedures connected with constraints. These tech-niques came up again when considering geometric quantization as a procedure toconstruct unitary irreducible representations for Lie groups by means of the orbitmethod [Ki76, Ki99].

We have already discussed a few instances in Sect. 1.2.10 of nonlinear systemsarising from linear ones. Of course if our concern is primarily with the equationsof motion we have to distinguish the various available descriptions: Newtonian,Lagrangian, Hamiltonian. Each description carries additional structures with respectto the equations of motion and one has to decide whether the reduction should beperformed within the chosen category or if the reduced dynamics will be allowed tobelong to another one.Wewill devote the rest of the chapter illustrating anddiscussingthe reduction of linear systems compatible with certain geometrical structures. We

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will start by reviewing the simplest example that shows how ‘nonlinearities’ arisefrom reduction of a linear system: the three dimensional free particle.

7.2.4 Reducing the Free System Again

We consider again the reduction of the free system on R3 discussed in Sect. 1.2.10.

Consider the equations of motion of a free particle of unit mass in Newtonian form:

r = 0 .

This system is associated to the second-order vector field in T R3, � = r · ∇ and has

as constants of motion r and � = r × r:

d

dt(r × r) = 0,

d

dtr = 0 .

By making use of constants of motion, we showed how to choose invariant subman-ifolds for � such that taking the restrictions of the dynamics on such submanifolds,we can associate with it an equation of motion involving only r = ‖r‖, r and some‘coupling constants’ related to the values of the constants ofmotion to get for instanceEq. (1.93):

r = α�2 + (1 − α)(2E − r2)r2

r3.

The geometrical interpretation of this is rather simple: we have selected an invari-ant submanifold ⊂ R

3 (the level set of a constant of motion), we have restrictedthe dynamics to it, and then we have used the rotation group to foliate into orbits.The reduced dynamics is a vector field acting on the space of orbits = /SO(3).It should be remarked that even if is selected in various ways, the choice we havemade is compatible with the action of the rotation group. It should be clear nowthat our presentation goes beyond the standard reduction in terms of the momentummap, which involves additional structures. Indeed this reduction, when carried outwith the canonical symplectic structure, would give us only the first solution in theexample above.

There is another way to undertake the reduction of the free system above. OnT ∗

R3 with coordinates (r, p), we can consider the functions:

ξ1 = 1

2〈r, r〉 , ξ2 = 1

2〈p, p〉 , ξ3 = 〈r, p〉 .

Here 〈a, b〉 denotes the scalar product a · b, but it can be extended to a non definitepositive scalar product.

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The equations of motion of the free system (1.90) using ξ1, ξ2 and ξ3 as coordinatefunctions become

d

dtξ1 = ξ3 ,

d

dtξ2 = 0 ,

d

dtξ3 = 2ξ2.

Note that any constant of motion of this system is then a function of ξ2 and (2ξ1ξ2 −ξ23 ). Consider first the invariant submanifold ξ2 = k/2 ∈ R. Then we find,

d

dtξ1 = ξ3 ,

d

dtξ3 = k ,

i.e., a uniformly accelerated motion in the variable ξ1. It may be described by theLagrangian

L(x, v) = 1

2v2 + kx,

where x = ξ1, v = ξ1 = ξ3.Had we selected a different invariant submanifold, for instance,

2ξ1ξ2 − ξ23 = �2,

the restricted dynamics would have been:

d

dtξ1 = ξ3 ,

d

dtξ3 = ξ23 + �2

ξ1.

A corresponding Lagrangian description is then provided by the function

L(x, v) = 1

2

v2

x− 2�2

x,

with x = ξ1 and x = v = ξ3.If we start with the dynamics of the isotropic harmonic oscillator, say r = p and

p = −r, we obtain in terms of the coordinates above:

d

dtξ1 = ξ3 ,

d

dtξ2 = −ξ3 ,

d

dtξ3 = 2(ξ2 − ξ1).

Therefore ξ1 + ξ2 is a constant of motion and if we introduce the functions η1 =ξ1 − ξ2, ξ3 and η2 = ξ1 + ξ2, we would get

η1 = 2ξ3, ξ3 = −2η1 ,

together with η2 = 0, i.e., we get a one dimensional oscillator.

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We would like to stress that the ‘position’ η1 of this reduced system is not afunction depending only on the initial position variables r.

Let us point out a general aspect of the example we just considered.We first noticethat the functions

ξ1 = 1

2

∑a

xa xa, ξ2 = 1

2

∑a

pa pa, and ξ3 =∑

a

xa pa ,

may be defined on any phase space R2n = T ∗

Rn , with R

n an Euclidean space. If weconsider the standard Poisson brackets, say

{xa, pb} = δab , {pa, pb} = 0 = {xa, xb}, a, b = 1, . . . , n,

we find that for the new variables we get:

{ξ1, ξ2} = ξ3, {ξ1, ξ3} = 2ξ1, {ξ2, ξ3} = −2ξ2. (7.12)

Thus the functions ξi we are considering close on the real Lie algebra sl(2, R)

(see Eq. (7.9)). The infinitesimal generators Xi = {ξi , ·} are complete vector fieldsand integrate to a symplectic action of SL(2, R) on R

2n endowed with its naturalsymplectic structure given by the identification of R

2n with T ∗R

n .Then, in the stated conditions there is always a symplectic action of SL(2, R) on

T ∗R

n R2n with a corresponding momentum map J : T R

n → sl(2, R)∗ ∼= R3.

If we denote again the coordinate functions on this three dimensional vectorspace by {ξ1, ξ2, ξ3}, with the Poisson brackets given by (7.12), then the momentummap J provides a symplectic realization of the Poisson manifold sl(2, R)∗. In thelanguage used by S. Lie we would say that the coordinate functions {ξ1, ξ2, ξ3}along with all the smooth functions of them { f (ξ1, ξ2, ξ3)} define a function group.The Poisson subalgebra of functions of F(T ∗

Rn) Poisson commuting with all the

functions f (ξ1, ξ2, ξ3), constitute the reciprocal function group, and all functions inthe intersection of both sets, say functions of the form F(ξ1ξ2 − 1

4ξ23 ), constitute the

indicial functions.By setting ξ1 = 1

2 , ξ3 = 0 we identify a submanifold in T Rn diffeomorphic with

T Sn−1, the tangent bundle of the (n − 1)-dimensional sphere. It is clear that thereciprocal function group is generated by functions Jab = pa xb − pbxa .

Thus, the reduced dynamics which we usually associate with the Hamiltonian

H = 1

2p2r + l2

2r2

is actually a dynamics on sl(2, R)∗ and therefore it has the same form independentlyof the dimension of the space T R

n we start with.Symplectic leaves in sl∗(2, R) are diffeomorphic to R

2 and pairs of conjugatedvariables may be introduced, for instance as μ = 1

2ξ1, ν = ξ3/ξ1, or μ = 12ξ2,


ν = −ξ3/ξ2, and then:

{μ, ν} = 1 .

Wesee in all these examples that the chosen invariant submanifold appears eventuallyas a ‘coupling constant’ in the reduced dynamics. Moreover, the final ‘second-orderdescription’ may be completely unrelated to the original one, that is, what we willcall now ‘position’ is actually a function of the old positions and velocities.

A few remarks are pertinent now.

Remark 7.1 We have not specified the signature of our scalar product on R3. It is

important to notice that the final result does not depend on it. However, because inthe reduced dynamics ξ1 appears in the denominator, when the scalar product is notpositive definite we have to remove the full algebraic variety 〈r, r〉 = 0 to get asmooth vector field. If the signature were (+,+,−), the relevant group would notbe SO(3) anymore but it would be replaced by SO(2, 1).

Remark 7.2 If we consider the Lagrangian description of the free particle, that is,we consider the Lagrangian:

L = 1

2〈r, r〉,

in polar coordinates it becomes

L = 1

2

(r2 + r2(n · n)

),

which restricted to the submanifold �2 = r4(n · n) would give

L = 1

2

(r2 + �2

r2

),

which is not the Lagrangian giving raise to the dynamics r = �2/r3. Therefore,we must conclude that the reduction, if done in the Lagrangian formalism, mustbe considered as a symplectic reduction in terms of the symplectic structure ofLagrangian systems, i.e., in terms of the symplectic Cartan 2-formωL and the energyfunction EL (recall Sect. 5.5.3).

Remark 7.3 The free particle admits many alternative Lagrangians, therefore oncean invariant submanifold has been selected, we have many alternative symplecticstructures to pull-back to and define alternative involutive distributions to quo-tient . The possibility of endowing the quotient manifold with a tangent bundlestructure has to be investigated separately because the invariant submanifold doesnot need to have a particular behaviour with respect to the tangent bundle structure.A recent generalization consists of considering that the quotient space may not have

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a tangent bundle structure but may have a Lie algebroid structure. Further examplesand additional comments on the previous examples may be found in [La91, Ma92].

We shall close now these preliminaries by recalling that another example is thegeneralization of this procedure to free systems on higher dimensional spaces dis-cussed in Sect. 1.2.12where theCalogero-Moser potential was obtained starting fromthe free motion on the space of 2 × 2 symmetric matrices.

Other examples, as the Toda system and other well-known systems, obtainedstarting with free or harmonic motions on the space of n × n Hermitian matrices,free motions on U (n), or free motions on the coset space GL(n, )/U (n, C), can befound in the literature (see for instance [OlPe81]).

We can summarize the previous discussion by saying that the reduction of thevarious examples that we have considered are based on the selection of an invariantsubmanifold and the selection of an invariant subalgebra of functions.

7.2.5 Reduction and Solutions of the Hamilton-Jacobi Equation

To illustrate the relation between reduction of linear systems and finding solutons tothe Hamilton-Jacobi equation we will choose the obtention of the Calogero-Mosersystem out of free motion on the space of symmetric 2 × 2 matrices performed inSect. 1.2.12.

Let us consider the space of symmetric matrices X and the Lagrangian function

L = 1

2Tr (X2) .

This Lagrangian gives raise to the Euler-Lagrange equations of motion Eq. (1.130):

X = 0 .

Moreover, the symplectic structure associated to it is defined by

ωL = Tr (d X ∧ d X) ,

and the energy is given by:EL = L .

The invariance of the Lagrangian under translations and rotations implies the con-servation of the linear momentum P = X and the angular momentum M = [X, X ].The corresponding explicit solutions of the dynamics are thus given by:

X (t) = X0 + t P0 . (7.13)

for given initial data X0 and P0.

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It is possible to find easily a solution of the corresponding Hamilton-Jacobi equa-tion. Indeed, by integrating the Lagrangian L along the solutions given by Eq. (7.13)or by solving

Pt d Xt − P0 d X0 = d S(Xt , X0; t)

we find that Hamilton’s principal function S is written as

S = 1

2tTr (Xt − X0)

2 .

By fixing a value �2 = 12Tr M2 we select an invariant submanifold . The corre-

sponding reduced dynamics gives the Calogero-Moser equations (1.134). Thereforeif we restrict S to those solutions which satisfy

1

2Tr (X2

t X20 − (Xt X0)

2) = �2 ,

we will find a solution for the Hamilton-Jacobi equation associated with the reduceddynamics [Ca07b, Ca09].

Remark 7.4 For any invertible constant symmetric matrix K , the Lagrangian func-tion L K = 1

2Tr (X K X) would describe again the free motion. More generally, forany monotonic function f , the composition f (L K ) would be a possible alternativeLagrangian. The corresponding Lagrangian symplectic structure ωL could be usedto find alternative Hamilton-Jacobi equations. For those aspects we refer to [Ca06].

7.3 The Geometrical Description of Reduction

We will provide first a geometric description of the various examples of reductiondiscussed before. We denote by M the manifold which is the carrier space for theequations of motion of our system that will be represented by a vector field �. Wesuppose that its flow gives raise to a one-parameter group of transformations

� : R × M → M .

Occasionally, when we want our map to keep track of the infinitesimal generator �

we will write �� or �� : R × M → M .To apply the general reduction procedure we need:

1. A submanifold ⊂ M , invariant under the evolution, i.e.,

�(R × ) ⊂ , or �t (m) = �(t, m) ∈ , ∀t ∈ R, m ∈ .

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7.3 The Geometrical Description of Reduction 501

2. An invariant equivalence relation among points of , that is we consider equiv-alence relation ∼ for which if m ∼ m′ then �t (m) ∼ �t (m′), for all t ∈ R.

The reduced dynamics � or ‘reduced evolution’ is defined on the manifold = /∼, of equivalence classes of points on (assumed to be endowed witha differentiable structure) by means of the natural definition:

�t(m) = �t (m) .

The invariance of the equivalence relation ∼ guarantees that �t is well defined, thatis will not depend on the choice of the representative m we choose in the class m.

One may also start the other way around: we could first consider an invariantequivalence relation ∼ on the whole manifold M and then select an invariant sub-manifold for the reduced dynamics �t defined on the quotient space M = M/∼,to further reduce the dynamical evolution.

In real physical situations the invariant submanifolds arise as level set of functions.These level sets were called invariant relations by Levi-Civita [Am26] to distinguishthem from level sets of constants of the motion. Usually, equivalence classes will begenerated by orbits of Lie group actions or leaves of involutive distributions. ‘Closedsubgroup’ theorems are often employed to guarantee the regularity of the quotientmanifold [Pa57].

We can collect the previous arguments in the form of a theorem. The theoremcannot provide in general conditions guaranteeing that the quotient spaces arising onit are smooth manifolds, something that must be assumed as part of the conditions oneach particular instance. The meaning of the theorem can be pictorially visualizedin Fig. 7.1.

Fig. 7.1 A pictorial representation of geometrical reduction: the ‘purple’ leaves in M representthe �-invariant equivalence relation while the ‘green’ submanifold is the �-invariant submanifold. We can reach the reduced space either by restricting to first and then quotienting by theinduced equivalence relation, or we can quotient first and then restrict to the submanifold


Theorem 7.2 (Geometrical reduction theorem) Let � be a dynamical system definedon the carrier manifold M. Let be a �-invariant submanifold and ∼ a �-invariantequivalence relation. Let us assume that the quotient spaces M = M/∼ and =/∼ are smooth manifolds and the projection maps πM : M → M, π : →

are smooth submersions. We denote by � the restriction of the dynamics � to . Thedynamical vector field � is πM -projectable and its projection on M will be denotedby �. Similarly, the vector field � is π-projectable and its projection to will bedenoted by � . Then is a �-invariant submanifold in M and the restriction of �

to it coincides with � , that is:

� = � | .

Exercise 7.2 Prove the previous theorem (the proof is an exercise in ‘diagramchasing’).

When additional structures are present, like Poisson or symplectic structures, itis possible to get involutive distributions out of a family of invariant relations. Theso-called ‘symplectic reduction’ is an example of this particular situation.

When the space is endowedwith additional structures, say a tangent or a cotangentbundle, with the starting dynamics being, for instance, second-order (in the tangentcase), we may also ask for the reduced one to be second order, once we ask thereduced space to be also endowed with a tangent space structure. This raises naturalquestions on how to find appropriate tangent or cotangent bundle structures on a givenmanifold obtained as a reduced carrier space. Similarly, we may start with a lineardynamics, perform a reduction procedure (perhaps by means of quadratic invariantrelations) and enquire about possible linear structures on the reduced carrier space.A simple example of this situation is provided by the Maxwell equations. Theseequations may be written in terms of the Faraday 2-form F encoding the electricfield E and the magnetic field B, as:

d F = 0 d ∗ F = 0 ,

when considered in the vacuum [Ma05]. We may restrict these equations to theinvariant submanifold

F ∧ F = 0, F ∧ ∗F = 0 .

Even though these relations are quadratic the reduced Maxwell equations provide assolutions the radiation fields and are still linear.

In conclusion, when additional structures are brought into the picture, we mayend up with extremely rich mathematical structures and quite difficult mathematicalproblems.

7.3 The Geometrical Description of Reduction 503

7.3.1 A Charged Non-relativistic Particle in a MagneticMonopole Field

This system was considered by Dirac [Di31] and a variant of it, earlier by Poincaré[Po96]. To describe it in terms of a Lagrangian, Dirac introduced a ‘Dirac string’. Thepresence of this unphysical singularity leads to technical difficulties in the quantiza-tion of this system. Several proposals have been made to deal with these problems.

Here we would like to show how our reduction procedure allows us to deal withthis system and provides a clear way for its quantization. In doing this we shall followmainly [Ba80, Ba83, Ba91].

The main idea is to replace R30 with R

40 described as the product R

40 = S3 × R+,

and to get back our space of relative coordinates for the charge-monopole by meansof a reduction procedure.

We set first x ·σ = rsσ3s−1, where r2 = x21 + x22 + x23 and s ∈ SU (2) (realized as2×2matrices of the defining representation; while {σ1, σ2, σ3} are the Pauli matricesEq. (10.68)). We write the Lagrangian function on R

40 as

L = 1

2mTr

(d

dt

(rsσ3s−1

))2

− k(Tr σ3s−1s

)2.

This expression for the Lagrangian shows clearly the invariance under the leftaction of SU (2) on itself and an additional invariance under the right U (1)-actions �→ seiσ3θ for θ ∈ [0, 2π). It is convenient to introduce left invariant 1-forms θa

by means of iσaθa = s−1ds and related left invariant vector fields Xa which are

dual to them θa(Xb) = δab . If � denotes any second-order vector field on R

40 we set

θa = θa(�), where, with some abuse of notation, we are using the same symbolfor θa on R

40 and its pull-back to T R

40. It is also convenient to use the unit vector n

defined by x = nr , i.e., n · σ = sσ3s−1.After some computations, the Lagrangian becomes

L = 1

2mr2 + 1

4mr2(θ21 + θ22 ) + kθ23 . (7.14)

It is not difficult to find the canonical 1- and 2-forms for the Lagrangian symplecticstructure. For instance θL = mr dr+ 1

2mr2(θ1 dθ1+θ2 dθ2)+2kθ3 dθ3; and of courseωL = −dθL . The energy function EL coincides with L .

If we fix the submanifold c by setting

c = {(r, v) ∈ T R

40 | θ3 = c

},

the submanifold turns out to be invariant because θ3 is a constant of the motion.On c, θL = mr dr + 1

2mr2(θ1 dθ1 + θ2 dθ2) + 2kc dθ3. If we then use thefoliation associated with XT

3 (the tangent lift of X3 to T R40), we find that ωL is the

pull-back of a 2-form on the quotient manifold because dθ3 = θ1 ∧ θ2, and hence

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contains X T3 in its kernel. The term dθ3 is exactly proportional to the magnetic field

of the magnetic monopole sitting at the origin. Thus on the quotient space of c bythe action of the left flow of X T

3 we recover the dynamics of the electron-monopolesystem on the (quotient) space T (S2 × R+) = T R

30. It is not difficult to show that

d

dt

(− i

2

[n · σ , mr2n · σ

]+ kn · σ

)= 0; k = eg

4π.

These constants of motion are associated with the rotational invariance and replacethe usual angular momentum functions.

This example shows that the reduction of theLagrangian system of Kaluza-Kleintype on T R

4 does not reduce to a Lagrangian system on T R3 but just to a symplectic

system.

7.4 The Algebraic Description

We can recapitulate the examples of reduction before and identify the two basicingredients present on all of them.

1. A � invariant subalgebra (of functions)R.2. A � invariant submanifold of the carrier space ⊂ M .

The evaluation map ev : M × F → R defined as (m, f ) �→ f (m), allows us todualize the basic ingredients from the manifold to the associative and commutativealgebra of functions on M , F , the set of observables.

We first notice that to any submanifold i : ↪→ M we can associate a shortexact sequence of associative algebras

0 �� I�� F

i∗ �� F�� 0 ,

where F is the associative and commutative algebra of functions in , defined interms of the identification map i : ↪→ M , m ∈ �→ m ∈ M . Here I isgiven by:

I = {f ∈ F | i∗( f ) = 0

}

Since i∗ is an algebra epimorphism, we find that I = ker i∗ is a bilateral ideal

in F . The algebra F is then the quotient algebra F/I .Any derivation � acting on the set of functions of F will define a derivation on

the set of functions F if and only if L�I ⊂ I , so that we can define an actionof � on the set of equivalence classes by LX ( f + I) = LX f + I , which definesa derivation on the reduced carrier space F .

A simple example illustrates the procedure. On T R3 we can consider the sub-

manifold defined by the vanishing of the funtions

7.4 The Algebraic Description 505

f1 = r · r − 1 f2 = r · v. (7.15)

Then I is the corresponding bilateral ideal.We get in this way the submanifold as T S2. The algebra of functions F is

obtained by pull-back fromF(T R3), i.e., simply by using in the argument of f (r, v)

the constraints (7.15), i.e., f1 = 0 = f2. A vector field X on T R3 will be tangent to

= T S2 if and only if

LX (r · r − 1) = α(r · r − 1) + βr · v,

for arbitrary functions α, β and also

LX (r · v) = α′(r · r − 1) + β ′r · v.

for arbitrary functions α′, β ′.It is not difficult to show that the module of such derivations is generated by

Rl =∑j,k

εl jk

(x j

∂

∂xk+ v j

∂

∂vk

); Vl =

∑i, j

εli j xi∂

∂v j

A �-invariant under � subalgebra in F , say F , for which I is an ideal, definesa �-invariant under � equivalence relation in M by setting

m ′ ∼ m′′ iff f (m ′) = f (m ′′), ∀ f ∈ F (7.16)

It follows that F defines a subalgebra inF and corresponds to a possible quotientmanifold of by the equivalence relation defined by F .

In general, a subalgebra of F , say FQ , defines a short exact sequence of Liealgebras

0 −→ Xv −→ XN −→ XQ −→ 0 (7.17)

whereXv is the Lie algebra of vector fields annihilatingFQ ,XN is the normalizer ofXv in X(M), and XQ is the quotient Lie algebra. This sequence of Lie algebras maybe considered a sequence of Lie modules with coefficients in FQ . In the previouscase,FQ would be the invariant subalgebra inF and the equivalence relation wouldbe defined by the leaves of the involutive distribution Xv (regularity requirementsshould be then imposed on FQ). See [La90] for details.

From the dual point of view it is now clear that reducible evolutions will bedefined by one-parameter groups of transformations which are automorphisms ofthe corresponding short exact sequences. The corresponding infinitesimal versionswill be defined in terms of derivations of the appropriate short exact sequence ofalgebras.


To illustrate this aspect, we consider the associative subalgebra of F(T R3) gen-

erated by {r · r, v · v, r · v}. For this algebra it is not difficult to see that the vectorfields

Xc =∑a,b

εabc

(xa ∂

∂xb+ va ∂

∂vb

)

generate Xv , while XN is generated by Xv and

r · ∇v, v · ∇, r · ∇, v · ∇v.

The quotient XQ , with a slight abuse of notation, can also be considered to begenerated by the vector fields

r · ∇v, v · ∇, r · ∇, v · ∇v,

which however are not all independent. Any combination of them with coefficientsin the subalgebra may be considered a ‘reduced dynamics’.

7.4.1 Additional Structures: Poisson Reduction

When a Poisson structure on the manifold M is available, we can further qualify theprevious picture. We can consider associated short exact sequences of Hamiltonianderivations, i.e., derivations given by Hamiltonian vector fields. Hence, a Poissonreduction can be formulated in the following way: we start with I , again an ideal inthe commutative and associative algebraF . We consider then the set of Hamiltonianderivations which map I into itself:

W (I) = { f ∈ F | { f, I} ⊂ I} .

Then we consider I ′ = I ∩ W (I) and get the exact sequence of Poisson algebras

0 −→ I ′ −→ W (I) −→ Q −→ 0 .

When the ideal I is given by constraint functions as in the Dirac approach,W (I) is the set of first-class functions while I ′

is that of the first-class con-straints.

We give here an example of an iterated reduction. We consider a parametrizationof T R

4 in terms of the identity matrix in dimension 2 σ0, and the 2×2 Pauli matricesEq. (10.68) as follows:

http://dx.doi.org/10.1007/978-94-017-9220-2_10


π = p0σ0 +3∑

a=1

paσa =(

p0 + p3 p1 − i p2p1 + i p2 p0 − p3

),

g = y0σ0 +3∑

a=1

yaσa =(

y0 + y3 y1 − i y2y1 + i y2 y0 − y3

).

A preliminary ‘constraint’ manifold is selected by requiring that

Tr (g†g) = 1, Tr (π†g) = 0.

Recall that (a ·σ )(b ·σ ) = b · a σ0 + (a × b) ·σ from where we easily obtain that

Tr g†g = y20 + y · y, Tr π†g = y0 p0 + y · p.

Therefore the sub manifold defined by the constraints is diffeomorphic to thetangent bundle of S3, i.e., T S3. The Hamiltonian

H = 1

2

(pμ pμ + yμyμ

)

defines a vector field tangent to the constraint manifold.Similarly for the ‘potential’ function V = 1

2 (y20 + y23 − y21 − y22). The Hamiltonianfunction H + V , when restricted to T S3 with a slight abuse of notation acquires thesuggestive form

H = 1

2

(p20 + p23 + y20 + y23

)+ 1

2

(p21 + p22 − y21 − y22

).

By using the relation y20 + y21 + y22 + y23 = 1 we may also write it in the form

H = 1

2

(p20 + p2

3 + 2(

y20 + y23

))+ 1

2

(p21 + p22

)− 1

2.

Starting now with T S3 we may consider a further reduction by fixing

K = {(yμ, pμ) ∈ T S3 | y0 p3 − p0y3 + y1 p2 − y2 p1 = K

},

and quotienting by the vector field

X = y0∂

∂ y3− y3

∂

∂ y0+ y1

∂

∂ y2− y2

∂

∂ y1+ p0

∂

∂ p3− p3

∂

∂ p0+ p1

∂

∂ p2− p2

∂

∂ p1,

which is tangent to the foliation because LX (y0 p3 − p0y3 + y1 p2 − y2 p1) = 0.The final reduced manifold will be T S2 ⊂ T R

3, with projection T S3 → T S2

provided by the tangent of the Hopf fibration π : S3 → S2, defined as


x1 = 2(y1y3 − y0y2) x2 = 2(y2y3 − y0y1) x3 = y20 + y23 − y21 − y22 .

The final reduced dynamics will be associated with the Hamiltonian function ofthe spherical pendulum.

The spherical pendulum is thus identified by

S2 ⊂ R3 = {

x ∈ R3 | 〈x, x〉 = x2

1 + x22 + x23 = 1}

T S2 ⊂ T R3 = {

(x, v) ∈ R3 × R

3 | 〈x, x〉 = 1, 〈x, v〉 = 0}

The dynamics is given by means of ω = ∑i dxi ∧ dvi when restricted to T S2,

in terms of E = 12 〈v, v〉 + x3. The angular momentum is a constant of motion

corresponding to the rotation around the Ox3 axis. The energy momentum map

μ : T S2 → R2 : (x, v) �→ (E(x, v), L(x, v))

has quite interesting properties as shown by [Cu97, Du80].

7.4.2 Reparametrization of Linear Systems

7.4.2.1 Reparametrization by Constants of Motion

Consider now, on a linear vector space E , dim E = n, with coordinates(x1, . . . , xn

),

a general, homogeneous linear dynamical system of the form

dx j

dt= A j

i xi , j = 1, . . . , n. (7.18)

With the matrix A ∈ End (E) we can associate, as already discussed, the (linear)vector field

�A = A ji xi ∂

∂x j. (7.19)

We can reparametrize the linear system (7.18) by using a constant of motion, andstill we will obtain an explicitly integrable nonlinear system. Let h be a constant ofmotion for the dynamical system (7.18), dh/dt = 0, or equivalently

A ji xi ∂h

∂x j= 0. (7.20)

Then if we assume h to be non zero in an open dense set of E the dynamicalsystem with the reparametrized evolution equations


dx j

dt= h (x) A j

i xi = C(x) ji xi (7.21)

has the same constants of motion. In fact a simple computation shows that now

d F

dt= h(x)A j

i xi ∂ F

∂x j, (7.22)

and hence, if h �= 0 in an open dense set of E , this implies that F is a constant ofthe reparametrized motion iff A j

i xi ∂ F/∂x j = 0 in an open dense set of E and bycontinuity, it vanishes on all E and therefore F is a constant of motion determined by(6.91). Let us recall that the constants of motion are an associative and commutativereal algebra, because if f and g are constants and λ,μ ∈ R, then

d

dt(λ f + μg) = λ

d f

dt+ μ

dg

dt= 0 (7.23)

and

d

dt( f g) = f

dg

dt+ g

d f

dt= 0. (7.24)

Thus we conclude that if h is a constant of motion for the system (7.18) and h �= 0in an open dense set of E , then the system (7.22) has the same algebra of constantsof motion as the original one.

This implies that if we denote by φt the flow of (7.22) and by γ (t) the solutionwith initial data x0, then

h(γ (t)) = h(φt (x0)) = h(x0), (7.25)

for all t ∈ R. Hence, the integral curve γ (t) of (7.22) starting at x0 will be(A = [

A ji])

γ (t) = exp (th (x0) A) x0. (7.26)

In fact, if γ (t) is given by (7.26),

d

dtγ (t) = d

dtexp(th(x0)A)x0 = h(x0)Aγ (t) = h(γ (t))Aγ (t). (7.27)

Denoting by � the dynamical system (7.18) and by �h = h� the system (7.22),if � has a maximal set of constants of motion the flow will be found by quadratures(see next chapter). If � is nonlinear, what will be interesting for us will be to findout whether there exists a constant of the motion h such that h� is linear (noticethat it can be linear with respect to the given linear structure or with respect to an

http://dx.doi.org/10.1007/978-94-017-9220-2_6


alternative linear structure, see the discussion of this point in one of the followingchapters).

The reparametrization procedure discussed above can be extended to a moregeneral setting that allows us to construct nonlinear systems whose solutions canbe found explicitly in terms of solutions of linear systems. Let � be the dynamicalsystem defined by the nonlinear equations

dx j

dt= C(x) j

i xi (7.28)

and such that the entries of the matrix field C(x) = [C ji (x)] are constants of motion

for �, i.e.:dC ji/dt = 0, for all i, j = 1, . . . , n. In other words, the functions C j

i

verify the partial differential equations

∂C ji

∂xkCk

l xl = 0, (7.29)

for all i, j . In such case, if γ (t) is the integral curve for � starting at x0, it is clearthat γ (t) is given by

(C = [

C ji])

γ (t) = exp (tC (x0)) x0. (7.30)

In fact, a simple computation shows that

d

dtγ (t) = C(x0)γ (t) = C(γ (t))γ (t), (7.31)

because C is a matrix of constants of motion for �.Thus, for each set of Cauchy data x0, C(x0) will be a numerical matrix and the

flow of our system, when restricted to the surface:

= {x ∈ E | C (x) = C (x0)} (7.32)

will be given by

x (t) = exp (tC (x0)) x0 (7.33)

Notice that the curve γ (t) above is contained in . It is clear that for given valuesof the constants of motion C j

i the surface can be empty.This situation is illustrated by the following example.

Example 7.3 On R4 with coordinates (x1, x2, v1, v2), consider the linear system �

corresponding to two uncoupled harmonic oscillators :


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dx1dt

= v1

dx2dt

= v2

dv1

dt= −x1

dv2

dt= −x2

(7.34)

We notice that as we have two copies of the same second-order differential equation,x + x = 0, the Wronskian of two solutions is a constant of the motion,

d

dt(x1v2 − v1x2) = 0, (7.35)

and hence x1v2−v1x2 is a constant of motion. According to the previously describedprocedure, we can construct the nonlinear system

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dx1dt

= v1

dx2dt

= v2

dv1

dt= −(x1v2 − v1x2)x1

dv2

dt= −(x1v2 − v1x2)x2

(7.36)

and for any initial conditions (x1 (0) , x2 (0) , v1 (0) , v2 (0)) our system will behavelike a linear one, because it corresponds to a time reparametrization of the latterby a constant of motion. However there is a difference with the reparametrizationdiscussed previously. In this case the vector field � is not modified by a product witha constant of motion but only its vertical part is. In other words, as pointed out before,we are multiplying not by a function but by a matrix.

7.4.2.2 Reparametrization by Arbitrary Functions

With reference to the previous section, it turns out that it is not strictly necessary that hbe a constant of motion for the dynamical system � to be used as a reparametrizationfunction. Consider again a linear equation of motion on the vector space E given by�, and an arbitrary function h, and the reparametrized equation �h = h �. We knowthat the integral curve of � starting from the point x0 is γ (t) = exp(t A) x0. We areinterested in finding a new parametrization given by t = ϕ(s) such that the curve

γh(s) = γ (ϕ(s)) (7.37)

be an integral curve of the vector field �h , i.e., that


d

dtγ (t) = �(γ (t)) =⇒ d

dsγh(s) = �h(γh(s)) .

Taking into account that

d

dsγh(s) = dϕ(s)

ds

(dγ (t)

dt

)t=ϕ(s)

= dϕ(s)

dsA γ (ϕ(s)) ,

we see that in order for γh be an integral curve of �h , the function ϕ should be suchthat

dϕ(s)

ds= h(γh(s)) ,

or when written in a different way,

dt

ds= h(γ (t)) ,

whose solutions are obtained by integrating along γ the function h, namely,

s − s0 =t∫

t0

dt ′

h(exp(t ′ A)x0), (7.38)

and by performing the integral we have

s − s0 = s(t, x0).

Notice that

ds

dt= 1

h(exp((t − t0)A)x0)�= 0, (7.39)

and thus we can solve this equation with respect to t , say t = t (s, x0). Then, we canwrite the integral curve of �h = h� through x0 as

γh(s) = γ (t (s, x0)) . (7.40)

In the particular case in which h is a constant of motion we have h((exp t A)x0) =h(x0) and therefore Eq. (7.39) yields

s − s0 = t − t0h(x0)

. (7.41)

It follows then that t − t0 = h(x0)(s − s0) and γ (t) = (exp(t − t0)A)x0 becomesx(s) = exp(h(x0)(s − s0)A)x0, according to what we have found in Eq. (7.26).


7.4.2.3 A Cubic Dynamical System

We will discuss here as an application of the generalized reparametrization methoddescribed above the explicit integrability of a given cubic vector field.

Let us consider R3 with the diagonal metric or pseudo-metric η with signature

(±,+,+). We will denote the product of two vectors q = (q0, q1, q2) and q′ =(q ′

0, q ′1, q ′

2) as

〈q, q′〉 = ηαβqα q ′β = ±q0q ′0 + q1q ′

1 + q2q ′2. (7.42)

Let us consider the Hamiltonian system (with phase space R6, then) associated with

the quartic Hamiltonian

H = 1

2

(〈p, p〉〈q, q〉 − 〈p, q〉2

). (7.43)

If the metric is Euclidean this can be written also as

H = 1

2(〈p × q, p × q〉) (7.44)

with ‘×’ denoting the usual exterior product (vector product) of vectors in R3. The

dynamical vector field is given by Hamilton’s equations of motion

q = ∂ H

∂p,

·p = −∂ H

∂q(7.45)

We can compute the previous equations easily by using the standard Poisson bracketson phase space, and we get

p = {p, H} = 〈q, p〉p − 〈p, p〉 q·q = {q, H} = 〈q, q〉 p − 〈q, p〉q (7.46)

Then,

q = {{q, H} , H} = −2H q (7.47)

We also find the following constants of motion:

{H, 〈q, p〉} = 0, {H, 〈p, p〉} = 0, {H, 〈q, q〉} = 0 .

Hence, the equations of motion can be written in the matrix form

d

dt

[qp

]=

[−〈q, p〉〈q, q〉−〈p, p〉〈q, q〉

] [qp

]= C(q, p)

[qp

](7.48)


with C a matrix of constants of motion. Then this system can be integrated byexponentiation using Eq. (7.30).

7.4.3 Regularization and Linearization of the Kepler Problem

We shall discuss here the Kepler problem taking the point of view of the ideasdeveloped in the previous sections, i.e., it will be shown that the 1-dimensionalKepler problem is related to a linear system and how this relation allows for animmediate and explicit integration of it. We discuss first the one dimensional Keplerproblem and we leave the discussion of the 2 and 3-dimensional Kepler problems toone of the following sections.

Using natural coordinates (x, v) in R+ × R, where R

+ is the set of positivereal numbers, we consider the dynamical system described by the second-orderdifferential equation:

� = v∂

∂x+ f (x, v)

∂

∂v(7.49)

where in our case f is Kepler’s force

f (x, v) = − k

x2, k > 0 ,

derivable from the potential function

V = V (x) = − k

x,

i.e., our dynamical vector field is

� = v∂

∂x− k

x2∂

∂v, (7.50)

which is well defined because we have removed the point 0 in the domain of �. Wedo not discuss at this moment the choice of the name for this system. We only pointout here that our system is not the 1-D reduction of the 3-D Kepler problem by usingconstants of motion and the natural equivalence relation defined by its symmetrygroup. The equations of motion for the 1-D Kepler problem are thus,

{x = v

v = − k

x2, (7.51)

which shows that they are singular at x = 0 (that is why we removed this point in thedefinition of the domain of �). These equations can be derived from the Lagrangian


L = 1

2v2 + k

x,

which gives raise to the energy

E(x, v) = 1

2v2 − k

x, (7.52)

that is a constant of motion.The usual Legendre transformation

p = ∂L

∂v= v

leads to the Hamiltonian function:

H(x, p) = 1

2p2 − k

x(7.53)

The symplectic structures to be used are either ωL = dx ∧ dv on the velocityphase space or ω0 = dx ∧ dp on the phase space, the configuration space being R

+.It is also obvious that the trajectories with energy E ≤ 0 drop toward x = 0

because of the attractive character of the force f and then, they will explode (thevelocity diverges) at x = 0 (see Fig. 7.2). In fact v ∼= x−1/2 as x → 0+.

Using the energy relation Eq. (7.52) it is immediate to integrate the equations ofmotion to get

t − t0 =x∫

x0

dx√2 E + (2 k/x)

.

Fig. 7.2 Left Trajectories of the 1d Kepler problem. Right the trajectories of the regularized flow


We have displayed the energy levels of E on Fig. 7.2 showing the existence ofthree types of trajectories. Those contained in the region E < 0 exit and returnto x = 0 with infinite velocity after reaching the maximum xmax = −k/E . Thetrajectories contained in the region E > 0 exit from x = 0 with infinite velocityand they escape to infinity with a residual escape velocity v∞ = √

2 E . Finally, thetrajectories separating these two regions correspond to E = 0 and they describe thesystem escaping to infinity with v∞ = 0.

All this suggests a natural way to obtain a better description of this system bymultiplying Eq. (7.52) by x . Then we obtain for a fixed E ,

k = 1

2x v2 − x E ,

which shows that in the level set E of the energy, if we redefine the velocity asw = √

x v, i.e., we define a new time function τ such that

dt

dτ= √

x ,

we obtain a new system whose equations of motion are

⎧⎨⎩

x ′ = w

w′ = 12w2

x− k

x

where x ′ = dx/dτ and w′ = dw/dτ .From the geometric viewpoint this reparametrization corresponds to first replacing

the vector field � given by (7.50) by a new vector field√

x �, i.e., multiplying theoriginal vector field � by the function h(x, v) = √

x , and second redefining thetangent structure in such a way that the new vector field is again a SODE withrespect to the new tangent structure.

In any case we see that unfortunately we have not removed the singular characterof the vector field �. An interesting problem is the study of the possibility of usingas a reparametrization function h(x, v) = xα , i.e., w = xα v in such a way thatthe new vector field free of singularities. There exists an appropriate choice forthe reparametrization of � determined by α = 1. This choice is frequently calledSundman regularization. The regularization we are studying is given multiplyingfirst the vector field by a function h(x, v) = xα and considering later the image ofsuch a vector field under a map φ : R

+ × R → R+ × R defined by

φ(x, v) = (x, xα v) ,

necessary for the image vector field to be a SODE.Wemust remark that such φ is notthe derivative of anymap ϕ : R

+ → R+ that allows us to restore the SODE character

of the field. We can compute the vector field � = �∗(xα �) obtained from the


reparametrized vector field X = xα � via this transformation by adirect computation,but we better consider the problem in a perspective that can easily be generalizedand will be shown to be useful in the future. We are looking for a vector field

� = w∂

∂x+ f (x, w)

∂

∂w

that is the image of X under the transformation φ. Notice that even if X is not aSODE anymore, � is again a SODE, or in other words, X can be seen as a SODEbut with respect to a different tangent space structure (see Sect. 5.6.3).

We recall that X and � are said to be φ-related if and only if

X (φ∗g) = φ∗(�g)

for any function g(x, w), and in particular, for g(x, w) = x and g(x, w) = w:

X (φ∗(x)) = φ∗(�x) = φ∗(w) ,

X (φ∗(w)) = φ∗(�w) = φ∗( f ) .

We find that

φ∗( f ) = αw2

x+ x2α f (x, w/xα),

therefore,

� = w∂

∂x+

(α

w2

x+ x2α f (x, w/xα)

)∂

∂w

whose associated system of differential equations is

⎧⎪⎨⎪⎩

dx

ds= w

dw

ds= α

w2


,

It is clear that w = xα v behaves like xα−1/2 when x → 0+ and therefore ifwe want v → 0+ we should choose α > 1/2. As we will see later on, the mostappropriate choice is α = 1. In such a case,

f (x, w) = w2

x+ x2 f (x, w/x) ,

and

� = w∂

∂x+

(w2

x+ x2 f (x, w/x)

)∂

∂w

http://dx.doi.org/10.1007/978-94-017-9220-2_5


and in our particular case, for which f (x, v) = −k/x2, the associated system ofdifferential equations is

⎧⎪⎨⎪⎩

dx

ds= w

dw

ds= w2

x− k

In spite of the appearance of the factor 1/x this system does not explode and it iseasily seen that w → 0 as x → 0+. See Fig. 7.2 for the trajectories of �.

It seems that we have found a better way of handling the 1-D Kepler problem,but, as we will see in a moment, we can do still better. We will show that, actually,for α = 1 this system is related to a linear one.

To show this we add an extra dimension, to be denoted h, i.e., we consider R+ ×

R × R with coordinates (x, w, h) and the vector field obtained extending trivially �

and that is denoted with the same symbol. The new equations of motion on M are,

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

dx

ds= w

dw

ds= α

w2


dh

ds= 0

which for α = 1 become:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

dx

ds= w

dw

ds= w2

x+ x2 f

dh

ds= 0

,

and we are interested in the case f = −k/x2. Now we select the invariant level setof constant energy given by

α ={(x, w, h) ∈ M | h = −V − w2

2x2α

}, f = −dV/dx ,

in the general case, or simply,

α ={(x, w, h) ∈ M; h = k

x− w2

2x2

},

in the Kepler case we are dealing with. On this surface our vector field � has theexpression, using (x, w) as coordinates on M ,


� = w∂

∂x+

(2α x2 α−1(−h − V ) + x2 α f (x, w/xα)

) ∂

∂w

and in the Kepler case:

� = w∂

∂x+ 2 α x2 α−1

(k/x − h − kx2α−2

) ∂

∂w

which shows that we obtain a linear system when α = 1, and in this case it becomes

� = w∂

∂x+ (k − 2 h x)

∂

∂w

Therefore we have been able to associate a linear non-homogeneous vector field withthe nonlinear equation describing the Kepler system in 1D. The equations of motion

⎧⎪⎨⎪⎩

dx

ds= w

dw

ds= k − 2hx

can be easily solved and we obtain for initial conditions x(0) = 0, w(0) = 0,

x(s) = k

2h

(1 − cos

√2hs

).

Using these solutions we can also find the parameter for the initial vector field

t =∫

x(s)ds = k

2h

(s − 1√

2hsin

√2hs

).

It is nowpossible to perform a further transformation suggested by the particular formof the solutions we have found. We consider now the transformation ψ : R

+ → R+

such that x = ψ(y) = y2, and we will extend it toR+ ×R → R

+ ×R asψ∗(y, u) =(y2, 2 y u), i.e., x = y2 and w = 2 y u. Hence we find the new vector field

Z = u∂

∂y− h

2y

∂

∂u

Therefore, for positive values of h, this vector field represents the harmonic oscil-lator with frequency w = √

h/2. Thus, after this long periple we have returned tothe dynamical system used to motivate this chapter, the 1-D harmonic oscillator,Equation.

Summarizing the previous discussion we conclude that we have reduced the inte-gration of the Kepler problem to the integration of an associated linear differentialequation. We should notice that the energy relation k/x − v2/2 = h in (x, v) coor-dinates or k/x − w2/2x2 = h in (x, w) coordinates, becomes


u2 − 1

2(k − hy2) = 0 ,

in (y, u) coordinates. Therefore, we could have started with a linear vector fieldon R

3 and after performing the various operations of reparametrization, nonlinearchanges of coordinates and restriction to surfaces described above we would obtainthe 1D Kepler problem.

7.5 Reduction in Quantum Mechanics

Having defined Poisson reduction we are now on good track to define a reductionprocedure for quantum systems. After all, according to deformation quantization thePoisson bracket provides us with a first order approximation to QuantumMechanics.We will use in the remaining the various formalisms to describe quantum dynamicalevolution presented in Chap.6, Sects. 6.6.2, 6.4. Simply recall here that the descrip-tion of quantum systems is done basically by means of either the Hilbert space ofstates, where we define dynamics bymeans of the Schrödinger equation, or bymeansof the algebra of observables, where dynamics is defined by means of the Heisen-berg equation. We may also consider other pictures like the Ehrenfest picture, thephase-space or Wigner picture, the tomographic picture, etc.

7.5.1 The Reduction of Free Motion in the Quantum Case

The description of the free quantum evolution is rather simple because the semi-classical treatment is actually exact [Es04]. In what follows we set � = 1 for sim-plicity.

The Hamiltonian operator for free motion in two dimensions, given by:

H = −1

2�, � = ∂2

∂x2+ ∂2

∂y2, (7.54)

written in polar coordinates (Q, φ) (x = Q sin φ, y = Q cosφ) it becomes:

H = −1

2

1

Q

∂

∂ QQ

∂

∂ Q− 1

Q2

∂2

∂φ2 .

By a similarity transformation H ′ = Q12 H Q− 1

2 (notice that Q > 0) we get ridof the linear term and we obtain:

H ′ = −1

2

[∂2

∂ Q2 + 1

Q2

(1

4+ ∂2

∂φ2

)].

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7.5 Reduction in Quantum Mechanics 521

Restricting H ′ to the subspace Sm ⊂ L2(R2) of square integrable functions ofthe form

Sm = {ψ = eimφ f (Q)},

we find that on this particular subspace

H ′ψ = −1

2

[∂2

∂ Q2 − 1

Q2

(m2 − 1

4

)]ψ .

This determines a Hamiltonian operator along the radial coordinate and setting g2 =m2 − 1

4 we have:

H = −1

2

∂2

∂ Q2 + 1

2

g2

Q2 .

Suppose that we consider now the space of Hermitean matrices X (of real dimen-sion n2) and the quantum free motion on it. Solutions of the free problem (7.54) aregiven by wave-packets formed out of ‘plane-waves’:

ψP (X) = AeiTr X P ,

where A is a normalization constant, chosen in such a way as to give a delta functionnormalization.

By decomposing X into a ‘radial’ part Q and an ‘angular’ part G, say X =G−1QG, like in the reduction of the classical free motion on matrices to get theCalogero-Moser system, Sect. 1.2.12, Eq. (1.131), we can write the wave function inthe form:

ψ(Q, G) = AeiTr (G−1QG P) = ψP (X).

In this particular case it is not difficult to show that I j (X, P) = Tr (P j ) are con-stants of motion in involution for the classical system and give raise to the operators

(−i) jTr(

∂∂ X

) j.

To perform specific computations let us go back to the two-dimensional situation.We consider ψP(X) = AeiTr P X and project it along the eigenspace Sm of the

angular momentum corresponding to the fixed value m.We recall that (in connection with the unitary representations of the Euclidean

group)

2π∫0

dφ eimφ ei P Q cosφ = 2π Jm(P Q),

where Jm is the Bessel function of order m. Thus we conclude

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ψP (Q) = 2π√

P Q Jm(P Q).

In the particular case we are considering free motion is described by a quadraticHamiltonian in R

2. Therefore the Green function becomes

G(Xt − X0, 0; t) = C

2tei Tr (Xt −X0)2

t .

The Green function can be written in terms of the action, that is, the solution of theHamilton-Jacobi equation and the Van Vleck determinant ([Es04], Appendix 4.B).

By using polar coordinates the kernel of the propagator is

G(Qt , Q; t) = √Qt Q0

2π∫0

dφeimφ K (Xt , X0; t)

= 1

2π i t

√Qt Q0 e

i(Q2

t +Q0)2

2t

2π∫0

dφ eimφe−i Qt Q0 cosφ

t

= 1

2π i t

√Qt Q0e

i(Q2

t +Q0)2

2t Jm

(Qt Q0

t

),

where the angle is coming from the scalar product of Xt with X0.

7.5.2 Reduction in Terms of Differential Operators

This simple example has shown that the reduction procedure in wave mechanicsinvolves differential operators (see Chap.10) and their eigenspaces. Let us thereforeconsider some general aspects of reduction procedures for differential operators.

In general, the Hamiltonian operator defining the Schrödinger equation onL2(D, dμ) is a differential operator, which may exhibit a complicated dependencein the potential. It makes sense thus to study a general framework for the reduction ofdifferential operators acting on some domain D, when we assume that the reductionprocedure consists in the suitable choice of some ‘quotient’ domain D′.

Let us recall that if we considerF = C∞(Rn), the algebra of smooth functions onR

n . A differential operator of degree at most r is defined as a linear map P : F → Fof the form

P =∑|σ |≤r

gσ

∂ |σ |

∂xσ

, gσ ∈ F (7.55)

where σ = (i1, · · · in), |σ | = ∑k ik and

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∂ |σ |

∂xσ

= ∂ |σ |

∂xi11 · · · ∂xin

n

This particular way of expressing differential operators relies on the generator of‘translations’, ∂/∂xk . Therefore, when the reduced space does not carry an action ofthe translation group this way of writing differential operators is not very convenient.There is an intrinsic way to define differential operators which does not dependon coordinates (see Chap.10, Appendix G, for more details). One starts from thefollowing observation

[∂

∂x j, m f

]= m ∂ f

∂x j

,

where m f is the multiplication by the function f operation, i.e., m f (g) = f g, withf, g ∈ F . It follows that

[P, m f ] =∑|σ |≤r

gσ

[∂ |σ |

∂xσ

, m f

],

is of degree at most r − 1. Iterating for a set of r + 1 functions f0, . . . , fk ∈ F , onefinds that

[. . . , [[P, f0], f1], . . . , fr ] = 0;

This algebraic characterization allows for a definition of differential operators on anymanifold.

The algebra of differential operators of degree 1 is a Lie subalgebra with respectto the commutator [·, ·] and splits into a direct sum

D1 = F ⊕ D1c

where D1c are derivations, i.e., differential operators of degree one which give zero

on constants. We can endow the set with a Lie algebra structure by setting

[( f1, X1), ( f2, X2)] = (X1 f2 − X2 f1, [X1, X2])

If we considerF as an Abelian Lie algebra, D1c is the algebra of its derivations and

then D1 becomes what is known in the literature as the ‘holomorph’ of F [Ca94].In this way the algebra of differential operators becomes the enveloping algebra ofthe holomorph of F .

The set of differential operators on M , denoted asD(M), can be given the structureof a graded associative algebra and it is also amodule overF . Notice that this propertywould not make sense at the level of abstract operator algebra. To consider theproblem of reduction of differential operators we consider the problem of reduction

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of first-order differential operators. Because the zeroth order ones are just functions,we restrict our attention to vector fields, i.e., the set D1

c .Given a projection π : M → N between smooth manifolds, we say that a vector

field X M projects onto a vector field X N if

LX M π∗ f = π∗(LX N f ) ∀ f ∈ F(N ).

We say thus that X M and X N are π -related.Thus if we consider the subalgebra π∗(F(N )) ⊂ F(M), a vector field is pro-

jectable if it defines a derivation of the subalgebra π∗(F(N )). More generally, for adifferential operator Dk , we shall say that it is projectable if

Dkπ∗(F(N )) ⊂ π∗(F(N )).

It follows that projectable differential operators of degree zero are elements inπ∗(F(N )). Therefore projectable differential operators are given by the envelopingalgebra of the holomorph of π∗(F(N )), when the corresponding derivations areconsidered as belonging to X(M).

Remark 7.5 Given a subalgebra of differential operators inD(M) it is not said that itis the enveloping algebra of the first-order differential operators it contains.When thishappens, we cannot associate a corresponding quotient manifold with an identifiedsubalgebra of differential operators. An example of this situation arises with angularmomentum operators when we consider the ‘eigenvalue problem’ in terms of Jz andJ 2. It is clear that this commuting subalgebra of differential operators cannot begenerated by its ‘first-order content’.

In the quantization procedure, this situation gives raise to anomalies [Al98].

7.5.3 The Kustaanheimo–Stiefel Fibration

In this section we would like to consider the reduction of differential operatorsassociated with the KS projection πK S : R

40 → R

30, where R

j0 = R

j − {0}, forj = 3, 4, and show that the Hydrogen atom operator may be obtained as a reductionof the operators associated with a family of harmonic oscillators.

Let us recall first how this map is defined. We first notice that R40 = S3 × R

+ andidentifying S3 with SU (2),R4

0 ∼ SU (2)×R+, whileR

30 = S2×R

+. By introducingpolar coordinates

g = Rs s ∈ SU (2), R ∈ R+,

if we take into account that sσ3s−1 is a traceless hermitian matrix, we can defineπK S : R

40 → R

30 as


πK S : g �→ gσ3g+ = R2sσ3s−1 = xkσk,

where {σk | k = 1, 2, 3} are Pauli matrices Eq. (10.68). In a Cartesian coordinatesystem one has

x1 = 2(y1y3 + y2y0) x2 = 2(y2y3 − y1y0) x3 = y21 + y22 − y33 − y20 ,

where g = ∑3i=1yiσ

i . Moreover,√

x j x j = r = R2 = yk yk .The KS projection defines a principal fibration with structure group U (1).By the definition of πK S it is easy to see that acting with eiλσ3 on SU (2) does

not change the projected point on R30. The associated fundamental vector field X3 is

the left-invariant infinitesimal generator associated with σ3, i.e., iX3s−1ds = iσ3. Incoordinates it reads

X3 = y0∂

∂y3− y3

∂

∂y0+ y1

∂

∂y2− y2

∂

∂y1

We consider the Lie algebra of differential operators generated by X3 andπ∗

K S(F(R30)). Projectable differential operators with respect to πK S are given by

the normalizer of this algebra in the algebra of differential operators D(R40). As we

already remarked this means that this subalgebra must map π∗(F(R30)) into itself. If

we denote this subalgebra by Dπ we may also restrict our attention to the operatorsin Dπ commuting with X3.

In order to explicitly construct this algebra of differential operators we use thefact that SU (2) × R+ is a Lie group and therefore it is parallelizable. Because theKS map has been constructed with the left-invariant vector field X3, we consider thegenerators of the left-action of SU (2), say right-invariant vector fields Y1, Y2, Y3, anda central vector field along the radial coordinate, say R. All these vector fields areprojectable and therefore along with π∗

K S(F(R30) generate a projectable subalgebra

of differential operators which covers the algebra of differential operators on R30.

This map is surjective and we can ask to find the ‘inverse image’ of the operator

H = −�3

2− k

r,

which is the operator associated with the Schrödinger equation of the hydrogenatom (�3 denotes the Laplacian in the three dimensional space). As this operatoris invariant under the action of so(4) ∼ su(2) ⊕ su(2), associated with the angularmomentum and the Runge-Lenz vector, we may look for a representative in theinverse image which shares the same symmetries. As the pull-back of the potentialk/r creates no problems,wemay concentrate our attention on the Laplacian. Becauseof the invariance requirements, our candidate for the inverse imagewill have the form

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D = f (R)∂2

∂ R2 + g(R)∂

∂ R+ h(R)�s

3 + c(R),

where R is the radial coordinate in R40, and f, g, h are functions to be determined.

We recall that in polar coordinates the Laplacian �3 has the expression

�3 = ∂2

∂r2+ 2

r

∂

∂r+ 1

r2�s

2,

where we denote by �sn the Laplacian on the n-dimensional sphere.

By imposing Dπ∗K Sφ = π∗

K S(H3φ) for any φ ∈ F(R30) we find that the repre-

sentative in the inverse image has the expression

H ′ = −1

2

1

4R2�4 − k

R2 .

This operator is usually referred to as the conformal Kepler Hamiltonian [Av05].Now, with this operator we may try to solve the eigenvalue problem

(−1

2

1

4R2�4 − k

R2

)ψ − Eψ = 0

It defines a subspace in F(R40) which coincides with the one determined by the

equation(

−1

2�4 − 4E R2 − 4k

)ψ = 0.

This implies that the subspace is given by the eigenfunctions of the Harmonicoscillator with frequencyω(E) = √−8E . We notice then that a family of oscillatorsis required to solve the eigenvalue problem associated with the Hydrogen atom. Tofind the final wave functions on R

3 we must require that LX3ψ = 0 in order to findeigenfunctions for the three dimensional problem. Eventually we find the correctrelations for the corresponding eigenvalues

E, = − k2

2(m + 1)2, m ∈ N.

Of course, dealing with Quantum Mechanics we should ensure that the operator

H ′ = −1

2

1

4R2�4 − k

R2

is essentially self-adjoint to be able to associate with it a unitary dynamics. One findsthat the Hilbert space should be constructed as a space of square integrable functionson R

40 with respect to the measure 4R2d4y instead of the Euclidean measure on R

4.


We shall not go into the details of this, but the problem of a different scalar product isstrictly related to the reparametrization of the classical vector field, required to turn itinto a complete one. This would be a good example for J. Klauder’s saying: ‘these areclassical symptoms of a quantum illness’ (see [Zu93]). Further details can be foundin [Av05, Av05b]. As for the reduction of the Laplacian in Quantum Mechanics seealso [Fe06, Fe07, Fe08].

7.5.4 Reduction in the Heisenberg Picture

The Heisenberg picture of QuantumMechanics relies on the non-commutative alge-bra of observables (see Sect. 6.6.2), therefore it is instructive to consider a reductionprocedure for non-commutative algebras.

The example of reduction procedure in a non-commutative setting that we aregoing to discuss reproduces the Poisson reduction in the ‘quantum-classical’ tran-sition and goes back to the celebrated example of the quantum SU (2) written byWoronowicz [Wo87] and is adapted from [Gr94].

We consider the space S3 ⊂ R4, identified with the group SU (2) represented

in terms of matrices. The �-algebra A generated by matrix elements is dense in thealgebra of continuous functions on SU (2) and can be characterized as the ‘maximal’unital commutative �-algebra A, generated by elements which we can denote asα, ν, α∗, ν∗ satisfying α∗α+ν∗ν = 1. This algebra can be generalized and deformedinto a non-commutative one by replacing some relations with the following ones:

αα∗ − α∗α = (2q − q2)ν∗ν ν∗ν − νν∗ = 0,

andνα − αν = qνα ν∗α − αν∗ = qν∗α.

This algebra reduces to the previous commutative one when q = 0. In this respectthis situation resembles the one on the phase-space where we consider ‘deformationquantization’ and the role of the parameter q is played by the Planck constant. Pursu-ing this analogy we may consider the formal product depending on the parameter q:

u �q v = uv +∑

n

qn Pn(u, v),

where Pn are such that the product �q is associative.Since the commutator bracket

[u, v]q = u �q v − v �q u

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is a biderivation (as for any associative algebra) and satisfies the Jacobi identity wefind that the ‘quantum Poisson bracket’ gives a Poisson bracket when restricted to‘first-order elements’

{u, v} = P1(u, v) − P1(v, u).

In general, we can write

limq→0

1

q[u, v]q = {u, v} (7.56)

From the defining commutation relations written by Woronowicz we get the cor-responding quadratic Poisson brackets on the matrix elements of SU (2):

{α, α} = 2νν, {ν, ν} = 0, {ν, α} = να, {ν, α} = να.

Passing to real coordinates, α = q2 + i p2 and ν = q1 + i p1, we get a purelyimaginary bracket whose imaginary part is the following quadratic Poisson bracket

{p1, q1} = 0, {p1, p2} = q1q2, {p1, q1} = −p1 p2,{q1, p2} = q1q2, {q1, q2} = −q1 p2, {p2, q2} = q2

1 + p21.

The functions q21 + q2

2 + p21 + p22 is a Casimir function for this Lie algebra.

By performing a standard Poisson bracket reduction we find a bracket on S3. Ifwe identify this space with the group SU (2) we get the Lie-Poisson structure onSU (2):

The vector field

X = −q1∂

∂ p1+ p1

∂

∂q1+ q2

∂

∂ p2− p2

∂

∂q2

selects a subalgebra of functions F by imposing the condition LX f = 0, ∀ f ∈ F .This reduced algebra can be regarded as the algebra generated by

u = −p21 − q21 + p22 + q2

2 , ν = 2(p1 p2 + q1q2), z = 2(p1q2 − q1 p2),

with brackets

{v, u} = 2(1 − u)z, {u, z} = 2(1 − u)v, {z, v} = 2(1 − u)u.

One finds that u2 + v2 + z2 = 1 so that the reduced space of SU (2) is the unitsphere S2 and the reduced bracket vanishes at the North Pole (u = 1, v = z = 0).

It may be interesting to notice that the stereographic projection from the NorthPole pulls-back the standard symplectic structure on R

2 onto the one associated withthis one on S2-{North Pole}.


It is now possible to carry on the reduction at the non-commutative level. Weidentify the subalgebra A′

q ⊂ Aq generated by the elements u = I − 2ν∗ν =α∗α − ν∗ν, w = 2ν∗α and w∗ = 2α∗ν. We have uu∗ + w∗w = I and the algebraA′

q admits a limit given by A′0 generated by the two-dimensional sphere S2. The

subalgebra A′q can be considered as a quantum sphere.

The quantum Poisson bracket on S2 is given by

[w, u] = (q2 − 2q)(1 − u)w, [w∗, u] = −(q2 − 2q)(1 − u)w∗,

and

[w,w∗] = −(2q2 − 2q)(1 − u) + (4q − 6q2 + 4q3 − q4)(1 − u)2.

Passing to the classical limit we find, by setting v = Re(w), z = −Im(w):

{v, u} = 2(1 − u)z, {u, z} = 2(1 − u)u, {z, v} = 2(1 − u)u,

which coincideswith the previous reduced Poisson bracket associatedwith the vectorfield X . In this case, the reduction procedure commutes with the ‘quantum-classical’limit.

In this same setting it is now possible to consider a ‘quantum dynamics’ and thecorresponding ‘classical’ one to see how they behave with respect to the reductionprocedure.

On the algebra Aq we consider the dynamics defined by the Hamiltonian

H = 1

2u = 1

2(I − 2ν∗ν) = 1

2(α∗α − ν∗ν).

This choice ensures that our Hamiltonian defines a dynamics on A′q . The resulting

equations of motion are

[H, ν] = 0, [H, ν∗] = 0, [H, α] = (q2 −2q)ν∗να, [H, α∗] = −(q2 −2q)ν∗να∗,

so that the dynamics written in the exponential form is

U (t) = ei tadH

and gives,

ν(t) = ν0, ν∗(t) = ν∗(0)

α(t) = ei t (q2−2q)ν∗να0, α∗(t) = e−i t (q2−2q)ν∗να∗0 .

Going to the ‘classical limit’ we find


H = 1

2(q2

2 + p22 − q21 − p2

1),

with the associated vector field on S3 given by [Li93]

� = 2(q21 + p21)

(q2

∂

∂ p2− p2

∂

∂q2

),

and the corresponding solutions are given by

q1(t) = q1(0), p1(t) = p1(0)

p2(t) = cos(2t

(q21 + p21

))p2(0) + sin

(2t

(q21 + p21

))q2(0),

q2(t) = − sin(2t

(q21 + p2

1

))p2(0) + cos

(2t

(q21 + p21

))q2(0).

If we remember (7.56), this flow is actually the limit of the quantum flowwhen wetake the limit of the deformation parameter q → 0 and hence q2/q → 0, q/q → 1.Indeed in this case ν∗ν = q2

1 + p21 and α = q2+ i p2. As the Hamiltonian was chosento be an element of A′

q we get a reduced dynamics given by

[H, w] = −1

2(q2 − 2q)(1 − u)w, [H, w∗] = −1

2(q2 − 2q)(1 − u)w∗.

The corresponding solutions for the endomorphism ei tadH become

w(t) = e−i t 12 (q2−2q)(1−u)w(0), w∗(t) = ei t 12 (q2−2q)(1−u)w∗(0).

Passing to the classical limit we find the corresponding vector field on R3 tangent

to S2

� = (1 − u)

(z

∂

∂v− v

∂

∂z

),

which is the reduced dynamics

du

dt= 0

dv

dt= 2(q2

1 + p21)(q2 p1 − p2q1) = (1 − u)z,

dz

dt= −2(q2

1 + p21)(p1 p2 + q1q2) = −(1 − u)v. (7.57)


By using the stereographic projection S2 → R2 given by (x, y) = 1

1−u (v, z) wefind the associated vector field on R

2

�(x, y) = 2

x2 + y2 + 1

(x

∂

∂y− y

∂

∂x

).

This example is very instructive because it provides uswith an example of reducedquantum dynamics that goes onto the corresponding reduced classical dynamics, i.e.,reduction ‘commutes’ with ‘dequantization’. Further details can be found in [Gr94].

7.5.4.1 Example: Deformed Oscillators

Another instance of a non-commutative algebra reduction is provided by the case ofthe deformed harmonic oscillator. Let us start thus by analyzing the case of deformedharmonic oscillators described in the Heisenberg picture. By including the deforma-tion parameter in the picture we can deal with several situations at the same time, aswe are going to see.

We consider a complex vector space V generated by a, a†. Out of V we constructthe associative tensorial algebra A = C ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ · · · . Adynamics on V , say

d

dta = −iωa,

d

dta† = iωa† ,

defines a dynamics on A by extending it by using the Leibniz rule with respect tothe tensor product

A bilateral ideal Ir,q of A, generated by the relation a†a − qaa† + r = 0, i.e.,the most general element of Ir,q has the form A(a†a − qaa† + r)B, with A, B ∈ A,is also invariant under the previously defined equations of motion. It follows thenthat the dynamics defines a derivation, a ‘reduced dynamics’ on the quotient algebraAr,q = A/Ir,q . When q = 1 and r = 0 the dynamics becomes a dynamics ona commuting algebra and therefore can be considered to be a classical dynamics.When q = 1 and r = � we get back the standard quantum dynamics of the harmonicoscillator. If we consider r to be a function of the ‘number operator’ defined asn = a†a we obtain many of the proposed deformations of the harmonic oscillatorexisting in the literature. In particular, these deformations have been applied to thedescription of the magnetic dipole [Lo97].

It is clear now that this reduction procedure may be carried over to any realizationor representation of the abstract algebra and the corresponding ideal Ir,q . In thisexample it is important that the starting dynamics is linear. The extension to theuniversal tensorial algebra gives a kind of abstract universal harmonic oscillator.The bilateral ideal we choose to quotient the tensor algebra is responsible for thephysical identification of variables and may arise from a specific realization of thetensor algebra in terms of functions or operators.


7.5.5 Reduction in the Ehrenfest Formalism

This picture of Quantum Mechanics is not widely known but it arises in connectionwith the so-called Ehrenfest theorem which may be seen from the point of viewof �-products on phase space (see [Es04]). Some aspects of this picture have beenconsidered by Weinberg [We89a] and more generally appear in the geometricalformulation of Quantum Mechanics [Ci90, Ci901, Ci91, Ci94].

We saw above how the Schrödinger picture assumes as a starting point the Hilbertspace of states and derive the observable as real operators acting on this space ofstates. The Heisenberg picture starts from the observables, enlarged by means ofcomplexification into a C

∗-algebra and derives the states as positive normalizedlinear functionals on the algebra of observables. In the Ehrenfest picture both spacesare considered jointly to define quadratic functions as

f A(ψ) = 1

2〈ψ, Aψ〉. (7.58)

In this way all operators are transformed into quadratic functions which are real-valued when the operators are Hermitian. The main advantage of this picture relieson the fact that we can define a Poisson bracket on the space of quadratic functionsby setting

{ fA, fB} := i f[A,B], (7.59)

where [A, B] stands for the commutator on the space of operators. By introducingan orthonormal basis inH, say {ψk}, we may write the function f A as

f A(ψ) = 1

2

∑jk

c j c∗k 〈ψ j , Aψk〉, ψ =

∑k

ckψk

and the Poisson bracket then becomes

{ f A, fB} = i∑

k

(∂ f A

∂ck

∂ fB

∂c∗k

− ∂ f A

∂c∗k

∂ fB

∂ck

).

This bracket can be used to write the equations of motion in the form

id fA

dt= { fH , f A},

where fH is the function associated to the Hamiltonian operator.While this way of writing the dynamics is very satisfactory because allows us to

write the equations of motion in a ‘classical way’, one has lost the associative productof operators. Indeed, the point-wise product (somehow a natural one for the func-tions defined on a real differential manifold) of two quadratic functions will not be


quadratic but a quartic function. To recover the associative product we can, however,get inspiration from the definition of the Poisson bracket (7.59) and introduce

( f A � fB)(ψ) := f AB(ψ) = 1

2〈ψ, ABψ〉. (7.60)

By inserting a resolution of the identity∑

j |ψ j 〉〈ψ j | = I (since there is a numer-able basis for H) in between the two operators in AB, say

〈ψ, A∑

j

|ψ j 〉〈ψ j |Bψ〉,

and writing the expression of ψ in terms of the basis elements ψ = ∑k ckψk we

find a product

( fA � fB)(ψ) =∑jkl

c j c∗l 〈ψ j , Aψk〉〈ψk, Bψl〉,

which reproduces the associative product of operators but now it is no longer point-wise.

As a matter of fact the Poisson bracket defines derivations for this product, i.e.

{ f A, fB � fC } = { fA, fB} � fC + fB � { f A, fC } ∀ fA, fB, fC .

Therefore it is an instance of what Dirac calls a quantum Poisson bracket [Di58]. Inthe literature it is known as a Lie-Jordan bracket [Em84, La98].

Using both products, the Ehrenfest picture becomes equivalent to Schrödingerand Heisenberg ones.

Let us consider now how the expressions of the products are written in termsof a different bases, namely the basis of eigenstates of the position operator Qor the momentum operator P . We have thus two basis {|q〉} and {|p〉} satisfyingQ|q〉 = q|q〉 and P|p〉 = p|p〉 and

∞∫−∞

|q〉dq〈q| = I =∞∫

−∞|p〉dp〈p|.

Now the matrix elements Akj = 〈ψ j , Aψk〉 of the operators in the definition ofthe � product above become

A(q ′, q) = 〈q ′, Aq〉 or A(p′, p) = 〈p′, Ap〉,

and the sum is replaced by an integral:

( fA � fB)(ψ) =∫

dq dq ′ dq ′′c(q ′)c∗(q ′′)A(q ′, q)B(q, q ′′). (7.61)


Thus this is a product of functions defined on Rn ×R

n or (Rn)∗ × (Rn)∗, i.e., twocopies of the configuration space or two copies of themomentum space. Following anidea of Dirac [Di45] one may get functions onR

n ×(Rn)∗ by using eigenstates of theposition operator on the left and eigenstates of the momentum operator on the right:

Al(q, p) = 〈q, Ap〉e− i�

qp,

or also interchanging the roles of position and momentum:

Al(p, q) = 〈p, Aq〉e i�

qp.

Without elaborating much on these aspects (we refer to [Ch05] for details) wesimply state that the �-product we have defined, when considered on phase space,becomes the standard Moyal product.

It is now clear that we may consider the reduction procedure in terms of non-commutative algebras when we consider the �-product. We shall give a simpleexample where from a �-product on R

4 we get by means of a reduction proce-dure a �-product on the dual of the Lie algebra of SU (2). Further details connectedwith their use in non-commutative geometry can be found in [Gr02].

7.5.5.1 Example: Star Products on su(2)

We are going to show how it is possible to define star products on spaces such assu(2) by using the reduction of the Moyal star product defined on a larger space (R4

in this case).Let us then consider the coordinates {q1, q2, p1, p2} for R

4, {x, y, w} for su(2)and the mapping π : R

4 → R3 ∼ su(2) defined as:

f1(q1, q2, p1, p2) = π∗(x) = 1

2(q1q2 + p1 p2)

f2(q1, q2, p1, p2) = π∗(y) = 1

2(q1 p2 − q2 p1)

f3(q1, q2, p1, p2) = π∗w = 1

4(q2

1 + p21 − q22 − p22)

It is useful to consider also the pull-back of the Casimir function of su(2), C =12 (x2 + y2 + w2), which becomes

π∗C = 1

32

(p21 + q2

1 + p22 + q22

)2.

To define a reduced star product on su∗(2) we consider the Moyal star product onthe functions of R

4, and select a �-subalgebra isomorphic to the �-algebra of su∗(2).To identify this subalgebra we need derivations of the �-product that annihilate the


algebra we are studying. We look then for a derivation H which is a derivation ofboth the point-wise algebra and the �-algebra, to ensure that reduction commuteswiththe ‘classical limit’. The commutative point-wise product condition will identify thequotient manifold, while the condition on the �-product identifies a star product onfunctions defined on the quotient. We consider thus a vector field H on R

4 satisfying

LH π∗x = 0 = LH π∗y = LH π∗w.

This condition characterizes the point-wise subalgebra of functions of R4 which

are projectable on functions of R3. Such a vector field can be taken to be the

Hamiltonian vector field associated to the Casimir function π∗C. It is simple tosee that the Poisson subalgebra generated by the functions {π∗x, π∗y, π∗w, fH }where fH = q2

1 + q22 + p21 + p22 is the Poisson commutant of the function fH (see

[Gr02]). And this set is an involutive Moyal subalgebra when we consider the Moyalproduct on them, i.e., for any functions F, G

{ fH , F} = 0 = { fH , G} ⇒ { fH , F � G} = 0.

The star product on su(2) is then defined as:

π∗(F �su(2) G) = π∗F � π∗G.

As an example we can consider the product:

x j �su(2) f (xi ) =(

x j − iθ

2ε jlm xl

∂

∂xm− θ2

8

[(1 + xk

∂

∂xk

)∂

∂x j− 1

2x j

∂

∂xk

∂

∂xk

])f (xi ).

The same procedure may be applied to obtain a reduced star product for allthree dimensional Lie algebras (see [Gr02]) and to deal with a non-commutativedifferential calculus [Ma06].

Another comment is in order. The reduction procedures within QuantumMechan-ics are most effective when they are formulated in a way such that the classical limitmay be naturally considered in the chosen formalism. We believe that this may beconsidered as an indication that Quantum Mechanics should be formulated in a waythat in some form it incorporates the so-called ‘correspondence principle’.

References

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anderen Anwendungen. Teubner, Leipzig (1893) (Edited and revised by G. Scheffers)


[Mm85] Marmo, G., Saletan, E.J., Simoni, A., Vitale, B.: Dynamical Systems: A DifferentialGeometric Approach to Symmetry and Reduction. Wiley, Chichester (1985)

[Ma83] Marmo, G.: Function groups and reduction of Hamiltonian systems. Att. Accad. Sci.Torino Cl. Sci. Fis. Mat. Nat. 117, 273–287 (1983)

[Ho85] Howe, R.: Dual pairs in physics: harmonic oscillators, photons, electrons and singletons.Lect. Appl. Math. 21, 179–207 (1985)

[Ch07] Chaturvedi, S., Ercolessi, E., Marmo, G., Mukunda, N., Simon, R.: Ray-space Riccatievolution and geometric phases for n-level quantum systems. Pramana J. Phys. 63, 317–327 (2007)

[OlPe81] Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite-dimensional systemsrelated to Lie algebras. Phys Rep. 71, 313–400 (1981)

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dynamical systems. Mod. Phys. Lett. A 6, 3445–3453 (1991)[Ma92] Man’ko, V.I., Marmo, G.: Generalized reduction procedure and nonlinear non stationary

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quantum mechanics. Int. J. Mod. Phys. 4, 1363–1403 (2007)[Ca09] Cariñena, J.F., Clemente-Gallardo, J., Marmo, G.: Towards a definition of quantum inte-

grability. Int. J. Geom. Meth. Mod. Phys. 6, 129–172 (2009)[Ca06] Cariñena, J.F., Gracia, X., Marmo, G., Martínez, E., Muñoz-Lecanda, M., Román-Roy,

N.: Geometric Hamilton-Jacobi theory. Int. J. Geom. Meth. Phys. 3, 1417–1458 (2006)[Am26] Amaldi, U., Levi-Civita, T.: Lezioni di Meccanica Razionale. N. Zanichelli, Bologna

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Soc. 22, 404 (1957)[Ma05] Marmo, G., Parasecoli, E., Tulczyjew, W.: Space-time orientations and Maxwell’s equa-

tions. Rep. Math. Phys. 56, 209–248 (2005)[Di31] Dirac, P.A.M.: Quantized singularities in the electromagnetic field. Proc. Roy. Soc. Lond.

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123, 530–533 (1896)[Ba80] Balachandran, A.P., Marmo, G., Skagerstam, B.S., Stern, A.: Magnetic monopoles with

no strings. Nucl. Phys. B 152, 385–396 (1980)[Ba83] Balachandran, A.P.,Marmo, G., Skagerstam, B.S., Stern, A.: Gauge Symmetries and Fiber

Bundles: Applications to Particle Dynamics, Lecture Notes in Physics, vol 188. Springer,New York (1983)

[Ba91] Balachandran, A.P., Marmo, G., Skagerstam, B.S., Stern, A.: Classical Topology andQuantum States. World Scientific, River Edge (1991)

[La90] Landi, G., Marmo, G.: Algebraic Differential Calculus for Gauge Theories. Nucl. Phys.B (Proc.Suppl.) 18, 171–206 (1990)

[Cu97] Cushman, R.H., Bates, L.M.: Global Aspects of Classical Integrable Systems. Birkhäuser,Basel (1997)

[Du80] Duistermaat, J.J.: On global action-angle variables. Commun. Pure Appl. Math. 33, 687–706 (1980)

[Es04] Esposito, G., Marmo, G., Sudarshan, G.: From Classical to QuantumMechanics: an intro-duction to the formalism. Cambridge University Press, Cambridge (2004)

[Ca94] Cariñena, J.F., Ibort, L.A., Marmo, G., Perelomov, A.M.: The Geometry of Poisson man-ifolds and Lie algebras. J. Phys. A Math. Gen. 27, 7425–7449 (1994)

[Al98] Aldaya, V., Guerrero, J., Marmo, G.: Quantization on a Lie group: higher order polar-izations. In: Gruber, B., Ramek, M. (eds.) Symmetries in Science X. Plenum Press, NewYork (1998)

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[Av05] D’Avanzo, A., Marmo, G.: Reduction and unfolding: the Kepler problem. Int. J. Geom.Meth. Mod. Phys. 2, 83–109 (2005)

[Zu93] Zhu, J.C., Klauder, J.R.: Classical symptoms of quantum illness. Am. J. Phys. 61, 605–611(1993)

[Av05b] D’Avanzo, A., Marmo, G., Valentino, A.: Reduction and unfolding for quantum systems:the hydrogen atom. Int. J. Geom. Meth. Mod. Phys. 2, 1043–1062 (2005)

[Fe06] Feher, L., Pusztai, B.G.: A class of Calogero type reductions of free motion on a simpleLie group. Lett. Math. Phys. 79, 263–277 (2007)

[Fe07] Feher, L., Pusztai, B.G.: Hamiltonian reductions of free particles under polar actions ofcompact Lie groups. Theor. Math. Phys. 155, 646–658 (2008)

[Fe08] Feher, L., Pusztai, B.G.: On the self-adjointness of certain reduced Laplace-Beltramioperators. Rep. Math. Phys. 61, 163–170 (2008)

[Wo87] Woronowicz, S.L.: Twisted SU (2) group: an example of a non-commutative differentialcalculus. Pub. Res. Inst. Math. Sci. 23, 117–181 (1987)

[Gr94] Grabowski, J., Landi, G., Marmo, G., Vilasi, G.: Generalized reduction procedure: sym-plectic and Poisson formalisms. Forts. Phys. 42, 393–427 (1994)

[Li93] Lizzi, F., Marmo, G., Sparano, G., Vitale, P.: Dynamical aspects of Lie-Poisson structures.Mod. Phys. Lett. A 8, 2973–2987 (1993)

[Lo97] López-Peña, R., Manko, V.I., Marmo, G.: Wigner problem for a precessing dipole. Phys.Rev. A 56, 1126–1130 (1997)

[We89a] Weinberg, S.: Precision tests of quantummechanics. Phys. Rev. Lett. 62, 485–488 (1989)[Ci90] Cirelli, R., Maniá, A., Pizzocchero, L.: Quantum mechanics as an infinite-dimensional

Hamiltonian system with uncertainty structure. Int. J. Math. 31, 2891–2897 (1990)[Ci901] Cirelli, R., Maniá, A., Pizzocchero, L.: Quantum mechanics as an infinite-dimensional

hamiltonian system with uncertainty structure II. J. Math. Phys. 31, 2898–2903 (1990)[Ci91] Cirelli, R., Maniá, A., Pizzocchero, L.: Quantum phase-space formulation of Schrödinger

mechanics. Int. J. Mod Phys A 6, 2133–2146 (1991)[Ci94] Cirelli, R., Maniá, A., Pizzocchero, L.: A functional representation for non-commutative

C∗-algebras. Rev. Math. Phys. 6, 675–697 (1994)

[Di58] Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press, Oxford (1958)[Em84] Emch, G.G.: Mathematical and Conceptual Foundations of 20th Century Physics. North-

Holland, Amsterdam (1984)[La98] Landsman. N.P.: Mathematical Topics Between Classical and Quantum Mechanics.

Springer, New York (1998)[Di45] Dirac, P.A.M.: On the analogy between classical and quantummechanics. Rev.Mod. Phys.

17, 195–199 (1945)[Ch05] Chaturvedi, S., Ercolessi, E., Marmo, G., Morandi, G., Mukunda, N., Simon, R.: Wigner-

Weyl correspondence in quantum mechanics for continuous and discrete time systems: aDirac inspired view. J. Phys. A Math. Gen. 39, 1405–1423 (2005)

[Gr02] Gracia-Bondia, J.M., Lizzi, F., Marmo, G., Vitale, P.: Infinitely many star products to playwith. J. High Energy Phys. 4, 025 (2002)

[Ma06] Marmo, G., Vitale, P., Zampini, A.: Noncommutative differential calculus for Moyal sub-algebras. J. Geom. Phys. 55, 611–622 (2006)

Chapter 8Integrable and Superintegrable Systems

Complex models are rarely useful (unless for those writing theirdissertations).

Vladimir Arnold, On teaching mathematics, 1997

8.1 Introduction: What Is Integrability?

There is no generally accepted definition of integrability that would include thevarious instances which are usually associated with the word “integrable”. Occasion-ally the word ‘solvable’ is also used more or less as synonymous, but to emphasizethe fact that the system need not be Hamiltonian. Any definition should, of course,include those systems that are usually termed completely integrable systems, orLiouville-Arnold integrable systems,moreover all systems that carry the qualification‘integrable’ have the characteristic property that their solutions can, in principle, beconstructed explicitly.

In this respect, also separability turns out to be instrumental for explicit integrabil-ity of the system. As a matter of fact, the decomposition of a matrix into semisimpleplus a nilpotent part is a way to introduce a preliminary separation. Afterwards, thespectral decomposition of the semisimple part allows us to separate the system intopairwise commuting ones, to separate the nilpotent part into pairwise commutingones, we restrict the analysis to those systems that are nilpotent of index two.

Decomposition into pairwise commuting subsystems formalizes the ‘compositionof independent motions’ that is commonly used at the level of Newton’s equations.

Spectral decomposition shows that one may search for particular coordinate sys-tems in which the equations acquire a particular simple form. By using the insightcoming from the fundamental theoremof algebra, onemay reduce the analysis to two-dimensional vector spaces and decompose the matrix associated with the subsysteminto a multiple of the identity plus a matrix proportional to σ3 or one proportional toiσ2 to deal with semisimple matrices, while the nilpotent part, with index two, maybe represented by σ+ = 1

2 (σ1 + iσ2). The systems represented by means of the Pauli


539

540 8 Integrable and Superintegrable Systems

matrices as explained will have associated flows representing hyperbolic motions,circular motions or straight line motion respectively. These may be considered tobe building blocks of generic systems with nilpotent part of index two. It shouldbe clear that this decomposition is defined when we only use the linear group toachieve diagonalization. If we allow nonlinear coordinate transformations, then wecan transform a semisimple system into a nilpotent one, which is the case for instancefor the Harmonic Oscillator when we go to action-angle variables.

On the other hand the separability procedure is the one always attempted for solv-ing Schrödinger equations or wave equations, usually under the name of separationof variables when we use the description in terms of differential operators. Moregenerally, it iswhatwe try to attemptwhenwedealwith diagonalization of theHamil-tonian operator (let us remark that the Hamiltonian operator of Quantum Mechanicsshould be thought of as a vector field not as the Hamiltonian function of classicalmechanics; as a matter of fact, if it is not Hermitian it need not be associated witha Hamiltonian function). Thus integrability in the previous meaning may be formal-ized by means of a decomposition of the vector field representing the dynamics aspairwise commuting vector fields, i.e., forming an Abelian algebra with coefficientwhich behaves like a numerical constant with respect to the vector fields enteringthe decomposition. Thus it imitates the low dimensions appearing in the spectraldecomposition of the matrix representing the starting vector field.

We could require that the vector fields entering the decomposition should bemaximally superintegrable, i.e. admitting a maximal number of constants of themotion. It should be stressed that this requirement may discriminate between dif-ferent decompositions which would be allowed by the ‘algebraic only’ restriction.At the quantum level this amounts to searching for an Abelian algebra of pairwisecommuting first-order differential operators (see Chap. 10) such that the Hamiltonianoperator would be an element of the enveloping algebra generated by the AbelianLie algebra.

The natural generalization originated by dropping the requirement of Abelianityof the algebra of vector fields entering the decompositionwill be examined in the nextchapter under the name of Lie–Scheffers systems. Again at the quantum level a Liealgebra of first-order differential operators would contain the Hamiltonian operatoras an element of the enveloping algebra.

In conclusion, the various procedures to identify ‘integrable systems’ all seemto rely on the construction of Abelian algebras of vector fields which allow for thedecomposition of the original dynamics.

As one-one tensor fields which are invariant under the dynamics allow, generi-cally, construction of commuting vector fields out of the starting one, the so-called‘recursion operator method’ uses this procedure. Often a recursion operator maybe obtained by composing invariant covariant and contravariant (0, 2)-tensors, thiswould mean composing a symplectic structure with a Poisson structure. Somerequirement of “integrability” or “flatness” of this tensor field (Nijenhuis condi-tion) would guarantee that the algebra one generates starting with the dynamicalvector field will be Abelian.

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8.1 Introduction: What Is Integrability? 541

For scattering theory, the role of the connecting intertwining operator would beplayed by theMoeller or wave operator; again one constructs a change of coordinatesconnecting a “free system” with the interacting one. This would be the analogue ofthe construction of action-angle variables for systems with bounded orbits.

In general the ‘comparison system’ may be chosen in such a way that it is easilyintegrable; as we have seen, that is the case when this comparison system may bedescribed in terms of matrices. The connecting map gives raise to the so-called Laxrepresentation.

Of course verymuch as in the case of the construction of action-angle variables, theidentification ofmaps connecting our systemwith onedescribedbymeans ofmatricesis highly non trivial, even though quite powerful; the search for Lax representations ofparticular integrable systems has been more of an ‘art’ than a technique generatingquite a few papers dealing with the construction of the map for relevant physicalsystems.

8.2 A First Approach to the Notion of Integrability: Systems withBounded Trajectories

As it is customary in this book, to pin-point the mathematical structures involved,without introducing additional technicalities, we shall first consider linear systemsin finite dimensions.

On a vector space V a dynamical linear system would have the form

d

dtv = Av,

with v a generic vector in V . With this linear system there is always associatedanother one on the dual linear space V ∗ and on any tensor space built out of thevector space V (recall the discussion in Sect. 2.2.5).

This equation is explicitly solved by the exponential map, as discussed at lengthin Sect. 2.2.2. Then we have � : R× V → V given by �(t, x0) = exp(t A)x0. Thusfor each initial condition x0 we know its evolution x(t) = �(t, x0) after a time t .From aminimalistic point of view this systemwould be ‘integrable’, nevertheless wedo not feel satisfied, for instance we would not know how to answer simple questionssuch as:

1. Are there periodic solutions?2. What happens when t → ±∞?3. If the system has some periodic orbits, are all the rest also periodic?4. What are the symmetries and the constants of motion?

One has the expectation, or prejudice, that integrability should also carry along asimple and direct answer to all previous questions. In some sense we would like to

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have a kind of ‘normal form’ for integrable systems so that previous questions wouldbe easily answerable.

In previous chapters we considered systems that allow for action-angle coordi-nates; these systems take the simple form

d

dtϕ j = ν j ,

d

dtν j = 0, j = 1, ..., n.

The flow in these coordinates takes a ‘linear form’ given by

ϕ j (t) = ϕ j (0) + tν j , ν j (t) = ν j (0).

In general this change of coordinates excludes some points of the carrier spacebecause they do not define a global diffeomorphism. The placement and the ‘exten-sion’ of these ‘offending points’ may change from system to system, making classi-fication very difficult if not impossible.

The previous list of questions can be extended now to this situation adding a fewmore:

1. Does the existence of this change of coordinates depends on the Poisson structureor can it be defined independently of it?

2. Are the trajectories periodic for any initial condition?3. If the system allows for many alternative invariant symplectic structures, are the

action-angle variables uniquely defined?

In the following we would like to address these questions and try to show thateven for simple systems the situation is rather composite.

8.2.1 Systems with Bounded Trajectories

In this section we would like to show to what extent a dynamical systemmay identifygeometrical structures without an a priori assumption that it preserves a symplecticor a Poisson structure.We assume that our system is linear and has all orbits bounded.

We have already encountered this example in connection with the existence ofinvariant Hermitian structures, as a particular instance of Hamiltonian systems, andit was discussed in Sect. 6.2.3. Here we would like to take up again this example toexhibit some properties of integrable systems which would suggest a normal formindependently of any invariant symplectic structure. Clearly, in finite-dimensionalvector spaces,we canuse anynorm todefineboundedness of the orbits. This conditionon the boundedness of the orbits has rather stringent consequences on the structureof the flow. The following proposition, which relies only on linearity of the vectorfield and boundedness of the orbits, extracts the relevant ingredients entering themathematical structures of this particular family of systems.

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8.2 A First Approach to the Notion of Integrability: Systems . . . 543

Proposition 8.1 Any finite-dimensional linear system, admitting only boundedorbits, allows for action-angle variables and admits an infinite number of constantsof motion and an infinite number of infinitesimal symmetries, moreover it admits aninfinite number of alternative Hamiltonian descriptions.

Exercise 8.1 Prove this proposition using the proof of Proposition6.15 and the ideassketched in the next paragraphs.

We shall see that even if two systems satisfy both requirements of this proposition,they need not be diffeomorphic, showing that an equivalence relation based on thediffeomorphism group is not adequate. In particularwith such an equivalence relationwe would find that systems with all orbits bounded would divide into an infinitenumber of equivalence classes of diffeomorphic systems. This result, surprising ornot, shows that without many more qualifications, the form

d

dtϕ j = ν j ; d

dtν j = 0, j = 1, . . . , n

is the best one may hope to achieve, if it exists at all.

8.2.1.1 Factorization Properties of the Flow

For any linear system, the flow φt = et A, may be factorized1 as et S et N with Na nilpotent and S a semi-simple matrix such that [S, N ] = 0. The requirement ofboundedness of the trajectories, i.e., ||et Ax(0)|| ≤ K ||x(0)|| for any x(0) ∈ V andany t ∈ R immediately requires N = 0 and S must have only imaginary eigenvalues.

If wemake now the further assumption that eigenvalues haveminimal degeneracy,we find that in the basis of eigenvectors for S, the matrix A acquires the form:

A =

⎛⎜⎜⎜⎜⎜⎝

0 ν1−ν1 0

. . .

0 νn

−νn 0

⎞⎟⎟⎟⎟⎟⎠

,

with (ν1, . . . , νn) ∈ Rn . The representation of the vector field � = X A describing

the dynamics gives:

� =n∑

j=1

ν j

(x j

∂

∂y j− y j

∂

∂x j

), ν j ∈ R, j = 1, . . . , n.

If we use complex coordinates z j = x j + iy j , j = 1, . . . , n, we have:

1 That is often quoted as ‘the composition of independent motions’.

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� =n∑

j=1

iν j

(z j

∂

∂z j− z∗

j∂

∂z∗j

),

and in terms of vector fields we deduce immediately (compare with the results onSect. 1.2.7):

1. Any function of|z1|2 = x21 + y21 , . . . , |zn|2 = x2n + y2n . (8.1)

is a constant of motion.2. The module generated by vector fields:

X j = x j∂

∂y j− y j

∂

∂x j, j = x j

∂

∂x j+ y j

∂

∂y j,

with coefficients in the algebra of constants of motion consists of infinitesimalsymmetries.

3. Many invariant symplectic structures are generated by (non-degenerate) linearcombinations of

f j (x2j + y2j )dx j ∧ dy j . (8.2)

4. Many invariant Poisson structures are generated by linear combinations of thebivector fields:

jk

(∂

∂x j∧ ∂

∂yk

), λ j X j ∧ j = λ j (x2j + y2j )

∂

∂y j∧ ∂

∂x j, λ j , jk ∈ R.

If we restrict our attention only to the semisimple part, the spectral decompositionof our matrix gives raise to a decomposition in terms of vector fields which arepairwise commuting; X S = a j Y j with [Y j , Yk] = 0, and LY j ak = 0 for any pair ofindices j and k.

Clearly, each Y j is maximally superintegrable if eigenvalues have minimal degen-eracy. It also follows that each one of the vector fields entering the decomposition isa complete vector field and gives raise to a one-parameter subgroup of linear trans-formations. As we are considering diagonalization over the complex numbers, thereal dimension of the vector space corresponding to each eigenvalue will be two. Atthe manifold level, to implement this property, we should introduce the notion of‘related’ vector fields. Roughly speaking we are dealing with a map from the carrierspace to another manifold of lower dimensions such that the algebra of functions onthis manifold, when pulled back to the starting carrier space, will be an invariant sub-algebra under the dynamical evolution. The restriction of the dynamical vector fieldto this subalgebra would play the role of the vector field Y j . The algebraic propertiesemerging from the spectral decomposition of the semisimple part may be recastedinto geometrical properties of integrable systems (see Sect. 8.3.1). It may be useful to

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8.2 A First Approach to the Notion of Integrability: Systems . . . 545

recall that Marcus andMeyer [MM74] have shown that generic Hamiltonian systemsare neither integrable nor ergodic.

8.2.1.2 Conditions on Frequencies

When further properties are satisfied by frequencies ν1, ν2, . . . , νn , additional prop-erties emerge for the dynamics. For instance, if we assume that ν j/νk is an irrationalnumber for any pair of indices j, k, it follows that the closure of each trajectorystarting with a generic initial condition (x1(0), x2(0), . . . , xn(0)) will be a torus ofdimension n. Under these hypotheses it is clear that there will not be any additionalconstant of motion which is functionally independent from the algebra of functionswhich is generated by {x2j + y2j | j = 1, 2, . . . , n}.

Infinitesimal symmetries however are linear combinations of the infinitesimalgenerators j , X j . Obviously, the coefficients in any combination may be functionsof the constants of motion.

Roughly speaking, the dynamical system has been split into a family of non-interacting systems, i.e., pairwise commuting. Each one� j = ν j X j generates a one-parameter subgroup of the torus group of dimension n acting on the 2n-dimensionalvector space V .

The symmetries j pairwise commute among themselves and also commute withthe action of the torus group. By using the differential forms θ j = x j dy j − y j dx jand functions f (x21 + y21 , . . . , x2n + y2n ) along with the infinitesimal symmetries j ,we can generate a rather large family of invariant tensor fields for �.

If we use the torus action generated by X j , all possible linear systems that share thesame invariant foliation by n-dimensional tori emerge as one-parameter subgroupsof toric transformations. When a given system has frequencies that are pairwise irra-tional, generic orbits are not closed and their closure coincides with the full torus.When particular initial conditions are chosen, say represented by vectors belongingto proper invariant subspaces, the corresponding orbits have a closure of dimensionequal to half the real dimension of the minimal invariant subspace containing thechosen initial condition. To say it plainly, if the frequencies are pairwise irrationaldoes not mean that the system has no closed trajectories. The invariance under dila-tions, generated by j , shows that on each plane from one periodic orbit we generatea new one.

8.2.1.3 Rationally Related Frequencies

We may assume now that some frequencies are rationally related, say ν j = νn j andνk = νnk , with the integers nk, n j relatively prime. In this case we have an additionalconstant of motion given by:

K = (z∗j )

nk (zk)n j .


If some of the frequencies are rationally related we find additional constants ofmotion and the system is superintegrable. The given system is said to be maximallysuperintegrable when the number of independent constants of motion is maximal,i.e., 2n − 1 for a system of dimension 2n. An important instance of a maximallysuperintegrable system, in action-angle variables is of the form:

� = ν

(∂

∂ϕ1+ · · · + ∂

∂ϕn

),

then, not only x2j + y2j are constants of motion but the differences ν j − νk provideadditional constants of motion.

When the frequencies satisfy ν j = νm j for all j and all pairs (m j , mk) arepairs of mutually prime integer numbers, any sequence of pairwise mutually primeintegers, say (m1, m2, . . . , mn) will identify a linear dynamical system which ismaximally superintegrable, however two systems associated with two sequencesof mutually prime integers which are also relatively prime as sequences, cannotrepresent diffeomorphic systems.

Thus even for maximally superintegrable systems, the classification under thediffeomorphism group would give an infinite number of equivalence classes.

8.3 The Geometrization of the Notion of Integrability

We have derived many of the previous properties for a linear system � with the onlyrequirement that orbits must be bounded. For a generic vector field the objects wemay be able to identify are:

1. The family of constants ofmotion under dynamical evolution: f (x(t)) = f (x(0))for any t ∈ R and any initial condition x(0).When a constant of motion is required to be at least continuous, this familycaptures the essence of the closure of the orbit for a generic initial condition.

2. The family of vector fields in the kernel of themap� = ( f1, . . . , fm) : M → Rm ,

where M denotes the carrier space of the dynamics �, usually a smooth finite-dimensional manifold, and the functions f1, . . . , fm are functionally independentconstants of motion.

If we consider differentiable constants of motion we may consider the algebra ofvector fields defined by:

LX f = 0,

for any f smooth enough such that L� f = 0. This algebra of vector fields is a Liemodule2 with coefficients constants of motion of � and was called the bicommutantin Sect. 3.2.4 where a stability theorem was proved, Lemma3.7, see also [IM12]).

2 That is a module M over an algebra A with a Lie bracket which isA-linear.

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8.3 The Geometrization of the Notion of Integrability 547

We might also consider the Lie module of vector fields Y such that LXLY f = 0for any constant of motion and all vector fields X such that LX f = 0, which is anatural way of defining the normalizer of the family of vector fields X . Actually thevector fields Y will take constants of motion into constants of motion.

The two Lie modules define a short exact sequence of Lie modules:

0 → X0 → XN → XQ → 0,

whereX0 is the Liemodule of vector fields defined byLX f = 0. The LiemoduleXN

is the normalizer of the previous Lie module and XQ is the quotient of the previoustwo.

We should stress that this sequence is canonically associated with any vector field�. The Lie module XQ is essentially related to the algebra of vector fields definedon the manifold associated with the constants of motion for �. With reference to theprevious example of linear systems with all orbits bounded, it would be generatedby the real spectrum of constants of motion built out of z∗

j z j and (znkj )∗z

n jk .

We could define an integrable system to be one for which the Lie modules XQ

and X0 have the same dimension, which would be superintegrable when dimX0 <

dimXQ . Maximally superintegrable systems will happen whenX0 is generated by �

itself and XQ has dimension 2n − 1. This picture has the advantage of disregardingthe parametrization of the dynamics.

The connectionbetween infinitesimal symmetries and constants ofmotion requirespreliminarily to associate vector fields and 1-forms. To associate vector fields with 1-forms we need a (0, 2)-tensor field. In the linear case it has been possible to constructthe elementary closed 2-forms f j (x2j + y2j )dx j ∧dy j , Eq. (8.2), by using the require-ment of simplicity of eigenvalues. For generic vector fields this would correspond tothe requirement that there exist projection maps, submersions: φ j : M → R

2j , such

that � is φ j -related to a vector field � j on R2j possessing a constant of motion and

preserving the volume. In this way the pull-back of the volume form on R2j and h j

would provide us with the analogue of f j (x2j + y2j )dx j ∧ dy j .In summary, the existence of an invariant non-degenerate closed 2-form, would

allow us to create a correspondence between a subalgebra of vector fields in X0 andclosed 1-forms generated by d f1, . . . , d fn .

Clearly it is conceivable that many such pairings are possible and therefore manyalternative Hamiltonian descriptions could exist. If we write the factorization of �

in the form:� = (d H),

it is now clear that any diffeomorphismwhich is also a symmetry for� would providea new Hamiltonian description when it is not canonical:

φ∗� = � = φ∗()d(φ∗ H).


We have already commented this situation in the case of Hamiltonian linear systems(Sect. 5.2.6). In what follows we shall further analyze what can be said when we startfrom the normal form:

� = ν j∂

∂ϕ j.

8.3.1 The Geometrical Notion of Integrabilityand the Erlangen Programme

Whenwemove to the geometrical picture of our dynamical system, it is rather naturalto consider all vector fields obtained by means of the diffeomorphism group actingon the starting one to represent the same abstract dynamical system. This groupmay be reduced, ‘broken’, to a smaller subgroup, by requiring that it preserves someadditional geometrical structure, say a symplectic structure. In this way we wouldobtain the group of “canonical transformations” to go from one Hamiltonian vectorfield to another equivalent to it.

After the Erlangen Programme written by Felix Klein with the collaboration ofSophus Lie (Lie visited Klein for 2months, just before the programme was written)it is by now accepted that a “geometry” or a “geometrical structure” on a manifoldM amounts to selecting a subgroup of the group of diffeomorphisms of M [Kl72,Kl92].

In Physics this correspondence is quite familiar, for instance we have the Poincarégroup for theMinkowski space-time in Special Relativity, the group Diff(R4) in gen-eral relativity, the group of symplectomorphisms in Hamiltonian dynamics, contacttransformations, the unitary group in Quantum Mechanics and so on. In all previ-ous cases, the subgroup of the group of diffeomorphisms determined by a geometrycharacterizes the geometry itself, for instance, if (M1, ω1) and (M2, ω2) are two sym-plectic manifolds and there is a group isomorphism � : Sp(M1, ω1) → Sp(M2, ω2)

of the corresponding groups of canonical transformations, i.e., symplectic diffeo-morphisms defined by each one, then the two symplectic manifolds are (up to aconformal constant) symplectically equivalent, i.e., there exists a diffeomorphismϕ : M1 → M2 such that ϕ∗ω2 = cω1 [Ba86, BG88]. Similar results were establishedby Grabowski for Poisson and Jacobi manifolds [Gr00]. Thus the automorphismsdetermined by some geometrical structures are essentially inner as it happens for thegroup of unitary transformations of a Hilbert space.

In this section we would like to identify the appropriate subgroup of Diff(M)

associated with an integrable or a superintegrable system �.Thus if we have an integrable Hamiltonian dynamical system � defined on a

symplectic manifold (M, ω) the naive thought that the subgroup of the group ofdiffeomorphisms determined by it should be a subgroup of the group of canonicaltransformations of ω is immediately shown to be inadequate because integrablesystems always admit alternative Hamiltonian descriptions and we would not knowwhich canonical transformations to consider.

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8.3 The Geometrization of the Notion of Integrability 549

An example of the application of the Erlangen programme is readily shown bythe geometrization of linear structures described in Sect. 3.3. Let us recall that alinear structure is characterized by a tensorial object (and the associated invariancesubgroup), i.e., a complete vector field such that:

1. There exists only one point, say m0 ∈ M , such that (m0) = 0.2. The eigenvalue problem: L f = 0 · f , f ∈ F(M) has only trivial solutions, i.e.,

f = constant if M is connected.3. The eigenvalue problem: L f = f , f ∈ F(M) has dim M = n independent

solutions fk such that d f1 ∧ · · · ∧ d fn �= 0.

Let us remark that the completeness condition on allows us to ‘generate’ allof M starting with a properly chosen transversal codimension-one submanifold in aneighborhood of m0 considered as a set of ‘initial conditions’. Any such vector field identifies a subgroup of Diff(M) by requiring ϕ∗() = . Thus the subgroupGL(M,) of Diff(M) of diffeomorphisms ϕ preserving is exactly the groupof linear isomorphisms GL(n,R) with n = dim M where we use the global chartprovided by the functions f j to identify M with R

n . In infinitesimal terms, vectorfields generators of linear transformations are solutions of [X,] = 0. Moreover ifwe have two linear structures (M1,1) and (M2,2) defined in two manifolds M1and M2, and there is a group isomorphisms � : GL(M1,1) → GL(M2,2), thenthere exists a diffeomorphism ψ : M1 → M2 such that ψ∗1 = 2.

As it was shown in Sect. 3.3 the notion of linear structure can be weakened byreplacing conditions 1–3 above. Specifically we may ask that the set of points in Msatisfying (m) = 0 define a smooth submanifold Z of M of dimension k insteadof condition 1. Condition 2 will be rephrased by asking that functions f such thatL f = 0 define an Abelian algebra whose spectrum is diffeomorphic to Z and,finally, condition 3 will be substituted by demanding that L f = f , define n − kfunctionally independent fibrewise-linear functions. Such a partial linear structure,as we have already seen, is actually equivalent to a vector bundle structure on themanifold M and the corresponding vector bundle autormophisms are selected by thecondition ϕ∗ = .

Another example in the same vein is concerned with possible tangent and cotan-gent bundle structures on a manifold M .

We recall (see Sect. 5.5.1) that a tangent bundle structure on M is identified bya pair (, S), where defines a partial linear structure on M , i.e., a vector bundlestructure, and S is a (1, 1)-tensor field [FL89] such that ker S = Im S (which impliesS2 = 0), S() = 0 and d2

S = 0 where the ‘twisted’ differential dS is defined by

dS f (X) = d f (S(x)). (8.3)

The vector field is required to define a vector bundle structure on M , i.e., apartial linear structure, that is to satisfy the modified conditions (1), (2) and (3)above with 2k = dim M . The vector bundle structure identified by functions in (3)becomes the tangent bundle structure of a manifold Q, so M = T Q. Notice that the

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(1, 1)-tensor S is the soldering form of T Q. In natural bundle coordinates (q, v),these tensor fields take the form:

S = dq ⊗ ∂

∂v, = v

∂

∂v.

The subgroup of Diff(M) identified by ϕ∗ = and ϕ∗S = S is the group oftangent bundle automorphisms. A single manifold can be equipped with alternativetangent bundle structures by considering diffeomorphisms ϕ : M → M such thatϕ∗S = S but not preserving , we consider 1 and 2 given as 1 = and2 = ϕ∗. Then M becomes a double vector bundle if [1,2] = 0 and M willcarry two tangent bundle structures T Q1 and T Q2.

Similarly one may define a cotangent bundle structure by means of a pair (, θ)

where again, is a partial linear structure on M , and θ is a particular 1-form suchthat dθ is a symplectic form, idθ = θ , and requiring that solutions of L f = 0be pairwise in involution and define a maximal Abelian subalgebra with respect tothe Poisson bracket defined by ω = dθ . Notice that is equivalent to asking that thesubmanifold Q defined by (m) = 0 is Lagrangian with respect to dθ . Thus M =T ∗Q and the canonical Liouville 1-form of T ∗ Q is just the 1-form θ above. Againas in the case of tangent bundle structures, alternative cotangent bundle structureson M can be constructed by choosing diffeomorphisms ϕ such that ϕ∗θ �= θ .

Thuswewill try to study the geometry of an integrable system� by determining itsassociated subgroup of the group of diffeomorphisms of the manifold by writing thesystem in “normal form” instead of determining such a subgroup from the subgroupof diffeomorphisms determined by some geometrical structure determined by it. Inparticular we shall consider systems with orbits possessing a compact closure, eventhough systems possessing unbounded orbits are relevant in scattering theory forinstance.

8.4 A Normal Form for an Integrable System

8.4.1 Integrability and Alternative Hamiltonian Descriptions

We have encountered in various occasions throughout this book systems exhibitingaction-angle variables. Let us review now the context in which action-angle variablesare exhibited inmost cases andwhich is related to the notion of complete integrabilityof a Hamiltonian vector field.

Definition 8.2 A vector field � on a 2n-dimensional symplectic manifold (M, ω)

such that i�ω = d H, is said to be completely integrable if there exist n functionallyindependent first integrals f1, . . . , fn, d f1 ∧ · · · ∧ d fn �= 0, such that:

{ f j , fk} = 0, {H, fk} = 0, ∀ j, k = 1, 2, . . . , n . (8.4)

8.4 A Normal Form for an Integrable System 551

The independence condition d f1 ∧ · · · ∧ d fn �= 0 may hold on some open densesubmanifold of M , or in a weaker form, on some open invariant submanifold. Wewill find this situation for instance when considering scattering problems where weprefer to remove closed regions composed of bounded orbits from phase space.

When the system � possesses more than n first integrals, we recall that the systemis said to be superintegrable. In this case,wehave f1, . . . , fn+k ,d f1∧· · ·∧d fn+k �= 0,and {H, f j } = 0, for all j = 1, 2, . . . , n +k. In the particular case of n +k = 2n −1the system is said to be maximally superintegrable.

To study the subgroup of diffeomorphisms of M appropriate for an integrablesystem, it is convenient to have a “normal form”. As it was discussed in the introduc-tion, in searching for normal forms, it is quite natural to ask which transformationsare allowed on the system. Thus for Hamiltonian systems, it would be natural touse canonical transformations, that is to consider transformations of the systemsbelonging to the closed subgroup of symplectic diffeomorphisms of the group of dif-feomorphisms of ourmanifold M . However, any integrable system admits alternativeHamiltonian descriptions, i.e., there are (ωa, Ha), a = 1, 2, such that

i�ω1 = d H1, i�ω2 = d H2.

In this case, which canonical transformations should we use, symplectic diffeomor-phisms with respect to ω1 or with respect to ω2?

In this respect it is amusing to recall a quarrel between Levi–Civita and Birkhoffaround this. Indeed in his 1934 paper “A general survey of the theory of adiabaticinvariants” [Le34] Levi–Civita feels the need to write a section entitled “Birkhoff’sseverity against canonical variables and methods. Apology for a milder attitude”(p. 430). From the point of view of Birkhoff, one should consider the orbit of vectorfields obtained by acting on � with Diff(M). All the vector fields in the same orbitshare the same properties. Thus, it makes sense to restrict our attention to just onerepresentative. Let us elaborate on this point in the particular case of linear systems.

We notice that a Hamiltonian vector field admits a factorization in terms of aPoisson tensor and an exact one-form:

� = (d H). (8.5)

Clearly, any vector field in the orbit of � will also be decomposable as above. More-over, if we consider diffeomorphisms ϕ ∈ Diff(M) such that ϕ∗� = �, and apply itto the decomposition above (8.5), we find:

� = ϕ∗()d(ϕ∗ H).

We conclude that any non-canonical transformation that is a symmetry for �, pro-duces alternative Hamiltonian descriptions. In infinitesimal form: if X is an infini-tesimal symmetry for the dynamics �, [�, X ] = 0, we get:

LX (i�ω) = dLX H,


theni�(LXω) = d(LX H),

and the previous equation provides an alternative Hamiltonian description for � ifLXω is nondegenerate. In relation with alternative descriptions, we notice that thereare additional ways to generate alternative Hamiltonian descriptions. Let us considerfor instance a (1, 1)-tensor T such that:

L�T = 0. (8.6)

Let us define as before Eq. (8.3), the twisted differential:

dT f (X) = d f (T X).

Then from any constant of motion F we will have the closed 2-form:

ωT,F = ddT F,

now if ωT,F is nondegenerate, it provides an alternative Hamiltonian description for�. In fact notice that:

i�ωT,F = L�dT F − d(i�dT F),

but L�dT F = 0 because of the invariance condition (8.6) and the fact that F is aconstant of motion. Hence −d F(T �) is a Hamiltonian function for � with respectto the symplectic structure ωT,F .

8.4.2 Integrability and Normal Forms

Because of the previous discussion we should expect, therefore, that for vector fieldswith a large group of symmetrieswewill find always alternativeHamiltonian descrip-tions. However we should stress that there are alternative Hamiltonian descriptionswhich are not generated by diffeomorphisms.

In conclusion, we should accept any diffeomorphism to reduce a vector field � toits normal form. Thus, if the orbit through � contains a completely integrable systemwe can concentrate our attention on the standard normal form we are familiar withwhen we construct action-angle variables, i.e., we could consider the form:

� = ν j (I )∂

∂ϕ j.

We should therefore study a normal form for integrable systems as emerging fromthe following conditions:


Definition 8.3 Given a vector field � we will say that � = ν j ( f1, . . . , fn)X j is anormal form for � (and that � is integrable) if:

i. There exist n functionally independent first integrals f1, . . . , fn, such that d f1 ∧· · · d fn �= 0.

ii. There exist n complete vector fields X1, . . . , Xn pairwise commuting [X j , Xl ] =0 and independent X1 ∧ · · · Xn �= 0, and,

iii. LX j fl = 0 for all j, l = 1, . . . , n.

We should notice that we have dropped the requirement for � to be Hamiltonianand consequently that the ‘frequencies’ ν j are derivatives of the Hamiltonian func-tion.

The usual Liouville–Arnold’s theorem becomes now a particular way to findfunctions (coordinates) which reduce � to normal form.

A few remarks are in order here.

1. All integrable systems have the same normal form, then what distinguishes onesystem from another if any such distinction exists?

2. Which aspects of the normal form above for a given integrable system are ableto discriminate integrable from superintegrable systems?

In connection with the first query, we immediately notice that many interestingaspects on the qualitative structure of the orbits of the system are to be extracted fromthe normal form because we know that usually specific integrable systems need notbe diffeomorphic among them. We may rephrase our questions by investigating howmany different orbits exist inX(M) under the diffeomorphism groupwhen each orbitis required to contain at least one element that is completely integrable.

To have an idea of the variety of situations wemight be facing we shall investigatesome variations on the theme of harmonic oscillators.

8.4.2.1 Hamiltonian Linear Systems

We may consider as a particular instance of the analysis performed in the previoussections the isotropic harmonic oscillator. Thus, let us consider M = R

2n = Cn and

the system � defined in cartesian coordinates (x j , y j ) by (recall Sect. 1.2.2):

d

dtx j = ωy j ,

d

dty j = −ωx j ,

then,

� =∑

j

ω

(y j

∂

∂x j− x j

∂

∂y j

).

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Introducing complex coordinates z j = y j + i x j , we obtain:

d

dtz j = iωz j ,

d

dtz j = −iωz j ,

The algebra of first integrals is generated by the quadratic forms zl zm and because� isproportional to the vector field defined by the complex structure onCn , we concludethat its group of linear symmetries is GL(n,C). For a given factorization � =(d H), the homogeneous space GL(n,C)/GL(n,C) ∩ Sp(2n,R) parameterizesalternative Hamiltonian descriptions, however not all of them.

What will happen then for a generic linear systems?Given a generic linear system,represented by the matrix A, it has a decomposition:

A = · H, (8.7)

with a non-degenerate skew-symmetric matrix and H a symmetric matrix if andonly if TrA2k+1 = 0, k = 0, 1, 2, . . . (see Sect. 4.2.3). For nongeneric matricesconditions guaranteeing the existence of the factorization (8.7) aremore cumbersomeand we refer to the paper [GM93] for a full discussion.

For instance, when A is generic all linear symmetries are generated by powers ofA. The non-canonical ones are given by even powers, therefore:

T = eλA2k, k = 1, 2, . . . ,

will be a non-canonical symmetry for any value of λ.For instance, two alternative descriptions for the isotropic harmonic oscillator in

R4 are given by:

1 = ∂

∂p1∧ ∂

∂q1+ ∂

∂p2∧ ∂

∂q2, H1 = 1

2ω(p21 + p22 + q2

1 + q22 ),

2 = ∂

∂p1∧ ∂

∂q2+ ∂

∂p2∧ ∂

∂q1, H2 = ω(p1 p2 + q1q2).

A particular invariant (1, 1)-tensor field is defined by:

T = dq1 ⊗ ∂

∂q2+ dq2 ⊗ ∂

∂q1+ dp1 ⊗ ∂

∂p2+ dp2 ⊗ ∂

∂p1.

We may then consider the 2-form ddT F with F = 14 (p21 + p22 + q2

1 + q22 )

2 and get:

ddT F = d(p21 + p22 + q21 + q2

2 ) ∧ d(p1 p2 + q1q2)

+ 2(p21 + p22 + q21 + q2

2 )(dq2 ∧ dq1 + dp1 ∧ dp2).

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We finally remark that the selection of a specific decomposition with the Hamil-tonian being positive definite gives a group of canonical symmetries which is theunitary group:

GL(n,C) ∩ O(2n,R) = U (n),

therefore the system may be thought of as a ‘quantum-like’ system [EM10] (seeSect. 6.2.1).

8.4.3 The Group of Diffeomorphisms of an Integrable System

If � is an integrable system possessing a normal form like in Definition8.3, thenbecause X1, . . . , Xn are complete and pairwise commuting we can define an actionof the Abelian group Rn onto M . Moreover, because LX j ν

k = 0, we could redefinethe vector field X j to Y j = ν j X j (no summation on j) and we would still havepairwise commuting vector fields [Y j , Yk ] = 0 for all j, k = 1, . . . , n. Notice thatthe completeness condition will not be spoiled by redefining the vector fields in thisform, i.e., the vector fields Yk will be complete (but with a different parametrization)and we would define an alternative action of Rn on M .

Thus what seems to matter is not the particular action ofRn but rather the integralleaves of the involutive distribution generated by X1, . . . , Xn . In those cases wherethe leaves are compact, say tori, we could require the choice of an action of Rn thatfactors to an action of Tn = R

n/Zn . Moreover we could select a particular basis ofvector fields such that X1, . . . , Xn each generate the action of a closed subgroup.With these particular prescriptions we decompose our dynamical vector field �. Letus denote the particularly chosen generators of closed subgroups by Z1, . . . , Zn ,then:

� = ω j Z j .

Thus when the closure of generic orbits of � are n-dimensional, the system doesnot have additional first integrals. If for some initial conditions the closure of theorbits does not coincide with the full torus, there are additional invariant relations.When for any initial condition, the closure of the orbits is some k < n dimensionaltorus then the system has additional first integrals and it is superintegrable.

When the closure is one dimensional for all initial conditions, the system is max-imally superintegrable.

It is now clear that the geometry of integrable and superintegrable systems withbounded flow is associated with a toroidal generalized bundle, i.e., projections onthe manifold M which have fibers diffeomorphic to tori of dimension going fromone to n. i.e. exactly the situation that happens if M is a compact 2n-dimensionalsymplectic manifold and the dynamics is invariant under the action of a torus groupT

n . Then because of Atiyah’s convexity theorem, the momentum map J : M → t∗is a surjective map onto a convex polytope P ⊂ t∗. The fibers of J are invariant tori

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that on points on the interior of P are n-dimensional. The fibers corresponding tothe boundary of the polytope are lower dimensional tori [At82].

Then the associated subgroup of diffeomorphisms of M determined by � is thesubgroup of bundle automorphisms of such generalized toroidal bundle. As we willsee later on, in connection with specific integrable or superintegrable systems, themost important obstruction to their beingdiffeomorphic is the energy-period theorem,that actually puts a restriction on the nature of the toroidal bundle of the system andin consequence on its group of diffeomorphisms.

8.4.4 Oscillators and Nonlinear Oscillators

To illustrate the previous situation we give now a few examples.We may consider the isotropic Harmonic oscillator with two degrees of freedom.

Say M = R4, ω = ∑

a dpa ∧ dqa , and H0 = 12

∑a(p2a + q2

a ). In this case onR40 = R

4 − {0}, an open dense submanifold, the dynamics generates orbits thatcoincide with their closure and they are one-dimensional. The toroidal bundle isprovided by:

S1 → R40 → S2 × R

+.

WehaveR40

∼= S3×R+ and the dynamics induces theHopf fibration S1 → S3 → S2

(see Sect. 5.4.5). The subgroup of diffeomorphisms of R40 which is selected by the

fibration is the group of projectable diffeomorphisms. Clearly this large group ofsymmetries, when applied to a chosen Hamiltonian description, will generate manymore alternative descriptions.

It should be remarked that the alternative Hamiltonian description provided bydp1 ∧ dq1 − dp2 ∧ dq2, and

H = 1

2(p21 + q2

1 ) − 1

2(p22 + q2

2 ),

cannot be derived from the standard one with the positive definite Hamiltonian H0because the diffeomorphism would preserve the signature of H (because it cannotmap compact energy levels, the ones of the Hamiltonian H0, into non-compact ones,the ones of H ).

The system is actually superintegrable, indeed the quotient manifold under theaction of the dynamics is three dimensional instead of two dimensional.

This example generalizes to any finite dimension and we have again for the n-dimensional isotropic harmonic oscillator the fibration:

S1 → R2n0 → CPn−1 × R

+.

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Again the symmetry group for the dynamics is the group of diffeomorphisms pro-jectable under the previous projection, hence it is diffeomorphic to the central exten-sion of the group of diffeomorphisms Diff(CPn−1 × R

+) by U (1).The 1-form

dτ = 1

n

∑k

(pkdqk − qkdpk

ω(p2k + q2k )

),

has the property that i�dτ = 1. Any closed 2-form on the quotient manifold and anyfunction on the quotient manifold without critical points on the invariant open densesubmanifold specified by p2k + q2

k �= 0, k = 1, 2, . . . , n, give raise to an alternativeHamiltonian description.

This example is a normal form for maximally superintegrable systems with one-dimensional closed orbits and constant period.

In higher dimensions we can consider the Hamiltonian H = ∑a ωa Ha , with

Ha = 12 (p2a + q2

a ), ωa ∈ R. The subgroup associated with the Hamiltonian vectorfields �a , i�a ω = d Ha , are closed subgroups. When all frequencies are pairwiseirrational, i.e., ωa/ωb is irrational, the closure of the generic orbit of � is the fulltorus, in this case there cannot be additional constants of motion. When some of thefrequencies are pairwise rational, the closure of a generic orbit is a torus of lowerdimensions. A particular example where the closure of the orbits goes from a one-dimensional torus to an n-dimensional one, depending on the initial conditions, isprovided by:

H =∑

a

±(Ha)2.

In this case we may also find invariant relations for particular values of the initialconditions.

This example gives raise to the so-called nonlinear oscillators and has been consid-ered in quantum mechanics to give interesting consequences at the level of Planck’sdistribution law and for alternative commutation relations [MM97, LM97].

8.4.5 Obstructions to the Equivalence of Integrable Systems

If we inquire about the obstructions to the existence of diffeomorphisms conjugat-ing two integrable systems, the energy-period theorem provides the first and mostimportant one. We refer to the literature [Ne02, Go69] for various versions of thistheorem, however a simple argument may be given as follows. On the carrier spaceof our dynamical system let us consider the 2-form dpk ∧ dqk − d H ∧ dt . Con-sider now the evolution t �→ t + τ , pk(t) = pk(t + τ), qk(t) = qk(t + τ) andH(q(t), p(t)) = H(q(t ′), p(t ′)) for all t ′ ∈ R. Because the evolution is canonical,we find d H ∧ dτ = 0 on the submanifold on which the period is a differentiable


function of (qk, pk). It follows that the period τ and the Hamiltonian H are func-tionally dependent.

It is well-known that the period of a dynamical system is an attribute of the vectorfield which is invariant under diffeomorphisms. It follows that if two Hamiltoniansystems have different periods, they cannot be connected via diffeomorphisms. Forinstance the isotropic Harmonic oscillator and the Kepler problem cannot be con-nected by a diffeomorphism. Indeed the map connecting solutions of the Harmonicoscillator and those of the Kepler problem, the Kustaanhneimo–Stiefel map, as wediscussed in Sect. 7.5.3, is a map defined on each energy level, i.e., for those orbitsthat have all the same period. The map changes from one energy-level to another(see for instance [DM05, DM05b]).

A more simple example is provided by H = 12 (p2 + q2) and H ′ = (p2 + q2)2.

For the first one, the frequency is independent of the energy, while for the second oneit depends on the initial conditions. The two systems cannot be diffeomorphic. Thiscircumstance was the main motivation to introduce the classification of dynamicalsystems up to parametrization, i.e., up to conformal factors [IM98].

We hope we have made clear that the geometrical picture we have derived is thebest one can do because each individual integrable systemwill give raise to infinitelymany different situations which cannot be classified in a meaningful way otherwise(i.e., identifying a finite or a countable family of equivalence classes).

8.5 Lax Representation

The origin of Lax representation is to be found in the theory of nonlinear partialdifferential equations. The Schrödinger operator

L = − ∂2

∂x2 + u(x, t),

has the remarkable property that for u rapidly decreasing at infinity, the spectrum ofL does not change with time if u satisfies the KdV equation [La68]. Lax suggestedthat the operator L(t) must be similar to a fixed one, say L(0), therefore L(t) =U (t)L(0)U (t)−1. This implies that ∂L

∂t = [A, L] where

A = dU

dtU−1.

There are no algorithmic procedures to construct Lax representations for “inte-grable” dynamical systems, in general they are constructed case by case with someeducated guesswork.

The so-called Laxmethod for the study of integrable systems is particularly usefulwhen dealing with integrable systems with an infinite number of degrees of freedom.The tangent bundle and the cotangent bundle picture are not immediately available

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8.5 Lax Representation 559

without further assumptions on the structure of the “configuration space”, an infinite-dimensional manifold. Normally the study of these systems require an extended useof the theory of partial differential equations, therefore it is out of the scope ofour present treatment. Nevertheless the main difference with the finite-dimensionalsituation is mostly technical rather than conceptual, therefore we shall restrict ourconsiderations to the finite-dimensional case.

The essential aspects of the Lax method rely on the following simple proposition:

Proposition 8.4 If A(t) and B(t) are two families of N × N matrices, with A(t)differentiable and B(t) bounded, assuming that they satisfy the differential equation:

d A

dt= [B, A], A(0) = A0, (8.8)

then all matrices A(t) have the same spectrum, with identical multiplicity (bothgeometrical and algebraic). We say that the map t �→ A(t) is isospectral.

Furthermore if ξk is an eigenvector of A0 belonging to the eigenvalue λ0, thevector U (t)ξk is an eigenvector of A(t) with the same eigenvalue when U (t) is thesolution of the equation:

dU (t)

dt= B(t)U (t), U (0) = I, U (t) �= 0. (8.9)

For the proof we notice that eigenvalues of A(t) are completely determined byknowing Tr (A − aI)k for a suitable choice of a and a sufficient number of choicesof the power k. Using the Leibnitz rule:

d

dtTr (A − aI)k =

k∑j=0

Tr

[(A − aI) j d A

dtTr (A − aI)k− j−1

]

=k∑

j=0

Tr[(A − aI) j [B, A]Tr (A − aI)k− j−1

]= 0

where for the last step we have used the cyclicity property of the trace, that is,Tr (M N ) = Tr (N M), and the derivation property of [B, ·].

Therefore if the equation of motion A = [B, A] is satisfied, we have found thatthe quantities:

fm = 1

mTr Am(t), m ∈ Z,

are constants of motion.We notice that also

f0 = log | det A(t)|



The statement concerning eigenvectors and eigenvalues is shown by using:

A(t) = U (t)A0U (t)−1,

with U (t) solving the Eq. (8.9). Because A(0) = A0 = A(0), the uniqueness of thesolution tells us that A(t) = A(t).

We see that from A0ξk = λkξk setting ξk(t) = U (t)ξk , we find:

A(t)ξk(t) = U (t)A0U (t)−1U (t)ξk = U (t)A0ξk = λkU (t)ξk = λkξk(t),

as we have stated.Now, as a corollary, we have:

Corollary 8.5 Let dx/dt = Y (x) be a dynamical system on some manifold M andϕ(t, x0) the solution with initial condition x0. If there exist two families of matricesA(x), B(x), such that

d

dtA(ϕ(t, x0)) = [B(ϕ(t, x0)), A(ϕ(t, x0))],

for all x0, then all eigenvalues of A(x) are constants of motion for our dynamicalsystem.

Remark 8.1 When thematrix B(t) is skew-symmetric thenmatricesU (t) are unitary.

Remark 8.2 Ifwewant to recover the notion of integrability in the sense ofLiouville–Arnold, we have also to show that the obtained constants of motion are in involutionand functionally independent.

It is quite clear that the Lax method may be considered to be a generalization ofthe Heisenberg form of the equations of motion in Quantum Mechanics. Here theequations are not restricted to observables, i.e., Hermitian operators:

id A

dt= [H, A],

whose exponential is given by unitary operators. In the Lax situation matrices areallowed to be generic matrices in the general linear group.

To find Lax families for a given dynamical system is rather non-trivial. Manyexamples, although in infinite dimensions, arise from geodetical motion on Liegroups (not restricted to be finite-dimensional). For simplicity we consider a Liegroup G realized as a group of matrices.

Geodetical motions are described by the equations of motion

d

dt

(g−1 dg

dt

)= 0;


or, equivalently: d/dt((dg/dt) g−1

) = 0.If B1, . . . , BN is a basis for the Lie algebra of the group G, the previous second-

order differential equation is equivalent to the family of first-order differential equa-tions: (

g−1 dg

dt

)= B j , j = 1, . . . , N .

It is not difficult to show that the second-order differential equations admit aLagrangian description by means of the Lagrangian:

L = 1

2Tr(g−1g)2 = 1

2Tr(gg−1)2,

with possible alternative Lagrangians

L K = 1

2TrK (g−1g)2,

with K any invertible matrix.By using the momentum map associated with the left or right action of the group

on itself, we find Lax-type equations on the dual of the Lie algebra expressed interms of Poisson brackets.

Going from g∗ (the dual of the Lie algebra) to g (the Lie algebra) by means of aninvariant pairing, we obtain equations of motion in the matrix Lax-type.

Many general aspects of dynamical systems associated with differential equationson Lie groups will be considered in Chap.9. Here we simply comment that byreplacing a matrix with a (1, 1)-tensor field we find Lax-type equations when aninvariant (1, 1)-tensor field can be factorized by means of a symplectic structureand a Poisson tensor. This approach is closely related to recursion operators andbiHamiltonian systems.

8.5.1 The Toda Model

The Toda model describes the motion of N particles on a line interacting amongthemselves with a repulsive force decreasing exponentially with the distance. To getmore symmetrical formulae, it is convenient to set:

q0 = −∞, qN+1 = +∞,

with q0, q1, . . . , qN , qN+1 the position coordinates. Hamilton equations are givenby:

qk = pk; pk = e−(qk−qk−1) − e−(qk+1−qk), k = 1, . . . , N .

These equations admit alternative Hamiltonian descriptions:

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The first one (the standard):

ω =N∑

k=1

dqk ∧ dpk, H(q, p) =N∑

k=1

(1

2p2k + e−(qk+1−qk)

),

but there is another one:

� =N∑

k=1

(e−(qk+1+qk)dqk ∧ dk+1 − pkdqk ∧ dpk

)+ 1

2

N∑j,k=1

ε( j − k)dpk ∧ dp j ,

with

K (q, p) =N∑

k=1

(1

3p3k + (pk+1 + pk)e−(qk+1+qk)

), ε(0) = 0, ε(p) = p

|p| (p �= 0).

It is possible to compose one symplectic structure with the inverse of the other todefine a (1, 1)-tensor field S = �−1 ◦ ω, we find:

S =[

B A−E B

],

with

Ai j = δi+1, j e−(qi+1−qi ) − δi, j+1e−(q j+1−q j ) Bi j = pi δi, j ; Ei, j = ε( j − i).

Constant of the motion are given by TrSk . We find for a few of them:

K1 = TrS = 2N∑

k=1

pk;

K2 = 1

2TrS2 =

N∑k=1

(p2k + 2e−(qk+1−qk));

K3 = 1

3TrS3 =

N∑k=1

(2

3p3k + (pk+1 + pk)e−(qk+1−qk)

);

K4 = 1

4TrS4

=N∑

k=1

(1

2p4k + (p2k + pk pk+1 + p2k+1)e

−(qk+1−qk) + e−2(qk+1−qk) + 2e−(qk+2−qk)

).

By setting:


Di j = (δi, j − δ j,i+1)e−(qi+1−qi ) + (δi, j − δ j,i−1)e

−(qi −qi−1)

we find the equations of motion in the Lax-type equations:

d A

dt= −[B, D], d B

dt= 1

2[E, D],

an alternative Lax representation is provided by:

d L

dt= [L , V ],

with:Li j = 1

2

(pi δi, j + δ j,i+1e−(qi+1−qi )/2 + δi, j+1e−(q j+1−q j )/2

),

andVi j = 1

2

(δi, j e−(qi+1−qi )/2 + δi, j+1e−(q j+1−q j )/2

),

with L symmetric and V skew-symmetric.

8.6 The Calogero System: Inverse Scattering

8.6.1 The Integrability of the Calogero-Moser System

In Sect. 1.2.12 we derived the Calogero-Moser system from a free systems by usingdifferent reduction techniques. In doing so, we know that our system is “integrable”,however it is not obvious how the properties of the Calogero-Moser system arerelated with the geometric notion of integrability that we have been discussing inthis chapter. In this section we will approach again the Calogero-Moser system froma different perspective enhancing some of the integrability properties that are moreappealing from the present discussion.

We consider N mass points, assumed to have all the same mass which we putequal to one for convenience, moving on a line. They interact with each other via arepulsive force proportional to the third power of the inverse of their mutual distance.Let us denote by qn the position of the nth point mass and by pn the correspondingmomentum, the Hamiltonian describing the system will be:

H(q, p) = 1

2

N∑n=1

p2n + V (q), V (q) =∑

1≤n<m≤N

1

(qn − qm)2.

The associated second-order equations of motion will be:

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d2qn

dt2= − ∂V

∂qn= 2

∑k �=n

qk − qn

(qk − qn)4, n = 1, . . . , N , (8.10)

and, pn = qn .A Lax-type description of Eq. (8.10) is provided by the following two matrix-

valued functions:

A(q, p)m,n = δm,n pn − iδm,n − 1

qm − qn, (8.11)

B(q, p)m,n = iδm.ndn + iδm,n − 1

(qn − qm)2, dn =

∑k �=n

1

(qn − qk)2. (8.12)

We have already seen that, for N = 2, 3, this system may be obtained as a suitablereduction of a free system on the space of Hermitian matrices. In particular, thisshows that the motion here is not bounded contrary to the analysis at the beginningof this chapter, see Sect. 8.2, and therefore not contained in a compact submanifold.As a result we cannot define action-angle variables for this system and the adequategeometrical notion of integrability goes beyond the methods of toric manifolds dis-cussed so far.

It is possible to obtain the general Calogero system by considering the free motiononHermitianmatrices and performing a reduction as we did for N = 2. However herewe would like to explain a different method which would apply when the conditionof boundedness of et A does not apply.

8.6.2 Inverse Scattering: A Simple Example

Definition 8.6 Given a second-order dynamics on a vector space of dimension N, we willsay that the Cauchy data x(0), x(0) define a scattering state (or scattering trajectory) if thetwo limits:

α±n = lim

t→±∞ xn(t); β±n = lim

t→±∞(xn(t) − tαn),

exist.

The inverse scattering method relies on the fact that, under suitable conditions,the transformations:

(p, x) �→ (α−, β−); (p, x) �→ (α+, β+),

are symplectic transformations with respect to the standard symplectic structure onT ∗

RN , or with respect to the standard symplectic structure on T ∗

R3N if we are

considering N particles moving in Euclidean three dimensional space R3, and the

functions αk are first integrals in involution pairwise or, in other words, the systemis asymptotically free.

8.6 The Calogero System: Inverse Scattering 565

Let us consider H(x, p) the Hamiltonian of the system we are considering. Letus denote by ϕt : ξ = (x, p) → ϕt (ξ) the flow associated with H and by ϕ0

t : ξ =(x, p) → ϕ0

t (ξ) the flow associated to a comparison Hamiltonian H0. We often takeas comparison Hamiltonian a ‘free’ motion system H0(x, p) = 1

2 pT Mp where M isa definite positive matrix (called the mass matrix of the system).

The asymptotic behavior of asymptotic trajectories express the fact that the limit:

�(ξ) := limt→∞(ϕ0−t ◦ ϕt )(ξ)

exists uniformly for ξ = (x, p) in any compact set (with similar considerationsapplying for t → −∞).

The transformation� turns out to be symplectic because it is the limit of symplec-tic transformations. More properly we should verify that also T (ϕ0−t ◦ ϕt ) convergesand this should be verified using the explicit form of the derivative. This conditionwill be obtained on each instance under appropriate hypotheses on the speed ofconvergence to zero of H − H0 when ||x || → ∞.

By construction � maps the Cauchy data onto ‘scattering data’ (α, β); in this wayα and β are differentiable functions of the coordinates (x, p) and are themselvescanonical variables because � is a symplectic transformation.

In particular the functions αk are N functions pairwise in involution and are func-tionally independent. They are defined only in the region of phase space correspond-ing to points for which scattering states exist in the asymptotic sense either fort → +∞ or t → −∞. The fact that our functions αk are constants of motion followsfrom the observation that in any scattering process:

ϕ0s ◦ � = � ◦ ϕs ,

and therefore H = H0 ◦ �. Then, being (α1, . . . , αN ) a set of first integrals for theflow associated with H0, we deduce that any αk ◦ � is a constant of motion for theflow associated to H .

Notice that any system whose trajectories have the asymptotic behavior wedescribed earlier has a carries space which is not compact, i.e., the assumption on theboundedness of the flow is not satisfied. The inverse scattering method may be con-sidered to be a completion to the action-angle variables representation of integrablesystems when the flow is not bounded.

8.6.3 Scattering States for the Calogero System

For the Calogero system, using the method of inverse scattering, we find for t → ∞:

qn(t) = α+n t + β+

n + O(t−1), pn(t) = α+n + O(t−1),


and for t → −∞:

qn(t) = α−N−n+1t + β−

N−n+1 + O(t−1), pn(t) = α−N−n+1 + O(t−1).

These statements follow from the form of the matrices in the Lax representationform, Eq. (8.11). In fact because

limt→±∞ pn(t) = α±

n , limt→±∞ ||qn(t) − qm(t)|| = +∞, n �= m,

then, because of Eq. (8.11):

limt→±∞ Am,n(q(t), p(t)) = lim

t→±∞

[δm,n pn(t) − i

δm,n − 1

qm(t) − qn(t)

]= δm,nα±

n .

Because the eigenvalues of the matrix A are first integrals, they will coincide withthe eigenvalues of the limit matrix, thus:

α±n = λ±

π(n)= λπ±(n), ∀n,

where n �→ π±(n) is a permutation. It follows that the functions α−n will differ from

the α+k at most for a permutation.

The limit of pn(t) and ||qn(t) − qm(t)|| follows from the following considerations.From the equations of motion we have:

qN − q1 = 2∑

n<N

1

(qN − qm)3+ 2

∑1<m

1

(qm − q1)3. (8.13)

The right-hand side of (8.13) is strictly positive for t finite. Indeed, material pointsmust keep their relative order on the line, due to the conservation of energy and thesingularity of the potential energy when two point masses coincide.

Integrating Eq. (8.13) between two finite times t1, t2 and using the fact that con-servation of energy implies that each one of the qn(t) is equibounded as functions oft , we have:

+∞∫−∞

(qN (t) − qn(t))−3dt < +∞, n < N ,

+∞∫−∞

(qn(t) − q1(t))−3dt < +∞, n > 1.

Using now:

qN−1 − q2 = 2∑

2<n<N−1

1

(qN−1 − qn)3+ 2

∑2<m<N

1

(qm − q2)3− 2

(qN − qN−1)3 − 2

(q2 − q1)3,

8.6 The Calogero System: Inverse Scattering 567

and taking into account that the last two terms in the previous equation, even thoughnot positive, admit a bounded integral to derive:

+∞∫−∞

(qN−1(t) − qn(t))−3dt < +∞, n < N − 1,

+∞∫−∞

(qn(t) − q2(t))−3dt < +∞, n > 2.

Thus, by induction we get:

+∞∫−∞

(qn(t) − qm (t))−3dt < +∞, ∀n �= m.

We finally deduce that:

qn(T ) = qn(0) +T∫0

∑n �=m

(qn(t) − qm (t))−3dt

and therefore the limits for qn(t) exist and are finite for t → ±∞ and the asymptoticbehavior is proved.

To show that αn and βn are symplectic coordinates and that their commutationrelations are the required ones is more cumbersome, and to rely on the reductionprocedure is much more convenient.

References

[MM74] Markus, L., Meyer, K.R.: Generic Hamiltonian dynamical systems are neither inte-grable nor ergodic. Mem. Am. Math. Soc. 144, 1–52 (1974)

[IM12] Ibort, A., Marmo, G., Rodríguez, M.A., Tempesta, P.: Nilpotent Integrability (Preprint)(2012).

[Kl72] Klein, F.: A comparative review of recent researches in geometry. Univ. of Erlangen(1872)

[Kl92] Klein, F.: A comparative review of recent researches in geometry. Bull. New YorkMath. Soc. 2, 215–249 (1892). arXiv: 0807.3161 v1[math.H0]

[Ba86] Banyaga, A.: On isomorphic classical diffeomorphism groups I. Proc. Am. Math. Soc.98, 113–118 (1986)

[BG88] Batlle, C., Gomis, J., Pons, J.M., Román-Roy, N.: Lagrangian and Hamiltonian con-straints for second-order singular Lagrangians. J. Phys. A: Math. Gen. 21, 2693–2703(1988)

[Gr00] Grabowski, J.: Isomorphisms of Poisson and Jacobi brackets. Poisson geometry.Banach Center Publ. 51, 79–85 (2000)

[FL89] de Filippo, S., Landi, G., Marmo, G., Vilasi, G.: Tensor fields defining a tangent bundlestructure. Ann. Inst. H. Poincaré 50, 205–218 (1989)

http://arxiv.org/abs/0807.3161


[Le34] Levi-Civita, T.: A general survey of the theory of adiabatic invariants. J. Math. Phys.13, 18–40 (1934)

[GM93] Giordano, M., Marmo, G., Rubano, C.: The inverse problem in the Hamiltonian for-malism: integrability of linear Hamiltonian fields. Inverse Prob. 9, 443–467 (1993)

[EM10] Ercolessi, E., Marmo, G., Morandi, G.: From the equations of motion to the canonicalcommutation relations. Riv. Nuovo Cim. 33, 401–590 (2010)

[At82] Atiyah, M.: Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14,1–15 (1982)

[MM97] Man’ko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: f -oscillators and nonlinearcoherent states. Phys. Scripta 55, 528–541 (1997)

[LM97] López-Peña, R., Manko, V.I., Marmo, G.: Wigner problem for a precessing dipole.Phys. Rev. A 56, 1126–1130 (1997)

[Ne02] Nekhoroshev, N.N.: Generalizations of Gordon theorem. Regular Chaotic Dyn. 7, 239–247 (2002)

[Go69] Gordon, W.B.: On the relation between period and energy in periodic dynamical sys-tems. J. Math. Mech. 19, 111–114 (1969)

[DM05] D’Avanzo, A., Marmo, G.: Reduction and unfolding: the Kepler problem. Int. J. Geom.Meth. Mod. Phys. 2, 83–109 (2005)

[DM05b] D’Avanzo, A., Marmo, G., Valentino, A.: Reduction and unfolding for quantum sys-tems: the hydrogen atom. Int. J. Geom. Meth. Mod. Phys. 2, 1043–1062 (2005)

[IM98] Ibort, A., Marmo, G.: A new look at completely integrable systems and double Liegroups. Contemp. Math. 219, 159–172 (1998)

[La68] Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm.Pure Appl. Math. XXI, 467–90 (1968)

Chapter 9Lie–Scheffers Systems

If only I knew how to get mathematicians interested intransformation groups and their applications to differentialequations. I am certain, absolutely certain, that these theorieswill some time in the future be recognized as fundamental. WhenI wish such a recognition sooner, it is partly because then Icould accomplish ten times more.

Sophus Lie, Letter to Adolf Mayer, 1884.

9.1 The Inhomogeneous Linear Equation Revisited

In 1893, Lie and Scheffers [Lie93] presented a result that has a deep implicationregarding the notion of integrability that we are describing, but that has been almostunnoticed since then (for two modern general references see [Ca00, Ca07b] and[CL11]).Wewill start by analysing the simplest case of linear (systemsof) differentialequations.

To start with we can consider the inhomogeneous linear differential equation

x = c0(t) + c1(t) x, (9.1)

which is well known to admit a solution in terms of two quadratures:

x(t) = exp

⎛⎝

t∫0

c1(t′) dt ′

⎞⎠⎡⎢⎣C +

t∫0

exp

⎛⎜⎝−

t ′∫0

c1(t′′) dt ′′

⎞⎟⎠ c0(t

′) dt ′

⎤⎥⎦ . (9.2)

It contains two simpler cases when either c0 or c1 identically vanish.The first remarkable point is that there is a superposition rule for such a differential

equation which is given by the function:

φ(x1, x2; k) = x1 + k (x2 − x1), (9.3)


569

570 9 Lie–Scheffers Systems

i.e., the differential equation is such that its general solution is given by

x(t) = φ(x1(t), x2(t); k) = x1(t) + k (x2(t) − x1(t)),

where x1(t) and x2(t) are two generic, but not proportional, particular solutions.A second interesting property is that its solutions are integral curves of the t-

dependent vector field in R

X = (c0(t) + c1(t)x)∂

∂x, (9.4)

which is a linear combination with t-dependent coefficients, X (t) = c0(t) X0 +c1(t) X1, of the two vector fields in R

X0 = ∂

∂x, X1 = x

∂

∂x, (9.5)

with the following commutation relation:

[X0, X1] = X0, (9.6)

i.e., they are closing on a two-dimensional real Lie algebra isomorphic to the Liealgebra of the affine group in one dimension, a(1, R). The t-dependent vector fieldin R for the two simpler particular cases, either when c0 = 0 or c1 = 0, reduces toa multiple by a t-dependent coefficient of the vector field X1 or the vector field X0,respectively. These cases admit simpler superposition rules, involving one particularsolution: φ(x; k) = k x , when c0 = 0 and φ(x; k) = x + k when c1 = 0.

Another useful property is that if we consider the set of transformations

x(t) �→ x(t) = α(t)x(t) + β(t), α(t) �= 0,∀t, (9.7)

we see that x(t) is a solution of the given equation if and only if x(t) is a solution ofa similar equation

˙x = c′0(t) + c′

1(t)x, (9.8)

with

c′0 = α c0 + β − β c1 − β

α

α, c′

1 = c1 + α

α.

The mentioned set of transformations is a group G isomorphic to the group of curvesin the affine group G in one dimension and is usually called the structure preservinggroup [Sc02, Nd08]. This way we obtain a group action of G on the set of inhomoge-neous linear differential equations, in the sense that the composition of the transfor-mations determined by first (α1(t),β1(t)) and second (α2(t),β2(t)) is the same asthe transformation corresponding to the product (α2(t)α1(t),α2(t)β1(t) + β2(t)).

9.1 The Inhomogeneous Linear Equation Revisited 571

Furthermore, one can see that there is an element (α(t),β(t)) such that the solutionfor a given initial condition x(0) = x0 is

x(t) = α(t) x0 + β(t). (9.9)

In fact, then

α(t) = exp

⎛⎝

t∫0

c1(t′) dt ′

⎞⎠ ,

β(t) = exp

⎛⎝

t∫0

c1(t′) dt ′

⎞⎠

t∫0

exp

⎛⎜⎝−

t ′∫0

c1(t′′) dt ′′

⎞⎟⎠ c0(t

′) dt ′. (9.10)

It is also to be remarked that the action of the group G can be used to reduce thegiven equation to one of the simpler cases. For instance, if x1(t) is a non-vanishingsolution of the given equation, then under the transformation x(t) = x1(t)+ x(t), as

x1(t) + ˙x(t) = c0(t) + c1(t)(x1(t) + x(t)) ,

the new equation becomes ˙x = c1(t)x , which is one of the two above mentionedsimpler cases.

9.2 Inhomogeneous Linear Systems

We recall that in Chap.2, Sect. 2.2 it was discussed in detail systems of differentialequations on vector spaces E associated with linear maps. They have associatedone-parameter subgroups of linear automorphisms of E . So, consider the system

dx

dt= A x, x ∈ E, (9.11)

with A being a given endomorphism of E . This system defines a flow φt whichcan be considered as a curve g(t) on the general linear group GL(n, R) given byg(t) = φt = eAt . Such flow satisfies (Eq. (2.5)):

dφt

dt= A φt , (9.12)

the one-parameter group property,φt1◦ φt2 = φt1+t2 , and the initial conditionφ0 = I .Note that choosing as initial condition each element of the canonical basis of R

n wefind that the columns of φt = eAt are a set of linearly independent solutions of the

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given system. Any other such a set defines a t-dependent matrix P(t) which is alsoa solution of the matrix differential equation

dP

dt= AP,

but for which P(0) is a regular matrix such that P(0) �= I . Then P(t) = exp(tA)P(0)

and the important point is that A can be recovered from P as A = (dP/dt) P−1.This implies that the flow φt can be determined from a fundamental set of solutions,i.e. when the n × n matrix X whose columns are the vectors defining the solutions{x(1)(t), . . . , x(n)(t)},

X (t) = (x(1)(t), . . . , x(n)(t)),

is an invertible matrix, then the equation

X (t) = etA X (0),

shows that the evolution operator etA is determined as etA = X (t)X (0)−1, i.e.,the fundamental system of solutions allows us to find the flow of our first-orderdifferential equation system. Moreover, the relation x(t) = eAt x(0) for a solutiontells us that the general solution can be written as a linear combination of n linearlyindependent solutions. In other words, the function � : R

n(n+1) → Rn given by

�(u1, . . . , un, c1, . . . , cn) = c1 u1 + · · · + cn un, (9.13)

is such that, if {x(1)(t), . . . , x(n)(t)} is a set of linearly independent solutions of thesystem, then x(t) = �(x(1)(t), . . . , x(n)(t), c1, . . . , cn) gives us the general solution.This superposition rule (9.13) for solutions is the same for all homogeneous linearsystems, no matter of the choice of the coefficients Ai

j .From here it is clear that with any linear differential equation on R

n we canassociate an equation on GL(n, R) by setting

g = Ag, (9.14)

or in other words, that is an equation for a curve in the group of automorphisms of E :

dg

dtg−1 = A. (9.15)

We recall that the matrix A is an element of the Lie algebra gl(n, R).Moreover, this way of finding the resolvent can also be used for time-dependent

systems

dxi

dt=

n∑j=1

Aij (t) x j , i = 1, . . . , n . (9.16)

9.2 Inhomogeneous Linear Systems 573

or in a shorter formdx

dt= A(t) x, (9.17)

as was shown in Sect. 2.2.4. The general theory of systems of differential equationsasserts that there exists a curve g(t) = φt in GL(n, R) such that x(t) = φt (x(0)),but now φt1 ◦φt2 = φt1+t2 no longer holds, but only the flow Eq. (9.12) and the initialcondition φ0 = id.

Now, if the matrix A is not constant, but we suppose that the matrix

B(t) =t∫

0

A(ζ) dζ, (9.18)

that satisfies B(0) = 0 anddB(t)

dt= A(t), (9.19)

is such that A(t) B(t) = B(t) A(t), what, for instance, happens if A(ζ1)A(ζ2) =A(ζ2)A(ζ1), for all couples of values ζ1 and ζ2, then, the commutativity of A(t) andB(t) implies that

d Bm(t)

dt= m A(t) Bm−1(t), (9.20)

and then,t∫

0

A(ζ) Bm−1(ζ) dζ = 1

mBm(t).

Therefore we find that the flow φt is given by

φt = exp B(t), (9.21)

because

d

dt

( ∞∑k=0

1

k! Bk(t)

)= A(t)

( ∞∑k=1

1

(k − 1)! Bk−1(t)

)= A(t)

( ∞∑k=0

1

k! Bk(t)

),

where use has been made of (9.20), and then

d

dtexp(B(t))x0 = A(t) (exp(B(t))x0) .

In the generic case of arbitrary dependence of A in the parameter t , we cannotexplicitly write the automorphism φt but normally is written using the time orderedexponential as explained in Sect. 2.2.4, Eq. (2.43)

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The linear superposition property for non-autonomous systems is still valid andthen we can take advantage of the knowledge of a fundamental set of solutionsto determine φt . In fact if X (t) is the matrix determined by the set of n linearlyindependent solutions of the equation, then from

X (t) = φt X (0)

we get φt = X (t)X (0)−1.We obtain in this way a curve g(t) = φt in GL(n, R), which is the solution of the

time-dependent equation

dg

dtg−1 = A(t),

with A(t) being a curve in gl(n, R), such that g(0) = I .The point we want to remark is that if the evolution preserves some geometrical

structure, then

A(t) = dφt

dt◦ φ−1

t

lies in certain subalgebra of GL(n, R) and the computation of φt becomes simpler.As in the preceding case of the linear equation, if we define a transformation

Y (t) = B(t)X (t), (9.22)

where B(t) is an invertible matrix, the new vectorial function Y (t) satisfies a similarsystem

dyi

dt=

n∑j=1

A′i (t) j ya j , i = 1, . . . , n ,

where

A′(t) = B(t) B(t)−1 + B(t) A(t) B(t)−1, (9.23)

and use can be made of this property to transform a given system into a simpler one,or to obtain information on the given system (see later on). It is worth noting thatthis shows that GL(n, R) is contained in the structure preserving group.

The critical fact is that the general solution is determined by a linear operator froma set of fundamental solutions, and that there is a linear superposition principle.

It is very natural to ask what happens when the vector field is nonlinear. Theanswer is that, at least in some cases, there is a kind of nonlinear superpositionrule, as it was proved by Lie [Lie93]. This nonlinear superposition rule is simplya generalization of the previous construction to those cases where the action of thegroup is not linear and R

n is replaced for a manifold M . A simple example wouldbe the inhomogenous linear system


dxi

dt=

n∑j=1

Aij (t) x j + ai (t) , i = 1, . . . , n. (9.24)

for which there is also a superposition rule involving now n + 1 solutions,

�(u1, . . . , un+1, c1, . . . , cn) = u1 + C1(u2 − u1) + · · · + cn(un+1 − u1) (9.25)

and then the general solution can be obtained from a fundamental set of n + 1particular solutions as

x(t) = �(x(1)(t), . . . , x(n+1)(t), c1, . . . , cn)

= x(1)(t) + c1(x(2)(t) − x(1))(t) + . . . + cn(x(n+1)(t) − x(1)(t)).

The properties of the general solution of a homogeneous linear system of first-order differential equations on R

n can be studied by means of the transformationprocedure pointed out before.

Given a linear system x = A(t)x , Eq. (9.11), we first can prove that the generalsolution is of the form x(t) = H(t)K , where K is a constant column matrix andH(t) is a square matrix whose columns are n linearly independent for each value oft particular solutions of x = A(t)x . Such a matrix H(t) is necessarily non-singular,and is called a fundamental matrix.

In fact, suppose we make a change of variables by

x = α(t)x, (9.26)

where α denotes a t-dependent invertible matrix. As indicated above, in terms of thenew variable the original equation reads ˙x = A(t)x , where

A = α α−1 + α A α−1. (9.27)

Let us first prove how we can choose α in such a way that A = 0. As far asuniqueness is concerned, note thatα1 andα2 are two suchmatrices if and only if thereexists a constant invertible matrix P such that α2 = Pα1. In fact, if α1 + α1A = 0then for each constant matrix P , α2 = Pα1 satisfies α2 + α2A = 0. Conversely, ifα1 + α1A = 0 and α2 + α2 A = 0, then, as dα−1

1 /dt = Aα−11 ,

d

dt(α2 α−1

1 ) = α2α−11 + α2

d

dtα−11 = −α2 A α−1

1 + α2 A α−11 = 0,

and therefore the matrix P = α2 α−11 is constant.

Suppose now that n particular solutions of (9.11), x(1), . . . , x(n), are known thatare linearly independent for each value of t , and define the matrix H(t) such that

H ij (t) = xi

( j)(t) , i, j = 1, . . . , n. (9.28)


This matrix is such that

dHij

dt= dxi

(j)

dt=

n∑k=1

Aik xk

( j) =n∑

k=1

Aik H k

j = (AH)ij

or in matrix form,d H

dt= AH.

As the matrix H is invertible, taking time derivative in H H−1 = I we obtain

dH

dtH−1 + H

dH−1

dt= 0 =⇒ dH−1

dt= −H−1 dH

dtH−1 = −H−1 A,

and consequently α = H−1 satisfies α + αA = 0. Therefore using (9.26) withα = H−1 produces A = 0 and then the general solution is x = K = x(t0), fromwhere we recover the form

x(t) = H(t)x(t0) = H(t)K = H(t)H−1(t0)x0. (9.29)

for the general solution of (9.11).Any other transformation (9.26) for which A is zero must be obtained by right

multiplication of the considered H by a constantmatrix. This corresponds to a changeof basis for the linear space of solutions of the system: a matrix H ′ determined byanother basis of the space of solutions for the system is related with H by right-multiplication with a constant invertible matrix: H ′(t) = H(t)P .

Let us now consider the above mentioned inhomogeneous linear system

x i =n∑

j=1

Aij (t)x j + bi (t) , i = 1, . . . , n. (9.30)

or written in matrix formx = A(t)x + b. (9.31)

In full similarity with the preceding case, if we introduce the change of variable

x = α(t)x + β(t), (9.32)

where α(t) denotes a t-dependent invertible matrix and β(t) a column matrix, withinverse transformation

x = α−1(t)(x − β(t)), (9.33)

then x satisfies˙x = A(t)x + b(t), (9.34)


where

A = α α−1 +α A α−1 , b(t) = αb + β − α α−1β −α A α−1β = αb + β − Aβ.

(9.35)Suppose that we know n linearly independent solutions of the associated homo-

geneous system, construct the matrix H and choose α = H−1 as before. Then, asα + αA = 0, the preceding equations reduce to

A = 0 , b = H−1b + β.

Therefore, in order to get b = 0 the column matrix β(t) must be chosen such thatH−1b + β = 0, which gives raise to the solution for β

β(t) = β(t0) −t∫

t0

H−1(t ′)b(t ′) dt ′.

With such choices for α and β, (9.34) turns out to be ˙x = 0, with solutionx(t) = x(t0), i.e.

x(t) = H(t)(x(t0) − β(t)) = H(t)

⎛⎝x(t0) − β(t0) +

t∫t0

H−1(t ′)b(t ′) dt ′⎞⎠ ,


x(t) = H(t)H−1(t0)x(t0) +t∫

t0

H(t)H−1(t ′)b(t ′) dt ′. (9.36)

and if we introduce the Green matrix G(t, t ′) = H(t)H−1(t ′) the preceding equationbecomes:

x(t) = G(t, t0)x(t0) +t∫

t0

G(t, t ′)b(t ′) dt ′. (9.37)

Note that the Green matrix G(t, t ′) does not depend on the choice for theset of independent solutions of the homogeneous system, because for any otherchoice H ′(t) = H(t)P , with P being a constant matrix and then G′(t, t ′) =H ′(t)H ′−1(t ′) = H(t)H−1(t ′) = G(t, t ′). Furthermore, the Green matrix G(t, t ′) issuch that

∂G(t, t ′)∂t

= H(t)H−1(t ′) = A(t)H(t)H−1(t ′) = AG(t, t ′),


and G(t, t) = I . Conversely, given a matrix satisfying these two properties, thesolution of (9.31) is given by (9.37), because taking derivatives with respect to t inthis expression we see that

dx

dt= ∂G(t, t0)

∂tx0 + G(t, t)b(t) +

t∫t0

∂G(t, t ′)∂t

b(t ′) dt ′

i.e.,

dx

dt= A(t)G(t, t0)x(t0) + A(t)

t∫t0

G(t, t ′)b(t ′) dt ′ + b(t)

and therefore x(t) given by (9.37) satisfies (9.31).

9.3 Non-linear Superposition Rule

We can now look for a group theoretical approach to the problem. First of all, system(9.17) can be seen as the one determining the integral curves of the vector field

X A =n∑

i, j=1

Aij (t) x j ∂

∂xi,

while system (9.24) determines the integral curves of

X(A,a) =n∑

i, j=1

Aij (t) x j ∂

∂xi+

n∑i=1

ai (t)∂

∂xi.

The vector field X A can be written as a linear combination

X A =n∑

i, j=1

Aij (t) Xi j

of the vector fields

Xi j = x j ∂

∂xi, i, j = 1, . . . , n, (9.38)

and the vector field X(A,a) as a linear combination

X(A,a) =n∑

i, j=1

Aij (t) Xi j +

n∑i=1

ai (t) Xi , (9.39)

9.3 Non-linear Superposition Rule 579

with Xi the vector fields

Xi = ∂

∂xi, i = 1, . . . , n. (9.40)

Notice that

[Xi j , Xkl ] =[

x j ∂

∂xi, xl ∂

∂xk

]= δil x j ∂

∂xk− δk j xl ∂

∂xi,

i.e.,[Xi j , Xkl ] = δil Xk j − δk j Xil , (9.41)

which means that the vector fields {Xi j | i, j = 1, . . . , n}, which are the only onesappearing in the case of an homogeneous system, close on an n2-dimensional Liealgebra isomorphic to the gl(n, R)-algebra. It suffices to compare these commutationrelations with those of the gl(n, R)-algebra. The latter is generated by the matricesEi j with elements (Ei j )kl = δikδ jl , which satisfy

[Ei j , Ekl ] = δ jk Eil − δil Ek j . (9.42)

Moreover,[Xi , Xk] = 0, ∀ i, k = 1, . . . , n, (9.43)

and[Xi j , Xk] = −δk j Xi , ∀ i, j, k = 1, . . . , n. (9.44)

Therefore, the Lie algebra generated by {Xi j , Xk | i, j, k = 1, . . . , n} is isomor-phic to the (n2 + n)-dimensional Lie algebra of the affine group.

The generalisation of such a property is obvious and it was stated by Lie as follows[Lie93]:

Theorem 9.1 (Lie–Scheffers theorem) Given a system

dxi

dt= Xi (x, t), i = 1, . . . , n, (9.45)

the general solution can be expressed in terms of m fundamental solutions if thereare r vector fields in R

n, Y1, …, Yr , such that the vector field X,

X =n∑

i=1

Xi (x, t)∂

∂xi(9.46)

can be expressed as a linear combination

X = b1(t)Y1 + · · · + br (t)Yr (9.47)


and furthermore the vector fields Yα, close on a finite-dimensional real (or complex)Lie algebra, with dimension r, i.e., there exist r3 real numbers cαβ

γ such that

[Yα, Yβ] =r∑

γ=1

cαβγ Yγ . (9.48)

Moreover, in this case mn ≥ r .

The examples of homogeneous and inhomogeneous linear systems mentionedabove are particular instances of this result. In the first case m = n, while in thesecond one m = n + 1. In the most general case m is arbitrary and the superpositionrule will be a function � : R

n(m+1) → R, x = �(x(1), . . . , x(m); k1, . . . , kn),

such that the general solution is

x(t) = �(x(1)(t), . . . , x(m)(t); k1, . . . , kn) ,

with {x(a)(t) | a = 1, . . . , m} being a set of particular solutions of the sys-tem and where k1, . . . , kn are real numbers. That is a generalisation of linearsuperposition rules for homogeneous linear systems (for which m = n and x =�(x(1), . . . , x(n); k1, . . . , kn) = k1 x(1) + · · ·+ kn x(n)) but the function � is nonlin-ear in the more general case.

As another example we can consider the Bernoulli differential equation

y = b1(t) y + b2(t) yn, n �= 0, 1, (9.49)

whose solutions are the integral curves of the t-dependent vector field

Y = b1(t)Y1 + b2(t) Y2 (9.50)

with

Y1 = y∂

∂y, Y2 = yn ∂

∂y, (9.51)

satisfying the commutation relations

[Y1, Y2] = (n − 1)Y2, (9.52)

and therefore Y2 and1

1 − nY1 satisfy the same commutation relations as the vector

fields X0 and X1 given in (9.5). This equation is therefore of Lie–Scheffers type withassociated Lie algebra a(1, R). There is only one action, up to conjugation by a dif-feomorphism, of a(1, R) on the real line, and consequently there is a diffeomorphismφ of R such that y = φ(x) and then y = φ(x) must be such that

1

1 − ny = x

dy

dx,

dy

dx= yn,

9.3 Non-linear Superposition Rule 581

therefore x and y must be related as follows:

x = y1−n

1 − n, (9.53)

the change used for reduction of the Bernoulli’s equation to the inhomogeneouslinear one, the factor 1/(1 − n) being irrelevant.

As the general solution of the linear equation x = a1(t) x + a0(t) can be writtenas a linear combination of two solutions x1 and x2, x(t) = k x1(t)+(1−k) x2(t), thegeneral solution of Bernoulli equation can be written as a function of two particularsolutions as follows:

y = [k y1−n1 + (1 − k) y1−n

2 ]1/(1−n) (9.54)

which is a superposition rule of solutions but it is not an affine superposition anymore.Another relevant example is to be studied in a later section, the Riccati equation,

of a high relevance in classical and quantum physics.We remark that in geometric terms, systems of first-order differential equations

appear as local expressions determining the integral curves of a vector field in amanifold M . Correspondingly, we call Lie–Scheffers system a t-dependent vectorfield on a manifold M for which there exists a superposition rule, which now can beseen as a map

� : Mm+1 = M × · · · × M → M.

Explicit examples will be given in the next sections.

9.4 Related Maps

Many of the properties of the systems that we are going to consider here have a com-mon framework. The idea is just to consider maps between differentiable manifoldsA : M1 → M2, B : M3 → M4, α : M1 → M3 and β : M2 → M4, in such a waythat B ◦ α = β ◦ A, i.e., the following diagram be commutative:

That is for instance the case when M1 = M2, M3 = M4 and a Lie group G actsboth on M1 and on M3 and A = �1g , B = �2g , with g ∈ G, in such a way that thediagramme is commutative for any g ∈ G, i.e., �2g ◦ α = β ◦ �1g, and then we willsay that the maps α and β interchange the actions of G on M1 and M3:


The simplest case is when G = R and A and B can be considered as the flows ofvector fields in M1 and M3 respectively, i.e., A = �1t and B = �2t .

Whenα = β = F the map F is said to be equivariant. The equivariance conditionmeans �2g ◦ F = F ◦ �1g.

Another example is when M2 = T M1, M4 = T M3 and β = T α. Then A andB can be chosen as being sections for the corresponding tangent projections, i.e.,vector fields, and in this case, B ◦ α = T α ◦ A, we will say that the vector fields Aand B are α-related.

In particular, when M1 = M3 = M , M2 = M4 = T M , A = �1 and B = �2 arevector fields, and α = φ, the vector �1 is φ-related with �2, i.e., �2 ◦ φ = T φ ◦ �1,if �1(φ

∗ f ) = φ∗(�2 f ), ∀ f ∈ C∞(M):

In this case the subalgebra φ∗(M) ⊂ C∞(M) is invariant under �1.The problem of ‘normal forms’ for linear maps studied in Sect. 3.6 is once again

of this kind. Now M1 = M2 = E and M3 = M4 = E ′ are linear spaces. Anequivalence between inear maps from E to E ′ is defined as follows: α and β areequivalent if there are invertible maps A and B relating α and β.

When E and E ′ coincide and A = B we will say that α and β are of the sameconjugacy class. The diagonalization of an endomorphism is just one of this classwith β being a diagonal matrix.

When we have related vector fields �1 and �2 with α being an differentiable andinjective, the integrability of �2 in the sense that we are able to determine the flow of�2 by algebraic manipulations and quadratures, implies that �1 is also integrable. In

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9.4 Related Maps 583

a similar way, if α is a submersion the integrability of �1 implies the integrability of�2. Note that γ is an integral curve of�1 if and only ifα◦γ is an integral curve of�2.

The framework of related maps is very general and by particularizing to somespecific cases, i.e., by restricting the possible choices for the ingredients, we findinteresting examples.

9.5 Lie–Scheffers Systems on Lie Groups and HomogeneousSpaces

Let now � : G × M → M be a left action of a Lie group G of dimension r on adifferentiable manifold M (see Sect. 3.2.5 for more details). If g ∈ G andm ∈ M , wedenote �m : G → M and �g : M → M the maps defined by �m(g) = �g(m) =�(g, m). If a is an element of the Lie algebra of G, a ∈ g, then the fundamentalvector field Xa is defined by

(Xa f )(m) = d

dtf (exp(−t a)m)|t=0, (9.55)

which, therefore, it can also be defined by

Xa(m) = T �m(e)(−a). (9.56)

Let us choose a basis {eα | α = 1, . . . , r} of the Lie algebra of the Lie group Gwith defining relations

[eα, eβ] =r∑

γ=1

cαβγ eγ,

and denote Xα the corresponding fundamental vector fields, Xα = Xeα , i.e., definedby

(Xα f )(m) = d

dtf (exp(−teα)m)|t=0.

We recall that such fundamental vector fields satisfy

[Xα, Xβ] =r∑

γ=1

cαβγ Xγ .

Another important property is that given two actions of the same group on twodifferent manifolds and a map F : M1 → M2, the fundamental vector fields in themanifold M1 are F-projectable on the corresponding ones on the second manifoldiff the map F is equivariant. Note that F is equivariant, i.e., �2g ◦ F = F ◦ �1g , if

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and only if F ◦ �1m = �2F(m), and therefore, having in mind that

F∗m(Xa(m)) = F∗m(�1m∗e(−a)) = (F ◦ �1m)∗e(−a)

we see that if F is equivariant,

F∗m(Xa(m)) = (�2F(m))∗e(−a)

i.e., the fundamental vector fields are F-related. One can also prove using the expo-nential map that if G is connected the converse property is also true.

It is well known that a Lie group acts on itself in a transitive and free way bothon the left and on the right

Lg : G → G , Rg : G → G, g′ �→ g g′ , g′ �→ g′ g.

The diffeomorphisms Lg and Rg allow us to define the left- and right-invariantvector fields determined by their values in the neutral element X L

g = Lg∗e Xe andX R

g = Rg∗e Xe.We use the shorter notation X Lα = X L

eαand X R

α = X Reα,α = 1, . . . , r ,

i.e., (X Lα )g = Lg∗e(eα), (X R

α )g = Rg∗e(eα). They generate, respectively, two Liealgebras

[X Lα , X L

β ] =r∑

γ=1

cαβγ X L

γ , [X Rα , X R

β ] = −r∑

γ=1

cαβγ X R

γ . (9.57)

We can also define a map ψ : G → G such that ψ(g) = g−1, which is an anti-homomorphism because ψ(g2g1) = ψ(g1)ψ(g2), and moreover, from the relationsLg ◦ ψ = ψ ◦ Rg−1 , Rg ◦ ψ = ψ ◦ Lg−1 , and the corresponding ones,

Lg∗ ◦ ψ∗ = ψ∗ ◦ Rg−1∗, Rg∗ ◦ ψ∗ = ψ∗ ◦ Lg−1∗,

we see that if X is a left-invariant vector field, then Y = ψ∗(X) is such that

Rg∗(Y ) = (Rg∗ ◦ ψ∗)(X) = (ψ∗ ◦ Lg−1∗

)(X) = ψ∗(X) = Y.

i.e., Y = ψ∗(X) is right-invariant.One can check that, as a consequence of the relation Rg2 ◦ �m=g1 = Rg1g2 ,

Rg2∗g1(Xa(g1)) = Rg2∗g1(�g1∗e(−a)) = Rg1g2∗e(−a) = Xa(g1g2)

and therefore the fundamental vector fields of the left action are right-invariant vectorfields in the Lie group G.

We remark that if we consider the left action of G on itself by left translations,�(g, g′) = gg′, then the fundamental vector fields Xa are right invariant because

9.5 Lie–Scheffers Systems on Lie Groups and Homogeneous Spaces 585

(Xa)(g) = �g∗e(−a) = Rg∗e(−a) = −(X Ra )(g),

where X Ra is the right-invariant vector field in G determined by its value at the neutral

element (X Ra )(e) = a.

The commutation relations (9.57) mean that we can consider a Lie–Schefferssystem on G given by

X(g, t) = −r∑

α=1

bα(t) X Rα (g), (9.58)

whose integral curves are the solutions g(t) of

g = −r∑

α=1

bα(t) X Rα (g),

where g(t) denotes the vector field along the curve g(t) defined by the tangent vectorat each point of the curve.

When applying Rg−1(t)∗g(t) to both sides we obtain the equation

g(t) g−1(t) = −r∑

α=1

bα(t)eα ∈ TeG, (9.59)

where g(t) g−1(t) is a short notation for Rg−1(t)∗g(t)g(t).This equation can be solved using a generalization of a method developed by

Wei and Norman for the linear case [WN63, WN64] for finding the time evolutionoperator for a linear systems of type dU (t)/dt = H(t)U (t), with U (0) = I , seealso [Ca98]. We should also mention that there exist alternative methods for solvingequation (9.59) by reducing the problem to a simpler one.

Both procedures are based on the following property [Ca01, CR02]: If g(t), g1(t)and g2(t) are differentiable curves in G such that g(t) = g1(t)g2(t), ∀t ∈ R, then,

Rg(t)−1 ∗g(t)(g(t)) = Rg1(t)−1 ∗g1(t)(g1(t)) + Ad (g1(t)){

Rg2(t)−1 ∗g2(t)(g2(t))}.

(9.60)

The generalization of this property to several factors is as follows. Let nowg(t) be a curve in G which is given by the product of other l curves g(t) =g1(t)g2(t) · · · gl(t) = ∏l

i=1 gi (t). Then, denoting hs(t) = ∏li=s+1 gi (t), for s ∈

{1, . . . , l − 1}, and applying (9.60) to g(t) = g1(t) h1(t) we have


Rh1(t)−1 ∗h1(t)(h1(t))}.

Simply iterating, and using that Ad (gg′) = Ad (g)Ad (g′) for all g, g′ ∈ G weobtain



Rg2(t)−1 ∗g2(t)(g2(t))}

+ · · · + Ad

(l−1∏i=1

gi (t)

){Rgl (t)−1 ∗gl (t)(gl(t))

}

=l∑

i=1

Ad

⎛⎝∏

j<i

g j (t)

⎞⎠{Rgi (t)−1 ∗gi (t)(gi (t))

}

=l∑

i=1

⎛⎝∏

j<i

Ad (g j (t))

⎞⎠{Rgi (t)−1 ∗gi (t)(gi (t))

}, (9.61)

where it has been taken g0(t) = e for all t .The generalizedWei-Normanmethod (see [Ca98] and references therein) consists

of writing the solution g(t) of (9.59) in terms of its second kind canonical coordinateswith respect to a basis {e1, . . . , er } of the Lie algebra g, for each value of t , i.e.

g(t) =r∏

α=1

exp(−vα(t)eα) = exp(−v1(t)e1) · · · exp(−vr (t)er ),

and transforming the differential equation (9.59) into a differential equation systemfor the vα(t), with initial conditions vα(0) = 0 for all α = 1, . . . , r . The minussigns in the exponentials have been introduced for computational convenience. Then,we use the result (9.61), taking l = r = dimG and gα(t) = exp(−vα(t)eα) for allα. Now, since Rgα(t)−1 ∗gα(t)(gα(t)) = −vα(t)eα, we see that (9.61) reduces to

Rg(t)−1 ∗g(t)(g(t)) = −r∑

α=1

vα

⎛⎝∏

β<α

Ad (exp(−vβ(t)eβ))

⎞⎠ eα

= −r∑

α=1

vα

⎛⎝∏

β<α

exp(−vβ(t)ad (eβ))

⎞⎠ eα,

where the identity Ad (exp(a)) = exp(ad (a)), has been used for all a ∈ g. Substitut-ing in Eq. (9.59) we obtain the fundamental expression of the Wei-Norman method

r∑α=1

vα

⎛⎝∏

β<α

exp(−vβ(t)ad (eβ))

⎞⎠ eα =

r∑α=1

bα(t)eα, (9.62)

with vα(0) = 0, α = 1, . . . , r . The resulting differential equation system for thefunctions vα(t) is integrable by quadratures if the Lie algebra is solvable [WN63,WN64], and in particular, for nilpotent Lie algebras.


Note also that the point that the t-dependent vector field (9.58) is right-invariantmeans that if g(t) is the integral curve of such a vector field starting from e, theng(t) = g(t) g0 is the integral curve of (9.58) starting from g0, and therefore thesuperposition rule involves only one solution, and is given by the following functionφ : G × G → G:

φ(g, g0) = g g0.

Assume that a transitive action� : G×M → M of the Lie group G on amanifoldM is given, i.e., M is a homogeneous space forG. Let x0 be an arbitrary but fixed pointin M and denote byGx0 the stability or isotopy subgroup of x0 ∈ M . Thiswill provideus an identification of G/Gx0 with M given by j (gGx0) = �(g, x0) (Sect. 3.2.5,Eq. 3.14). The important point is (see e.g., [Ca01]) that the natural projection mapτ : G → G/H is equivariant, with respect to the left action of G on itself by lefttranslations and the natural left-action on G/H , and consequently, the fundamentalvector fields corresponding to the two actions are τ -related. Therefore, the right-invariant vector fields X R

α , which are the fundamental vector fields corresponding tothe natural left action of G on itself, are τ -projectable and the τ -related vector fieldsin M are the fundamental vector fields −Xα = −Xaα corresponding to the naturalleft action of G on M , τ∗g X R

α (g) = −Xα(gH). In this way system (9.58) will havean associated Lie–Scheffers system on M :

X =r∑

α=1

bα(t) Xα.

The converse property is true in the following sense: Given such a Lie–Schefferssystem in a manifold M defined by complete vector fields Xα and with associatedLie algebra g, we can see these as fundamental vector fields relative to an actiongiven by integrating the vector fields. Then, the restriction to an orbit will provide ahomogeneous space of the above type. The choice of a point x0 in the homogeneousspace allows us to identify the homogeneous space M with G/H , where H is thestability group of x0.Different choices for x0 will lead to conjugate subgroups [Ca01].

Hence we reduce the problem of integrating the given Lie–Scheffers system inM to studying the curve g(t) on the Lie group itself such that g(0) = e and is anintegral curve of the t-dependent right-invariant vector field

XG(g, t) = −r∑

α=1

bα(t) X Rα (g)

i.e., a solution g(t) of

g = −r∑

α=1

bα(t) X Rα (g).

Using the right translation Rg−1 we find

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dg

dtg−1 = −

r∑α=1

bα(t) eα . (9.63)

Then, the given t-dependent vector field X on M is related to a t-dependent vectorfield in the group, X G , and the integral curves of this latter vector field provide usthe integral curves of the initial system.

The relatedness property of the t-dependent vector fields X G and X , for eachfixed value of t , is displayed in the next diagram:

Even if the Eq. (9.63) may be difficult to solve directly, it can be explicitly solvedwhen a fundamental set of particular solutions of the system of differential equationsis known (see e.g., [Ca98]) and this property leads to a superposition rule allowing usto find the general solution of the system in terms of any fundamental set of solutionsby using a (nonlinear) superposition rule.

In this way the solution of the given system for any initial condition is reducedto only one higher dimensional system of first-order linear equations with initialcondition g(0) = e. Once the curve g(t) has been found, the solution of the givensystem determined by the initial condition x(0) = x0 is �(g(t), x0).

A set x1, . . . , xm , of solutions is said to be a fundamental system of solutions, if

x1(t) = �(g(t), x1(0)) = F(g(t), x1(0)). . . = . . . . . .

xm(t) = �(g(t), xm(0)) = F(g(t), xm(0)) (9.64)

is a minimal set allowing us to solve for g(t) via the implicit function Theorem. Ifthis can be done we get

g(t) = G(x1(t), . . . , xm(t); x1(0), . . . , xm(0)),

and then any other solution can be written as

x(t) − F(G(x1(t), . . . , xm(t); x1(0), . . . , xm(0)), x(0)) = 0.

Therefore the left hand-side of this relation defines a constant of motion.From the operational point of view, starting with the action � : G × M → M

we should find the minimal integer number m such that the isotopy group ofthe action of G on the product M × · · · × M (m times), extended from � by�m(g, x1, . . . , xm) = (�(g, x1), . . . , �(g, xm)), reduces to the neutral elementfor a generic point. When expressed in terms of vector fields, that means that the


corresponding diagonal extensions of the fundamental vector fields Xα do not vanishin a generic point. The solution then is found by adding a new component and lookingfor constants of motion.

9.6 Some Examples of Lie–Scheffers Systems

In this section we present some specific examples illustrating how the theory works.The procedure is first illustrated with one interesting example for the simplest casen = 1 and afterwards we analyse other examples arising in classical and QuantumMechanics.

9.6.1 Riccati Equation

For the simplest case n = 1, according to Lie’s Theorem we should look for afinite-dimensional real Lie algebra of differential operators

Xα = fα(x)∂

∂x.

It can be shown that the only finite-dimensional Lie algebra that can be foundfrom vector fields in one real variable are sl(2, R) and its subalgebras. The uniquelydefined (up to a change of variables) realization of sl(2, R) is given by

X0 = ∂

∂x, X1 = x

∂

∂x, X2 = x2

∂

∂x,

with commutation relations

[X0, X1] = X0 , [X1, X2] = X2 , [X2, X0] = −2 X1 .

More explicitly (see [Ca99]), let us consider the action of G = SL(2, R) onM = R = R ∪ ∞, the real line completed with a point at the infinity, given by

�(A, x) = αx + β

γx + δ,

when

A =(

α βγ δ

)

with the understanding that �(A,∞) = α/γ and �(−δ/γ) = ∞.


If we consider x0 = ∞, then the stability subgroup Gx0 is the affine group in onedimension, made up by the matrices

A =(

α β0 1/α

)

with α �= 0. On the other side, if x0 = 0 the stability group is that of matrices

A =(

α 0γ 1/α

),

which is conjugate to the previous one.The actions of the one-parameter subgroups of SL(2, R) are, respectively, given

by

x �→ x + t, x �→ et x, x �→ x

1 − t x,

and they give raise to the corresponding fundamental vector fields X0, X1 and X2,given above.

The Lie algebra sl(2, R) of the Lie group SL(2, R) is generated by the tracelessmatrices

e0 =(0 10 0

), e1 = 1

2

(1 00 −1

), e2 =

(0 0

−1 0

)(9.65)

with commutation relations

[e1, e0] = e0 , [e1, e2] = −e2 , [e2, e0] = 2 e1.

The time-dependent vector field on SL(2, R) given by

X = [b0(t)X R0 + b1(t)X R

1 + b2(t)X R2 ],

gives raise on M = SL(2, R)/A1 ≈ R to the t-dependent vector field

X (t) = −[

b0(t)∂

∂x+ b1(t)x

∂

∂x+ b2(t)x2

∂

∂x

],

whose integral curves are solutions of the Riccati equation

x + b0(t) + b1(t)x + b2(t)x2 = 0. (9.66)

In this case one can prove that m = r = 3, i.e., the superposition rule involvesthree different solutions, and the superposition principle comes from the relation

9.6 Some Examples of Lie–Scheffers Systems 591

x − x1x − x2

: x3 − x1x3 − x2

= k, (9.67)

or in other words, the general solution of (9.66) is

x(t) = x1(t)(x3(t) − x2(t)) + k x2(t)(x1(t) − x3(t))

(x3(t) − x2(t)) + k (x1(t) − x3(t)). (9.68)

The value k = ∞ must be accepted, otherwise we do not obtain the solution x2.The solutions are obtained from the curve g(t) in SL(2, R) such that g(0) = e

and is a solution of the equation

gg−1 = b0(t) e0 + e1(t)a1 + b2(t) e2,

as follows, x(t) = �(g(t), x(0)).Let us check now that the number m of fundamental solutions is m = 3 and

determine the superposition rule: First we note that the determinant

∣∣∣∣∣∣1 ξ1 ξ211 ξ2 ξ221 ξ3 ξ23

∣∣∣∣∣∣ = (ξ1 − ξ2)(ξ2 − ξ3)(ξ3 − ξ1)

is generically different from zero, while the system given by the two equationsa+b ξ1+c ξ21 = 0, a+b ξ2+c ξ22 = 0 alwayshas a non trivial solution.Consequently,m = 3 in this case.

Recall that for obtaining the general solution we should define the vector fields

V0 = ∂

∂x+ ∂

∂x(1)+ ∂

∂x(2)+ ∂

∂x(3),

V1 = x∂

∂x+ x(1)

∂

∂x(1)+ x(2)

∂

∂x(2)+ x(3)

∂

∂x(3),

V2 = x2∂

∂x+ x(1)

2 ∂

∂x(1)+ x(2)

2 ∂

∂x(2)+ x(3)

2 ∂

∂x(3)(9.69)

and look for a solution of the system

V0 f = V1 f = V2 f = 0.

This system of partial differential equations is integrable, because the vector fieldsV0, V1 and V2 close a Lie algebra, and therefore they define an integrable distribution.

The first equation V0 f = 0 tells us that the function f depends only on thedifferences u1 = x − x(1), u2 = x(1) − x(2) and u3 = x(2) − x(3), because thecharacteristic system is

dx

1= dx(1)

1= dx(2)

1= dx(3)

1,


and it has as first integrals the differences u1 = x − x(1), u2 = x(1) − x(2) and u3 =x(2) − x(3). Now, if f (x, x(1), x(2), x(3)) = ϕ(u1, u2, u3), the condition V1 f = 0 iswritten

u1∂ϕ

∂u1+ u2

∂ϕ

∂u2+ u3

∂ϕ

∂u3= 0,

i.e., the function ϕ should be homogeneous of degree zero, and therefore it only candepend on the quotients v1 = u1/u2 and v2 = u3/u2, ϕ(u1, u2, u3) = φ(v1, v2).Finally, the condition V2 f = 0 can be written in these coordinates, after a longcomputation, as

v1(v1 + 1)∂φ

∂v1− v2(v2 + 1)

∂φ

∂v2= 0.

The corresponding characteristic system is

dv1

v1(v1 + 1)= − dv2

v2(v2 + 1),


∫dx

x(x + 1)= log

x

x + 1,

we obtain that the constant of motion f should be a function of

ζ = v1

v1 + 1

v2

v2 + 1,

and therefore,

(x − x(1))(x(2) − x(3))

(x − x(2))(x(1) − x(3))= k

provides the nonlinear superposition rule giving x(t) as a function of three indepen-dent solutions

x = (x(1) − x(3))x(2)k + x(1)(x(3) − x(2))

x(1) − x(3))k + (x(3) − x(2)).

Let us remark that for the n = 1 case there is only one nonlinear differentialequation family satisfying Lie’s theorem: the Riccati equation. Of course, propersubalgebras of sl(2, R) lead to the linear inhomogeneous equation when b2 = 0or Lie homogeneous equation when b2 = b3 = 0 (remember also the Bernoulliequation studied before). However, for n = 2 in addition to SL(3, R), O(3, 1) andO(2, 2), we can realize families of Lie algebras with arbitrary large Abelian ideals.


The importance of Lie–Scheffers systems in Supersymmetric Quantum Mechan-ics (see for instance [Co95] and references therein) is based on the above mentionedfact that Riccati equation can be considered as a Lie–Scheffers system with groupSL(2, R). Recall, for instance, that the condition for the determination of the super-potential W in the factorization of a Hamiltonian H in such a way that

H − c =(

− d

dx+ W

)(d

dx+ W

),

is a Riccati equation. Actually, if

H = − d2

dx2+ V (x),

then W can be found from V as a solution of the Riccati equation

W ′ = W 2 − (V − c).

Moreover, a similar equation plays a relevant role in the search for Shape Invariantpotentials using the so-called Infeld and Hull Factorization method [In51] (see also[CR00] for a modern approach). Consequently, the Riccati equation plays a rele-vant role in the theory of intertwining operators and shape invariance in QuantumMechanics (see e.g., [CRF01] and [CR00b]).

Finally, as indicated in Sect. 2.3.3, a linear system in two variables can be reducedto a Riccati equation. Therefore as a homogeneous linear second-order differentialequation,

d2z

dx2+ b(x)

dz

dx+ c(x)z = 0 , (9.70)

can be written in the form{

z′ = v

v′ = −b(x)v − c(x) z; (9.71)

the quotient v/z satisfies a Riccati equation.In other words, the Riccati equation is particularly important because it appears

as a consequence of the reduction theory when taking into account that dilations aresymmetries of linear second-order differential equations [Ca98]. Actually, the homo-geneous linear second-order differential equation (9.70) admits as an infinitesimalsymmetry the vector field X = z ∂/∂z generating dilations in the variable z, whichis defined for z �= 0. According to Lie theory we should change the coordinate z to anew one, u = ϕ(z), such that X = ∂/∂u. This change is determined by the equationXu = 1, which leads to u = log |z|, i.e., |z| = eu . In both cases of regions withz > 0 or z < 0 we have:

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dz

dx= z

du

dx, and

d2z

dx2= z

(du

dx

)2+ z

d2u

dx2,

so Eq. (9.70) becomes

d2u

dx2+ b(x)

du

dx+(

du

dx

)2+ c(x) = 0,

and as the dependent variable does not explicitly appear, the order can be loweredby introducing the new variable w = du/dx . We arrive to the following Riccatiequation for w,

dw

dx= −w2 − b(x)w − c(x). (9.72)

Notice that w = z−1dz/dx , which corresponds to w = v/z in (9.71), and that thisrelation together with (9.72) is equivalent to the original second-order equation. Inthe particular case of the one-dimensional time-independent Schrödinger equation

−d2φ

dx2+ (V (x) − ε)φ = 0,

the reduced Riccati equation for W = φ−1dφ/dx is

W ′ = −W 2 + (V (x) − ε), (9.73)

which is the equation that W must satisfy in the previously mentioned factorizationof H = −d2/dx2 + V (x).

As another example of Lie–Scheffers system, not an equation but with n = 2, wecan apply the previous procedure to rotation group SO(3, R) acting transitively onthe sphere M = S2. We start by considering the linear action of SO(3, R) on R

3.The fundamental vector fields are:

X1 = x2∂

∂x3− x3

∂

∂x2, X2 = x3

∂

∂x1− x1

∂

∂x3, X3 = x1

∂

∂x2− x2

∂

∂x1,

satisfying the commutation relations

[Xi , X j ] = −εi jk Xk .

The function x21 + x22 + x23 is invariant under such fundamental vector fields, i.e.,Xi (x21 + x22 + x23 ) = 0, i = 1, 2, 3, and therefore the restrictions of the fundamentalvector fields on the sphere of radius one, M = S2, are tangent to the sphere.

There we will consider a t-dependent vector field � on R3 of the form,


� = b1(t)X1 + b2(t)X2 + b3(t)X3,

and its restriction to M . The integral curves of such restriction are the integral curvesof γ starting from a point of the sphere.

Note that the vector field in R3 is a linear vector field associated with a skew-

symmetric matrix matrix.A fundamental system of solutions is made by two solutions. In fact, if Y (t) =

(y1(t), y2(t), y3(t)) and Z(t) = (z1(t), z2(t), z3(t)) are two solutions, i.e., curves onS2, there is only one rotation R(t) such that Y (t) = R(t)Y (0) and Z(t) = R(t)Z(0).For instance, we can consider the case in which Y (0) and Z(0) are orthogonal,because if

〈Y (0), Z(0)〉 = 0,

as the matrix defining the evolution is orthogonal, we find that

〈Y (t), Z(t)〉 = 0.

We then find for a generic solution,

X (t) = 〈Y (0), X (0)〉Y (t)+〈Z(0), X (0)〉Z(t)+〈Y (0)∧ Z(0), X (0)〉(Y (t)∧ Z(t)).

9.6.2 Euler Equations

We recall that given a Lie group G, each g ∈ G defines an innner automorphismi(g) : G → G by i(g)g′ = gg′g−1, and therefore, as i(g2 g1) = i(g2) ◦ i(g1), thereis an action of G on itself defined by φ(g2, g1) = i(g2)(g1). This action allows usto define a linear representation of G with its Lie algebra as representation space,as follows: Ad (g) = i(g)∗. Then it provides us a linear representation of its Liealgebra, called adjoint representation, and denoted ad, i.e.

Ad (exp(a)) = exp(ad (a)), a ∈ g.

As an instance we can consider G = GL(E), where E is a linear space. Now letB0 be an arbitrary but fixed endomorphism of E , choose another endomorphism Aand define the curve in End(E),

B(t) = eAt B0 e−At . (9.74)

Such a curve satisfies B(0) = B0 and

d B

dt= [A, B] = ad (A)B.


That is an equation for a curve in End(E) which will be called, by similarity withthe quantum case, the Heisenberg description of the dynamical system. The solutionstarting from B0 is given by (9.74).

Remark that ad (A) is an inner derivation of the Lie algebra End(E) of the endo-morphisms of E . This motivates our interest in characterizing derivations of such aLie algebra that are inner derivations and study the evolution equations in End(E)

of the type B = D(B).More generally, the adjoint representation of a Lie group G provides a natural

realization of G in the linear space of the Lie algebra g. In the associated action ofG on g the fundamental vector field defined by a ∈ g is given by Xa = ad (a),i.e., Xa(b) = [a, b]. Consequently, the integral curves of the vector field Xa aregiven by c = [a, c]. that is a Lax-like equation, a very special kind of equations thatplay an important rôle in the study of integrable systems [La68, CI85, CM94]. It iswell known that the trace of the elements c in the curve is constant along the curve,because the trace of a commutator is always zero. Furthermore, when G is a subgroupof the general linear group, i.e., the elements of G are invertible matrices, then theelements of its Lie algebra g are also matrices, but in general they are singular.

It is to be remarked that the different powers ck of the curve solution of theequation c = [a, c] satisfy the same equation as c itself, and therefore their tracesare also constant along the curve. In other words, not only the trace of c but also allthe coefficients of its characteristic polynomial are constants of motion [CI83].

The case of Lie–Scheffers systems is when instead of having a fundamental vectorfield ad (a) defined by an element a ∈ gwe have a linear combination, with functionsof t as coefficients,

X =r∑

α=1

bα(t) ad (aα),

of such vector fields. In this case, according to the recipe established in the generaltheory of Lie systems for finding the general solution, we should look for the curveg(t) in G, starting from the neutral element, g(0) = e and satisfying the equation

g(t) g−1(t) =r∑

α=1

bα(t) ad (aα).

Once such a curve has been found, the solution determined by the initial conditionc(0) is given by

c(t) = Ad (g(t))c(0),

and in the particular case in which G is a subgroup of the general linear group itreduces to

c(t) = g(t) c(0) g−1(t) .


9.6.3 SODE Lie–Scheffers Systems

Newtonian equations of motion have the following form:

x i = Fi (x, x), i = 1, . . . , n. (9.75)

An associated system of 2n first-order differential equations is:

{x i = vi

vi = Fi (x, v)(9.76)

which can be understood as the system determining the integral curves of a vectorfield in T R

n:

X = vi ∂

∂x+ Fi (x, v)

∂

∂vi.

We can say that the system of SODEs (9.75) is a SODE Lie–Scheffers system ifthe system of first-order differential equations (9.76) obtained by adding the newvariables vi = x i , with i = 1, . . . , n, to system (9.75), is a Lie–Scheffers system.

One of the simplest examples is today called the Milne-Pinney equation:

|x = −ω2(t)x + k

x3, x ∈ R+ = (0,∞), (9.77)

where k is a constant which in the interesting cases in physics is k < 0. Such anequationwas introduced by aUkrainianmathematician of the nineteenth century,V.P.Ermakov, as a way of looking for a first integral for the time-dependent harmonicoscillator [Er80] (the case k = 0 corresponds to the harmonic oscillator and theinterval (0,∞) is replaced by the real line). TheMilne-Pinney equation describes thetime-evolution of an isotonic oscillator [Cal69, Pe90] (also-called pseudo-oscillator),i.e., an oscillator with an additional inverse quadratic potential. This oscillator shareswith the harmonic one the property of having a period independent of the energy[Ch05], i.e., they are isochronous systems, and in the quantum case they have anequispaced spectrum [ACMP07].

As usual, we can relate the Milne-Pinney equation to a system of first-orderdifferential equations defined in T R+ by introducing a new auxiliary variable v ≡ x .Such a system is given by

{x = v

v = −ω2(t)x + k

x3

where x ∈ R+ and (x, v) ∈ Tx R+. The associated t-dependent vector field overT R+,


X = v∂

∂x+(

−ω2(t)x + k

x3

)∂

∂v,

is a Lie–Scheffers system because X can be written as

X = L2 − ω2(t)L1,

where the vector fields L1 and L2 are given by

L1 = x∂

∂v, L2 = k

x3∂

∂v+ v

∂

∂x,

and are such that

[L1, L2] = 2L3, [L3, L2] = −L2, [L3, L1] = L1

with

L3 = 1

2

(x

∂

∂x− v

∂

∂v

),

i.e., they span a 3-dimensional real Lie algebra g isomorphic to sl(2, R). More detailscan be found in [CL08, CLr08] and [CdLS12]. Other interesting examples, such asthe second-order Riccati equation, have recently been studied [CL11].

9.6.4 Schrödinger–Pauli Equation

As an example of Lie–Scheffers systems in Quantum Mechanics, we can considerthe non-relativistic dynamics of a spin 1/2 particle, when only the spinorial part isconsidered [CGM01]. The dynamics of such a particle in a time-dependent magneticfield B is described by the so-called Schrödinger–Pauli equation:

i �d |ψ〉

dt= H |ψ〉 = −μ B · S |ψ〉,

with the coefficient μ being proportional to the Bohr magneton, B = (B1, B2, B3)

the t-dependent magnetic field, and Si = �

2 σi . The matrices −iσ1, −iσ2 and −iσ3

generate the real Lie algebra of traceless skew-Hermitian 2 × 2 matrices, the Liealgebra of the group SU (2, C) and therefore of SO(3, R).

As a consequence of the theory developed in this chapter, in order to find thegeneral solution of the evolution equation, it suffices to determine the curve R(t) inSO(3, R) such that R R−1 = B3M3 + B2M2 + B1M1 and R(0) = I , where


M1 =⎛⎝0 0 00 0 −10 1 0

⎞⎠ , M2 =

⎛⎝ 0 0 1

0 0 0−1 0 0

⎞⎠ , M3 =

⎛⎝ 0 −1 01 0 00 0 0

⎞⎠ .

Such a curve gives us the general solution for the dynamics

|ψ(t)〉 = R(t)|ψ(0)〉,

where R is an element in SU (2, C) corresponding to R.

9.6.5 Smorodinsky–Winternitz Oscillator

As another example consider the differential equation of an n-dimensionalSmorodinsky–Winternitz oscillator of the form [WSUP67, CLS13]

⎧⎨⎩

xi = pi ,

pi = −�2(t)xi + k

x3i,

i = 1, . . . , n,

which describes the integral curves of the t-dependent vector field on T ∗R

n

Xt =n∑

i=1

[pi

∂

∂xi+(

−�2(t)xi + k

x3i

)∂

∂ pi

].

Note that Xt can bewritten as Xt = X2+�2(t)X1 with X1, X2 and X3 = −[X1, X2]being given by

X1 = −n∑

i=1

xi∂

∂ pi, X2 =

n∑i=1

(pi

∂

∂xi+ k

x3i

∂

∂ pi

), X3 =

n∑i=1

1

2

(xi

∂

∂xi− pi

∂

∂ pi

).

Therefore, Xt is a Lie–Scheffers system, because X1, X2 and X3 close on a sl(2, R)

algebra:

[X1, X2] = −2X3, [X1, X3] = −X1, [X2, X3] = X2.

Moreover, the preceding vector fields are Hamiltonian vector fields with respect to

the usual symplectic form ω0 =n∑

i=1

dxi ∧ dpi with Hamiltonian functions

h1 = 1

2

n∑i=1

x2i , h2 = 1

2

n∑i=1

(p2i + k

x2i

), h3 = 1

2

n∑i=1

xi pi ,


which satisfy that

{h1, h2} = 2h3, {h1, h3} = h1, {h2, h3} = −h2.

Consequently, every curve ht that takes values in the three-dimensional Lie algebra(W, {·, ·}) spanned by h1, h2 and h3 gives raise to a Lie–Scheffers system in T ∗

Rn

which is Hamiltonian with respect to the symplectic structure ω0 in such a way thatthe t-dependent vector field is given by

Xt = X2 + �2(t)X1 = ω−10 (dh2 + �2(t)dh1),

i.e., the Hamiltonian is ht = h2 + �2(t)h1.This suggests that we consider Lie–Scheffers systems admitting a similar Hamil-

tonian description.

9.7 Hamiltonian Systems of Lie–Scheffers Type

An important case of Lie–Scheffers systems is when (M,�) is a symplectic manifoldand the vector fields in M arising in the expression of the t-dependent vector fielddescribing a Lie–Scheffers system are Hamiltonian vector fields closing on a finite-dimensional real Lie algebra g [CLS13]. When these vector fields are complete, theyare fundamental vector fields of a symplectic action of a Lie group G with Lie algebrag on the symplectic manifold (M,�).

The corresponding Hamiltonian functions hα, defined by i(Xα)� = −dhα, do notclose in general on the same Lie algebra g when the Poisson bracket is considered,but we can only assure that

d({hα, hβ} − h[α,β]

) = 0 ,

i.e., that there exist real constants λαβ such that

{hα, hβ} − h[α,β] = λαβ .

Therefore, the functions hα span a Lie algebra extension of the original one.The situation in QuantumMechanics is quite similar: we have analyzed the geom-

etry of a Hilbert space H when considered as a real manifold with a global chart inSect. 6.4.4. The tangent space TφHR at any point φ ∈ H can be identified with HR

itself, the isomorphism associating ψ ∈ HR with the vector Xψ ∈ TφHR being givenby:

(Xψ f )(φ) := d

dtf (φ + tψ) |t=0, ∀ f ∈ C∞(H).

http://dx.doi.org/10.1007/978-94-017-9220-2_6

9.7 Hamiltonian Systems of Lie–Scheffers Type 601

We also recall that the Hilbert space HR as a real manifold is endowed with asymplectic 2-form � defined by

�φ(ψ, ψ′) = 2 Im〈ψ|ψ′〉.

A vector field is just a map A : HR → HR; in particular a linear operator A onH is aspecial kind of vector field. Given a smooth function a : H → R, its differential daφ

at φ ∈ H is an element of the (real) dual H′Rgiven by

〈daφ, ψ〉 :=(

d

dta(φ + tψ)

)|t=0

.

Actually, the skew-self-adjoint linear operators −i A in H, that is, A is a self-adjointoperator, define Hamiltonian vector fields, the Hamiltonian function of −i A beinga(φ) = 1

2 〈φ, Aφ〉. Therefore, the Schrödinger equation plays the rôle of Hamiltonequations, because it determines the integral curves of the vector field −i H , whereH is the Hamiltonian of the system [BCG91].

In particular, the theory of Lie–Scheffers systems applies in the previous frame-work of Quantum Mechanics when we have a t-dependent quantum Hamiltonianthat can be written as a linear combination, with t-dependent coefficients, of Hamil-tonians Hα, with α = 1, . . . , r , closing on a finite-dimensional real Lie algebra underthe commutator bracket. However, note that as indicated above this Lie algebra doesnot necessarily coincide with that of the corresponding classical problem, but it maybe a Lie algebra extension of the latter.

For the illustration of these classical and quantum Lie–Scheffers systems ofHamiltonian type we consider an important example: the time-dependent classicaland quantum quadratic Hamiltonians.

The configuration space for the classical system is the real line R, the corre-sponding phase space T ∗

R being endowed with its canonical symplectic structureω0 = dq ∧ dp, and the time-dependent classical Hamiltonian function is

H = α(t)p2

2+ β(t)

qp

2+ γ(t)

q2

2+ δ(t) p + ε(t) q. (9.78)

The dynamical vector field solution of the dynamical equation i(�H ) ω0 = d H isgiven by

�H =(

α(t) p + 1

2β(t) q + δ(t)

)∂

∂q−(1

2β(t) p + γ(t) q + ε(t)

)∂

∂ p, (9.79)


�H = α(t) X1 + β(t) X2 + γ(t) X3 − δ(t) X4 + ε(t) X5,

where the vector fields


X1 = p∂

∂q, X2 = 1

2

(q

∂

∂q− p

∂

∂ p

), X3 = −q

∂

∂ p, X4 = − ∂

∂q, X5 = − ∂

∂ p,

satisfy the following commutation relations:

[X1, X2] = X1, [X1, X3] = 2X2, [X1, X4] = 0, [X1, X5] = −X4,

[X2, X3] = X3, [X2, X4] = −1

2X4, [X2, X5] = 1

2X5, (9.80)

[X3, X4] = X5, [X3, X5] = 0, [X4, X5] = 0,

and therefore they close on a five-dimensional real Lie algebra. Consider the abstract,five-dimensional, Lie algebra g such that in a basis {a1, a2, a3, a4, a5}, the Lie prod-ucts are analogous to that of (9.80). Then, the Lie algebra g is a semi-direct sumof the Abelian two-dimensional Lie algebra generated by {a4, a5} with the sl(2, R)

Lie algebra generated by {a1, a2, a3}, i.e., g = R2

� sl(2, R). The corresponding Liegroup will be the semi-direct product G = T2 � SL(2, R) relative to the linear actionof SL(2, R) on the two-dimensional translation algebra. When computing the flowsof the previous vector fields Xα, we see that they correspond to the affine action ofG on R

2, and therefore, the vector fields Xα can be regarded as fundamental fieldsassociated to the previous basis of the Lie algebra with respect to the affine action.

In order to find the integral curves of the time-dependent vector field (9.79), wecan solve first the corresponding equation in the Lie group G and then use the affineaction of G on R

2. We should find the curve g(t) in G such that

gg−1 = −5∑

i=1

bi (t)ai , g(0) = e, (9.81)

with b1(t) = α(t), b2(t) = β(t), b3(t) = γ(t), b4(t) = −δ(t), and b5(t) = ε(t). Theexplicit calculation can be carried out by using the generalized Wei-Norman method[WN63, WN64], i.e., writing g(t) in terms of a set of second class canonical coordi-nates, for instance,

g(t) = exp(−v4(t)a4) exp(−v5(t)a5) exp(−v1(t)a1) exp(−v2(t)a2) exp(−v3(t)a3),

and then, equation (9.81) leads to the system

v1 = b1 + b2v1 + b3v21, v2 = b2 + 2b3v1, v3 = ev2b3,

v4 = b4 + 1

2b2v4 + b1v5, v5 = b5 − b3v4 − 1

2b2v5,

with initial conditions v1(0) = · · · = v5(0) = 0.For some specific choices of the functions α(t), . . . , ε(t), the problem becomes

simpler and it may be enough to consider a subgroup, instead of the whole Lie groupG, to deal with the arising system. For instance, the only non-vanishing coefficientsof the classical Hamiltonian


H = p2

2m+ f (t) q,

are α(t) = 1/m and ε(t) = f (t), and then the problem is reduced to one in a three-dimensional subalgebra, generated by {X1, X4, X5}. The associated Lie group willbe the subgroup of G generated by {a1, a4, a5}. Themain point is that such a subgroupis solvable and therefore the problem can be integrated by quadratures.

Another remarkable property is that the Hamiltonian functions hα correspondingto the Hamiltonian vector fields X1, . . . , X5, defined by i(Xα)ω = −dhα, i.e.

h1(q, p) = − p2

2, h2(q, p) = −1

2q p, h3(q, p) = −q2

2, h4(q, p) = p, h5(q, p) = −q,

have almost the same Poisson bracket relations as the vector fields Xα, but they donot coincide because of {h4, h5} = 1, instead of [X4, X5] = 0. In other words, they closeon a Lie algebra which is a central extension of R

2� sl(2, R) by a one-dimensional

algebra.Let us now consider the quantum case [Wo80], with applications in a number of

physical problems, as for instance, the quantum motion of charged particles subjectto time-dependent electromagnetic fields (see, e.g., [FMM94]), and connects withthe theory of exact invariants developed by Lewis and Riesenfeld (see [LR69] andreferences therein).

A generic time-dependent quadratic quantum Hamiltonian is given by

H = α(t)P2

2+ β(t)

Q P + P Q

4+ γ(t)

Q2

2+ δ(t)P + ε(t) Q + φ(t)I, (9.82)

where Q and P are the position andmomentum operators satisfying the commutationrelation

[Q, P] = i I.

The previous Hamiltonian can be written as a sum with t-dependent coefficients

H = α(t)H1 + β(t)H2 + γ(t)H3 − δ(t)H4 + ε(t)H5 − φ(t)H6,

of the Hamiltonians

H1 = P2

2, H2 = 1

4(Q P + P Q), H3 = Q2

2, H4 = −P, H5 = Q, H6 = −I,

which satisfy the commutation relations

[i H1, i H2] = i H1 , [i H1, i H3] = 2 i H2 , [i H1, i H5] = −i H4 , [i H2, i H3] = i H3 ,

[i H2, i H4] = − i

2H4 , [i H2, i H5] = i

2H5 , [i H3, i H4] = i H5 , [i H4, i H5] = i H6 ,


and [i H1, i H4] = [i H3, i H5] = [i Hα, i H6] = 0, α = 1, . . . , 5. That is, the skew-self-adjoint operators i Hα generate a six-dimensional real Lie algebra which is a centralextension of the Lie algebra arising in the classical case, R

2� sl(2, R), by a one-

dimensionalLie algebra. It canbe identified as the semi-direct sumof theHeisenberg–Weyl Lie algebra h(3), which is an ideal in the total Lie algebra, with the Liesubalgebra sl(2, R), i.e. h(3) � sl(2, R). Sometimes this Lie algebra is referred to asthe extended symplectic Lie algebra hsp(2, R) = h(3) � sp(2, R). The correspondingLie group is the semi-direct product H(3) � SL(2, R) of the Heisenberg–Weyl groupH(3) with SL(2, R), see also [Wo80].

The time-evolution of a quantum system can be described in terms of the evolutionoperator U (t) which satisfies the Schrödinger equation (see, e.g., [CDL77])

idU

dt= H(t)U , U (0) = Id ,

where H(t) is the Hamiltonian of the system. In our current case, the Hamiltonianis given by (9.82), and therefore the time-evolution of the system is given by anequation of the type

gg−1 = −6∑

α=1

bα(t) aα, g(0) = e, (9.83)

with the identification of g(t) with U (t), e with Id, i Hα with aα for α ∈ {1, . . . , 6} andthe time-dependent coefficients bα(t) are given by

b1(t) = α(t) , b2(t) = β(t) , b3(t) = γ(t),

b4(t) = −δ(t) , b5(t) = ε(t) , b6(t) = −φ(t).

We would like to remark that time-dependent quantum Hamiltonians are seldomstudied, because it is generally difficult to find their time evolution. However, inthe case where the system could be treated as a Lie–Scheffers system in a certainLie group, the calculation of the evolution operator is reduced to the problem ofintegrating the system appearing after application of the Wei-Norman method. Inthe case where the associated Lie group is solvable, the integration can be made byquadratures, leading to an exact solution of the problem.

The calculation of the solution of (9.83) can be carried out by using the generalizedWei–Norman method, i.e. writing g(t) in terms of a set of second class canonicalcoordinates. We take, for instance, the factorization

g(t) = exp(−v4(t)a4) exp(−v5(t)a5) exp(−v6(t)a6)

× exp(−v1(t)a1) exp(−v2(t)a2) exp(−v3(t)a3)

and therefore, the Eq. (9.83) leads in this case to the system


v1 = b1 + b2v1 + b3v21 , v2 = b2 + 2b3v1, v3 = ev2b3,

v4 = b4 + 1

2b2v4 + b1v5, v5 = b5 − b3v4 − 1

2b2v5,

v6 = b6 + b5v4 − 1

2b3v

24 + 1

2b1v

25,

with initial conditions v1(0) = · · · = v6(0) = 0.Analogously to what happened in the classical case, special choices of the time-

dependent coefficient functions α(t), . . . , φ(t), may lead to problems for which theassociated Lie algebra is a subalgebra of that of the complete system, and similarlyfor the Lie groups involved. For example, we could consider as well the quantumHamiltonian linear in the positions

H = P2

2m+ f (t) Q,

which in the notation of (9.82) has the only non-vanishing coefficients α(t) = 1/m

and ε(t) = f (t). This problem can be regarded as a Lie–Scheffers system associatedto the four-dimensional Lie algebra generated by {i H1, i H4, i H5, i H6}, which is alsosolvable, and hence the problem can be solved by quadratures.

9.8 A Generalization of Lie–Scheffers Systems

The theory of Lie–Scheffers systems has been generalized to deal with more generalcases, as for instance the Abel equation:

x = A0(t) + A1(t)x + A2(t)x2 + A3(t)x3.

Note that then the linear space

VAbel(R) = 〈 ∂

∂x, x

∂

∂x, x2

∂

∂x, x3 ∂

∂x〉

is not a finite-dimensional real Lie algebra, because

[x2

∂

∂x, x3

∂

∂x

]= x4

∂

∂x.

Note also that the vector fields of the Lie algebra in the hypothesis of the Lie–Scheffers Theoremplay a double role, because on one side they generate a Lie algebrawith an associated Lie group G, and on the other one, they explicitly appear in thedynamics.

The important fact is that the group G of curves in G transforms an element ofthe Lie family of systems into another one, and this fact may be used to simplifythe problem. For instance, the transformed vector field may be included in a Lie


subalgebra of the given Lie algebra and therefore it is a simpler ‘Lie–Schefferssystem’.

This fact admits the following generalization:

• The dynamics is, as before, X = ∑r0 bα Xα, but the vector fields Xα do not close a

finite-dimensional Lie subalgebra of vector fields anymore but only span a linearspace

• There is a Lie subalgebra W of (complete) vector fields {Ya | a = 1, . . . , l} such that

[Ya , Xα] =r∑

β=1

caαβ Xβ .

In this case the group G of curves in the group corresponding to the Lie algebra alsotransforms each element of the Lie family of systems into another one. Appropriatecurves can lead to simpler systems, for instancewhen the image of the given dynamicsis included in a finite-dimensional Lie algebra, reducing the system to aLie–Schefferssystem

As an instance, for the Abel equation [CLR11] we can consider the Lie algebra

WAbel(R) = span

{∂

∂x, x

∂

∂x

}.

which leaves invariant VAbel(R) and then the subgroup of transformations

x(t) = α(t) x(t) + β(t), α(t) �= 0,

is a structure preserving group and we can take advantage of such transformationsto transform a given equation into another simpler one.

Under such a transformation the given Abel equation becomes a new Abel equa-tion

˙x = A0(t) + A1(t)x + A2(t)x2 + A3(t)x3,

with

A3(t) = A3(t)α2(t),

A2(t) = α(t)(3A3(t)β(t) + A2(t)),

A1(t) = 3A3(t)β2(t) + 2A2(t)β(t) + A1(t) − α(t)α−1(t),

A0(t) = α−1(t)(

A3(t)β3(t) + A2(t)β2(t) + A1(t)β(t) + A0(t) − β(t)).

The action of the group of curves in the affine group produces orbits, i.e., equivalenceclasses of Abel equations, with the same properties of integrability or existence ofsuperposition rules. For instance, Abel equations of an orbit containing an integrableby quadratures Riccati equation are also integrable by quadratures. The orbits arecharacterized by different values of invariant functions.

One must determine the Lie algebras contained in VAbel(R) reachable by the set oftransformations (the coefficient of X3 cannot be transformed to be zero). It is possible

9.8 A Generalization of Lie–Scheffers Systems 607

to check that the only subalgebra of dimension three is the one of the Riccati equationwhich is not reachable by the considered set of transformations.

There is a one-parameter family of two-dimensional subalgebras of VAbel(R) reach-able from a proper Abel equation, those generated by

(−2μ3 + 3μx2 + x3)∂x , (μ + x)∂x , μ ∈ R.

These are integrable by two quadratures. For instance the case μ = 0 corresponds toBernoulli equation. Finally, the Lie algebras generated by constant vector fields

(d0 + d1x + d2x2 + x3)∂x , di ∈ R

lead to separable differential equations, therefore integrable by one quadrature.One can proceed in a similar way with the Gambier equation [CGL13], which

can be described as the coupling of two Riccati equations:

⎧⎪⎨⎪⎩

dy

dt= − y2 + a1y + a2,

dx

dt= a0x2 + nyx + σ,

where n is an integer, σ is a constant, which can be scaled to 1 unless it happens tobe 0, and a0, a1, a2 are certain functions depending on time.

If n �= 0, we can eliminate y between the two equations above, which gives raiseto the one referred to as second-order Gambier equation:

d2x

dt2= n − 1

xn

(dx

dt

)2

+ a0(n + 2)

nx

dx

dt+ a1

dx

dt− σ

(n − 2)

nx

dx

dt

− a20

nx3 +

(da0dt

− a0a1

)x2 +

(a2n − 2a0

σ

n

)x − a1σ − σ2

nx.

Particular examples are the second-order Riccati equation (n = 1 and σ = 0):

d2x

dt2= (a1 + 3a0x)

dx

dt− a2

0 x3 +(

da0dt

− a0a1

)x2 + a2x,

and the second-order Kummer–Schwarz equation (n = −2, a1 = σ = 0):

d2x

dt2= 3

2x

(dx

dt

)2

− 2c0x3 + 2ω(t)x .

Coming back to the general case, the vector fields on T R0, with R0 ≡ R − {0},involved are:

Y1 = v∂

∂x, Y2 = v2

x

∂

∂v, Y3 = xv

∂

∂v, Y4 = v

∂

∂v, Y5 = v

x

∂

∂v,

Y6 = x3∂

∂v, Y7 = x2

∂

∂v, Y8 = x

∂

∂v, Y9 = ∂

∂v, Y10 = 1

x

∂

∂v,


to be completed with the vector field

Y11 = x∂

∂x.

Then,

X =11∑

α=1

bα

(a0,

da0dt

, a1, a2,σ, n

)Yα,

where b11 = 0 and

b1 = 1, b2 = n − 1

n, b3 = a0

n + 2

n, b4 = a1, b5 = −σ

n − 2

n,

b6 = −a20

n, b7 = da0

dt− a0a1, b8 = a2n − 2a0

σ

n, b9 = −a1σ, b10 = −σ2

n.

Here the important Lie algebra W is the one generated by Y4, Y8 and Y11:

[Y4, Y8] = −Y8, [Y4, Y11] = 0, [Y8, Y11] = −Y8.

More details can be found in [CGL13].

References

[Lie93] Lie, S., Scheffers, G.: Vorlesungen über continuierliche Gruppen mit geometrischenund anderen Anwendungen. Teubner, Leipzig (1893) (Edited and revised by G. Schef-fers)

[Ca00] Cariñena, J.F., Grabowski, J., Marmo, G.: Lie-Scheffers Systems: A GeometricApproach. Bibliopolis, Napoli (2000)

[Ca07b] Cariñena, J.F., Grabowski, J., Marmo, G.: Superposition rules, Lie theorem, and partialdifferential equations. Rep. Math. Phys. 60, 237–258 (2007)

[CL11] Cariñena, JF., de Lucas, J.: Lie systems: theory, generalisations, and applications. Dis-sertationes Mathematicae 479, Institute of Mathematics, Polish Academy of Sciences,Warszawa (2011)

[Sc02] Schwarz, F.: Equivalence classes, symmetries and solutions of linear third-order dif-ferential equations. Computing 69, 141–162 (2002)

[Nd08] Ndogmo, J.C.: A method for the equivalence group and its infinitesimal generators. J.Phys. A: Math. Theor. 41, 102001 (2008)

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[WN64] Wei, J., Norman, E.: On global representations of the solutions of linear differentialequations as a product of exponentials. Proc. Amer. Math. Soc. 15, 327–334 (1964)

[Ca98] Cariñena, J.F., Marmo, G., Nasarre, J.: The nonlinear superposition principle and theWei-Norman method. Int. J. Mod. Phys. A13, 3601–3627 (1998)

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twined Hamiltonians. Ann. Phys. 292, 42–66 (2001)[CR00b] Cariñena, J.F., Ramos, A.: The partnership of potentials in Quantum Mechanics and

shape invariance. Mod. Phys. Lett. A15, 1079–1088 (2000)[La68] Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm.

Pure Appl. Math. 21, 467–490 (1968)[CI85] Cariñena, J.F., Ibort, L.A.: Geometrical setting for Lax equations associated to dynam-

ical systems. Phys. Lett. 107A, 356–358 (1985)[CM94] Cariñena, J.F., Martínez, E.: A new geometric setting for Lax equations. Int. J. Mod.

Phys. A9, 4973–4986 (1994)[CI83] Cariñena, J.F., Ibort, L.A.: Non-Noether constants of motion. J. Phys. A: Math. Gen.

16, 1–7 (1983)[Er80] Ermakov, V.P.: Second-order differential equations. Conditions of complete integra-

bility, Univ. Isz. Kiev Series III 9, 1–25 (1880) (translation by A.O. Harin)[Cal69] Calogero, F.: Solution of a three body problem in one dimension. J. Math. Phys. 10,

2191–2196 (1969)[Pe90] Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras.

Birkhauser, Basel (1990)[Ch05] Chalykh, O.A., Vesselov, A.P.: A remark on rational isochronous potentials. J. Nonlin.

Math. Phys. 12(Suppl. 1), 179–183 (2005)[ACMP07] Asorey,M., Cariñena, J.F., Marmo, G., Perelomov, A.M.: Isoperiodic classical systems

and their quantum counterparts. Ann. Phys. 322, 1444–1465 (2007)[CL08] Cariñena, J.F., de Lucas, J.: A nonlinear superposition rule for solutions of the Milne-

Pinney equation. Phys. Lett. A372, 5385–5389 (2008)[CLr08] Cariñena, J.F., de Lucas, J., Rañada, M.F.: Recent applications of the theory of Lie

systems in Ermakov systems. SIGMA 4, 031 (2008)[CdLS12] Cariñena, J.F., de Lucas, J., Sardón, C.: A new Lie systems approach to second-order

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Chapter 10Appendices

Appendix A: Glossary of Mathematical Terms

In this appendix we summarize the most fundamental and common mathematicalnotions that appear everywhere on the book refereeing the reader to the quotedliterature for further information.

A.1 Glossary of Algebraic Terms

Groups

Definition 10.1 A group G is a set together with a binary composition law ◦: G ×G → G such that it has a neutral element e (e ◦ g = g ◦ e = g), is associative(g ◦ h) ◦ k = g ◦ (h ◦ k), and every element g ∈ G has an inverse g−1 (g ◦ g−1 =g−1 ◦ g = e), for all g, h, k ∈ G.

A group G is Abelian or commutative if g ◦ h = h ◦ g for all g, h ∈ G. It isfrequent to omit the symbol ◦ (unless there is risk of confusion) when writing thecomposition g ◦ h of two elements g, h ∈ G that will written in what follows as gh.

Examples of groups are provided by the collection S(X) of all bijective mapsα : X → X of an arbitrary set X with the binary operation the composition law ofmaps. If X is finite with n elements S(X) is called the group of permutations of ordern and denoted by Sn .

A set S with a composition law ∗ verifying only the associativity and the existenceof a neutral element is called a semigroup.1

1 A fundamental algebraic structure extending the notion of group (even though it will not be usedin this book) is the notion of groupoid that we will skip here.


611

612 10 Appendices

Definition 10.2 A subgroup H of a group G is a subset H ⊂ G such that hk ∈ Hand h−1 ∈ H for any h, k ∈ H . A subgroup H of G is called normal if ghg−1 ∈ Hfor any g ∈ G and h ∈ H .

Given a group G and a subgroup H , the quotient G/H denotes the family ofsubsets gH = {gh | h ∈ H}, g ∈ G, called (right) cosets. If H is a normalsubgroup of G the set G/H becomes a group with the induced composition law(gH) ◦ (g′H) = (gg′)H .

Definition 10.3 Given two groups G, G ′, a group homomorphism from G to G ′ isa map ϕ : G → G′ such that ϕ(gg′) = ϕ(g)ϕ(g′), for all g, g′ ∈ G.

The set {g ∈ G | ϕ(g) = e′} is called the kernel of the homomorphism ϕ andis always a normal subgroup of G. It will be denoted by ker ϕ. The range of thehomomorphism ϕ is a subgroup of G ′ and it will be denoted by Im ϕ.

Definition 10.4 A group homomorphism ϕ : G → G ′ is called a monomorphism ifit is injective; epimorphism if it is surjective and isomorphism if it is bijective. If ϕ ifa group isomorphism the inverse ϕ−1 is a group homomorphism too, hence a groupisomorphism. The set of all group isomorphisms ϕ : G → G is called the group ofautomorphisms of G and denoted Aut(G).

Given a group homomorphism ϕ : G → G ′, if we denote by π the canonicalprojection π : G → G/ ker G given by π(g) = g ker ϕ, by ϕ : G/ ker ϕ → Imϕ, themap given by ϕ(g ker ϕ) = ϕ(g), and by i : Imϕ → G′ the canonical inclusion, thenπ is a group epimorphism, ϕ is a group isomorphism, i is a group monomorphism,and ϕ = i ◦ ϕ ◦ π (First theorem of group isomorphisms).

Rings and Fields

Definition 10.5 A ring R is an Abelian group (with the group operation denotedadditively as ‘+’), equippedwith a further operation, denotedmultiplicatively, whichis associative, (ab)c = a(bc), and satisfies the distributive laws (a+ b)c = ac+ bcand a(b + c) = ab + ac, for all a, b, c ∈ R.

A subring Y of a ring R is a subgroup Y ⊂ R of the group structure such thatxy ∈ Y for any x, y ∈ Y .

The ring R is commutative if ab = ba, ∀a, b ∈ R. A unit element in the ringR is an (unique) element 1 ∈ R satisfying: 1a = a1, for all a ∈ R. The symbol 0stands for the neutral element of the (Abelian) group structure ofR.

Given a ring R we can always consider that it has unit element because of thefollowing construction. Suppose that R doesn’t have a unit, then consider the set2

R = R×Z of pairs (x, n)where n ∈ Z, with the composition laws: (x, n)+(y,m) =(x+ y, n+m) and (x, n) · (y,m) = (xy+mx+ny, nm)where x, y ∈ R, n,m ∈ Z,

2 We may use any ring with unit instead of Z like R or C as it will be done in the case of algebras(see later).

Appendix A: Glossary of Mathematical Terms 613

and nx = x + · · ·+ x , n times. The setR with the previous operations is a ring withunit element (0, 1). The subring of elements of the form (x, 0) is isomorphic (seebelow) to the original ring R. In what follows we will consider that all rings have aunit 1.

A familiar example of a ring is provided by the setZ of the integers. Other relevantexamples are:

(i) The ring R[x] of the polynomials with coefficients in R.(ii) The set of all real-valued functions f : X → R on a set X is a ring, de-

noted as F(X), under the usual definition of pointwise addition and multiplicationof functions.

(iii) The collection Mn(Z) of all n×n matrices with integer entries. Similarly thematrices Mn(R)with entries in a ringRwith the standard addition andmultiplicationof matrices (A · B) jk =∑n

l=1 A jl Blk .(iv) Given an Abelian group (A,+) the family of all group homomorphisms

ϕ : A → A, is a ring denoted byEnd(A)with the addition (ϕ+ψ)(a) = ϕ(a)+ψ(a),and the composition of maps (ϕ ◦ψ)(a) = ϕ(ψ(a)), for all a ∈ A. The unit elementis the identity map.

(v) The set of all the differential operators (Appendix G) with constant (real) coef-ficients. The Fourier transform exhibits an isomorphismwith the ring of polynomialsin ‘momentum space’.

Definition 10.6 Given a ring R a left (right) ideal J is a subgroup J ⊂ R of theAbelian group structure such that ax ∈ J (xa ∈ R) for all a ∈ R, x ∈ J . AnAbelian subgroup which is both a left and a right ideal is called a bilateral ideal. IfR is a commutative ring any left or right ideal is a bilateral ideal (or ideal for short).

The subgroups {0} and R are ideals, called improper ideals of the ringR.The abelian subgroup pZ of integers multiples of p is an ideal of Z. Given a

polynomial p(x) its multiples are again a ideal on R[x]. Given a set X and a subsetY ⊂ X the set of functions vanishing at Y is an ideal on F(X).

Given an ideal J of the ring R, the quotient Abelian group R/J is again a ringwith the induced product (x + J )(y + J ) = xy + J .

Given a subset S ⊂ R the set of finite linear combinations∑

k ak xk (∑

k xkak)of elements xk ∈ S and ak ∈ R is a left (right) ideal, called the left (right) idealgenerated by S and denoted by RS (SR). In what follows we will omit ‘left’ or‘right’ when referring to ideals if the definitions and properties stated hold whensubstituting left for right and vice versa.

The intersection of two ideals is again and ideal. The ideal generated by S is thesmallest ideal containing it, that is, the intersection of all ideals containing it.

Definition 10.7 An ideal J of the ring R is called maximal if is proper and theredoesn’t exist an ideal I such that J ⊂ I ⊂ R (all inclusions strict). Equivalently,J is a maximal ideal if for any x /∈ J the ideal generated by J ∪ {x} isR.

Definition 10.8 Given two rings Ra , a = 1, 2, a ring homomorphism � is a grouphomomorphism � : R1 → R2 of the corresponding Abelian groups, such that

614 10 Appendices

�(xy) = �(x)�(y) for all x, y ∈ R1. Given a ring homomorphism � the ker-nel of � as a group homomorphism is an ideal of R1 and it will be denoted byker�.

A ring homomorphism will be called mono-, epi- or isomorphism if the corre-sponding group homomorphism is mono-, epi- or isomorphism.

Definition 10.9 A division ring is a ring K such that its non zero elements form agroup under multiplication. A field is a commutative division ring.

The only ideals on a field K are {0} and K, and this property characterizes fields.Standard examples of fields are the set Q of the rational numbers and the sets R andC of the real and complex numbers.

If J is a maximal ideal of R, then R/J is a field an conversely. For instance ifp is a prime number the ideal pZ is maximal and the congruence classes of integersmod p, Zp = Z/pZ is a field.

The ideal Jx of all real-valued functions on a set X vanishing at the point x ∈ Xis maximal. The quotient F(X)/Jx is just the field R.

Definition 10.10 The family of allmaximal ideals of the ringR is called themaximalspectrum (or simply, the spectrum) ofR. It will be denoted by Spec(R).

Modules, Linear Spaces and Algebras

Definition 10.11 A left R-module M over a ring R is an Abelian group (M,+)

together with a product ◦: R ×M → M, (x,m) → x ◦ m, such that the mapαx : M → M defined by αx (m) = x ◦ m, for all x ∈ R and m ∈ M, is a ringhomomorphism fromR to the ring of endomorphisms End(M) ofM. It is also saidthat R is represented in the ring of endomorphisms of M as an Abelian group. Asusual we will use a multiplicative notation for ◦.

RightR-modules are defined in a similar way. A setMwhich is a left and a rightR-module will be called andR-bimodule. In what follows, unless stated otherwise,we will just call ‘module’ a leftR-module.

An ideal J of a ring R is a left (right) R-module with the operation x ◦ a = xa(a ◦ x = ax) for all a ∈ J and x ∈ R. The Cartesian product J n = J × · · · × Jis a leftR-module with the operation x(a1, . . . , an) = (xa1, . . . , xan).

Let M = F(X,Rn) be the space of functions on the set X with values in

Rn . Then M is a left (right) F(X)-module with the operation f ◦ (σ1, . . . , σn) =

( f σ1, . . . , f σn), with f any real-valued function on X and σ = (σ1, . . . , σn) anyelement inM.

A submodule of the moduleM is a subgroup (necessarily Abelian)N ⊂M suchthat xn ∈ N for any x ∈ R and n ∈ N . The quotient (Abelian) group M/N withthe operation x(m +N ) = xm +N is a module.

The intersection of two submodules of a module is again a module. Given a setS ⊂ M the module generated by S is the smallest module in M containing S, or


equivalently the set of all finite linear combinations∑

k xkmk , with xk ∈ R andmk ∈M. The set S will be called a generating system ifM is the module generatedby S. The module M is finitely generated if it possess a finite generating set.

We will say that M is a free module with B as a basis if every m ∈ M canbe written uniquely as a finite sum: m = ∑

k xkmk , mk ∈ B, xk ∈ R. Given anyset S the module M/S) defined as the family of finite formal linear combinations∑

k xksk , sk ∈ S and xk ∈ R, is called the free module generated by S. Clearly S isa basis for M/S). A finitely generated free R-module has a finite basis. Basis neednot be unique.

Definition 10.12 A homomorphism of modules � : M1 → M2 is a group homo-morphism such that �(xm) = x�(m) for all x ∈ R, m ∈ M1. A homomorphismof modules is a mono-, epi- or isomorphism if is so as a group homomorphism.

Definition 10.13 A linear space (or vector space) E is a left K-module where K is afield, tipically R or C, in which case it will be called a real or complex (linear) spacerespectively.

Any linear space is a K-bimodule with the right K-module structure given by theobvious operation v ◦ λ = λv, λ ∈ K, v ∈ E . Any linear space is a free module,hence it has a basis, sometimes called a Hamel basis if its cardinal is non-finite. Thedimension of a linear space is the cardinal of any basis on it. A linear subspace V ofa linear space E is a submodule of E .

If E , F are linear spaces, a module homomorphism� : E → F is called a linearmap. The space of linear maps from E to itself is a linear space and a ring denoted byEnd(E). The overall structure of End(E) is called an algebra (see Definition 10.14)and, accordingly, it is called the algebra of endomorphisms of E . The group of allisomorphisms from E into itself is called the general linear group of the space E andis denoted by GL(E).

If� : E → F is a linearmap, there is a canonical linear isomorphism� : E/ ker�→ �(E) ⊂ F which constitutes the extension to linear spaces of the first isomor-phism theorem for Abelian groups.

Associative Algebras and Lie Algebras

Definition 10.14 Analgebra is a ring that is at the same time a vector space (A,+, ◦)and such that: λ(u ◦ v) = (λu) ◦ v = u ◦ (λv) and ∀λ ∈ K.

Any ringR can be made into an algebra over the fieldK by considering the familyof finite linear combinations of the form

∑k λk xk , λk ∈ K and xk ∈ R, with the

obvious addition and multiplication operations. Such algebra is denoted byR⊗K.The notion of subrings, ideals, ring homomorphisms, and all other related conceptsextend without change to the class of algebras. Thus, in what follows we will justdiscuss algebras.

616 10 Appendices

All examples of rings discussed before are algebras or can be made into algebrasby using the construction before. For finite-dimensional algebras, that is, possessinga finite-dimensional basis, sayB = {e1, . . . , en}, we can write, ei ◦e j =∑n

k=1 cki j ek ,

ci jk ∈ K. The n3 constants cki j are called the structure constants of the algebra relative

to the basis B.A map� : A1 → A2 is an algebra homomorphism if it is a linear map and a ring

homomorphism. The group of algebra isomorphisms from A into itself is the calledthe automorphisms group of A, sometimes denoted by Aut(A).

The algebras wewill be dealingmost in this book are associative and Lie algebras.

Definition 10.15 An algebra (A,+, ◦) is associative if (u ◦ v) ◦ w = u ◦ (v ◦ w)

for all u, v, w ∈ A. An associative algebraA is commutative if u ◦ v = v ◦ u for allu, v ∈ A.

A Lie algebra is an algebra (L,+, [·, ·]) such that: [u, v] = −[v, u], and[u[v,w]] + [v, [w, u]] + [w, [u, v]] = 0 (Jacobi identity), for all u, v, w ∈ L.

With any associative algebraA we can associate a Lie algebra by defining a newproduct as, [u, v] = u◦v−v◦u called the commutator. It is straightforward to checkthat this product does indeed satisfy the Jacobi identity. Obviously, if the algebra Ais commutative, the commutator always vanish and the Lie algebra is trivial.

Typical examples of Lie algebras are provided by the linear transformationsEnd(E) of a vector space E with the Lie algebra product defined by the commutatorof linear maps. Ado’s theorem [Ja79] states that any finite-dimensional Lie algebra isisomorphic to a Lie subalgebra of the Lie algebra End(E)with E a finite-dimensionallinear space.

Definition 10.16 Given an algebraA, a derivation is a linear map D : A→ A suchthat: D(a ◦ b) = a ◦ D(b)+ D(a) ◦ b (Leibnitz’ identity), for all a, b ∈ A. The setof derivations of A will be denoted as Der(A).

The set of derivations ofA can be made into a Lie algebra by defining the productof derivations as,3

[D1, D2](a) = D1(D2(a))− D2(D1(a)), ∀D1, D2 ∈ Der(A), a ∈ A, (10.1)

We can consider derivations on the algebra of derivations, i.e., linear maps,D : Der(A)→ Der(A), satisfying,

D([D1, D2]) = [D(D1), D2] + [D1,D(D2)]. (10.2)

Any derivation D ∈ Der(A) defines a derivationDD , called inner, of the Lie algebraof derivations:DD(D1) = [D, D1].With this definition, the Jacobi identity is nothingbut the assertion that DD is a derivation on the Lie algebra Der(A) for any D.

3 Notice that the composition of derivations is not a derivation in general.


Graded Algebras, Graded Lie Algebras and Graded Derivations

Definition 10.17 An algebraA is a graded algebra if, as a linear space,A is a gradedlinear space over the Abelian group H , that is,A =⊕

r∈H Ar , andAr ◦As ⊆ Ar+s .A Lie algebra L is a graded Lie algebra if L = ⊕

r∈H Lr is a graded algebraover the Abelian group H , i.e., [Lr ,Ls] ⊆ Lr+s , it is graded commutative: [x, y] =−(−1)rs [y, x], for all x ∈ Lr , y ∈ Ls , and satisfies the graded Jacobi identity:

(−1)r1r3 [x1, [x2, x3]]+ (−1)r2r1 [x2, [x3, x1]]+ (−1)r3r2 [x3, [x1, x2]]

= 0, xi ∈ Lri , i = 1, 2, 3 .

Definition 10.18 Let A = ⊕rAr be a graded algebra. A linear map D : A→ A issaid to be a (graded) derivation of degree k if D(Ar ) ⊂ D(Ar+k) and, for any pair ofhomogeneous elements,α ∈ Ar andβ ∈ As , D(α◦β) = (Dα)◦β+(−1)krα◦(Dβ).

It should be remarked that, if D1 and D2 are derivations of a graded Lie algebraof degrees k1 and k2 respectively, then the graded commutator defined by,

[D1, D2] = D1 ◦ D2 − (−1)k1k2 D2 ◦ D1 (10.3)

is a derivation of degree k1 + k2. On the other hand, for H = Z, it can be shownthat, if A1 generates Ar for all r ≤ 1, then any derivation of A will be determinedby giving the images of the elements of degree zero and one.

A.2. Topology: A Brief Dictionary

Topological Spaces: Hausdorff and Separable Spaces

Definition 10.19 A topological space is a set X and a family of subsets O of Xcontaining ∅ and X , and such that if Oα , α ∈ A is any collection of subsets in O,then

⋃α Oα belongs to O, and the intersection of any two elements in the family

belongs to the family. The familyO is called a topology (for the space X ) and subsetsO in the familyO are called open sets. The open sets ∅, X are called improper opensubsets. A proper open set is an open set different from ∅ and X .

Given a point x ∈ X , a subset N ⊂ X is said to be a neighborhood of x it itcontains and open set O that contains x , i.e., x ∈ O ⊂ N . Given a subset S ⊂ X , apoint x ∈ S is said to be in the interior of S if S is a neighborhood of x . The interiorof S is the union of all its interior points. The interior of S will be denoted by Int(X).

Given two topological spaces X and Y its Cartesian product X×Y is a topologicalspace too, its topology defined by the family of arbitrary unions of sets of the formOX × OY where OX is open in X and OY open in Y . This topology is called theproduct topology (of the topologies in X and Y ).

618 10 Appendices

The set of real numbers R with the topology obtained by considering arbitraryunions of open intervals, including the improper subsets, is a topological space. Wewill call this topology the natural topology in R and unless stated otherwise this willbe the topology that will be considered on it. The Cartesian product Rn of n copies ofR is a topological space with the product topology (again the only topology that willbe considered). In particular the identification of C with R

2 by means of z = x + iyallows us to consider C as a topological space.

Definition 10.20 A topological space X is called Hausdorff, also said to satisfy theT2 separationproperty, if given twodifferent points x �= y, there exists neighborhoodsNx of x and Ny of y such that Nx

⋂Ny = ∅.

The topological space Rn is Hausdorff.

Definition 10.21 Let (X,O) be a topological space. A subset F ⊂ X is said to beclosed if X\F is open. Notice that the improper subsets ∅ and X are always openand closed in any topology.

It is clear that the intersection of an arbitrary family of closed sets is closed. Givena subset S ⊂ X , the intersection of all closed sets containing S is called the closureof S and is by construction the smallest closed set containing S. The closure of Swill be denoted by S. The set S\Int(S) is called the boundary of S and denoted by∂S.

Definition 10.22 A subset S ⊂ X of the topological space X is said to be denseif S = X . A topological space satisfies the second countability axiom, also-calledseparable, if it contains a countable dense subset.

Continuous Maps and Homeomorphisms

Definition 10.23 Amap ϕ : X → Y from the topological space X to the topologicalspace Y is continuous if ϕ−1(O) is open in X for any open set O ⊂ Y .

The composition ψ ◦ ϕ of two continuous maps ϕ : X → Y , ψ : Y → Z iscontinuos.

Definition 10.24 A bijective continuous map ϕ : X → Y such that the inverse mapϕ−1 : Y → X is continuous too, is called a homeomorphism. The family of allhomeomorphisms from X into itself form a group.

If f, h : X → R are real-valued continuous functions, then the pointwise sum( f + h)(x) = f (x)+ h(x), the pointwise product ( f · h)(x) = f (x)h(x), and thefunction (λ · f )(x) = λ( f (x)) where λ ∈ R, are continuous. The set of real-valuedcontinous functions on X is an algebra denoted by C(X) (in particular a linear spaceover the field R).

In a similar way the set of all continuous complex-valued functions f : X → C

is an algebra. Notice that the real and imaginary parts u, v of f given respectively as


u(x) = Re( f (x)), v(x) = Im( f (x)), are such that f = u + iv and f is continuousiff u and v are continuous. The space of complex-valued continuous functions onX is a complex algebra, that is an algebra which is a complex linear space. Thelinear complex linear space of complex-valued continuous functions on X is thecomplexification of the real space of real-valued continuous functions on X and itwill be denoted either as C(X;C) or C(X)C.

Fundamental examples of topological spaces are linear spaces E equipped witha topology such that the addition and multiplication maps +: E × E → E and·C × E → E are continuous. Such linear spaces will be called topological linearspaces.

Compactness

A covering of a subset S of a topological space X is a family C of subsets Cα ⊂ X ,labelled by a set α ∈ A such that S ⊂⋃

α Cα . The covering C is called open if all itselements are open sets. Given a covering C of S a sub covering is a subfamily C′ ofC (that is there is a subset A′ ⊂ A such that Cα belongs to C′ if α ∈ A′ ⊂ A) whichis still a covering of S.

Definition 10.25 A subset K ⊂ X of the topological space X is compact if givenan open covering C of K there exists a finite sub covering.

Bolzano’s theorem states that any real-valued continuous function f on a compactspace X reaches its maximum and minimum, in other words, f (X) = [a, b] wherea = min f and b = max f .

Heine-Borel theorem states that a subset of Rn is compact iff it is closed andbounded (see later on Appendix A.2, Metric spaces).

Given a covering C of a given subset S ⊂ X , a covering C′ of S such that for anyset C ′

α′ in C′ there exists a set Cα in C such that C ′α′ ⊂ Cα is called a refinement of C.

Definition 10.26 A subset S of the topological space X is called paracompact ifgiven any open covering C of it, there exists a refinement C ′ which is locally finite,that is by that that given any point x ∈ X there exists only a finite number of sets inC′ containing x .

Subspaces, Conectedness

Given a subspace S ⊂ X of the topological space X , the family of sets S ∩ O withO an open set in X defines a topology on S called the trace topology. The subsetS ⊂ X equipped with the trace topology is said to be a topological subspace of X .

Definition 10.27 A subset S of a topological space is said to be connected if itdoesn’t have proper subsets which are open and closed simultaneously with respectto the trace topology.

620 10 Appendices

A continuous curve on the topological space X is a continuous map γ : I → X .A subset S ⊂ X is said to be arc-connected if given any two points x, y ∈ S thereexists a continuous curve γ : [a, b] → S such that γ (a) = x and γ (b) = y. Then wewill say that the continuous curve γ joins x and y. If S is open and arc-connected, itis connected.

Let S be topological subspace, then two points x, y ∈ S are in the same connectedcomponent of S if there exists a continuous curve joining x and y. The relation ‘beingin the same connected component’ is an equivalence relation and a connected com-ponent of S is any of the corresponding equivalence classes. A connected componentof S is open, hence connected.

The Fundamental Group of a Topological Space

Definition 10.28 Given x, y ∈ X , two continuous curves γ1, γ2 : [0, 1] → X arehomotopic if there exists a continuous map H : [0, 1] × [0, 1] → X such thatH(0, t) = γ1(t), H(1, t) = γ2(t) for all 0 ≤ t ≤ 1, and H(s, 0) = x , H(s, 1) = yfor all 0 ≤ s ≤ 1. The points x, y are called the end-points of the curves. The relationγ is homotopic to γ ′ is an equivalence relation in the set of all continuos maps form[0, 1] to X with fixed end-points x, y.

Given a point x ∈ X , the family of homotopy equivalence classes of continuouscurves with coinciding end-points x is a group with respect to the composition law:

(γ1 � γ2) ={γ2(2t), if 0 ≤ t < 1/2γ1(2t − 1), if 1/2 ≤ t < 1

This group will be called the fundamental group of the topological space at the pointx and denoted by π1(X, x).

If x, y lie in the same connected components of X the corresponding fundamentalgroups are isomorphic. If the space X is arc-connected then the fundamental groupsat all points are isomorphic. The equivalence class of homotopy groups defined byπ1(X, x),x an arbitrary point in X will be called the fundamental group of X , thefirst homotopy group of X of the Poincarre group of X , and will be denoted simplyas π1(X).

Definition 10.29 A connected and arc-connected topological space X will be saidto be simply connected if π1(X) = 0 or, in other wordis, if any closed continuouscurve is homotopic to a constant continuous curve.

Metric Spaces: Completeness

Definition 10.30 A metric space is a set M and a map d : M × M → R+ where

R+ denotes the set of non-negative real numbers, such that d(x, x) = 0 iff x = 0,


d(x, y) = d/y, x) and d(x, y) ≤ d(x, z) + d(z, y) (“triangle inequality”), for allx, y, z ∈ M . Then the map d is called a metric (or distance) on M .

Given a pont x and a positive real number r the subset {y ∈ M | d(x .y) < r iscalled the open ball or radius r and center x in M , and it will be denoted as Br (x).We construct a topology on a metric space (M, d) by defining open sets as arbitraryunion of open balls or, equivalently, sets O such that for any x ∈ O , there existsr > 0 such that Br (x) ⊂ O . Such topology is called the metric topology.

Twometrics d1, d2 on M are equivalent if the identitymap i : M toM is continuouswith respect to both metric topologies in M . Equivalently, d1 is equivalent to d2 ifthere exists two positive real numbers B > A > 0 such that for any pair of pointsx, y ∈ M , we have Ad1(x, y) ≤ d2(x, y) ≤ Bd1(x, y).

The set Rn is a metric space with respect to the metrics dp(x, y =

(∑n

i=1 |xi .yi |p)1/p for all p ≥ 1. Moreover we define d∞(x, y) = max{|xi − yi | |i = 1, . . . , n}. The metrics dp, 1 ≤ p ≤ ∞ are equivalent.

Definition 10.31 A sequence |{xn}, n = 1, 2, . . ., is said to be convergent if thereexists lim xn , that is, there exists x ∈ M such that for every ε > 0, there existsN ∈ N such that if n ≥ N , then d(xn, x) < ε. A sequence |{xn}, n = 1, 2, . . ., issaid to be Cauchy, if given ε > 0, there exists N ∈ N such that if n,m ≥ N , thend(xn, xm) < ε. It is immediate to show that any convergent sequence is Cauchy.

Definition 10.32 A metric space (M, d) is complete if any Cauchy sequence isconvergent. We will also say that the metric topology is complete.

The metric space Rn with respect to any metric dp is complete.

The space of real-valued continuous functions on a compact space is a completemetric space with respect to the metric d∞( f, h) = max{| f (x)| | x ∈ X}.

Let X = Rn and consider the subset of continuous real-valued functions such that

the following integral is convergent:

dp( f, h) =⎛⎝∫Rn

| f (x)− h(x)|pdn x

⎞⎠

1/p

<∞

The function dp defines a metric on the corresponding subspace, however it is notcomplete for any 1 ≤ p <∞.

A.3. A Concise Account of Differential Calculus

Banach and Hilbert Spaces

Definition 10.33 Let E be a real or complex linear space. A norm on E is a function|| · || : E → R

+ such that ||x || = 0 iff x = 0, ||λx || = |λ||x ||, and ||x + y|| ≤||x || + ||y|| for all λ ∈ C, x, y ∈ E .

622 10 Appendices

An inner product on a complex linear space E is a map 〈·, ·〉E × E → C suchthat 〈x, x〉 = 0 iff x = 0, 〈x, λy + μz〉 = λ〈x, y〉 + μ〈x, z〉 and 〈x, y〉 = 〈y, x〉 forall λ,μ ∈ C, x, y, z ∈ E .

It is obvious that if || · || is a norm, the function d(x, y) = ||x − y|| defines ametric on E . The topology on E defined by the metric d is called the norm topology,and the linear space E equipped with this topology is a topological vector space.

If 〈·, ·〉 is an inner product then it is always satisfied the following fundamentalCauchy-Schwartz inequality: |〈x, y〉| ≤ ||x ||||y|| for all x, y ∈ E . Then it is clearthat if 〈·, ·〉 is an inner product the function ||x || = +√〈x, x〉 defines a norm thatwill be called the norm associated to the inner product 〈·, ·〉.Definition 10.34 A complex linear space E equipped with a norm will be called anormed space. A Banach space is normed space E which is complete with respectto the norm topology.

Similarly, a complex linear space E equipped with an inner product is called apre-Hibert space. A Hilbert space is a pre-Hilbert space H which is complete withrespect to the topology defined by the norm associated to the inner product. EveryHilbert space is a Banach space with respect to the norm associated to the innerproduct.

Examples of Banach spaces are provided by the completion of the spaces ofcontinuous functions onR

m with respect to the norm || f ||p = dp( f, 0), 1 ≤ p ≤ ∞.Such spaces are denoted by L p(Rn). In the particular instance of p = 2, the norm|| · ||2 is associated to the inner product 〈 f, h〉2 =

∫f (x)h(x)dn x and L2(R2) is a

Hilbert space with respect to this product.The natural extension ofC

n to infinite dimensions is the space l2(C) consisting oninfinite sequences z = {zk}, k = 1, 2, . . .. of complex numbers such that

∑k |zk |2 <

∞. It is a Hilbert space with respect to the inner product 〈z, w〉 =∑k zkwk .

Definition 10.35 Let E be a Banach space. A linear map T : E → E is said tobe continuous (or bounded) if it is continuous with respect to the norm topology.The space of bounded linear maps from E to E form an algebra. The function||T || = sup{||T x || | ||x || = 1} defines a norm on the complex linear space B(E) ofbounded linear maps from E to E sometimes called the operator norm.

The space B(E) is a Banach space with the operator norm.Given a Hilbert spaceH and a bounded linear operator T , its adjoint operator T †

is the (unique) operator defined by 〈T †x, y〉 = 〈x, T y〉, for all x, y ∈ H.

Definition 10.36 Let H be a Hilbert space. An operator U : H → H is said to beunitary if 〈U x,U y〉 = 〈x, y〉 for all x, y ∈ H. Notice that an unitary operator isnecessarily bounded and ||U || = 1.

A bounded operator T : H→ H is called self-adjoint (or Hermitean) if 〈T x, y〉 =〈x, T y〉, or all x, y ∈ H, that is T = T †.


Definition 10.37 Let T be a (non-necessarily bounded) linear operator defined ona subspaceD ⊂ H. We will say that λ is in the spectrum of T if the operator T − λI

is not invertible. The spectrum of T will be denoted by σ(T ). A complex number λwill be said to be an eigenvalue of T if there exists a non-zero vector x ∈ D suchthat T x = λx . If λ is an eigvenvalue of T then λ ∈ σ(T ).

Theorem 10.38 (Easy spectral theorem) Given a bounded self-adjoint operator Ton a Hilbert space T , then σ(T ) ⊂ R is compact. Moreover, there exists a Borelmeasure E on R with values in the space of projectors onH such that T = ∫

λE(dλ).

A similar theorem holds for unbounded (normal) self-adjoint operators.

C∗-Algebras

Definition 10.39 A Banach algebra B is an algebra with a complete norm || · || suchthat ||xy|| ≤ ||x ||||y|| for all x, y ∈ B.

Standard examples of Banach algebras are provided by the space C(K ) of con-tinuous functions on a compact space K and the norm the supremum norm || · ||∞.Morevoer the space B(E) of bounded operators on a Banach space is a Banach alge-bra. In particular the space of finite-dimensional square matrices Mn(C) is a Banachalgebra.

We can associate with each Banach algebra B the space Spec(B) of closed propermaximal ideals on it called the maximal spectrum (or just the spectrum) of B.

IfB is a commutative Banach algebra with unit, thenmaximal ideals are in one-to-one correspondence with the kernel of algebra morphisms ρ : B → C. The spectrumof B is compact in a natural topology and there is a natural map from B into thespace of continuous functions C(Spec(B)), called the Gelf’and transform, definedby x ∈ B → x , with x(ρ) = ρ(x) which is a homeomorphism.

Definition 10.40 A �-algebra is a complex algebra A possessing a complex anti-linear map � : A → A, x → x�, such that �2 = I, and (xy)� = y�x�, for allx, y ∈ A.

A C∗-algebra is a �-algebraA carrying a complete norm || · || such that (A, || · ||)is a Banach algebra and ||x�x || = ||x ||2 for all x ∈ A.

The space of continuous complex-valued functions on a compact topologicalspace with the map f �(x) = f (x) and the supremum norm is a commutative C∗-algebra. Given a complex Hilbert spaceH the algebra of all bounded operatorsB(H)

is a C∗-algebra with the operation � : T → T †. The GNS (Gel’fand-Naimark-Segal)Theorem, establishes that given aC∗-algebra, there exists a complex Hilbert spaceHand an invectivemorphism ofC∗-algbrasπ : A→ B(H), that is, anyC∗-algebra canbe faithfully represented as a C∗-subalgebra of the C∗-algebra of bounded operatorson a Hilbert space.

As a particular instance of the GNS theoremwe obtain that any finite-dimensionalC∗-algebra can be represented faithfully as an algebra of finite-dimensional complexmatrices where A� is the Hermitean conjugate of the matrix A.

624 10 Appendices

Differential Calculus on Banach Spaces

Most of the standard results of calculus in several variable stand in the general settingof Banach spaces (see for instance [La85]). We will succintly review them.

Definition 10.41 Let f : O ⊂ E → F be a from the open set O in the Banach spaceE to the Banach space F . Let x be a point in O . We will say that f is differentiableat x if there exists a continuous linear map L : E → F such that the following limitexists and is zero:

limh→0

|| f (x + h)− f (x)− L(h)||||h|| = 0.

In such case we will say that L is the derivative (or differential) of f at x and will bedenoted either as f ′(x), d f (x) of D f (x). If f is differentiable for any x ∈ O wewillsay that f is differentiable in O . In such case we have a map d f : O → B(E, F)

from the open set O into the Banach space B(E, F) of bounded linear operatorsfrom E to F . If the map d f is continuous we will say that f is of class C1 on O .

We will say that f is of class Ck , k > 1 on O if Dk−1 f is differentiable forall x ∈ O and Dk f = D(Dk−1 f ) is continuous on O . The collection of functionsof class Ck on O with values in F will be denoted Ck(O, F). We will say that afunction f is of class C∞ or smooth if it is of class Ck for all k.

Clearly the composition addition andmultiplication by complex numbers of func-tions of class Ck are again of class Ck and the standard rules apply: D( f + g) =D f + Dg, D(λ f ) = λD f and if f, h : O ⊂ E → A with A a normed algebra, thenD( f · h) = D f · h + f · Dh (Leibnitz’ rule).

Standard examples are the spaces Ck(O,R) (denoted simply by Ck(O)) withO ⊂ R

n . In such cases even if Ck(O) is a linear space, the topologies induced bythe norms || · ||p, 1 ≤ p <∞, are not complete (even if O is compact).

There is a natural topology on C∞(O) which is being discussed in the main text.We will establish without proofs the main theorems of differential calculus on

Banach spaces:

Theorem 10.42 (Inverse function theorem) Let E, F be a Banach spaces, O ⊂ Eand open set, and f : O → F a function of class C1 such that the differentialD f (x0) : E → F at the point x0 ∈ O is a topological isomorphism, that is, D f (x0) isinvertible and continuous and its inverse is continuous too. Then there exists an openneighborhood V of x0 in O such that the restriction of f to V is a diffeomorphism ontoits image, i.e., the map f : V → f (V ) is invertible and its inverse f −1 : f (V )→ Vis C1.

Theorem 10.43 (Implicit function theorem) Let E, F be Banach spaces, E is thedirect sum of the Banach spaces E1 and E2 and πa : E → Ea, a = 1, 2, denotethe corresponding projections. Let � : O → F, O ⊂ E an open set, be a func-tion of class C1 such that �(x0, y0) = 0 at the point (x0, y0) ∈ O, x0 ∈ E1,


y0 ∈ E2. We will suppose that the differential D�(x0, y0) : E → F is surjectiveand, if we denote by �x0 : π2(O)→ F the map �x0(y) = �(x0, y), that the differ-ential D�x0(y0) : E2 → F is a topological isomorphism. Then, there exists an openneighborhood W of x0 in π1(O) ⊂ E1 and a C1 map ψ : W → π2(O) ⊂ E2, suchthat y0 = ψ(x0) and �(x, ψ(x)) = 0 for all x ∈ W . The function ψ is called theimplicit function determined by � at the point (x0, y0).

We have the following corollary in finite dimensions.

Corollary 10.44 Let � : O ⊂ Rn+m → R

m be a C1 function, and S = {x ∈ O |�(x) = 0}. Suppose that the components ϕi of � = (ϕ1, . . . , ϕm) are functionallyindependent on S, then for every point x ∈ S one can choose a subset (xi1 , . . . , xin )

of the coordinates (x1, . . . , xn+m) and associate with any point x ∈ S an openneighborhood Ux in S and an open set Vx in R

m with coordinates (xi1 , . . . , xin ) thatcan be mapped difeomorphically one onto each other.

Appendix B: Tensor Algebra

The Tensor Algebra of a Linear Space

Let E and F be linear spaces (we shall consider only real or complex linear spaces).The tensor product of the linear spaces E and F , denoted by E ⊗ F , can be simplydefined as the vector space generated by elements of the form u ⊗ w,4 u ∈ E andw ∈ F with relations (u+u′)⊗w = u⊗w+u′⊗w, u⊗(w+w′) = u⊗w+u⊗w′,(λu) ⊗ w = u ⊗ (λw), for all u, u′ ∈ E , w,w′ ∈ F and λ ∈ K. In other words,E⊗ F is the quotient space of the vector space generated by all elements of the formu⊗w by the subspace whose elements have the form (u+u′)⊗w−u⊗w−u′ ⊗w,u⊗ (w+w′)−u⊗w− u⊗w′ or, (λu)⊗w− u⊗ (λw). The tensor product E ⊗ Fis sometimes called the algebraic tensor product of E and F .

If the vector spaces involved in the tensor product are finite-dimensional, thetensor product of both is still finite dimensional. It is easy to check that if {ui },i = 1, . . . n, is a basis for E and {wk}, k = 1, . . . ,m, is a basis for F , then {ui ⊗wk},i = 1, . . . , n, k = 1, . . . ,m is a basis for E ⊗ F , hence dim E ⊗ F = dim E dim F ,and no ambiguities arise when equipping it with a norm.

However, in the infinite-dimensional case, the construction of norms on the alge-braic tensor product of two normed spaces becomes an extremely subtle question.We will just mention here that if E, F are separable Hilbert spaces, there is a canon-ical norm on E ⊗ F associated to the inner product defined by 〈u ⊗ w, u′ ⊗ w′〉 =〈u, u′〉E 〈w,w′〉F for all u ⊗ w, u′ ⊗ w′ ∈ E ⊗ F . The completion of the algebraictensor product E⊗ F with respect to the norm associated to the inner product beforeis denoted by E⊗F .

4 The symbol ‘⊗’ is conventional.

626 10 Appendices

There are natural isomorphisms E⊗F ∼= F⊗E and (E⊗F)⊗G ∼= E⊗(F⊗G),for any three linear spaces E, F and G.

The first natural space associated to a linear space E is the linear space ofK-linearmaps from E to K, denoted E∗ and called the algebraic dual of E . If the space Eis an infinite-dimensional topological space, the natural notion of dual space is thespace of continuous linear maps from E toKwhich is called the topological dual anddenoted by E ′. Of course in finite dimensions E∗ = E ′ and if E is a Hilbert space,because of Riesz duality theorem, E ′ ∼= E . In what follow we will just considerfinite-dimensional spaces and then E∗∗ ∼= E .

Using as building blocks E and E∗ it is possible to construct the linear space,

T rs (E) = E ⊗ · · · ⊗ E ⊗ E∗ ⊗ · · · ⊗ E∗,

with r factors E and s factors E∗. Elements T ∈ T rs (E) are called homogeneous

tensors of type (r, s). It is also said that T is an r -contravariant, s-covariant homoge-neous tensor. Notice that if {ui }, i = 1, . . . , n is a basis for E and {uk}, k = 1, . . . , ndenotes its dual basis, i.e., the basis in E∗ defined by the formula: uk(u j ) = δk

j , than

T = T j1,..., jrk1,...,ks

u j1 ⊗ · · · ⊗ u jr ⊗ uk1 ⊗ · · · ⊗ uks

(where Einstein’s convention of summation over repeated indices at different levelsis used from now on). Notice that the space of linear maps from E to E is isomorphicto E∗⊗E , that is, to the space of homogeneous tensors of type (1, 1) on E . Similarlythe linear space of linear maps from E∗ to E is isomorphic to E ⊗ E and so on.

The direct sum,

T •• (E) =⊕r,s≥0

T rs (E), (10.4)

carries a graded associative algebra structure under the tensor multiplication ⊗ andis called the tensor algebra over E .

The group GL(E) can be realized (or represented) on T (E) as follows: letϕ : E → E be a linear isomorphism, thenwe define the linearmap T s

r (ϕ) : T sr (E)→

T sr (E), by T s

r (ϕ)(u j1 ⊗ · · · ⊗ u jr ⊗ uk1 ⊗ · · · ⊗ uks ) = ϕ(u j1) ⊗ · · · ⊗ ϕ(u jr ) ⊗(ϕ−1)∗(uk1) ⊗ · · · ⊗ (ϕ−1)∗(uks ), for any basis element in T s

r (E) and whereϕ∗(α)(u) = α(ϕ(u)), for all u ∈ E and α ∈ E∗. We will call T s

r ϕ the natural(or tensorial) extension of ϕ to the space of homogeneous tensors of type (r, s). It isa simple exercise to chek that T s

r (ϕ ◦ ψ) = T sr (ϕ) ◦ T s

r (ψ) for all ϕ,ψ ∈ GL(E).In what follows, unless there is risk of confusion, we denote T s

r ϕ simply as ϕ. It iseasy to check that ϕ(T ⊗ S) = ϕ(T )⊗ ϕ(S), for all tensors T, S ∈ T •• (E).

Appendix B: Tensor Algebra 627

The Exterior Algebras of Forms and Multivectors

The tensor algebra over E , has two distinguished (graded) subalgebras: the algebrasT•(E) = ⊕

s T 0s (E) and T •(E) = ⊕

r T r0 (E) of fully covariant and contravariant

tensors on E , respectively. Additional subalgebras can be considered by consideringcomplete symmetric or antisymmetric tensors in T 0

s (E) or T r0 (E) for instance.

Defining the operators Sn (symmetrizer) andAn (anti-symmetrizer) on T 0n (E) by

the formulas:

Sn(u1 ⊗ · · · ⊗ un) = 1

n!∑π∈Sn

uπ(1) ⊗ · · · ⊗ uπ(n) (10.5)

and

An(u1 ⊗ · · · ⊗ un) = 1

n!∑π∈Sn

σ(π)uπ(1) ⊗ · · · ⊗ uπ(n), (10.6)

(where the π ’s are permutations of n objects that define the symmetric group Sn , andσ(π) is the signature of π ). The range Sr (T 0

n (E)) of the linear map S are the com-pletely symmetric covariant tensors of order n in E and S•(E) =⊕

n≥0 Sn(T 0n (E))

is called the covariant symmetric algebra over E .If, instead, we use the anti-symmetrizer An we obtain the space An(T 0

k (E))

of completely anti-symmetric covariant tensors of order k, commonly denoted by�k(E). Clearly, the dimension of �k(E) is

(dim Ek

), and �k(E) = 0 if k > dim E .

Then the direct sum⊕dim E

k=0 �k(E) is called the exterior algebra of E , has dimension2dim E , and is denoted as �•(E).

Similarly, starting with T r0 (E) we get the algebras S•(E) = ⊕

r Sr (E) whereSr (E) = Sr (T r

0 (E)), and V •(E) = ⊕r V r (E) where V r (E) = Ar (T r

0 (E)). Ele-ments on V •(E) are called multivectors and elements on�•(E) are called forms. Ifa multivector u belongs to Ar (E) it is said to be of degree r and the degree is denotedalso by |u| (similar notions hold for linear forms).

Both �•(E) and V •(E) inherit a graded associative algebra structure from thegraded algebra structure of the tensor algebra. The corresponding products are givenby α ∧ β = A(α ⊗ β) in the case of forms and V ∧U = A(V ⊗U ) in the case ofmultivectors.

Finally, we will point it out that a tensor in T rs (E) can be represented as

a multilinear mapping, T : E∗ ⊗ · · · ⊗ E∗ ⊗ E ⊗ · · · ⊗ E → K, defined asT (u j1⊗· · · u jr ⊗uk1⊗· · ·⊗uks ) = T j1,..., jr

k1,...,ks. Thus the tensor T defines a multilinear

map that is precisely s-multilinear in E and r -multilinear in E∗.

628 10 Appendices

Pull-Back and Push-Forward of Forms and Multi-vectors

Even if only the group GL(E) acts in the tensor algebra, we can define in a naturalway the pull-back of forms and the push-forward of multi-vectors5 along a linearmap ϕ : E → F .

Thus let α be a k-form on F . Define ϕ∗α as the k-form on E by the formula:

ϕ∗α(v1, . . . , vk) = α(ϕ(v1), . . . , ϕ(vk)), ∀v1, . . . , vk ∈ E .

It is easy to check that (ϕ ◦ ψ)∗ = ψ∗ ◦ ϕ∗, and ϕ∗(α ∧ β) = ϕ∗(α) ∧ α∗(β).Similarly we define the push-forward of a k-multi-vector V on E along ϕ as the

k-multi-vector ϕ∗V on F defined as:

ϕ∗V (α1, . . . , αk) = V (ϕ∗(α1), . . . , ϕ∗(αk)), ∀α1, . . . , αk ∈ F∗.

Now we have (ϕ ◦ ψ)∗ = ϕ∗ ◦ ψ∗.

The Algebra of Polynomials Over a Linear Space

Concentrating now on the subalgebra of r -covariant tensors, we see that evaluationof P ∈ T r

0 (E) ‘along the diagonal’, i.e., for arguments u1 = · · · = ur = u, providesus with functions of the form,

P (u) = pi1...ir ui1 . . . uir (10.7)

with distinguished homogeneity properties with respect to the given linear structurein E .

To begin with, linear functions correspond to tensors α ∈ T 10 (E) (commonly

denote as T ∗E) that maps α : E → R. Linear maps are continuous homogenousfunctions of degree one.

Tensors b ∈ T 20 (E), are bilinear maps b : E × E → R and define continuous

quadratic forms Q(u) = b(u, u) which are homogeneous of degree 2, Q(λ u) =λ2Q(u). Moreover, the symmetric part of the bilinear map b is recovered from theexpression:

1

2(Q (x + y)− Q (x)− Q (y)) (10.8)

We can repeat the same procedure to obtain cubic forms from trilinear maps t : E ×E × E → R as C(u) = t (u, u, u). Cubic forms are clearly homogeneous of degree3: C (λu) = λ3C (u), and the totally symmetric part of the trilinear form t can be

5 The definitions extend without change to covariant tensors and contravariant tensors respectively.

Appendix B: Tensor Algebra 629

recovered from C by a formula similar to Eq. (10.8). In general, multilinear maps mof degree r will define ‘m’-tic forms or homogeneous polynomials of degree r onE . It is an interesting exercise to recover the totally symmetric part of m from thecorresponding homogeneous polynomial.

We denote by Pr the space of homogeneous polynomial of degree r in E . Theyform a linear subspace on the space of all polynomials P(E),

P(E) = P0 ⊕ P1 ⊕ P2 ⊕ · · · ⊕ Pr ⊕ · · · =⊕r≥0

Pr (10.9)

The space of polynomialsP(E) is also a subspace of the linear space of all real-valuedfunctions on E . This subspace is also a commutative algebra because the product oftwo polynomial functions is again a polynomial function and, as a vector space, isan infinite-dimensional real space. On the contrary, as an algebra, it is generated bylinear functions because any polynomial function is a finite sum of finite products ofthe linear functions xi . Thus the commutative algebra P is finitely generated.

Appendix C: Smooth Manifolds: A Standard Approach

In themain textwe proceed by introducing first exterior differential calculus on finite-dimensional linear spaces and, afterwards we jumped to the notion of a differentiablespace as the space defined by the quotient of the algebra of smooth functions on someR

n by a closed ideal. This way of proceeding was a fast track towards the possibilityof using dynamics on manifolds without expending too much time in discussingthe theory of smooth manifolds in abstract terms. Actually, as it was discussed anysmooth manifold can be described in this way, and then the use of local coordinatesand all the apparatus of exterior differential calculus comes directly from the R

n inwhich our manifold is embedded.

However, in order to help readers that want to ‘taste’ amore orthodox introductionto smooth manifolds, we have summarized the main aspects of the theory in thisAppendix. Actually, we decided to discuss them again as submanifolds in some R

n

in order to make easier the connection with the formalism of differentiable spacesused in the main text.

At this stage, capitalizing on what has been done in Sects. 2.3 and 2.4 of Chap. 2,we can try to extend the notions already established to the case in which differentialcalculus needs to be done on ‘surfaces’ that are (with all the technical qualificationsthat will given below) subsets of R

n that are ‘curved’ in some sense (i.e., for thetime being, that cannot be endowed with a linear space structure). These are thesubmanifolds of (or immersed in) R

n . In this connection we may quote again afamous theorem by Whitney stating that any finite-dimensional manifold can beviewed as a submanifold of R

n for n large enough.

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630 10 Appendices

Theorem 10.45 (Whitney’s embedding theorem) [Whit44] For n > 0, every para-compact hausdorff n-manifold embeds in R

2n. Moreover it immerses in R2n−1 if

n > 1.

Wewill limit ourselves inwhat follows towhatwewill call ‘regular’ submanifolds,i.e., those that can be defined as surface levels of functions (againwith some regularityproperties to be specified), although most of what we will say applies also to moregeneral cases.

Regular Submanifolds of Rn as Level Surfaces of Functions

We will introduce first the notion of a regular level surface.Consider amapF : R

n → Rk , k = n−m, withm ≤ n−1, andF = ( f 1, . . . , f k),

with f i : Rn → R, i = 1, . . . , k = n − m , smooth functions. The subset of R

n

defined by S = F−1(c) with c ∈ Rk in the range of F with the topology inherited

from Rn , is called a level surface of F. The level surface S = F−1(c) will be called

a regular level surface if c is a regular value of F, that is the Jacobian matrix of f :

J f (x) = ∂( f 1, . . . , f k)

∂(xi , . . . , xn), (10.10)

must be of maximal rank k on any x ∈ S. Equivalently, the values ε j1... jki1...ik

∂ f 1

∂x j1. . .

∂ f k

∂x jkof the minor determinants of rank k of the Jacobian are not all zero, or because

d f 1 ∧ · · · ∧ d f k = 1

k!∂( f 1, . . . , f k)

∂(xi , . . . , xn)dxi1 ∧ · · · ∧ dxik , (10.11)

the functions f i are functionally independent along S, that is:

(d f 1 ∧ · · · ∧ d f k) |S �= 0 . (10.12)

Under these conditions, the Implicit Function Theorem, Theorem 10.43 andits Corollary 10.44, assures that for every point x ∈ S we can choose a subset(xi1 , . . . , xim ) of m = n − k of the coordinates (x1, . . . , xn) and associate with apoint x ∈ S an open neighborhood Ux in S and an open set Vx in R

m with coordi-nates (xi1 , . . . , xim ) that can be mapped diffeomorphically one onto each other. Wesay then that S can be ‘locally modeled on R

m’ and as m independent coordinatessuffice to parametrize the neighborhood of every point, this justifies saying that S isan m-dimensional surface immersed in R

n .

Example 10.1 Let be n = 3 and k = 1 with x = (x1, x2, x3) and F such thatF(x) = ‖x‖2 − 1. Then, the zero level set of F , S = F−1(0), is the two-sphereS2, which can be covered by the six open sets U±

i = {x ∈ S2 | ±xi > 0}, on each

Appendix C: Smooth Manifolds: A Standard Approach 631

one of which points on S2 are parametrized by a pair of their coordinates in R3. For

example, on U±1 we have the maps:

φ±1 : U+1 → D1 = {(x2, x3) ∈ R

2 | (x2)2 + (x3)2 < 1} ,

via φ±1 (x1, x2, x3) = (x2, x3), that can be inverted as:

(φ±1 )−1 : D1 → U+1 , (φ±1 )−1(x2, x3) = (±

√1− (x2)2 + (x3)2, x2, x3) .

It is left as an exercise to define the remaining homeomorphisms φ±i (i = 2, 3)between half-spheres without the boundary and open disks in some R

2.This literal use of the Implicit Function Theorem is admittedly not the most

efficient way of modeling open subsets of S2 on open sets of R2. A different and

maybe more elegant way is offered by the stereographic projections from the twopoles onto C ∼= R

2, namely,

(i) Stereographic (N): Here UN = {S2 − (0, 0, 1)} and φN : UN → C is definedby:

φN (x1, x2, x3) = z = x + i y = x1 + i x2

1− x3, (10.13)

that can be inverted as:

x1 = z + z

|z|2 + 1, x2 = i

z − z

|z|2 + 1, x3 = |z|2 − 1

|z|2 + 1(10.14)

(ii) Stereographic (S): Here US = {S2 − (0, 0,−1)} and φS : US → C is givenby

φS(x1, x2, x3) = z = x + i y = x1 + i x2

1+ x3, (10.15)

that can be inverted as:

x1 = z + z

1+ |z|2 , x2 = iz − z

1+ |z|2 , x3 = 1− |z|21+ |z|2 . (10.16)

Remark 10.1 As S2 is compact it cannot, for obvious topological reasons, bemappedhomeomorphically onto a single open set in R

2. That is the reason why we need atleast two open sets, as in the stereographic projections, to represent the whole of S2.

632 10 Appendices

C.1. Charts and Atlases: Submanifolds

What the Implicit Function Theorem is telling us is that we can cover our surfaceS with a collection {Ui } of open sets each one of which can be mapped homeomor-phically on (i.e., ‘modeled on’) some open subset of R

m via an associated collection{φi } of maps. When this happens, we say that each pair (Ui , φi ) is a chart for S, andthe open set Ui is said to be the domain of the chart.

It can happen that some points of S belong to the intersection of the domains oftwo charts, {Ui , φi } and {U j , φ j } say, and hence, they have different parametrizationsin Ui ∩U j . The problem then arises of how to compare these different descriptions.

Consider again the example of the previous Subsection, namely the two-sphereS2, and, e.g., the two charts: {U+

1 , φ+1 }, and {U+3 , φ+3 }. Hence, U+

1 ∩ U+3 =

{x ∈ S2, x1, x3 > 0} and D1 and D3 is the unit open disks in the planes{x2, x3} and {x1, x2} respectively. So,φ+1 (x1, x2, x3) = (x2, x3), (φ+1 )−1(x2, x3) =(√1− (x2)2 + (x3)2, x2, x3),whileφ+3 (x1, x2, x3) = (x1, x2) and (φ+3 )−1(x2, x3)

= (x1, x2,√1− (x1)2 + (x2)2). Combining them together we find that on U+

1 ∩U+3 , φ+3 ◦ (φ+1 )−1 : D1 → D3 is given by

(x2, x3) →(√

1− (x2)2 + (x3)2, x2)

, (10.17)

and similarly, φ+1 ◦ (φ+3 )−1 : D3 → D1 is defined by

(x1, x2) →(

x2,√1− (x1)2 + (x2)2

). (10.18)

Quite similarly, in the case of the stereographic projections, denoting the two chartsas (Ui , φi ), i = N , S, UN (S) is the sphere without the N (S) pole, and UN ∩ US

projects onto the punctured plane C − {0} ∼= R2 − {0}. Denoting by z and z′ the

projections associated with φN and φS respectively, then:

φS ◦ φ−1N : z → z′ = 1

z, (10.19)

(the bar denoting complex conjugation). This is also a C∞ diffeomorphism of thepunctured plane onto itself. In particular, it maps diffeomorphically the exterior ofthe unit disk onto its punctured interior.

The functions φ+3 ◦ (φ+1 )−1 and φ+1 ◦ (φ+3 )−1, or φS ◦ φ−1N and φN ◦ φ−1S ,(that are the inverse of each other) or, more generally, φi ◦ φ−1j : φ j (Ui ∩ U j ) →φi (Ui ∩ U j ) on a general surface S, are called transition functions, and we see onthe example of the sphere that they are all smooth diffeomorphisms. We state herewithout proof that the same holds true in general whenever the conditions of theInverse Function Theorem are fulfilled. We will abstract the situation depicted aboveto get the following definition.


Definition 10.46 Let S be a topological space. A collection of open sets and maps{(Ui , φi )} such that

⋃i Ui = S, i.e., the family {Ui } is a open covering of S, φi :

Ui → φi (Ui ) ⊆ Rm is a homeomorphism onto its image, and the maps φi ’s are such

that φi ◦ φ−1j : φ j (Ui ∩ U j ) → φi (Ui ∩ U j )) are smooth for all i, j , is called an

smooth atlas6 (or just an atlas for short). A chart is compatible with an atlas whenthe union of the atlas with the chart is an atlas. Two different atlases are said to becompatible iff their union is once again an atlas. Given an atlas and considering theunion of all compatible atlas we will find a compatible maximal atlas. A maximalatlas on a topological space is called a differentiable structure.

Exercise 10.2 Show that the two atlases for the two-sphere that have been introducedbefore are compatible.

We get immediately:

Proposition 10.47 A regular level surface can be equipped with a differentiablestructure.

Definition 10.48 The regular level surface S equipped with its differentiable struc-ture is said to be a regular submanifold of R

n .

Again, the previous statements can be made into a general definition.

Definition 10.49 A(smooth)manifold M modeled on the realBanach space E (also-called a Banach manifold) is a Hausdorff and second countable topological spacewith an (smooth) atlas with charts (Ui , φi ), φ : Ui → E defining its differentiablestructure. If E is finite dimensional, say R

n , then we say that M has dimension n.

When S is a subset of Rn for some n, together with it goes a natural map, the

so-called canonical identification (or inclusion) map, i : S → Rn by which points on

S are identified with points in Rn , i : x ∈ S → x ∈ R

n . We will make a systematicuse of the identification map in what follows. It can also be shown that the inducedtopology is the weakest topology under which the identification map is a continuousmap.

From now on we will work on the class of smooth manifolds, even though all theideas can be particularized to the situation where the manifolds are just regular levelsurfaces of R

n .

C.2. Tangent and Cotangent Bundle: Orientability

What we want to show briefly here is that the existence of a differentiable structureallows us to extend all of the machinery of differential and exterior calculus fromlinear spaces as it was developed in Sect. 2.4 to manifolds, and that the fulfillment

6 Cr -atlas are defined similarly.

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634 10 Appendices

of the compatibility conditions on charts ensures the internal consistency of theextensions.

We say that a curve c : I → M , where I is an interval in R, is a (smooth)differentiable curve if, for any chart (U, φ), the curve: c = φ ◦ c : I → R

m is a(smooth) differentiable curve in R

m . If (U ′, φ′) is another chart with U ∩ U ′ �= ∅,the fact that φ′ ◦φ−1 is a C∞-diffeomorphism ensures that the equivalent image c′ =φ′ ◦ c ≡ (φ′ ◦ φ−1) ◦ c, enjoys the same properties, so the notion of differentiabilityof a curve is established once a given differentiable structure has been establishedon M . In what follows we will consider that curves are always smooth and that 0 isin the interior of their domains.

Quite similarly, a function f : M → R is said to be (smooth) differentiable if thefunction f : (φ−1)∗ f = f ◦ φ−1 : φ(U ) → R is (smooth) differentiable for anychart (U, φ), and once again the compatibility conditions ensure the consistency ofthe definition, i.e., that it depends only on the differentiable structure and not on theparticular chart that has been used for the definition of f .

In this way we can define the associative commutative algebra F(M) of smoothfunctions on M (sometimes denoted asC∞(M), even thoughwewill stickmost of thetime to the previous notation). The sum and the product of two functions f1 and f2 arethe functions defined by ( f1+ f2)(x) = f1(x)+ f2(x) and ( f1 f2)(x) = f1(x) f2(x),and the product by a scalar λ by (λ f )(x) = λ f (x) for all x ∈ M (and clearly all ofthem are smooth if f1 and f2 are smooth).

Given the point x ∈ M , we may define an equivalence relation on the family ofsmooth functions as follows: we will say that the functions f and g are equivalent atx if there exists an open neighborhoodU of x such that f |U= g |U . The equivalenceclasses of functions at x are called germs of functions at x and will be denoted as[ f ]x . The spaceFx of germs of functions at x inherits the structure of a commutativeassociative algebra fromF(M) by means of [ f ]x +[g]x = [ f +g]x and [ f ]x [g]x =[ f g]x . The algebra Fx has a natural maximal ideal given by all germs of functionsvanishing at x that will be denoted by mx .

Proceeding further, we can, much as in the linear case (see Sect. 2.3.7), define theequivalence of curves at a point x ∈ M as follows: Two differentiable curves c1 andc2 through the point x , i.e., c1(0) = c2(0) = x , are tangent at x if in a given chart(U, φ) (hence in all), with x ∈ U , they satisfy d

dt (φ ◦ c1)|t=0 = ddt (φ ◦ c2)|t=0.

The relation ‘c1 is equivalent to c2 if they are tangent at x’ is indeed an equivalencerelation and that it does not depend on the choice of the chart.

Let us denote by [c]x an equivalence class of curves at x , c = c(t) being any oneof its representatives with c(0) = x . Each equivalence class [c]x defines linear mapon the algebra of germs at x , by means of [c]x ([ f ]x ) = d

dt ( f ◦ c(t)) |t=0. Hencethe space of equivalence classes of curves at x becomes a linear space that will bedenoted by Tx M and its elements, the classes [c]x will be denoted by vx or simplyv ∈ Tx M and will be called tangent vectors to the manifold M at the point x . Thespace Tx M consequently will be called the tangent space to M at x .

If M is a manifold of dimension n, given x an a chart (U, φ) such that x ∈ U ,there is a natural basis of Tx M induced from the standard basis {ei } in R

n , bymeans of the family of curves ci (t) = φ−1(φ(x) + tei ), t ∈ (−ε, ε) for ε small

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enough. Then the tangent vector corresponding to the class [ci ]x will be denotedas ∂

∂xi |x . Notice that given any tangent vector v = [c]x in Tx M , then because ofddt φ ◦ c(t) |t=0=∑n

i=0 vi ei , we get that v = vi ∂∂xi .

It is clear that [c]x determines a unique vector v in Rm (the common tangent

at φ(x) of all the image curves). In turn, any vector v ∈ Rm determines the curve

c : t → φ−1(φ(x) + v t) in S, and hence an equivalence class. We denote thisbijection by Txφ:

Txφ : {[c]x } ←→ Rm by [c]x ←→ v = d

dt(φ ◦ c)|t=0 (10.20)

Exercise 10.3 Determine how Txφ transforms under a change of charts and provethat

Txφ′ = d

dt(φ′ ◦ φ−1) Txφ . (10.21)

Much like as in the linear case on can endow the set {[c]x } with a real linearstructure defined by

α[c1]x + β[c2]x = (Txφ)−1

(α

d

dt(φ ◦ c1)|t=0 + β

d

dt(φ ◦ c2)|t=0

)(10.22)

where α, β ∈ R. Again, it is left as an exercise to show that the right-hand side doesnot depend on the choice of the chart. Then,

Definition 10.50 The real vector space TxS ={[c]x } is called the tangent space toS at the point x .

Besides considering real-valued functions on S, we will need in the followingalso to consider, more generally, maps between manifolds. Let then S and � be(sub)manifolds with dim(S) = m and dim(�) = k, and consider a map betweenthem F : S → �.

Let (U, φ) be a chart on S containing the point x0 and (V, ψ) a chart of �

such that F(x0) ∈ V . The function F is a differentiable map in x0 if the mapψ ◦ F ◦ φ−1 : φ(U ) → ψ(V ) is differentiable at the point φ(x0). The function isdifferentiable in an open set of S when it is differentiable in each point of the openset. The map F induces in an obvious way a map Tx0(F) called the derivative, or thedifferential of F between the tangent spaces at x0 ∈ S and F(x0) ∈ �, i.e.

Tx0(F) : Tx0S → TF(x0)� (10.23)

by means of

[c]x0 → [F ◦ c]F(x0) (10.24)

636 10 Appendices

From the definition (see above) of Tx0(φ) (and of TF(x0)(ψ)) one obtains imme-diately,

Tx0(F) = (TF(x0)(ψ))−1 ◦ D(ψ ◦ F ◦ φ−1) ◦ Tx0(φ) (10.25)

where D is the differential of a function of Rm into R

n , from where it is evident (butthat could have been inferred directly from the definition) that Tx0(F) is a linear mapbetween tangent spaces. Of course D(ψ ◦ F ◦ φ−1), the ‘local representative’ of F ,is nothing but the Jacobian matrix of the ‘local representative’ ψ ◦ F ◦ φ−1 of F onthe charts.

Let us go back now to tangent spaces. Tangent vectors are also-called “contravari-ant vectors’ to distinguish them from covariant ones (see below).

Noting that, given ux0 = [c]x0 ∈ Tx0S and f ∈ F(S), the composition mapf ◦ c can be defined entirely in terms of its image on the range of a chart, i.e.,that f ◦ c ≡ f ◦ φ−1 ◦ φ ◦ c ≡ f ◦ c, the proof that tangent vectors at x0 act asderivations at the point x0 on the algebra of smooth functions (and vice versa) canbe extended once more from the context of linear spaces to that of (sub)manifoldswithout essentially no significant changes, and we do not repeat it here. Then, if thecurve c is a representative of the vector ux0 at the point x0,

d

dt( f ◦ c) = d

dt( f ◦ c) (10.26)

and if {ei | i = 1, . . . ,m} is a basis of unit vectors in Rm (and hence also a basis for

Tφ(x0)Rm), the set of inverse images

[ci ]x0 = (Tx0φ)−1(ei ) , i = 1, . . . ,m, (10.27)

in turn defines a basis in Tx0S, more explicitly [ci ]x0 = [φ−1(φ(x0) + ei t)]x0 , andhence

d

dt( f ◦ ci )|t=0 =

(∂ f

∂xi

)|φ(x0)

, (10.28)

where (x1, . . . , xm) are the local coordinates associated with the chart (U, φ) and

ux0 f = ∂ f

∂xi

dci

dt.

It is then natural to introduce, in the domain of a chart, the notation

(∂

∂xi

)|x0= [ci ]x0 = (Tx0φ)

−1(ei ) . (10.29)

Every tangent vector ux0 at the point x0 is then represented in the form


ux0 = ui(

∂

∂xi

)|x0

(10.30)

(and Tx0φ(ux0) = ui ei ). In general, just as in the linear case,

ux0( f ) = ui(

∂ f

∂xi

)|φ(x0)

(10.31)

(and in particular ui = ux0(xi )).

Note that the vector ux0 can be seen as a linear map ux0 : F(S) → F(S) suchthat satisfies the Leibnitz rule,

ux0( f1 f2) = f1(x0)ux0 f2 + f2(x0)ux0 f1 (10.32)

Exercise 10.4 Prove that, if (x ′1, . . . , x ′m) are the local coordinates associated witha different chart (U ′, φ′), then,

(∂

∂x ′i

)|x0=

(∂xk

∂x ′i

)|φ′(x0)

(∂

∂xk

)|φ(x0)

. (10.33)

and hence that

u′i =(∂x ′i

∂xk

)|φ(x0)

uk . (10.34)

Remark 10.2 We are employing here an apparently cumbersome notation. This isto stress the fact that, while local bases can be defined rather easily at every pointof a (sub)manifold, there is no ‘natural’ or ‘canonical’ relationship between themat different points, and that unless some additional structure such as some notion of‘parallel transport’ of vectors on the manifold has been established, what we will doonly in a later section. However, in all the cases in which no ambiguities can arise,we use the simplified notation {(∂/∂xi )|x0} for the natural bases in the domain of achart.

We next go from the tangent to the cotangent space at a point much in the sameway as we went in the linear case, namely:

Definition 10.51 The cotangent space T ∗x0S at the point x0 ∈ S is the dual spaceof Tx0S. Any ω ∈ T ∗x0S is called covariant vector, or covector for short, at the pointx0, and is a linear functional on Tx0S.Definition 10.52 The differential of a function f ∈ F(S) at the point x0 is theunique covector d f (x0) such that

d f (x0)(u) = u( f ) , ∀u ∈ Tx0S . (10.35)

638 10 Appendices

Now, if {(∂/∂xi )|x0 | i =1, . . . ,m} is the basis of Tx0S associated to a given chartand we choose for f a coordinate function, e.g., f : x → xi , then, the previousdefinition gives

dxi (x0) ((∂/∂x j )|x0) = δij , (10.36)

and hence: {dxi (x0) | i = 1, . . . ,m} is a basis of T ∗x0S dual to the natural basisin Tx0S that is called the cobasis associate to the given chart. Any covector ω atthe point x0 is locally expressed in a unique way in the form ω = ωi dxi (x0) andω(u) = ωi ui , for any pair of elements ω ∈ T ∗x0(S) and u ∈ Tx0(S). In particular,

d f (x0) =(

∂ f

∂x j

)|φ(x0)

dx j (x0) , (10.37)

for any f ∈ F(S). Under a change of chart the associated bases are related by

dx ′i (x0) =(∂x ′i

∂xk

)|φ(x0)

dxk(x0) (10.38)

and therefore the relation among coordinates of a covector in both bases is

ω′i =(∂xk

∂x ′i

)|φ′(x0)

ωk . (10.39)

The transformation law explains the nomenclature that has been introduced for vec-tors and covectors, because the matrix appearing in the preceding relation is theinverse of that of relation (10.34).

Vector fields and 1-forms are defined here too as smooth assignments on S oftangent and cotangent vectors respectively. In the domain of a chart that amounts topromoting the ui ’s and/or the ωi ’s to smooth functions, transforming appropriatelywhen moving from a chart to an overlapping one. Out of vector fields and forms wewill be able eventually to construct tensor fields of arbitrary rank as well.

The unions

TS =⋃x∈S

TxS , T ∗S =⋃x∈S

T ∗x S (10.40)

are called the tangent and cotangent bundle respectively. They have been definedhere only as set-theoretic unions. It can be shown that there is a canonical way toendow both of them with a differentiable structure inherited from the one on S.


Pull-Back of Covariant Tensors and Forms from Rn to Submanifolds

The operation of taking the pull-back of a map has been already defined mainly inconnectionwith functions in the linear context.Wewant to extend it here tomanifoldsand to general

(0r

)-type (i.e., totally covariant) tensor fields. Let F : S → � be

a differentiable (smooth, actually) map between (sub)manifolds. In analogy withthe linear case, the pull-back (to S) of a function g : � → R can be defined asF∗g = g ◦ F . Let now t ∈ T 0

r (�) be a totally covariant tensor field of rank r on �.Then,

Definition 10.53 The pull-back by F on tensors of type (0, r) is the map:

F∗ : T 0r (�)→ T 0

r (S) (10.41)

defined by:

(F∗t)x (u1, . . . ,ur ) = tF(x)(Tx F(u1), . . . , Tx F(ur )) (10.42)

where u1 , . . . ,ur ∈ TxS.

Note that (F∗t)x has the same symmetry properties under interchange of indicesas tx , and therefore, the pull-back of symmetric or skew-symmetric tensors are alsosymmetric or skew-symmetric. In particular, forms of arbitrary rank may be pulled-back, a possibility of whichwewill make large use, as well as other covariant tensors.

In particular, consider the case in which � ≡ Rn and F is the identification map

F = i : S ↪→ Rn . Assume that a Riemannian metric (see below Appendix E) g has

been assigned on Rn (for example the standard Euclidean metric). Then,

g = i∗g (10.43)

is aRiemannianmetric onS. Sowe obtain that ifS is a regular immersed submanifoldof R

n for some n it can be always endowed with a Riemannian metric.

Example 10.5 ConsiderR3 (withCartesian coordinates (x, y, z)) with the Euclideanmetric g = (dx)2+ (dy)2+ (dz)2. If we choose polar coordinates on the two-sphereS2 (actually the two-sphere with the poles removed, which is homeomorphic to acylinder), given by x = sin θ cosφ, y = sin θ sin φ, z = cos θ ,θ ∈ (0, π) , φ ∈[0, 2π ], then S2 inherits7 the Riemannian metric:

g = (dθ)2 + sin2 θ (dφ)2 , (10.44)

i.e., the Riemannian structure is defined by,

7 See below, page 647.

640 10 Appendices

gθθ = 1 , gφφ = sin2 θ, gθφ = gφθ = 0. (10.45)

Note however that the 1-forms dθ and dφ are well defined only on the cylinder (i.e.,on the sphere without the poles) and are not globally defined on the sphere (see alsothe discussion in Remark 2.7 above). This has to do with the fact that the sphere S2

is not ‘parallelizable’.

Orientability and Orientations

Consider Rn to start with. Any two coordinate systems (xi ), (x ′i ), i = 1, . . . , n

on Rn (or on any open subset of it) will be said to define the same orientation iff

their Jacobian determinant: J = ∂(x ′1,...,x ′n)∂(x1,...,xn)

is everywhere positive. As the sign ofthe Jacobian can be changed by just changing the sign of one of the coordinates,this implies that R

n and its open subsets can have only two different orientations.Comparing the standard volume-forms associated with � = dx1 ∧ · · · ∧ dxn and�′ = dx ′1 ∧ · · · ∧ dx ′n , we know that:�′ = J ·�, so the two volume-forms will beproportional (as they should) with an everywhere positive proportionality factor.

The obvious way to extend this notion to submanifolds of Rn is given by the

following:

Definition 10.54 A submanifold S of Rn will be orientable iff it can be given an

oriented atlas, i.e., an atlas {(Ui , φi )} such that for any pair of overlapping charts,(Ui , φi ), (U j , φ j ), such that Ui ∩ U j �= ∅, det[D(φi ◦ φ−1j )] > 0 holds, where

D(φi ◦φ−1j ) is (cfr. Sect. 5.4.5.) the Jacobianmatrix of themap: φi ◦φ−1j : φ j (U j )→φi (Ui ).

For example, on the two-sphere S2, the atlas defined by the stereographic projec-tion (see Sect. 1.1.1) has: D(φS ◦ (φN )−1) = −1/|z|2 and is not an oriented atlas.However,the atlas defined using the coveringwith the six (open) half-spheresU±

i has,

e.g.: D(φ+3 ◦ (φ+1 )−1) = x3/√1− (x2)2 − (x3)2 > 0 and so do the other Jacobians.

The atlas is therefore an oriented one and S2 is orientable. This carries over to thegeneral case, namely it can be shown that all regular submanifolds (obtained as levelsets of functions) of R

n are orientable.Orientability has to dowith the global existence of volume-forms (i.e., of nowhere

vanishing forms of maximal rank). Actually, it can be proved [Ab78, Ab88] that:

Proposition 10.55 A manifold S admits a volume-form iff it is orientable.

Exterior Differential Calculus

We have seen that there are manifolds that arise as submanifolds of some Rn , and

we have mentioned also Whitney’s theorem [Whit44], Theorem 10.45, according towhich any manifold may be seen as a sumbanifold of R

n when n is large enough.

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http://dx.doi.org/10.1007/978-94-017-9220-2_5


We may now take advantage of the intrinsic differential calculus introduced sofar on R

n to induce a differential calculus on submanifolds.From the conceptual point of view, this means that we are not required to go over

again through the definition of vector fields and 1-forms as we have already doneit in Sect. 2.4. The main idea here is to capitalize on our presentation in Sect. 2.4.2,i.e., expressing the exterior derivative and the Lie derivative in terms of appropriatelichosen vector fields and 1-forms.

Let us go back to the bare essentials involved in our presentation of the externaldifferential calculus on R

n . We have expressed vector fields and 1-forms as,

X = f j ∂

∂x j; d f = ∂ f

∂x jdx j (10.46)

When the x j ’s are globally defined so are the vector fields ∂/∂x j and 1-formsdx j , j = 1, . . . , n. They provide then us with a basis of the modules of vectorfields and 1-forms respectively, the coefficients being functions on R

n .From the point of view of the differential equations associated with vector fields

we notice that by integrating the differential equations associated with each oneof the ∂/∂x j ’s we obtain a global action of the Abelian translation group R

n . Thecorresponding 1-forms dx j provide us with the dual version, in terms of covariantfields, of the same action, because they give raise to a one-to-one map between ourmanifold E (the vector space manifold we are dealing with) and R

n . This one-to-onemap allows us to ‘export’ on E all the structures that are available on R

n .When the submanifold of E we are considering is ‘curved’ in some way, for

instance if it is a sphere, it is reasonable to expect that a global action of the Abeliantranslation group will not exixt on it. Our strategy to define an external differentialcalculus will be to replace the ∂/∂x j ’s with other vector fields ‘adapted’ to thesubmanifold, i.e., a basis of them tangent to the submanifold itself. At the same timewe will consider an ‘adapted’ basis of 1-forms.

The most simple example is provided by the submanifold of R2, defined by the

equation,

a2x2 + b2y2 = 1 (10.47)

which represents an ellipse.An adapted basis of vector fields and 1-forms on R

2 − {0} is given by,

R = a

bx

∂

∂y− b

ay

∂

∂x, � = x

∂

x∂+ y

∂

∂y(10.48)

and,

θ = abxdy − ydx

a2x2 + b2y2, α = a2xdx + b2ydy

a2x2 + b2y2(10.49)

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642 10 Appendices

We have excluded the origin from R2 because R and � are zero and θ and α would

not be defined there.We find,

θ (R) = 1, θ (�) = 0α (R) = 0, α (�) = 1

(10.50)

These relations allow us to write a (1, 1) tensor field that represents the identity mapon the tangent or on the cotangent bundle as,

TI = θ ⊗ R + α ⊗� ≡ I (10.51)

Any vector field, say X , may be written in the form,

X = TI (X) = θ (X) R + α (X)� (10.52)

and similarly for any 1-form β,

β = TI (β) = β (R) θ + β (�) α (10.53)

which, for exact 1-forms, becomes,

d f = θLR ( f )+ αL� ( f ) (10.54)

which replaces the expression:

d f = dx∂ f

∂x+ dy

∂ f

∂y(10.55)

Stated otherwise, we are writing the exterior differential d in the adapted basis as,

d = θ ∧ LR + α ∧ L� (10.56)

On a generic 1-form, say, β = g d f , we have,

d (g d f ) = (θLR (g)+ αL� (g)) ∧ (θLR ( f )+ αL� (g))= (θ ∧ α) (LR (g)L� ( f )− LR ( f )L� (g))

(10.57)

If: iS : S1 ↪→ R2 is the identification map immerging the ellipse in R

2, we seeimmediately that from: i∗Sα = 0 there follows:

i∗SdR2 f = dS1 i∗S f (10.58)

wherewe havewritten dR2 and dS1 to emphasize that they are the exterior differentialson R

2 and S1 respectively.


A generalization of our simple example to many dimensions would require an‘adapted basis’ of vector fields, say,

{R1, R2, . . . , Rk; Y1, Y2, . . . ,Yn−k} (10.59)

and corresponding 1-forms,

{θ1, θ2, . . . , θk;α1, α2, . . . , αn−k} (10.60)

such that,

θ j (Rl) = δjl , αm (Rl) = 0

θ j (Ym) = 0, αm (Yn) = δmn

(10.61)

The corresponding decomposition of the identity,

TI =k∑

l=1θ l ⊗ Rl +

n−k∑j=1

α j ⊗ Y j (10.62)

allows us to decompose 1-forms and vector fields as,

β =k∑

l=1β (Rl) θ

l +n−k∑j=1

β(Y j

)α j (10.63)

and,

X =k∑

l=1θ l (X) Rl +

n−k∑j=1

α j (X) Y j (10.64)

Of course, these decompositions extend to any tensor field. On exact 1-forms, wefind,

d f =k∑

l=1

(LRl ( f ))θ l +

n−k∑j=1

(LY j ( f ))α j (10.65)

We have not said anything on the commutation relations:[Rl , R j

], [Rl , Yn] and

[Ym, Yn] among the vector fields of the adapted basis, as well as on the correspondingrelations on 1-forms, that must be of the form,

dθ l = clmnθ

m ∧ θn + aljmθ j ∧ αm + bl

rsαr ∧ αs (10.66)

with similar relations for the dα j ’s.

644 10 Appendices

Qualifying these relations identifies special bases. Clearly, the Abelian situationwe started from in R

n in terms of the ∂/∂x j ’s and dx j ’s uses commuting vectorfields and exact 1-forms. In a generic situation, for generic submanifolds, it is notgranted that we can be able to find globally an adapted basis. For instance, the two-dimensional sphere does not admit, for topological reasons, a global basis of 1-forms(nor of vector fields).

When such a basis exists, the submanifold is said to be parallelizable. Thus, ingeneral, a manifold will be said to be parallelizable if it admits of a global basis of1-forms and vector fields considered as modules with coefficients smooth functions.A large class of parallelizable manifolds is provided by the Lie groups.

The exterior differential calculus we have considered here may be thought of asarising from the Abelian Lie group structure of R

n . Generalizations are provided byreplacing the Abelian vector group R

n with a generic Lie group G.When the (sub)manifold we are considering is not parallelizable, but is a homo-

geneous space of a Lie group G, it is still possible to take advantage of the exteriordifferential calculus associated with G by considering the algebra of functions onthe submanifold as a subalgebra of functions in F(G). We shall make sense of thesestatements by considering the parallelizable submanifold S3 ⊂ R

4 (the group man-ifold of SU (2)) and the submanifold S2 ⊂ R

3, which is not parallelizable but maybe considered a homogeneous space of SU (2).

Differential Calculus on S3 ⊂ R4

We recall that a convenient parametrization of R4 is provided by the complex, Her-

mitian matrices,

h =(

x0 + x3 x1 − i x2x1 + i x2 x0 − x3

)(10.67)

which, in terms of Pauli matrices σ = (σ 1, σ 2, σ 3

)given by:

σ 0 =(1 00 1

), σ 1 =

(0 11 0

), σ 2 =

(0 −ii 0

), σ 3 =

(1 00 −1

), (10.68)

may be written as

h = x0 σ0 + x · σ (10.69)

With this choice of parametrization by Hermitian matrices we find,

det (h) = x20 − x21 − x22 − x23 (10.70)

which makes it appropriate when we deal with Special Relativity.


If we consider a different parametrization using anti-Hermitian matrices,

g =(

x0 + i x3 i x1 + x2i x1 − x2 x0 − i x3

)= x0σ

0 + ix · σ (10.71)

we find,

det (g) = x20 + x21 + x22 + x23 (10.72)

In this parametrization matrices with det (g) = 1 represent the unit sphere S3 ⊂R4. We may set g = Rs, with: s ∈ SU (2) ≈ S3, to find det (g) = R2. Hence,

s = 1√x20 + x21 + x22 + x23

(x0 + i x3 i x1 + x2i x1 − x2 x0 − i x3

)(10.73)

We notice also that: g† = Rs† = Rs−1.To find a basis of 1-forms adapted to S3 we may set,

g−1dg = θ0σ0 + iθaσa (10.74)

(a = 1, 2, 3) and we find,

R−1s−1d (Rs) =(

R−1d R)σ0 + s−1ds (10.75)

The basis of 1-forms is given specifically by,

θ0 = x0 dx0 + x1 dx1 + x2 dx2 + x3 dx3x20 + x21 + x22 + x23

(10.76)

θ1 = x0 dx1 − x1 dx0 + x3 dx2 − x2 dx3x20 + x21 + x22 + x23

(10.77)


(10.78)


(10.79)

The corresponding basis of vector fields is given by,

Y0 = xμ∂

∂xμ

, (μ = 0, 1, 2, 3) (10.80)

646 10 Appendices

Y1 = x0∂

∂x1− x1

∂

∂x0+ x3

∂

∂x2− x2

∂

∂x3(10.81)

Y2 = x0∂

∂x2− x2

∂

∂x0+ x1

∂

∂x3− x3

∂

∂x1(10.82)

Y3 = x0∂

∂x3− x3

∂

∂x0+ x2

∂

∂x1− x1

∂

∂x2(10.83)

Again, the adapted decomposition of the identity is given by,

TI = θ0 ⊗ Y0 + θa ⊗ Ya (10.84)

The exterior differential of a function is simply given by,

d f = (LY0 ( f ))θ0 + (LYa ( f )

)θa (10.85)

It can be extended to any form by using the mutual relations between the Lie, innerand exterior derivatives.

As an example, we write explicitly the exterior derivative of a 1-form α. Weconsider first: α = α (Y0) θ

0 + α (Ya) θa , and then,

dα = (dα(Y0)) ∧ θ0 + α (Y0) dθ0 + (dα(Ya)) ∧ θa + α (Ya) dθa (10.86)

We notice that, with respect to the use of bases with closed 1-forms, we get hereadditional terms due to the fact that the basis is ‘non-holonomic’, this meaning that:dθa �= 0.

In our specific case: dθ0 = 0 and: dθa + (1/2) εabcθ

b ∧ θ c = 0, and we get,

dα = (LYa (α (Y0))− LY0 (α (Ya)))θa ∧ θ0

+ (LYb (α (Yc))− LYc (α (Yb))− εabcα (Ya)

)θb ∧ θ c (10.87)

A volume-form can be written as,

� = θ0 ∧ θ1 ∧ θ2 ∧ θ3 (10.88)

while the Euclidean metric tensor will be,

g = θ0 ⊗ θ0 + θ1 ⊗ θ1 + θ2 ⊗ θ2 + θ3 ⊗ θ3 (10.89)

It is also possible to derive dx0, . . . , dx3 by noticing that,

dg = dx0 σ0 + i dx · σ (10.90)


and therefore,

dx0 = 1

2Tr (dg σ0) = 1

2Tr

{g(θ0σ0 + iθaσa

)σ0

}(10.91)

and,

dxa = 1

2Tr

(g(θ0σ0 + iθaσa

)σa

). (10.92)

Also, and more directly, using Eq. (10.84) (cfr. also Eq.10.85),

dxμ = TI(dxμ

) = (LY0

(xμ

))θ0 + (LYa

(xμ

))θa, μ = 0, 1, 2, 3 (10.93)

Differential Calculus on S2 ⊂ R3

For S2 ⊂ R3 it is not possible to find a global adapted basis, because S2 is not

parallelizable. For instance, the 1-forms dθ and dφ used in Example (10.5) cannotbe smooth on the whole of S2. As a matter of fact, they are well defined only on(−π, π)× [0, 2π ].

To deal with a smooth exterior differential calculus with globally defined fieldswe may consider the projection map,

π : SU (2) ≡ S3 −→ S2 ⊂ R3

s → s−1σ3s = x · σ (10.94)

with det(s−1σ3s

) = −1 = − (x21 + x22 + x23

)and

xi = 1

2Tr

{σi s

−1σ3s}, i = 1, 2, 3 (10.95)

The algebra of functions on S2 may be pulled back to a subalgebra ofF (SU (2))simply replacing f (x) with f

(s−1σ3s

).

The vector fields Y1, Y2 and Y3 act on this subalgebra as derivations. They are left-invariant and generate right translations, commuting therefore with the infinitesimalgenerator of the left action s → eiϕσ3s, giving,

s−1σ3s =(

eiϕσ3s)−1

σ3

(eiϕσ3s

). (10.96)

The dual 1-forms are well defined on S3 but not on S2, because the projectionson S2 of Y1, Y2 and Y3 are not independent. In this way we may write,

d(π∗ f ) = (LY1 f)θ1 + (LY2 f

)θ2 + (LY3 f

)θ3 (10.97)

648 10 Appendices

which may be expressed then in terms 1-forms that are globally defined on S3, butcannot be expressed in terms globally defined 1-forms on S2.

Therefore, the use of Y1, Y2, Y3 and θ1, θ2, θ3 allows for an exterior differentialcalculus on tensorial objects on S2 by considering them as objects in the subalgebraof tensors on S3.

Appendix D: Differential Concomitants: Nijenhuis,Schouten and Other Brackets

Along themain textmany computationswith tensors involve the use of various brack-ets with strong algebraic flavor. We will summarize in this appendix the definitionsand main properties of some of them for the ease of the reader. Mostly, we followthe conventions in [Ab88], trying to avoid the proliferation of factorial factors.

The Lie Derivative and the Lie Bracket

As in the main text M will denote a smooth manifold, X(M) will denote the Liealgebra of smooth vector fields on M and �·(M) the algebra of differential formson M . The smooth tensors of type (p, q) over M will be denoted as T p

q (M) and thebigraded tensorial algebra over M with the tensor product ⊗, T (M).

Given two tensors t1 of type (p1, q1) and t2 of type (p2, q2), we will definethe contraction of t1 with t2, p1 ≤ q2 and p2 ≤ q1, as the tensor it1 t2 of type(0, q2 − p1 + q1 − p2) obtained by contracting the contravariant indices of t1 withthe first p1 covariant indices of t2, and the contravariant indices of t2 with the firstp2 contravariant indices of t1.

In particular the contraction of a vector field X with a k-form α is the k − 1-formiXα:

iXα(X1, . . . , Xk−1) = α(X, X1, . . . , Xk−1) .

The contraction of A of type (1, 1) with a k form α will be the k-form:

i Aα(X1, . . . , Xk) =k∑

i=1α(X1, . . . , A(Xi ), . . . , Xk).

We have already shown that in the algebra of differential forms �•(M) we havethe differential operator d of degree 1 satisfying (see Appendix G):

1. d2 = 0.2. d(α ∧ β) = dα ∧ β + (−1)kα ∧ dβ, for all α, β ∈ �•(M), α ∈ �k(M).

Appendix D: Differential Concomitants: Nijenhuis, Schouten and Other Brackets 649

The Lie derivative is the degree 0 operator on the algebra�•(M) defined by usingCartan’s formula:

LXα = iX dα + diXα.

The Lie derivative satisfies the following properties:

1. LX dα = dLXα.2. LX iY α = iYLXα + i[X,Y ]α.3. L f Xα = f LXα + d f ∧ iXα.4. L[X,Y ]α = LXLYα − LYLXα.5. LX (α ∧ β) = (LXα) ∧ β + α ∧ (LXβ).

The Lie derivative of a vector field Y along the direction of X is precisely the Liebracket [X, Y ], that is: LX Y = [X, Y ]. Hence, we denote the Lie derivative of anycontravariant skew symmetric tensor A as:

LX A = [X, A].

Then it is easy to check that for any skew symmetric contravariant tensor A, we have

iLX Aα = iX d(iAα)− iAdiXα.

The Nijenhuis Bracket

Wemay consider if it possible to extend the bracket [X, A] defined above for a vectorfield and a contravariant tensor to all skew symmetric contravariant tensors in sucha way that it coincides with the Lie derivative on vector fields and making it into thegraded analog of a Lie algebra. The answer is positive and was studies by Nijenhuis[Ni55].

Let A, B be skew symmetric homogeneous contravariant tensors of degrees qand p respectively. Then we define [A, B] to be a q + p − 1 homogeneous skewsymmetric contravariant tensor such that when contracted with any q+ p−1–closedform we get:

i[A,B]α = (−1)[(q + 1)piAdiBα + (−1)qiBdiAα,

moreover,

[A, B] = (−1)qp[B, A],

650 10 Appendices

and, it satisfies the graded Jacobi identity:

(−1)qp[A, [B,C]] + (−1)pr [C, [A, B]] + (−1)rq [B, [C, A]] = 0

with C homogeneous of degree r .The space of skew symmetric contrariant tensors on M becomes a graded Lie

algebra with degree the degree of the tensors minus 1 (that is the graded degree of avector field is 0) [Ni64].

The Schouten-Nijenhuis Bracket

Finally, it is very useful a bracket defined on (1, 1) tensors and that gives a (1, 2)tensor, skew symmetric in the covariant part. This tensor is actually the “differentialconcomitant” defined for (1, q) tensors, skew symmetric in the covariant part. Inother words, we may define a bracket on vector-valued forms on M [Ni66].

Thus if T, R are two (1, 1) tensors, we define [T, R] as a 2-form with values invector fields as:

[T, R](X, Y ) = 1

2([T X, RY ] + [R X, T Y ] + (T R + RT )[X, Y ]

−T ([X, RY ] + [R X, Y ])− R([X, T Y ] + [T X, Y ])).

it follows from the previous definition that [T, R] = [R, T ]. If T = R, the previousexpression reduces to:

[T, T ](X, Y ) = [T X, T Y ] + T 2[X, Y ] − T [X, T Y ] − T [T X, Y ],

which is just the well known torsion operator used to characterize the integrabilityof a almost complex structure.

The extension of this bracket to the graded algebra of vector-valued forms provi-dent with a graded Lie algebra structure.

Appendix E: Covariant Calculus

The Covariant Derivative

The exterior derivative calculus we have been developing up to now is applied toskew–symmetric forms built out of functions, f, g, . . ., and the exterior differentialoperator d.We have f , dg, h dg, dg∧d f , and so on. Alongwith d we have developedthe Lie derivative along a vector field X , LX , and the contraction with respect to X ,

Appendix E: Covariant Calculus 651

iX . These operations, which are graded derivations, satisfy the relation,

iX d + diX = LX (10.98)

This calculus, however, does not enable us to differentiate general tensors fields.We assume here again that our carrier space is an affine space M of dimension n.We have seen that for a function f , d f is a covariant vector field which in a givencoordinate system is expressed by,

d f =n∑

i=1

∂ f

∂xidxi (10.99)

In other words, the partial derivatives of f with respect to xi are the componentsof a covariant vector field. This process of generating new vector fields by taking thepartial derivatives of the components of a tensor field with respect to a coordinatesystem cannot be extended to tensor fields of higher orders. Let us consider a con-travariant vector field X whose local components in two coordinate systems xi andx i are Ai and Ai respectively, so that we have,

X =n∑

i=1Ai ∂

∂xi=

n∑i=1

Ai ∂

∂ x i(10.100)

or,

Ai =n∑

j=1

∂ x i

∂x jA j (10.101)

Partial differentiation gives,

∂ Ai

∂ x k=

n∑j,l=1

∂ x i

∂x j

∂xl

∂ x k

∂ A j

∂xl+

n∑j,l=1

∂2 x i

∂x j∂xl

∂xl

∂ x kA j (10.102)

which, because of the second terms on the right-hand side, shows that the ∂ Ai/∂x j ’sare not the components of a tensor field of type (1, 1).

To remedy this situation we introduce the notion of affine connection.

Definition 10.56 An affine connection is given by a set of linear differential 1–formsωi

j , with i, j = 1, . . . , n, in each coordinate system, in such a way that when goingfrom one coordinate system to another we have,

d Pij +

n∑k=1

ωik Pk

j =n∑

k=1ωk

j Pik (10.103)

652 10 Appendices

where,

Pij = ∂ x i

∂x j(10.104)

Notice that Eq. (10.101) can be rewritten as,

Ai =n∑

j=1Pi

j A j (10.105)

It is also remarkable that such set of ωij is a matrix–valued 1-form.

We have to show that this definition is consistent. By this we mean that if we

consider three coordinate systems, xi , x i ,˜xi , the relation amongst xi , and

˜xi follows

from those of xi with x i , and those of x i with˜xi .

Indeed from,

d

(∂ x i

∂x j

)+

n∑k=1

ωik∂ x k

∂x j=

n∑k=1

∂ x i

∂xkωk

j (10.106)

and,

d

(∂˜xk

∂x j

)+

n∑k=1

˜ωi

k∂˜xk

∂ x j=

n∑k=1

∂˜xi

∂ x kωk

j (10.107)

we get,

d(

∂˜xi

∂x j

)+∑n

k=1˜ωi

k∂˜xk

∂x j = d(∑n

k=1 ∂˜xi

∂ x k∂ x k

∂x j

)+∑n

k,l=1˜ωi

l∂˜xk

∂ x k∂ x k

∂x j

=∑nk=1 ∂ xk

∂x j

[d(

∂˜xi

∂ x k

)+∑n

l=1˜ωi

l∂˜xl

∂ x k

]+∑n

k=1 ∂˜xi

∂ x k d(

∂ x k

∂x j

)

=∑nk,l=1 ∂ xk

∂x j∂˜xi

∂ xl ωl

k +∑nk=1 ∂˜xi

∂ x k d(

∂˜xk

∂x j

)

=∑nk=1 ∂˜xi

∂ x k

[∑nl=1 ωk

l∂ x l

∂ x j + d(

∂˜xk

∂x j

)]

=∑nk,l=1 ∂˜xk

∂ x k∂ x k

∂xl ωl

j =∑nl=1 ∂˜xi

∂xl ωl

j

(10.108)

which proves the consistency of the condition.Condition (10.103) can be put into an equivalent form using the quantities,

Qik = ∂xi

∂ x k(10.109)


Indeed, matrix–valued functions Q and P are inverses to each other, i.e.,

n∑j=1

Pij Q j

k =n∑

j=1Qi

j P jk = δi

k (10.110)

Differentiating any one of these equations we find the equivalent condition,

d Qij +

n∑k=1

ωik Qk

j =n∑

k=1Qi

kωk

j (10.111)

In fact,

n∑j=1

d Qkj P j

i =n∑

j,l=1Qk

j

(−ω j

l Pli + P j

lωl

i

)= 0 (10.112)

and then,

n∑j=1

d Qkj P j

i −n∑

j,l=1Qk

j ωj

l Pli + ωk

i = 0 (10.113)

which implies that,

d Qki +

n∑j=1

ωkj Q j

i =n∑

j=1Qk

j ωj

i (10.114)

Connections exist on any affine space. This can be easily shown by consideringa second-order dynamics associated with a non-degenerate quadratic in velocities.This will be seen in a later section.

We assume that on our affine space an affine connection has been selected and tryto study the properties of M arising from this connection.

By differentiating (10.101) and using (10.103) we get,

d Ai =n∑

j=1Pi

j d A j+n∑

j=1d Pi

j A j =n∑

j=1Pi

j d A j+n∑

j,k=1(Pi

kωk

j−ωik Pk

j )A j

(10.115)which can be rewritten as,

d Ai +n∑

k=1ωi

k Ak =n∑

j=1Pi

j

(d A j +

n∑k=1

ω jk Ak

)(10.116)

654 10 Appendices

We define,

D Ai = d Ai +n∑

k=1ωi

k Ak =n∑

j=1Ai ; jdx j (10.117)

where the Ai ; j ’s turn out to be the components of the (1,1) tensor field,

Ai ; j = ∂ Ai

∂x j+

∑k

�ik j Ak; ωi

k =∑

j

�ik j dx j (10.118)

because D Ai transform as the components of a vector field.Writing (10.116) in the form,

D Ai =n∑

j=1Pi

j (D A j ) (10.119)

we see that the D A j ’s are linear differential forms transforming like contravariantvectors. They are said to define the covariant differential of the vector field X .

More generally, let, for instance, Ai jk be the components of a tensor field of type

(2, 1), relative to coordinates (xi ). Under a change of coordinates they transform inthe following way,

Ai jk =

n∑l,m=1

Pil P j

m Qrk Alm

r (10.120)

Making use of (10.103) and (10.115) we get in a similar way as above,

D Ai jk =

n∑l,m=1

Pil P j

m Qrk D Alm

r (10.121)

where,

D Ai jk =

n∑l=1

D Ai jk;l dxl = d Ai j

k +n∑

l=1(ωi

l Al jk + ω j

l Ailk − ωl

k Ai jl)

(10.122)It follows that the components Ai j

k;l define a tensor field of type (2, 2). Wedeclare D Ai j

k to be the covariant differential of the tensor field Ai jk .

Now, it should be clear how these considerations extend to a tensor field of anytype. In particular, for a scalar function f we define D f = d f .


Under tensor multiplication and contraction of tensor fields we have, for instance,

D

(n∑

l=1Ai jk

l Bli

)=

n∑l=1

(D Ai jkl)Bl

i +n∑

l=1Ai jk

l(DBli ) (10.123)

as it may be verified by direct computation. In a given coordinate system we mayintroduce the quantities �i

jk via the relation,

ωik =

n∑j=1

�ijk dx j (10.124)

and then (10.117) becomes,

D Ai = d Ai +n∑

j=1�i

jk dx j Ak (10.125)

Conditions (10.103) can be written as,

�ijk =

n∑l=1

∂2 xl

∂x j∂xk

∂xi

∂ x l+

n∑l,m,q=1

�lmq

∂ xm

∂x j

∂ xq

∂xk

∂ x i

∂xl(10.126)

The numbers �ijk are called the Christoffel symbols of the connection.

Second Order Differential EquationsAssociated with a Connection

Definition 10.57 Let γ : I ⊂ R → M be a smooth curve with parameter t . Then,a vector field along γ is a map V : I → T M such that V (t) ∈ Tγ (t)M .

For instance, if W ∈ X(M)we can define V = W ◦γ . Notice that not every vectorfield along γ can be factorized in such way. For instance, if there are two values t1and t2 in I such that γ (t1) = γ (t2), vector fields along γ such that V (t1) �= V (t2)cannot be factorized in such away.However, the factorization is possible if we restrictourselves to a small enough neighborhood of a t0.

Another way of constructing a vector field along γ is by considering a vectorfield in I and defining the image under the tangent map T γ . As a very importantexample of it, we can consider the vector field in I given by d/dt , because then thecorresponding vector field along γ is the tangent vector in each point of the curve,γ , defined by,

656 10 Appendices

γ : f → d

dtf (γ (t)) =

n∑i=1

∂ f

∂xi

dxi

dt(10.127)

Definition 10.58 Let γ : I ⊂ R → M be a smooth curve with parameter t and V avector field along γ with coordinates V i (t) in a particular coordinate system. Then,the vector field along γ given in the same coordinate system by,

DV i

dt= 〈DV i (V (t), γ (t)〉, (10.128)

is called absolute derivative or covariant derivative along γ of the vector field alongγ , V : I → T M .

Definition 10.59 Let γ : I ⊂ R → M and V be as in the preceding definition.Then V is said to be a covariantly constant vector field along γ if,

DV i

dt= 0 (10.129)

Analytically the condition can be written as,

DV i

dt= dV i

dt+

n∑j=1

�ik j V k dx j

dt= 0 (10.130)

In particular, if γ : R → M is a smooth curve, then the tangent vector field along

γ associated with the curve γ , γ , is parallel along γ if Dγ i

dt = 0 along γ .

Definition 10.60 A parametrized curve is said to be auto-parallel if its tangent vec-tors are parallel along γ .

The preceding equations (10.130) can be considered as a differential equationsystem forV i (t), therefore there exists one andonly one solutionV i (t) correspondingto any initial values V i (t0). We interpret this geometrically by saying that given acurve γ : I → R

n and a vector V i (t0) at the point γ (t0), the solution of thedifferential equation system (10.130) provides a vector field along γ (for t enoughclose to t0), V (t), and we usually say that the vector V i (t0) is parallelely transportedalong γ . Moreover, as the differential equation system is linear, the vector field alongγ obtained by parallel transport from the vector V (t0)+ W (t0), is the sum of thoseobtained from V (t0) and W (t0), respectively, and for any real number α, the vectorfield along γ obtained from αV (t0) is αV (t).

By setting V i = dxi

dt we get the second-order differential equation for autoparallelcurves (also-called geodesics),

d2xi

dt2+

n∑j,k=1

�ijk

dx j

dt

dxk

dt= 0 (10.131)


In this sense, geodesics are the generalization of straight lines, because for affinelyparametrized straight lines, its tangent vector is parallelely transported along thestraight line.

It is also remarkable that the equation is invariant under affine reparametrization;in other words, for any couple a, b, of real numbers, with a �= 0, if γ is a geodesiccurve then γ defined by γ (t) = γ (at + b) is a geodesic too.

There is one, and only one, solution xi (t) corresponding to any given initial

values xi (t0) and dxi

dt (t0), which is the geodesic characterized by the mentionedinitial conditions.

It is clear from the preceding formula that any second-order differential equationwith forces quadratic in the velocities defines an affine connection.

Definition 10.61 Let γ : I ⊂ R → M be a smooth curve with parameter t . Then,a covariant vector field along γ , or a 1–form along γ , is a map α : I → T M suchthat α(t) ∈ T ∗γ (t)M .

Notice that a 1–form along a curve γ , α, can be contracted with a vector fieldalong the same curve, V , giving raise to a map f : I → R by,

f (t) = 〈α(t), V (t)〉. (10.132)

Then the covariant derivative along γ can be defined by using the relation,

d

dt〈α(t), V (t)〉 =

⟨D

dtα(t), V (t)

⟩+

⟨α(t),

DV

dt

⟩, (10.133)

and we will obtain,

Dαi

dt= dαi

dt−

n∑j,k=1

�jikα j

dxk

dt(10.134)

and then we will say that the 1–form along the curve γ , α, is constant along γ , orparallel along γ , if

dαi

dt−

n∑j,k=1

�jikα j

dxk

dt= 0 , (10.135)

and therefore to define Dαi by

Dαi = dαi −n∑

j,k=1�

jikα j dxk . (10.136)

658 10 Appendices

The components of Dαi are given by

Dαi =n∑

j=1αi; j dx j . (10.137)

E.2. Torsion and Curvature

The covariant differential of a tensor field can be introduced in a similar way. Given atensor field, the covariant differential gives a new tensor fieldwith onemore covariantindex. For instance, we obtain Ai

jk;l out of Aijk . Applying this process again, we

see that the components Aijk;l;m defined by,

D Aijk;l =

n∑m=1

Aijk;l;m dxm (10.138)

are those of a tensor field of type (1, 4).It will be of importance to find a formula for the difference Ai

jk;l;m − Aijk;m;l .

Consider first a function f . We find,

f;i = ∂ f

∂xi(10.139)

and,

D f;i =n∑

j=1

∂2 f

∂xi∂x jdx j −

n∑j,k=1

�ki j

∂ f

∂xkdx j (10.140)

and then,

f;i; j = ∂2 f

∂xi∂x jdx j − �k

i j∂ f

∂xk(10.141)

It follows that,

f;i; j − f; j;i =n∑

k=1(�k

ji − �ki j )

∂ f

∂xk(10.142)

Defining,

T ki j = �k

ji − �ki j (10.143)


and having in mind (10.126), we see that the components T ki j transform as the coor-

dinates of a (1, 2) tensor, because of the first term in the right-hand side of (10.126),which destroys the tensorial character, is symmetric in the interchange of indices iand j , and therefore the T k

i j ’s transform as the components of a tensor of type (1, 2).It depends alone of the affine connection and it is called the torsion tensor. It vanishesif and only if the components of the Christoffel symbols are symmetric under theinterchange of the two lower indices.

We can also see that for any function f the components,

f;i; j − f; j;i =n∑

k=1T k

i j∂ f

∂xk(10.144)

are those of a tensor field of type (0, 2), which is the contraction of the torsion tensorand the differential of f .

Next we consider a contravariant vector field with components Ai . We find, bydefinition,

Ai; j =

∂ Ai

∂x j+

n∑l=1

�ijl Al (10.145)

and,

Ai; j;k =

n∑l=1

(∂�i

jl

∂xk+

n∑m=1

�ikm�m

jl

)Al −

n∑l=1

�lk j Ai ;l + ( j ⇔ k) (10.146)

It follows that,

Ai; j;k − Ai

;k; j =n∑

l=1

(T l

jk Ai;l + Ri

jlk Al)

(10.147)

where,

Rijlk =

∂�ik j

∂xl− ∂�i

l j

∂xk+

n∑m=1

(�i

lm�mkj − �i

km�ml j

)(10.148)

The Rijkl ’s are the components of a tensor field of type (1, 3), skew–symmetric in

its last two indices. It is called the curvature tensor.

Remark 10.3 The derivation of similar formulae for general tensor fields does notgive raise to new tensors out of the affine connection.

Up to now we have been working in coordinates (xi ) and we have used a basis of1–forms given by (dxi ) and a basis of vector fields given by ∂/∂xi .

660 10 Appendices

In general we could use a different frame at each point, say, we could useX1, X2, . . . , Xn , independent vector fields,which in previous coordinate basiswouldbe,

Xi =n∑

k=1Ak

i∂

∂xk(10.149)

implying,

det ||Akj || �= 0 (10.150)

because of independence. By introducing the matrix Bkj such that,

n∑k=1

Bkj Ak

i = δji (10.151)

we may introduce the dual basis of 1–forms given by,

θ j =n∑

k=1Bk

j dxk (10.152)

In a different coordinate system,

Xi =n∑

k=1Ak

i∂

∂ x k=

n∑k=1

Aki

∂

∂xk(10.153)

and,

θ j =n∑

k=1Bk

j d xk =n∑

k=1Bk

j dxk (10.154)

By considering the Aki as independent variables (introducing the space of frames

at each point of M) we have that,

D Aki = d Ak

i +n∑

l=1ωk

l Ali (10.155)

considered as linear differential frames in (xi , Akj ). We also get,

D Aki =

n∑l=1

∂ x k

∂xlD Al

i (10.156)


which using the preceding expressions gives,

n∑j=1

Bkj D A j

i =n∑

j=1B j

k D A ji (10.157)

They represent differential forms in the space of frames.We set,

θki =

n∑j=1

B jk D A j

i =n∑

j=1B j

k D A ji (10.158)

These n + n2 linear differential 1–forms, θ i and θ ki are linearly independent and

we find,

θ i =n∑

j=1B j

i dx j , θki =

n∑j=1

B jk

(d A j

i +n∑

l=1ωi

l Ali

)(10.159)

or,

dxi =n∑

j=1Ai

j dx j ,

n∑j=1

Akjθ

ji = d Ak

i +n∑

l=1ωk

l Ali (10.160)

Applying exterior differentiation on these equations and simplifying we get,

dxi =n∑

j,k=1T i

jk dx j ∧ dxk (10.161)

dωkj −

n∑l=1

ωlj ∧ ωk

l =n∑

l,m=1

1

2Rk

jlmdxl ∧ dxm (10.162)

or,

dθ i −n∑

k=1θ k ∧ θ i

k =1

2

n∑l,m=1

Pilmθ l ∧ θm (10.163)

dθ ij −

n∑l=1

θ lj ∧ θ i

l =1

2

n∑l,m=1

Sijlmθ l ∧ θm (10.164)

662 10 Appendices

where,

1

2Pi

lm =n∑

k,p,q=1Bk

i A pl Aq

m T kpq ; Si

jlm =n∑

k,p,q,r=1Bk

i A pk Aq

l Arm Rk

pqr

(10.165)Theses equations are called equations of structure.When we use a holonomic frame, i.e.,

Akl = Bl

k = δkl (10.166)

we find,

−n∑

k=1dxk ∧ ω j

k =n∑

l,m=1T j

lmdxl ∧ dxm (10.167)

and,

dω ji −

n∑l=1

ωli ∧ ω j

l = 1

2

n∑l,m=1

R jilmdul ∧ dum (10.168)

It is sometimes convenient to introduce the exterior quadratic differential forms,

� j = 1

2

n∑l,m=1

P jlmθ l ∧ θm , �

ji =

1

2

n∑l,m=1

S jilmθ l ∧ θm (10.169)

and write the previous equations in the form,

dθ j −n∑

k=1θ k ∧ θ

jk = � j , �

ji =

n∑k=1

dθ ji −

n∑k=1

θ ki ∧ θ

jk (10.170)

Exterior differentiation of this system gives,

d� j −n∑

k=1�k ∧ θ

jk −

n∑k=1

θ k ∧�jk = 0 (10.171)

d� ji +

n∑k=1

�ki ∧ θ

jk −

n∑k=1

θ ki ∧�

jk = 0 (10.172)

which are called Bianchi identities.Summarizing, an affine connection on M gives raise to n2+n linearly independent

1–forms θ i and θki on the space of frames on M whose exterior derivatives satisfy

equations (10.171, 10.172). Viceversa, any set of n2 + n linearly independent 1–


forms on the space of frames whose exterior derivatives satisfy equations (10.171,10.172) gives raise to an affine connection on M .

Riemannian Connections

Given a Riemannian metric g, it is possible to define a scalar product of two vectorswith the same origin as the contraction of their tensor product with the metric at thepoint. If gi j are the components of the metric tensor with respect to the natural frameof a coordinate system (xi ) and Ai and Bi are the components of the vectors X andY the scalar product is given by,

X · Y =n∑

i, j=1gi j Ai B j (10.173)

The length of a vector X is the positive square root of the product X2 = X · X .A Riemannian metric also enables us to define the arc length of a parametrized

curve. If γ : R → M is a curve with components xi (t) = γ i (t), we have,

ds2 = gi j dxi dx j (10.174)

and,

s =t1∫

t0

√gi j

dxi

dt

dx j

dtdt (10.175)

defines the arc length between the points γ (t0) and γ (t1).The angle between vectors is also given by,

cos θ = X · Y√X · X

√Y · Y

(10.176)

Definition 10.62 A map F : M → M ′ between Riemannian manifolds is said tobe isometric if it preserves distances between points, and therefore it also preservesangles between curves.

Oneof themain results ofRiemanniangeometrywas the discoveryof the dominantrole played by the Levi–Civita connection.

664 10 Appendices

E.3. The Levi–Civita Connection

Theorem 10.63 On a Riemannian manifold there is one affine connection with thefollowing properties

(i) The torsion tensor is zero(ii) The scalar product of two vectors remains unchanged when they are displaced

parallelely along a curve.

One such torsionless connection is called a metric connection.

Proof 10.1 This theorem can be easily proved in coordinates (xi ). We consider aparametrized curve, xi = xi (t) and along it the vector fields with components Ai (t)and Bi (t). The conditions for parallelism along the curve are,

d Ai

dt+

n∑j,k=1

�ik j Ak dx j

dt= 0 (10.177)

and,

d Bi

dt+

n∑j,k=1

�ik j Bk dx j

dt= 0 (10.178)

for i = 1, . . . , n.�

Property (ii) in the statement of the theorem is equivalent to,

d

dt

n∑i, j=1

gi j Ai B j = 0 (10.179)

Whenever these conditions are satisfied we find,

d

dt

n∑i, j=1

gi j Ai B j =n∑

i, j=1

(dgi j

dt−

n∑l=1

gil �lk j

dxk

dt−

n∑l=1

gl j �lik

dxk

dt

)Ai B j = 0

(10.180)with the requirement that this holds for all parametrized curves and all vector fieldsalong the curve, therefore,

∂gi j

∂xk=

n∑l=1

(gil�

ljk + gl j�

lik

), i, j, k = 1, . . . , n. (10.181)

On the other side, the vanishing of the torsion can be expressed by,

�ijk = �i

k j i, j, k = 1, . . . , n (10.182)


or by introducing �ki j =∑nl=1 gli , by: �ki j = � j ik . With these Christoffel symbols

of first kind, (10.181) can be rewritten as,

∂gi j

∂xk=

n∑l=1

(� j ik + �i jk) (10.183)

Permuting cyclically n i, j, k we get,

∂g jk

∂xi = � jki + �k ji∂gki∂x j = �ki j + �ik j , i, j, k = 1, . . . , n

(10.184)

Since the �i jk’s are symmetric in the first and third indices, these two equationstogether with previous one give,

�i jk = 1

2

(∂gi j

∂xk+ ∂g jk

∂xi− ∂gik

∂x j

)(10.185)

or,

�ijk =

1

2

n∑l=1

gil(∂gkl

∂x j+ ∂g jl

∂xk− ∂g jk

∂xl

)(10.186)

when gi j are defined by,

n∑j=1

gi jgjk = δk

i (10.187)

It follows that the affine connection is completely determined. The components�i jk and �i

jk are called respectively the Christoffel symbols of the first and secondkind, respectively. This particular connection associatedwith ametric tensor is calledthe connection of Levi–Civita.

Remark 10.4 If we consider other quadratic forms constructed out of two vectorfields and a (0,2)–type tensor, say,

n∑i, j=1

αi j Ai B j (10.188)

we may say that a connection is an α–connection if,

d

dt

n∑i, j=1

αi j Ai B j = 0 (10.189)

666 10 Appendices

For instance, α could be a symplectic structure, a non positive definite product, anHermitean structure and so on.

When the connection is ametric connection there are further properties on theRie-mann tensor. In terms of the structure forms, and using themetric tensor to lower someindices, we find θi , θi j , �i j , Si jkl out of θ i , θ

ji , �

ji , S j

ikl respectively.Then the following properties hold,

θi j + θ j i = 0 , �i j +� j i = 0 (10.190)

and θi , θi j (with i < j) are linearly independent.These forms satisfy the equations,

dθi =n∑

j=1θ j ∧ θ j i (10.191)

dθi j =n∑

k=1θik ∧ θk j +�i j (10.192)

d�i j +n∑

k=1�ik ∧ θk j −

n∑k=1

θik ∧�k j = 0 (10.193)

and the functions Si jkl have the following symmetry properties,

Si jkl = −S jikl = −Si jlk (10.194)

Si jkl = −Skli j (10.195)

Si jkl + Sikl j + Sil jk = 0 (10.196)

E.4. Properties of Connections and Comparisonwith Other Approaches

A linear space M can be endowed with two o more linear connections. Notice that if�(1) and�(2) are the Christoffel symbols of two such connections, then the differencesymbols �(1) − �(2) transform under changes of coordinates as the components ofa (1,2) tensor field, even if the symbols of each connection do not transform as atensor field due to the presence of the first term in the right-hand side of (10.126).However, when taking the differences�(1)−�(2), the first terms cancel and thereforethe components of the difference �(1)−�(2) transform as the components of a (1,2)tensor field. Conversely, given a connection with Christoffel symbols �(1) and a


tensor field L of type (1,1) we can define a new set of Christoffel symbols �(2) bywriting in any coordinates,

�(2)ijk = �

(1)ijk + Li

jk (10.197)

because if the �(1)ijk satisfy the property of changing as (10.126) when going from

one set to another of coordinates, the same is true for �(2)ijk , because L being a

(1, 1)–tensor field, its coordinates transform as,

Lijk =

n∑m,p,q=1

Lmpq

∂ x p

∂x j

∂ xq

∂xk

∂xi

∂ xm(10.198)

Now, given two affine connections, any convex combination gives a new connec-tion, because the symbols �i

jk = λ�(1)ijk + (1 − λ)�

(2)ijk , with 0 ≤ λ ≤ 1 transform

properly when going from one coordinate system to another. If we change the lowerindices in the symbols we also obtain a new connection because the first term in(10.126) is symmetric under such interchange of indices. Therefore, if �(1)i

jk are the

symbols of a connection, then �(2)ijk = �

(1)ik j is a new connection in which the torsion

changes the sign and the convex combination of both with λ = 1/2 gives raise to asymmetric connection. We notice that geodesics are indifferent to torsion.

Let us now consider a curve σ : I → M , with 0 ∈ I , and a vector u ∈ Tσ(0)M .We have seen before that we can transport this vector all along the curve by paralleltransport and we obtain in this way a curve in T M , σ h : I → T M , which projectson σ . The curve so obtained will be called the horizontal lift of σ .

The same can be done for vectors. So, given a vector u ∈ Tx M , we can lift it to avector tangent to T M atw, for everyw ∈ Tx M , just by choosing a curve σ : I → Msuch that σ(0) = x and σ (0) = u and taking the tangent vector to the horizontal liftσ h of such a curve starting at w ∈ Tx M at t = 0. The vector so obtained is said tobe horizontal and the lift so defined does not depend on the choice of the curve σ .

In fact, if W (t) is a vector field along σ such that W (0) = w, then the conditionof the curve σ h of being obtained by parallel transport means that,

W i (t)+n∑

j,k=1�i

jk(x)Wj uk = 0 (10.199)

and therefore the vector tangent to σ h at t0 is,

n∑i=1

(σ i (0)

∂

∂xi + W i (0)∂

∂vi

)=

n∑k=1

uk

⎛⎝ ∂

∂xk −n∑

i, j=1�i

jk(x)wj ∂

∂vi

⎞⎠ (10.200)

668 10 Appendices

We see in this expression for the lifting of a vector that the lifting of a vectordepends linearly on it. Moreover, the lifting of a vector is zero if and only if thevector is null. In other words, the linear map of lifting is injective.

The preceding expression for the lifting of a vector displays clearly that we canchoose a basis of horizontal vector fields formed by,

Hi = ∂

∂xi−

n∑j,k=1

�kji (x)v

j ∂

∂vk, i = 1, . . . , n (10.201)

These fields determine a distribution in T M which in general is not integrablebecause,

⎡⎣ ∂

∂xi−

n∑k,l=1

�kliv

l ∂

∂vk,

∂

∂x j−

n∑h,g=1

�hg jv

g ∂

∂vh

⎤⎦

= −n∑

h,g=1

∂�hg j

∂xivg

∂

∂vh+

n∑k,l=1

∂�kli

∂x jvl ∂

∂vk

+n∑

k,l,h=1vl�k

li�hk j

∂

∂vh−

n∑j,k,l,g=1

vg�lg j�

kli

∂

∂vk

=n∑

j,k,l,g=1vl

[∂�k

li

∂x j− ∂�k

l j

∂xi+ �

gli�

kg j − �l

g j�kli

]∂

∂vk(10.202)

which can be rewritten as,

⎡⎣ ∂

∂xi−

n∑k,l=1

�kliv

l ∂

∂vk,

∂

∂x j−

n∑g,h=1

�hg jv

g ∂

∂vh

⎤⎦ =

n∑k,l=1

Rkl jiv

l ∂

∂vk(10.203)

and therefore, the horizontal distribution is integrable if and only if the curvature ofthe connection vanishes.

The horizontal distribution can also be defined as the kernel of the 1-forms,

ηi = dvi +n∑

j,k=1�i

jkvj dxk (10.204)

because,

ηi (Hj ) = (dvi +n∑

l,k=1�i

lkvl dxk)(

∂

∂x j −n∑

l,k=1�k

l j (x)vl ∂

∂vk ) =n∑

m=1�i

m jvm −

n∑k=1

�ik jv

k = 0

(10.205)


Conversely, if the vector field,

X =n∑

i=1

(Ai ∂

∂xi+ Bi ∂

∂vi

)(10.206)

is annihilated by the 1–forms ηi , then as,

ηi (X) = Bi +n∑

j,k=1�i

jkvj Ak (10.207)

we see that X is of the form,

X =n∑

i=1Ai

⎛⎝ ∂

∂xi−

n∑j,k=1

�kjiv

j ∂

∂vk

⎞⎠ (10.208)

and then it lies in the horizontal distribution. Another very remarkable property ofthe linear connections we are considering is that if V h is the horizontal lift of avector field V in the base manifold, then [�, V h] = 0, i.e., the lifting so defined ishomogeneous of degree zero in velocities.

Let us now mention the relationship between the theory of connections as ex-plained here, which is based in the ideas of Levi–Civita, and other alternative butequivalent approaches.

We recall that the affine connection provide a method of transporting a vectorat a point x ∈ M to another point along any chosen curve σ . That means that ify ∈ M is another point of the given curve the affine connection provides a map:Yy,x : Tx M → Ty M , where mention is not made to the curve σ for the sake ofnotational simplicity. Such a map Y satisfies that for any triplet of points in σ(I ),Yy,z ◦ Yz,x = Yy,x . Obviously: Y x,x = I dTx M (along a degenerate curve reduced

to a point), and: Y x,y =(Yy,x

)−1, when the curves we consider are the oppositeone to another.

We remark that as the transport rule is defined by solving the ordinary differentialequation system,

dV i

dt+

n∑j,k=1

�ijk V j dσ k

dt= 0 (10.209)

which is linear , the map Yy,x so defined is linear . Moreover, when we considertwo curves σ and σ ′ which are related by a reparametrization, i.e., there exists adiffeomorphism h : I ′ → I , such that σ ′ = σ ◦ h, we see that σ ′ = σ h and takinginto account that V becomes V ′ = V ◦ h, we also see that,

670 10 Appendices

dV ′i

dt ′= h

dV i

dt(10.210)

and therefore the mapsYx,y obtained for the two curves coincide. In other words, thetransport rule is defined for each path, i.e., a curve freed from its parametrization.

Conversely, the affine connection can be introduced by giving an appropriate ruleof parallel transport in M , i.e., a rule of parallel transport is given for each path inM . Additional smoothness conditions have to be imposed.

So, the rule allows us to say that a vector field along a path is a parallel field whenit may be obtained by parallel el transport of a given vector at some point of thecurve.

The rule of parallel transport can be used to define the absolute o covariant deriv-ative along a curve σ of a vector field along σ , as follows:

If V (t) ∈ X(σ ), we can define DV /Dt ∈ X(σ ) by,

DV

Dt= lim

δ→0

1

δ

[V (t + δ)‖ − V (t)

](10.211)

where V (t + δ)‖ is the vector in Tσ(t) obtained by transporting parallelely V (t + δ)

from σ(t + δ) to σ(t),

V (t + δ)‖ = Yσ(t),σ (t+δ)V (t + δ) (10.212)

With this definition we can easily check that,

D

Dt(U + V ) = DU

Dt+ DV

Dt,

D

Dtf V = f

DV

Dt+ d f

dtV, ∀ f ∈ F(I ) (10.213)

and that DV /Dt = 0 if and only if V is parallel along the curve σ .We will assume, as an additional requirement on our transport rule the two fol-

lowing properties:PT1. The value of the absolute derivative along a curve σ at a point σ(t) depends

only on σ (t): if σ and σ ′ are curves such that σ(0) = σ ′(0) = x and σ (0) = σ ′(0) =v, and if V is a vector field defined in a neighborhood of x ∈ M , then the absolutederivatives of V at the point x along σ and σ ′ are equal.

PT2. For any vector field V defined in a neighborhood of x ∈ M , the mapTx M → Tx M assigning to each u ∈ Tx M the absolute derivative of V along anycurve whose tangent vector is at x in u is linear.

With these two properties for the rule of parallel transport we can define theconcept of covariant derivative along a vector u ∈ Tx M of a vector field defined ina neighborhood of x ∈ M as follows:

If V ∈ X(M) y v ∈ Tx M , with v �= 0, let us choose a curve σ : I → M such thatσ(0) = x and σ (0) = v, and then, by assumption, (DV/dt)(0) does not depend onthe choice of the curve σ , but only on x and v, and we will denote this vector ∇vV .When v = 0, by definition, ∇0V = 0, for any vector field V .


Given two different vector fields U, V ∈ X(M), we can define a new one to bedenoted ∇U V ∈ X(M) by means of,

(∇U V )(x) = ∇Ux V (10.214)

In this way we have found a map ∇ : X(M) × X(M) → X(M) given by∇(U, V ) = ∇U V such that the following properties hold,

(i) ∇U+V W = ∇U W +∇V W (10.215)

(i i) ∇U (V + W ) = ∇U V + ∇U W (10.216)

(i i i) ∇ f U V = f ∇U V, ∀ f ∈ F(M) (10.217)

(iv) ∇U ( f V ) = (U f )V + f ∇U V (10.218)

In this approach to connections the Christoffel symbols arise directly from ∇. Infact, if xi are arbitrary coordinates, then:

∇∂/∂xi

(∂

∂x j

)=

n∑k=1

�kji

∂

∂xk(10.219)

Therefore, in arbitrary coordinates, ∇U V will be expressed by:

∇U V =n∑

i,k=1U k

⎛⎝∂V i

∂xk+

∑j=1

�ijk(x)V

j

⎞⎠ ∂

∂xi(10.220)

because if,

U =n∑

i=1U i ∂

∂xi ,

n∑i=1

V i ∂

∂xi (10.221)

then,

∇∑ni=1 Ui ∂/∂xi

∑nj=1 V j ∂

∂x j

=∑ni, j=1 U i

(∂V j

∂xi∂

∂x j + V j∇∂/∂xi∂

∂x j

) (10.222)

where use is made of the preceding properties, and therefore, using (10.219) we find(10.220).

672 10 Appendices

From this expression we find that,

∇vV

(D(V ◦ σ)

dt

)t=0

=∑

i,k=1vk

(∂V i

∂xk+ �i

jk(x)Vj (x)

)∂

∂xi(10.223)

If v is a vector in the point x ∈ M , the covariant derivative of ∇vα of a 1-formα can be defined in a similar way by means of (D(α ◦ γ )/Dt)t=0, γ being a curvesuch that γ (0) = x and γ (0) = v. This definition does not depend on the choice ofthe curve. If U is a vector field in M , we can define ∇Uα by,

(∇Uα)(x) = ∇Ux α (10.224)

We find in this way a map ∇ : X(M)×�1(M)→ �1(M) such that,

(i) ∇U+V α = ∇Uα + ∇V α (10.225)

(i i)∇U (α + β) = ∇Uα + ∇Uβ (10.226)

(i i i)∇ f Uα = f ∇Uα, ∀ f ∈ F(M) (10.227)

(iv)∇U ( f α) = (U f )α + f ∇Uα (10.228)

Finally, the relation,

d

dt〈α, V 〉 =

⟨Dα

Dt, V

⟩+

⟨α,

DV

Dt

⟩, (10.229)

translates to,

W 〈α, V 〉 = 〈∇Wα, V 〉 + 〈α,∇W V 〉. (10.230)

As far as the torsion and curvature is concerned, we should remark that,

T (X, Y ) = ∇X Y − ∇Y X − [X, Y ] , X, Y ∈ X(M) (10.231)

is such that T (X, Y ) = −T (Y, X) and for each function f in M ,

T ( f X, Y ) = f T (X, Y ) (10.232)

because,

T ( f X, Y ) = f ∇X Y − (Y f )X − f ∇Y X − [ f X, Y ]= f ∇X Y − (Y f )X − f ∇Y X − f [X, Y ] + (Y f ) X = f T (X, Y )

(10.233)


Consequently, T defines a (1, 2) tensor field. In local coordinates, the componentsof T are given by,

T

(∂

∂xi,

∂

∂x j

)= ∇ ∂

∂xi

(∂

∂x j

)− ∇ ∂

∂x j

(∂

∂xi

)=

n∑k=1

(�kji − �k

i j )∂

∂xk(10.234)

and then T is but the torsion tensor we introduced before.In a similar way, if R(X, Y )Z , is defined by,

R(X, Y )Z = ∇X (∇Y Z)− ∇Y (∇X Z)− ∇[X,Y ]Z = 0 (10.235)

then it satisfies R(X, Y )Z = −R(Y, X)Z , together with,

R( f X, Y )Z = f R(X, Y )Z = R(X, Y )( f Z) (10.236)

∀ f ∈ F(M), because,

R( f X, Y )Z = f∇X (∇Y Z)− (Y f )∇X Z − f∇[X,Y ]Z + (Y f )∇X Z = f R(X, Y )Z(10.237)

and,

R(X, Y )( f Z) = ∇X ((Y f )Z + f∇Y Z)− ∇Y ((X f )Z + f∇X Z)

− f∇[X,Y ]Z − (([X, Y ]) f )Z = f R(X, Y )Z(10.238)

R can then be seen as a map from X(M) × X(M) × X(M) into X(M) whichis F(M)-multilinear and therefore it defines a tensor of type (1, 3). Its componentsRa

bcd in a coordinate system are,

R

(∂

∂xk,

∂

∂xi

)∂

∂x j=

n∑i=1

Rijkl

∂

∂xi(10.239)

and then,

Rijkl =

∂�ijl

∂xk − ∂�ijk

∂xl +n∑

m=1(�m

jl �imk − �m

jk �iml) (10.240)

Therefore, the tensor R is the curvature tensor introduced before.If the connection is torsionless, T = 0, when applying ∇X on both sides of the

equation T (X, Y ) = 0 we will obtain,

∇X (∇Y Z)− ∇X (∇Z Y )−∇X ([Y, Z ]) = 0 (10.241)

674 10 Appendices

and permuting cyclically the vector fields X, Y, Z and adding the correspondingexpressions we find that,

∇X (∇Y Z)−∇X (∇Z Y )− ∇X ([Y, Z ])+∇Y (∇Z X)−∇Y (∇X Z)− ∇Y ([Z , X ])+∇Z (∇X Y )− ∇Z (∇Y X)− ∇Z ([X, Y ]) = 0 (10.242)

We can now use the relations∇X (∇Y Z)−∇X (∇Z Y )+ R(X, Y )Z+∇[X,Y ]Z andanalogously for the other cyclic permutations and then we get,

R(X, Y )Z +∇[X,Y ]Z − ∇Z ([X, Y ])+ R(Y, Z)X +∇[Y,Z ]X − ∇X ([Y, Z ])+R(Z , X)Y +∇[Z ,X ]Y −∇Y ([Z , X ]) = 0

(10.243)In the torsionless case we are considering the terms in which R does not appear

cancel, because of Jacobi identity and we obtain the relation,

R(X, Y )Z + R(Y, Z)X + R(Z , X)Y = 0 (10.244)

which is known as Ricci or first Bianchi identity.Koszul’s approach to the theory of connections was based in relations (10.215) to

(10.218). In other words, we can consider a connnection as a map ∇ : X(M) ×X(M) → X(M) given by ∇(U, V ) = ∇U V satisfying properties (10.215) to(10.218). In this case the connection 1-forms can be recovered by,

ωij (V ) =

⟨dxi ,∇V

∂

∂x j

⟩, (10.245)

and Christoffel symbols are then defined by (10.124).In the case in which (M, g) is a Riemannian manifold, given a connection in the

Koszul approach by means of the ∇ map, we can see that the connection is a metricconnection if and only if for any three vector fields X, Y, Z ∈ X(M), the followingrelation holds,

X (Y · Z) = (∇X Y ) · Z + Y · (∇X Z) (10.246)

In fact, if∇ satisfies thementioned property, then given a curve in M , γ : I → M ,starting from x = γ (0), for any couple of vectors u, v ∈ Tx M , letU, V be the vectorfields along γ obtained by parallel transport of u and v respectively. If f : I → R

is the function given by f (t) = U (t) · V (t), then,

d f

dt= Tγ (t) f = (∇T U (t)) · V (t)+U (t) · (∇T V (t)) (10.247)

where T is a vector field in M extending to dγdt , and as U and V are constant along

the curve, we see that d fdt = 0.


Conversely, let assume that the connection is metric. Given any three vector fields,X, Y, Z ∈ X(M), for any point x ∈ M , let γ : I → M be the geodesic curve startingfrom x and with tangent vector Xx . If {u1, . . . , un} is an orthonormal basis of Tx MandUi ∈ X(γ ) denote the vector fields along γ obtained by parallel transport of ui , asby assumption the connection is metric, the set of vector fields along γ , U1, . . . ,Un ,determine an orthonormal basis of the tangent spaces at each point of the curve. Letthe coordinate expressions of the vector fields along γ , Y ◦ γ and Z ◦ γ , be,

Y (t) =n∑

i=1Y i (t)Ui , Z(t) =

n∑i=1

Zi (t)Ui (10.248)

and then Y (t) · Z(t) =∑ni, j=1 gi j Y i (t)Z j (t), where gi j are constant. Let T denote

as before a vector field in M extending dγdt , and then,

Tγ (t)(Y (t) · Z(t)) =n∑

i, j=1gi j

d

dt

(Y i (t)Z j (t)

)(10.249)

The vector fields along γ Ui are constant, and then,

∇T Y =⎛⎝ d

dt

n∑j=1

Y j (t)

⎞⎠U j , ∇T Z =

⎛⎝ d

dt

n∑j=1

Z j (t)

⎞⎠U j (10.250)

Therefore,

(∇T Y · Z)+ (Y · ∇T Z) =n∑

i, j=1gi j

d

dt

(Y i (t)Z j (t)

)(10.251)

Particularizing for the value t = 0, and having in mind that by construction of γ ,T (0) = Xx , we obtain the relation (10.246).

The explicit expression of the torsionless metric connection of Levi–Civita isgiven by the so-called ‘Koszul formula’,

2(∇X Y · Z) = X (Y · Z)+ Y (Z · X)− Z(X · Y )

− (X · [Y, Z ])+ (Y · [Z , X ])+ (Z · [X, Y ]) (10.252)

In a local coordinate system xi in M , we can see that the Koszul formula yields,

2

(∇∂/∂xi

(∂

∂x j

))· ∂

∂xk= ∂g jk

∂xi+ ∂gik

∂x j− ∂gi j

∂xk(10.253)

676 10 Appendices

and using the definition of Christoffel symbols,

(∇∂/∂xi

(∂

∂x j

))· ∂

∂xk=

n∑l=1

�lj iglk (10.254)

Therefore,

�kji =

1

2

n∑l=1

gkl(∂g jl

∂xi+ ∂gil

∂x j− ∂gi j

∂xl

)(10.255)

Similarly, given a skew–symmetric in the lower indices tensor of type (1, 2), T ,there is only one metric connection with such a torsion tensor T which is given bythe generalization of the Koszul formula,

2 (∇X Y ) · Z = X (Y · Z)+ Y (Z · X)− Z(X · Y )− X · [Y, Z ] + Y · [Z , X ]+ Z · [X, Y ] + X · T (Y, Z)− Y · T (Z , X)+ Z · T (X, Y )

(10.256)

On the other hand, we have been able to associate to the connection an hori-zontal distribution satisfying an additional property of homogeneity. If we supposethat, conversely, we give as an input an horizontal distribution, with appropriatesmoothness conditions, i.e., we choose in every tangent vector to M a n-dimensionalsubspace complementary to the vertical subspace, we can use it to lift vector fieldsin M to horizontal vector fields in T M , i.e., the lift of the vector field V ∈ X(M)

is the unique horizontal vector field such that T τ(V h) = V . More specifically, in agiven coordinate system, the horizontal lifting of ∂/∂qi is,

(∂

∂qi

)h

= ∂

∂qi−

n∑j=1

�ji

∂

∂v j(10.257)

If the distribution verifies the mentioned homogeneity condition [�, V h] = 0,from the relation,

⎡⎣ n∑

k=1vk ∂

∂vk,

∂

∂qi−

n∑j=1

�ji

∂

∂v j

⎤⎦ =

n∑j,k=1

(−vk ∂�

ji

∂vk+ �k

i

)∂

∂v j= 0 (10.258)

from where we see that,

n∑k=1

vk ∂�ji

∂vk= �

ji , i, j = 1, . . . , n (10.259)


and therefore the functions �ji are homogeneous of degree one. If they are smooth

at v = 0, they should be linear, i.e., there will be of the form,

�ji =

n∑k=1

�jkiv

k (10.260)

Once we have been able to lift vectors, we can also lift curves in the base, i.e.,given a curve in the base there exists one curve in T M starting from each vector andsuch that the tangent vector to the curve is horizontal. With these horizontal lifts ofcurves we can do the parallel transport from a vector at a point of M along a curvein M just by using the horizontal lift of the curve.

Given a symmetric linear connection on a affine space M , one can define a vectorfield � on T M whose value at v ∈ T M is its horizontal lift to Tv(T M). In localcoordinates the expression of this vector field is,

� =n∑

i=1vi

⎛⎝ ∂

∂qi−

∑j,k

�ijk

∂

∂v jvk

⎞⎠ (10.261)

This vector field is called the geodesic spray of the connection and its integralcurves are just the natural lift of geodesics in M , which are determined by (10.131).Notice that, by construction, the natural lift of a geodesics is a horizontal curve.As we see, the geodesic spray is a second-order differential equation vector fieldsatisfying an additional condition of degree–one homogeneity which corresponds tothe affine reparametrization property of geodesics. More generally, a second-orderdifferential equation vector field is said to be a spray if the set of its integral curvesis invariant under any affine reparametrization: these curves are called geodesics ofthe spray. Given a spray �, there is a symmetric connection whose geodesic sprayis �; and conversely, the connection is fully determined by its geodesic spray, thatcan also be used to construct the exponential map exp : Tx M → M . A Riemannianstructure on M determines one symmetric connection, the Levi-Civita connection,and consequently a Riemannian spray.

Riemannian and Pseudo-Riemannian Metricson Linear Vector Spaces

Let again E be an n-dimensional vector space equipped with coordinates x1, . . . , xn ,and let g ∈ S2(E) = S(T 0

2 (E)) be a fully covariant symmetric tensor field of ranktwo. In coordinates,

g = gi j dxi ⊗ dx j (10.262)

678 10 Appendices

where: gi j = g j i and gi j ∈ F(E), i.e., the gi j ’s are smooth functions on E . Thetensor g induces the maps:

g : X(E)× X(E)→ F(E) (10.263)

where X (E) is the set of smooth sections of the tangent bundle,8 via: (X, Y ) →g(X, Y ), and:

g : X(E)→ �k(E) = X∗(E) (10.264)

by means of,

X = Xi ∂

∂xi→ g(X) = Xi dxi ; Xi = gi j X j (10.265)

Of course: g(X, Y ) = gi j Xi Y j = iX g(Y ) = iY g(X). The tensor g will be said tobe non-degenerate if g(X) = 0 when and only when X = 0, i.e., if,

g(X, Y ) = 0, ∀Y −→ X = 0 (10.266)

Definition 10.64 A fully covariant symmetric and non-degenerate second-rank ten-sor field g will be said to define a pseudo-riemannian metric on E . If the strongercondition,

g(X, X) = 0↔ X = 0 (10.267)

holds, the metric will be said to be Riemannian.

Example 10.6 In Rn , the tensor:

gi j = δi j dxi ⊗ dx j (10.268)

defines the (Riemannian)Euclideanmetric associatedwith a given coordinate system.

Example 10.7 Let E = R4 with coordinates: x0, x1, x2, x3. The Lorentz metric on

R4 is the pseudo-riemannian metric,

gi j = dx0 ⊗ dx0 −3∑

i=1dxi ⊗ dxi (10.269)

Remark 10.5 If g is a Riemannian metric, then:

8 i.e. The set of vector fields on E .


Proposition 10.65 g(X, X) has constant sign on E.

So, if that is the casewemay always assume,without loss of generality: g(X, X) >

0 for all X �= 0.

Remark 10.6 If g is nondegenerate, then the map of Eq. (10.265) becomes actually abijection. In other words, and again in local coordinates, this implies that the matrix||gi j || is invertible. Defining then the inverse matrix (with contravariant indices)||gi j ||, such that,

gi jg jk = δik (10.270)

we can also define the map,

g = g−1 : X∗(E)→ X(E) (10.271)

via,

ω = ωi dxi → g−1(ω) = ωi ∂

∂xi; ωi = gi jω j (10.272)

Hence, g (or g and/or g) can be used to “raise” and “lower” indices. We canassociate with ||gi j || the fully contravariant symmetric second-rank tensor g definedby,

g = gi j ∂

∂xi⊗ ∂

∂x j(10.273)

The right-hand side ofEq. (10.263), being bilinear , symmetric and nondegenerate,will define for us (pointwise) the scalar product of vector fields (which need not bepositive if the metric is not Riemannian, of course), also denoted as (X, Y )g , orsimply as (X, Y ) for short if the metric g has been selected once and for all. Quitesimilarly, the scalar product of two one-forms: ω = ωi dxi and: η = ηi dxi will bepointwise defined as,

(ω, η)g = gi jωiη j (10.274)

Let’s denote now by G the determinant of the matrix ||gi j ||,

G = det ||gi j || = εi1...in1...n g1i1 . . . gnin (10.275)

Under a change of coordinates: xi → xi ,

gi j → gi j = ghk∂xh

∂xi

∂xk

∂x j(10.276)

680 10 Appendices

and hence,

G → G = G[∂(x1, . . . , xn)

∂(x1, . . . , xn)]2 (10.277)

(proof left as an exercise). Therefore, if we limit ourselves to transformations witha positive jacobian (orientation-preserving changes of coordinates), then,

√|G| →√|G| = √|G|∂(x1, . . . , xn)

∂(x1, . . . , xn)(10.278)

i.e.,√|G| transforms (with respect to transformations with a positive jacobian)

as the coefficient of a form of maximal rank. Being nowhere vanishing:

Proposition 10.66 If g is a (pseudo) Riemannian metric, then,

� = √|G|dx1 ∧ · · · ∧ dxn (10.279)

is a volume-form on E.

The volume-form can be written also as,

� = 1

n!�i1···in dxi1 ∧ · · · ∧ dxin (10.280)

where,

�i1···in =√|G|ε1···ni1···in (10.281)

are the totally antisymmetric components of �.

Appendix F: Cohomology Theories of Lie Groupsand Lie Algebras

In many instances, some of them found in this book, there appear functions definedon groups that must satisfy equations deeply related to the group structure. Suchequations are often related to the cohomological structure of the group and theirunderstanding provides the clues to solve the problems from which they emerge.

Groups often appear as represented on certain modules or acting on given spaces,we will be interested in cohomology theories for groups G with coefficients in G-modules M (for instance the algebra of smooth functions on a manifold where thegroup G acts, or the module of symmetric tensors over G, or the Lie algebra of Gitself, etc.)

Appendix F: Cohomology Theories of Lie Groups and Lie Algebras 681

We will be mainly interested in Lie groups. We always have the de Rham coho-mology of the group as a smooth manifold on one side, where the group structure isdisregarded, and the so-called Eilenberg-Maclane cohomology on the other, whereG is just an algebraic group.Wewill discuss in this appendix the relation among bothnotions and the relation with a few other cohomology theories that appear naturallyalong the road.

In this context, both Lie algebras and associative algebras arise naturally, andtheir cohomology theories will be also succintly discussed as well as their relationto deformation theory.

F.1. Eilenberg-MacLane Cohomology

LetG be an algebraic group andM aG-module (that is, there is amorphismρ : G →Aut(M) or, in other words, G acts on M). We can define the set of n-cochains ofG with coefficients on M as the set of normalized maps f : G × n· · · × G → M,f (g1, . . . , gn) = 0 if gi = e for some index i . We will denote the Abelian group ofn-cochains by Cn(G,M) (the group operation induced pointwise from the Abeliangroup structure of M as a module. We can form in this way the Abelian groupC•(G,M) = ⊕

n≥0 Cn(G,M), with C0(G,M) = M. We may define now thegraded homomorphism of degree 1, δn : Cn(G,M)→ Cn+1(G,M), as follows:

δn f (g1, . . . , gn+1) = ρ(g1) f (g2, . . . , gn+1)

+n∑

i=1(−1)i f (g1, . . . , gigi+1, . . . , gn+1)

+(−1)n+1 f (g1, . . . , gn), (10.282)

for all gi ∈ G, i = 1, . . . , n+1. A direct computation shows that δn+1 ◦ δn = 0. Theextension δ to C•(G,M) of the homomorphisms δn , defined by δ |Cn(G,M)= δn ,satisfies δ2 = 0. Hence we have defined a cochain complex (C•(G,M), δ) whosecohomology H •(G,M) =⊕

n≥0 H n(G, A) is a graded Abelian group whose n-thfactor, called the n-th Eilenberg-MacLane cohomology group Hn(G,M) is givenby [ML75]:

H n(G,M) = ker δn/Imδn−1 .

If we call n-cocycles the n-cochains f such that δn f = 0 and we denote them byZn(G,M) and we called n-coboundaries the n-cochains f such that there existsa n − 1-cochain h such that f = δn−1h and we denote them by Bn(G;M), thenBn(G,M) is a subgroup of Zn(G,M) and Hn(G,M) = Zn(G,M)/Bn(G,M).

682 10 Appendices

Mackey-Moore and Bargmann-Mostow Cohomologies

Let G be now a topological group. Then we considerM to be a topological moduleand the representation ρ of G on M will be assumed to be strongly continuous,i.e., ρ(g) is a homeomorphism for each g ∈ G and ρ(g)x is continuous for eachx ∈ M. In that case we will say that M is a topological G-module. We denote byCn

b (G,M) the space of Borel measurable n-cochains in G with values inM. Then itis obvious that the Eilenberg-MacLane cohomology operator preserves the Boreliancochains, then it induces a cohomology operator inC•

b(G,M) =⊕n≥0 Cn

b (G,M).The cohomology of the cochain complex (C•

b (G,M), δ) is called theMackey-Moorecohomology [Mo64, Mac57], it will be denoted by H•

b (G,M) and it is useful whendealing with topological groups not too wild, for instance Polish groups (that is,topological groups which are separable, metrizable and complete). In such cases, forinstance, H2

b (G,M)measures the space of topological extensions of G byM (theyare actually the same if G and M are locally compact [Mac76]).

Even if it is not known like that in the literature, we will call the Bargmann-Mostow cohomology [Ba54, Ho77], the cohomology defined on a topological groupG by considering the cochain complex (C•

0(G,M), δ) of continuous cochains onG with values in M, where M is as before a topological G-module and δ is therestriction of the Eilenberg-MacLane cohomology operator to continuous cochains.We will denote such cohomology as H•

c (G,M) and it is trivial to see that if G isdiscrete, then:

H•(G,M) = H•b (G,M) = H•

c (G,M).

In general there are just natural morphisms between the three previous cohomologies(the sequence is not exact):

H •c (G,M)→ H•

b (G,M)→ H•(G,M).

However the most appealing cohomology studied by Bargmann is not the contin-uous cohomology but a local cohomology that we describe now. Let Cn

loc(G,M) bethe space of germs of continuous n-cochains from G toM, that is [ f ] ∈ Cn

loc(G,M)

is the equivalence class containing the continuous n-cochain f with respect to theequivalence relation, ‘ f ≡ h if there is a neighborhoodU of e such that f |U= h |U ’.Now it is easy to check that the Eilenberg-MacLane cohomology operator is compat-ible with the preceding equivalence relation, thus it induces an operator, denoted withthe same symbol, in C•

loc(G,M) =⊕n≥0 Cn

loc(G,M). We denote by H•loc(G,M)

the corresponding cohomology. It is remarkable that if G is a Lie group, on eachcocycle class [ f ] there exists a smooth representative f [Ba54]. Thus if G is a Liegroup, we may consider H•

loc(G,M) as the cohomology of the cochain complex ofgerms of smooth cochains from G toM.


F.2. Smooth Cohomologies on Lie groupsand de Rham Cohomology

Under the name of smooth cohomologies we put together all those cohomologiesdefined on Lie groups that use explicitly the smooth structure of the group. Thesecohomologies play an important role in various places along the main text, as theycharacterize either properties of the action of Lie groups on dynamical systems, orthey ‘measure’ the obstructions for some properties of the group to hold true. In anycase we are interested in differentiable objects and that is the main reason we restrictourselves to consider them.

We have already seen that H•loc(G,M) is basically a smooth cohomology (and

as we will see later on, it coincides with the cohomology of the Lie algebra of G).A natural extension of H•

loc(G,M) is to consider the cohomology defined on M-valued smooth cochains on G. This cohomology will be denoted as H•

diff(G,M) andis related to the structure of a maximal subgroup as we will discuss at the end of thisappendix.

From a completely different perspective, that is, not considering the Eilenberg-MacLane cochain complex on G, but starting afresh by considering that G is asmooth manifold, we may define the de Rham cohomology of G with values inM.Let�p(G,M) be the set ofM-valued smooth p–forms on G. In the graded algebra�∗(G,M) = ⊕

p≥0 �p(G,M) we define the differental operator of degree 1 as[Ch48] :

dαg(X1, . . . , Xk+1) =k+1∑i=1

(−1)i+1 d

dsρ(φ

(i)s (g))

((φ

(i)−s)

∗αg(X1, . . . , Xi , . . . , Xk+1))

+∑i< j

(−1)i+ jαg([Xi , X j ], X1, . . . , X, . . . , X j , . . . , Xk+1), (10.283)

where Xi ∈ X(G), i = 1, . . . , k + 1 and φ(i)s is the local flow of the vector field Xi

at the point g ∈ G. It can be shown that d2 = 0 (because it is the natural extension ofthe exterior differential in G), and its associated cohomology, denoted H•

dR(G,M),will be called the de Rham cohomology of G with values inM.

IfM = Rwith the trivialG-action, H•dR(G,R)becomes the standard deRhamco-

homology of G, isomorphic to the singular (topological) cohomology of G, H•(G).The complex �•(G,M) has various interesting subcomplexes: the subcomplex

of equivariant forms, the subcomplex of left- (or right-) invariant forms and that ofbi-invariant forms.

First of all notice that the group G acts on itself by left- (right-) translations,then we may consider the subcomplex of left- (right-) invariant forms denoted by�•L(G,M) (�•

R(G,M)).The group G acts also on M, then we may consider equivariant forms, that is

forms α satisfying g∗α = g · α, where g · α means ρ(g)(α(X1, . . . , Xk)) for anyvector fields Xi on G. Notice that if the form α is equivariant, then:

684 10 Appendices

d

dsρ(φ(i)

s (g))((φ

(i)−s)

∗αg(X1, . . . , Xi , . . . , Xk+1))= 0,

and formula (10.283) becomes simply:

dαg(X1, . . . , Xk+1) =∑i< j

(−1)i+ jαg([Xi , X j ], X1, . . . , X, . . . , X j , . . . , Xk+1),

(10.284)

It is clear that d maps equivariant forms into equivariant forms, thuswe have defined asub complex E•(G,M) =⊕

k≥0 Ek(G,M)where Ek(G,M), k ≥ 1, E0 =MG ,denotes the set of equivariant k-forms. The cohomology of the equivariant sub-complex will be denoted by H•

E (G,M).The cohomology operator d restricted to the subcomplex of left-invariant forms

�•L(G,M) becomes:

dα(X1, . . . , Xk+1) =k+1∑i=1

(−1)i+1ρ(ξ)(α(X1, . . . , Xi , . . . , Xk+1)

)

+∑i< j

(−1)i+ jα([Xi , X j ], X1, . . . , X, . . . , X j , . . . , Xk+1),

(10.285)

where ξi = Xi (e). The cohomology defined in the subcomplex of left-invariant formscoincide with the cohomology of the Lie algebra of G (see later Appendix F.3) andif the group G is compact, connected and M = R, then all the cohomologies wehave described coincide. The cohomology defined over the subcomplex �•

L (G,M)

will be denoted by H•L(G,M).

A relevant question at this point is how the smooth cohomologies H•loc(G,M)

and H•diff(G,M) are related with the de Rham like cohomologies H•

dR(G,M)

and H•L(G,M). We will see in the Appendix F.3 that H•

dR(G,M) is the sameas H •

L(G,M), but H•diff(G,M) is related to H•

L(G,M) by means of a maximalcompact subgroup of G.

F.3. Chevalley Cohomology of a Lie Algebra

Let us consider now a Lie algebra g and a representation ρ of g on a moduleM, thatis, a Lie algebras homomorphism ρ : g → End(M). We consider now the gradedalgebra of M–valued skew-symmetric forms on g, �•(g,M) = ⊕

k≥0 �k(g,M),with

�k(g,M) = �k(g)⊗M = {c : g× k· · · × g | c skew-symmetric}.


Let us define the graded operator of degree one dC : �k(g,M)→ �k+1(g,M) as:

dC c(ξ1, . . . , ξk+1) =k+1∑i=1

(−1)i+1ρ(ξi )c(ξ1, . . . , ξi , . . . , ξk+1)

+∑i< j

(−1)i+ j c([ξi , ξ j ], ξ1, . . . , ξ, . . . , ξ j , . . . , ξk+1),

(10.286)

for all ξi ∈ g, i = 1, . . . , k + 1. It can be checked that d2C = 0 and the cohomology

definedbydC is called theChevalley cohomologyofg associated to the representationρ with values in A and denoted byH•

C (g,M) [Ch48].If g is the Lie algebra of a Lie group G and ρ is the representation of g defined

by a representation ρ : G → Aut(M), then it is obvious that we may identify thecochains of the complex�•(g,M)with the left-invariant cochains on G with valueson M, �•

L(G,M). Then, the Chevalley operator dC is just the restriction of theexterior differential d on G and ξ L is the unique left-invariant vector field on Gdetermined by the Lie algebra element ξ . Then H∗

L (G.M) ∼= H•C(g,M).

Again, it is not hard to show that H•loc(G,M) is isomorphic to H •

C(g,M) becauseeach cochain f ∈ Hk

loc(G,M) defines a unique left-invariant k-form on G by simplytaking the differential at e of all factors. Then we get H•

loc(G,M) ∼= H •C (g,M)

[Ba54, Ho80].On the other hand, the global case is more complicated. Basically, the situation we

are facing is as follows. If G is a non-compact Lie group and K is a maximal compactsubgroup, then K is a strong retraction ofG and the deRhamcohomology ofG and Kare the same. Then it remains to analyze theEilenberg-MacLane cohomology comingfrom the quotient G/K . We may define relative cohomology groups Hn(G, K ,M)

and it can be shown that Hn(G, K ,M) ∼= Hndiff(G,M) [Ho80].

Along the main text, only the cohomologies H•C (g,M) and H∗(G.M) will be

used. In general, unless it is necessary to bemore specific,wewill omit the subindexesL , C and dR when writing the corresponding cohomology groups.

F.4. Cohomology Theory of Associative Algebras

In this section we wil introduce the basic notions of cohomology theory for associa-tive algebras and its relation with the deformation theory for associative algebras,introduced by Gerstenhaber [G64].

Let A be an associative algebra over a field K and M be a A-bimodule, i.e., Mis a module that is the carrier space for a linear representation � of A and a linearanti-representation � ′ of A that commute. The action of an element a ∈ A on anelement m ∈ M will be denoted either am or ma depending if we are consider theleft- or right action.

686 10 Appendices

A n-cochain on A with values in M, is a n-linear map α : A × · · · × A → M(n times). The set of such n-cochains,which can be regarded as an additive group,willbe denoted Cn(A,M), and for every n ∈ N we introduce the coboundary operator(compare with Eq. 10.282) δn : Cn(A,M)→ Cn+1(A,M), by means of [GH46]:

(δnα)(a1, . . . , an+1) := a1α(a2, . . . , an+1) +n∑

i=1(−1)iα(a1, . . . , ai ai+1, . . . , an+1)

+ (−1)n+1α(a1, . . . , . . . , an)an+1 .

For instance,whenn = 1weobtain (δα1)(a1, a2) = a1α1(a2)−α1(a1a2)+α1(a1)a2,and for n = 2,

(δα2)(a1, a2, a3) = a1α2(a2, a3)− α2(a1a2, a3)+ α2(a1, a2a3)− α2(a1, a2)a3 .

It is now an easy but cumbersome task to check that the linear maps δn satisfyδn+1 ◦ δn = 0.

We can form now the cochain complex C•(A,M) = ⊕n≥0 Cn(A,M) and the

natural extension of the operators δn to it, will allow us to define the correspondingcohomology H •(A,M) = ⊕

n≥0 H n(A,M), called the Hochschild cohomologyof the associative algebra A with coefficients inM, whose nth cohomology groupsis defined as H n(A,M) = Zn(A,M)/Bn(A,M), with Zn(A,M) the group ofn-cocycles, δnα = 0, and Bn(A,M) the subgroup of n-coboundaries, α = δn−1βwith β ∈ Cn−1(A,M).

A simplest example happens when M is the additive group of A itself, and thenthe A-module structure is given by left and right multiplication. In this case, ifF : A×A→ A is a bilinear map, it defines a 2-cochain and then

(δF)(a, b, c) = aF1(b, c)− F1(ab, c)+ F(a, bc)− F1(a, b)c .

and therefore δF = 0 reduces to the condition (10.288) below.

F.5. Deformation of Associative Algebras

Let R = K[λ] denote the ring of power series in one variable λ, and K = K (λ) thefield of power series of R. If V is the underlying linear space of A, let VK denotethe linear space obtained from V by extending the coefficient domain from K to Ki.e., VK = V ⊗K K . Let suppose that fλ : VK × VK → VK is a bilinear functionexpressible as

fλ(a, b) = F0(a, b)+ λ F1(a, b)+ λ2F2(a, b)+ · · · , (10.287)

where F0(a, b) = a ∗ b and Fk are bilinear functions over V .


The map fλ can be used to define an algebra structureAλ on VK . The conditionsfor fλ to be associative, i.e.

fλ(a, fλ(b, c)) = fλ( fλ(a, b), c) ,

are

k∑j=0

Fj (a, Fk− j (b, c)) =k∑

j=0Fj (Fk− j (a, b), c) ,

for any k, with the convention F0(a, b) = a ∗ b. Notice that when k = 0, it reducesto the associativity condition on A. For k = 1, the condition reads:

a ∗ F1(b, c)+ F1(a, b ∗ c) = F1(a ∗ b, c)+ F1(a, b) ∗ c , (10.288)

which can be written as δF1 = 0, i.e., in terms of Hochschild cohomology, thefunction F1, which is called infinitesimal deformation, is an element of the groupZ2(A,A) of 2-cocycles of A with coefficients on A.

Coming back to the new deformed algebra, even if the original algebra wasAbelian, the deformed one is, in general, no longer commutative, but

fλ(a, b)− fλ(b, a) = λ[F1(a, b)− F1(b, a)] + ϑ(λ2) ,

where ϑ(λ2) is of degree higher than one in λ, i.e.

limλ→0

1

λ

(ϑ(λ2)

)= 0 .

This allows us to define a new bilinear map in A, to be denoted {·, ·} by

{a, b} = limλ→0

1

λ[ fλ(a, b)− fλ(b, a)] = F1(a, b)− F1(b, a) ,

which, by construction, is a Lie algebra structure. Moreover,

{a, b ∗ c} = F1(a, b ∗ c)− F1(b ∗ c, a) , b ∗ {a, c} = b ∗ (F1(a, c)− F1(c, a)),

and

{a, b} ∗ c = (F1(a, b)− F1(b, a)) ∗ c ,

688 10 Appendices

we see that

{a, b ∗ c} − b ∗ {a, c} − {a, b} ∗ c = F1(a, b ∗ c)− F1(b ∗ c, a)

−b ∗ [F1(a, c)− F1(c, a)]−[F1(a, b)− F1(b, a)] ∗ c ,

and therefore, using that

b ∗ [F1(a, c)− F1(c, a)] = F1(b ∗ a, c)− F1(b, a ∗ c)+ F1(b, a) ∗ c

− F1(b ∗ c, a)+ F1(b, c ∗ a)− F1(b, c) ∗ a

we find that

{a, b ∗ c} − b ∗ {a, c} − {a, b} ∗ c = F1(a, b ∗ c)− F1(b ∗ c, a)

− [F1(b ∗ a, c)− F1(b, a ∗ c)+ F1(b, a) ∗ c

− F1(b ∗ c, a)+ F1(b, c ∗ a)− F1(b, c) ∗ a]− F1(a, b) ∗ c + F1(b, a) ∗ c ,

and simplifying terms and replacing

F1(a, b ∗ c)− F1(a, b) ∗ c = F1(a ∗ b, c)− a ∗ F1(b, c) ,

we will arrive to

{a, b ∗ c} − b ∗ {a, c} − {a, b} ∗ c = F1(a ∗ b, c)− a ∗ F1(b, c)

− F1(b ∗ a, c)+ F1(b, a ∗ c)

− F1(b, c ∗ a)+ F1(b, c) ∗ a ,

which can be reordered as

{a, b ∗ c} − b ∗ {a, c} − {a, b} ∗ c = F1(a ∗ b, c)− F1(b ∗ a, c)+ F1(b, c) ∗ a

−a ∗ F1(b, c)+ F1(b, a ∗ c)− F1(b, c ∗ a) ,

which clearly shows that if the original algebra is commutative, then

{a, b ∗ c} − b ∗ {a, c} − {a, b} ∗ c = 0 ,

i.e., the map {a, ·} is a derivation in the commutative and associative algebra (A, ∗).This leads us to introduce the general concept of Poisson bracket and Poisson algebra.

Definition 10.67 A Poisson algebra is a set A endowed with a commutative asso-ciative algebra (A,+, ·, ∗), and a Lie algebra structure defined by the compositionlaw {·, ·} such that


{a, b ∗ c} = b ∗ {a, c} + c ∗ {a, b} .

The element {a, b} is called the Poisson bracket of the elements a ∈ A and b ∈ A.

This means that the composition law {·, ·} defines a Lie algebra structure in thecommutative and associative algebra (A,+, ∗)

{a, b + λc} = {a, b} + λ{a, c}, {a, b} + {b, a} = 0 ,

and for any triple of elements of A

{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0 .

Given a Poisson algebra (A,+, · , ∗, {·, ·}), a Poisson subalgebra is a subset Bwhich is invariant under the composition law, and, therefore, when endowed withthe composition lawss which are the restrictions onto B of the corresponding onesin A is a Poisson subalgebra. A subalgebra I is said to be a Poisson ideal when

AI ⊂ I , {A, I} ⊂ I .

In such a case, the equivalence relacion defined by I is compatible with thecomposition laws and then A/I can be endowed with a Poisson algebra structure.

A linear map D : A→ A is said to be a derivation of the Poisson algebra when

D(a ∗ b) = (D A) ∗ b + a ∗ (Db) , D{a, b} = {Da, b} + {a, Db} ,

and the set of all derivations of a Poisson algebra is a Lie algebra when the followingcomposition law is defined:

[D1, D2] = D1 ◦ D2 − D2 ◦ D1 .

In particular, for each element a ∈ A, the map Da : A→ A defined by

Da(b) = {a, b} ,

is a derivation. Such derivations are called inner derivations.The elements a ∈ A of the Poisson algebra such that

{a, b} = 0 , ∀b ∈ A ,

are called Casimir elements of the Poisson algebra.The most interesting case is when A is the algebra of differentiable functions on

a vector space, or, even more generally, on a differentiable manifold.Let us now consider the case in which g is a Lie algebra rather than an associative

one. In similarity what we did with associative algebras, let g be a Lie algebra and a

690 10 Appendices

be a g-module, i.e., a is a module that is the carrier space for a linear representation� of g, i.e., � : g→ End a is such that

�(a)�(b)−�(b)�(a) = �([a, b]) ,

we will call n-cochain to a n-linear alternating mapping from g× . . .× g (n times)into a. We denote by Cn(g, a) the space of n-cochains. For every n ∈ N we defineδn : Cn(g, a)→ Cn+1(g, a) by Cariñena and Ibort [CI88]

(δnα)(a1, . . . , an+1) :=n+1∑i=1

(−1)i+1�(ai )α(a1, . . . , ai , . . . , an+1)

+∑i< j

(−1)i+ jα([ai , a j ], a1, . . . , ai , . . . , a j , . . . , an+1) ,

where ai denotes, as usual, that the element ai is omitted.In particular, if α : g→ a is a linear map, then

δα(a1, a2) = �(a1)α(a2)−�(a2)α(a1)− α([a1, a2]) ,

and if β ∈ C2(g, a),

δβ(a1, a2, a3) = �(a1)β(a2, a3)−�(a2)β(a1, a3)+�(a3)β(a1, a2)

−β([a1, a2], a3)+ β([a1, a3], a2)− β([a2, a3], a1) .

The linear maps δn satisfyδn+1 ◦ δn = 0 .

The proof is a simple but cumbersome checking.The linear operator δ on C(g, a) := ⊕∞

n=0 Cn(g, a) whose restriction to eachCn(g, a) is δn , satisfies δ2 = 0. We will then denote

Bn(g, a) := {α ∈ Cn(g, a) | ∃β ∈ Cn−1(g, a) such thatα = δβ} = Im δn−1 ,Zn(g, a) := {α ∈ Cn(g, a) | δα = 0} = ker δn .

The elements of Zn are called n-cocycles, and those of Bn are called n-coboundaries. Since δ2 = 0, we have Bn ⊂ Zn . The n-th cohomology groupHn(g, a) is defined as

H n(g, a) := Zn(g, a)

Bn(g, a),

and we will define B0(g, a) = 0, by convention.


In particular, a linear map α : g → a is a coboundary if there exists a ∈ a suchthat

α(a1) = �(a1)a ,

and β ∈ C2(g, a) is a cocycle (δ-closed) when

�(a1)β(a2, a3)−�(a2)β(a1, a3)+�(a3)β(a1, a2)− β([a1, a2], a3)

+ β([a1, a3], a2)− β([a2, a3], a1) = 0 ,

and a coboundary (δ-exact) when there exist a linear map α : g→ a such that

β(a1, a2) = �(a1)α(a2)−�(a2)α(a1)− α([a1, a2]) .

As a first example, let a := R, �(a) := 0 , ∀a ∈ g. We can also replace R forany trivial g-module. In this case, the operator δ reduces to

(δα)(a1, a2, . . . , an+1) =∑i< j

(−1)i+ jα([ai , a j ], a1, . . . , ai , . . . , a j , . . . , an+1) .

In particular, if α ∈ C1(g,R), then (δα)(a1, a2) = −α([a1, a2]). Thus Z1(g,R) isthe set

Z1(g,R) = {α ∈ g∗ | α([a1, a2]) = 0, ∀a1, a2 ∈ g } ,

and B1(g,R) = 0, by convention. If β ∈ C2(g,R), then β is a 2-cocycle if

β([a1, a2], a3)+ β([a2, a3], a1)+ β([a3, a1], a2) = 0 ,

and β ∈ B2(g,R) if there exists a linear map τ : g→ R such that

β(a1, a2) = τ([a1, a2]) .

As a second example we can consider the case in which g is the Lie algebraof vector fields, g = X(M), and the cohomology corresponding to its action onthe set of functions in M . In other words, we consider a := C∞(M) and define�(X) f := LX f , which is a linear representation of g, because of

[�(X) ◦�(Y )−�(Y ◦�(X)] f = (LXLY −LYLX ) f = L[X,Y ] f = �([X, Y ]) f.

The δ-operator takes the following form: if α ∈ Cn(g,C∞(M)),

692 10 Appendices

(δnα)(X1, . . . , Xn+1) =n+1∑i=1

(−1)i+1LXi α(X1, . . . , Xi , . . . , Xn+1)

+∑i< j

(−1)i+ jα([Xi , X j ], X1, . . . , Xi , . . . , X j , . . . , Xn+1) .

In particular, if α ∈ C1(g,C∞(M)),

(δα)(X, Y ) = LXα(Y )− LY α(X)− α([X, Y ]) .

The elements of B1(g,C∞(M)) are those α for which ∃β ∈ �p(M) with

α(X) = LXβ ,

while the elements of Z1(g,C∞(M)) are linear maps α : g→ C∞(M) satisfying

LXα(Y )− LY α(X) = α([X, Y ]) .

We have seen that the divergence of a vector field is the generalization of thetrace and that the set of all divergence-free vector fields is an infinite-dimensionalLie Algebra. We want to remark that the association of X(E) with F (E) given byX → div X , is a 1-cocycle, i.e.,

LXdiv Y − LY div X = divx [X, Y ] , (10.289)

and therefore, the set ker div ⊂ X(E) is the Lie subalgebra which generalizes isl(n).We notice that this time it is not possible to decompose a vector field like in (4.29).

However, it is possible to consider a new vector space E ×R, and a new volume

� = dx1 ∧ dx2 ∧ · · · ∧ dxn ∧ ds = � ∧ ds (10.290)

such that we can associate to any vector field X a new vector field

X = X + (div X) s∂

∂s,

which is divergence-free with respect to �. In fact,

L X � = (LX�) ∧ ds −� ∧ d((div X)s) = (LX�) ∧ ds − (div X)� ∧ ds

= −d((div X)�)) = 0.

Coming back to the theory of deformations, when the algebraA is a Lie algebra grather than an associative one, and we consider the one-parameter deformation givenby (10.287), the conditions for fλ = [·, ·]λ to define a Lie algebra structure,

http://dx.doi.org/10.1007/978-94-017-9220-2_4


fλ(a, b) = − fλ(b, a) ,

fλ( fλ(a, b), c)+ fλ( fλ(b, c), a)+ fλ( fλ(c, a), b) = 0 ,

are translated to

Fk(a, b) = −Fk(b, a) ,k∑

j=0Fj (Fk− j (a, b), c)+

k∑j=0

Fj (Fk− j (b, c), a)+k∑

j=0Fj (Fk− j (c, a), b) = 0 .

For k = 0 these conditions only reduce to the conditions for math f rakg to be aLie algebra, with respect to the composition law given by F0, i.e., [a, b] = F0(a, b),while for k = 1, we get

F1([a, b], c)+ F1([b, c], a)− F1([a, c], b)− [a, F1(b, c)] + [b, F1(a, c)]− [c, F1(a, b)] = 0 ,

(10.291)

together with F1(a, b) + F1(b, a) = 0, or in other words, when considering g asa g-module by means of the adjoint representation of g and F1 as an element ofC2(g, g), the conditions only mean that F1 is again a 2-cocycle, δF1 = 0.

For k = 2 we have

F0(F2(a, b), c)+ F0(F2(b, c), a)+ F0(F2(c, a), b)+ F1(F1(a, b), c)+ F1(F1(b, c), a)

+F1(F1(c, a), b)+ F2(F0(a, b), c)+ F2(F0(b, c), a)+ F2(F0(c, a), b) = 0 .

Very often we consider deformations in which Fk = 0 for k ≥ 2, i.e. (10.287)reduces to

[a, b]λ = [a, b] + t F1(a, b) , (10.292)

and in this case the condition for k = 2 is

F1(F1(a, b), c)+ F1(F1(b, c), a)+ F1(F1(c, a), b) = 0 ,

i.e., [a, b]1 = F1(a, b) also defines a Lie algebra structure. The condition (10.291)can be written as δF1 = 0, with δ denoting the coboundary operator in the complexwith coefficients in the adjoint representation.

In fact, in the casewe are considering, g can be considered as a g-module bymeansof the adjoint representation, i.e., � : g→ End g is given by �(a)(b) = [a, b], andthen, if β ∈ Cn(g, g), then

694 10 Appendices

δβ(a1, . . . , an+1) =n+1∑i=1

(−1)i+1[ai , β(a1, . . . , ai , . . . , an+1)]

+∑i< j

(−1)i+ jβ([ai , a j ], a1, . . . , ai , . . . , a j , . . . , an+1) .

In particular, a 1-cochain is given by a linear map A : g→ g. The coboundary ofsuch 1-cochain is

δA(a1, a2) = [a1, A(a2)] − [a2, A(a1)] − A([a1, a2]) .

Note that the linear map A is a derivation of the Lie algebra g if and only if δA = 0.The coboundary of a 2-cochain ζ : g× g→ g is

δζ(a1, a2, a3) = [a1, ζ(a2, a3)] − ζ([a1, a2], a3)+ [a2, ζ(a3, a1)]−ζ([a2, a3], a1)+ [a3, ζ(a1, a2)] − ζ([a3, a1], a2) .

Then, the Jacobi identity in the Lie algebra can be written δζ = 0 where ζ is justthe bilinear map defining the composition law in the Lie algebra, ζ(a, b) = [a, b].

The skew-symmetric bilinear map F1 : g× g→ g appearing in (10.287) definesa 2-cochain and

(δF1)(a, b, c) = [a, F1(b, c)] − [b, F1(a, c)] + [c, F1(a, b)]−F1([a, b], c)+ F1([a, c], b)− F1([b, c], a) .

Therefore the condition (10.291) can be written as δF1 = 0.

Definition 10.68 Adeformation Tλ is said to be trivial if there exists a linear operatorA such that Tλ = I + λ A and

Tλ[a, b]λ = [Tλa, Tλb] , ∀ a, b ∈ g ,

where [a, b]λ = [a, b]+λ[a, b]1, i.e., F1 in (10.292) is given by F1(a, b) = [a, b]1.Taking into account that

Tλ[a, b]λ = (I + λ A)([a, b] + λ [a, b]1) = [a, b] + λ (A[a, b] + [a, b]1)+ λ2 A[a, b]1 ,

and

[Tλa, Tλb] = [a+λ Aa, b+λ Ab] = [a, b]+λ([Aa, b]+ [a, Ab])+λ2[Aa, Ab] ,

we see that Tλ is a trivial deformation if and only if

[a, b]1 = [Aa, b] + [a, Ab] − A[a, b] , (10.293)

A[a, b]1 = [Aa, Ab] . (10.294)


The first condition can be expressed in terms of the coboundary operator in thecomplex with coefficients in the adjoint representation as follows:

F1 = δA .

We should also remark that using in the second relation the expression for [a, b]1given in the first one we obtain the following relation for the linear operator A:

[Aa, Ab] − A[Aa, b] − A[a, Ab] + A2[a, b] = 0 .

A linear map A satisfying this condition is called a Nijenhuis map.In general, the Nijenhuis torsion of A is defined as the (1,2)-tensor such that

NA(a, b) = [Aa, Ab] − A[Aa, b] − A[a, Ab] + A2[a, b] .

The linear map A is a Nijenhuis map when its Nijenhuis torsion vanishes.When such a condition is satisfied, the bilinear map [a, b]1 defines an alternative

Lie bracket, which is ususally denoted

[a, b]A = [Aa, b] + [a, Ab] − A[a, b] ,

and then the preceding condition can we written as

A([a, b]A) = [Aa, Ab] ,

i.e. A : (g, [·, ·]A)→ (g, [·, ·]A) is a homomorphism of Lie algebras.In fact, in order to prove the Jacobi identity for [·, ·]A, it suffices to note that for

any three elements, a1, a2, a3 ∈ g,

[a1, [a2, a3]A]A + [a3, [a1, a2]A]A + [a2, [a3, a1]A]A

= [A(a1), [A(a2), a3]]+ [A(a1), [a2, A(a3)]] − [A(a1), A ([a2, a3])] + [A(a3), [A(a1), a2]]

+[A(a3), [a1, A(a2)]] − [A(a3), A ([a1, a2])] + [A(a2), [A(a3), a1]]

+[A(a2), [a3, A(a1)]] − [A(a2), A ([a3, a1])] + [a1, A ([A(a2), a3])]+A ([a2, A(a3)])− A2 ([a2, a3])] + [a3, A ([A(a1), a2])]+ A ([a1, A(a2)])−A2 ([a1, a2])] + [a2, A ([A(a3), a1])]+ A ([a3, A(a1)])− A2 ([a3, a1])]

and using the Jacobi identity for [·, ·], we finally get

[a1, [a2, a3]A]A + [a3, [a1, a2]A]A + [a2, [a3, a1]A]A= [a3, NA(a1, a2)]+ [a2, NA)(a3, a1)]+ [a1, NA(a2, a3)] .

696 10 Appendices

Therefore, we see that if A is a Nijenhuis map, NA = 0, then [·, ·]A satisfiesJacobi identity and therefore it defines a new Lie algebra bracket. This is a sufficient,but not necessary, condition for δA to define a new Lie algebra bracket.

As indicated above, we remark that the vanishing of the Nijenhuis torsion of A,NA = 0 also implies that A : (g, [·, ·]A)→ (g, [·, ·]) is aLie algebra homomorphism,because

A ([a1, a2]A)− [A(a1), A(a2)] = −NA(a1, a2) = 0 .

In summary, the knowledge of a Nijenhuis map A allows us to define a new Liealgebra structure on g, such that the map A is a homomorphism of Lie algebrasA : (g, [·, ·]A)→ (g, [·, ·]).

A particularly important case is that of g = X(M), the Lie algebra of vector fieldson a manifold M . Then the linear maps are given by (1, 1)-tensor fields T in M .Given a (1, 1)-tensor field T , the Nijenhuis torsion of T , NT is defined by

NT (X, Y ) = T ([T (X), Y ] + [X, T (Y )])− T 2([X, Y ])− [T (X), T (Y )] ,

for any pair of vector fields X, Y ∈ X(M).A Nijenhuis structure on a manifold M is a (1, 1)-tensor field T with vanishing

Nijenhuis torsion,NT (X, Y ) = 0 ,

i.e.T ([T (X), Y ] + [X, T (Y )])− T 2([X, Y ])− [T (X), T (Y )] = 0 .

Note that, from the relations

(LT (X)T )(Y ) = [T (X), T (Y )] − T ([T (X), Y ]) ,

and(T ◦ LX T )(Y ) = T ([X, T (Y )] − T 2([X, Y ]) ,

we see that the condition for the (1, 1)-tensor field T to be a Nijenhuis tensor can bewritten as

T ◦ LX T = LT (X)T , ∀X ∈ X(M) .

A Nijenhuis structure allows us to define an alternative Lie algebra structure onX(M) with the new Lie algebra bracket

[X, Y ]T = [T (X), Y ]+ [X, T (Y )]− T ([X, Y ]) .

Moreover, as a consequence of the vanishing of NT , the linear map


T : (X(M), [·, ·]T )→ (X(M), [·, ·])

is a Lie algebra homomorphism.

F.6. Poisson Algebras and Deformation Quantization

As an instance of an abstract associative algebra, given a n × m matrix J , we canconsider an associative algebra in the set of m × n matrices by means of a newcomposition law

A ∗ B = AJB.

When m = n the matrix J is a square matrix and the composition law A ∗ B is analternative associative composition law in the set of square n × n matrices.

We can also consider not only one alternative structure but all a family dependingon a parameter λ, and the composition law will be written A ∗λ B. This problemwas recently studied [CGM00] and it may be relevant in the study of alternativequantum descriptions for a given system. For instance, in the linear space Mn ofn × n matrices the matrix J can be replaced by exp(λK ) with λ being an arbitraryparameter, and K any n × n matrix, and then

A ∗λ B = A eλK B .

Consequently, the Lie algebra structure associated to this new associative algebrastructure is

[A, B]λ = A ∗λ B − B ∗λ A ,

which also satisfies the compatibility condition

[A, B ∗λ C]λ = [A, B]λ ∗λ C + B ∗λ [A,C]λ .

Note that if we define the map of the algebra in itself given by

φλ(A) = e12λK A e

12λK ,

we see that φλ is a linear map φλ :Mn →Mn such that

φλ(A ∗λ B) = e12λK A eλK Be

12λK ,

698 10 Appendices

and then

φλ(A ∗λ B) = φλ(A) φλ(B) ,

i.e., φλ : (Mn, ∗λ) → (Mn, ·) is a homomorphism of associative algebras, and, inan analogous way, as

φλ([A, B]λ) = [φλ(A), φλ(B)] ,

φλ is also a Lie algebra homomorphism- The image of the neutral element isφλ(1) = eλK , and the neutral element for this new product is 1λ = e−λK . Notethat φλ(e−λK ) = 1.

Many of the algebras we will use are algebras of functions on some set. If weconsider a linear space E , the set of real functions defined in E is an algebra withthe usual point-wise composition laws. But the important remark is that it is possibleto show [RR84] that as far as a new associative local algebra structure (E, ∗), therewill exist a real function h in E such that

f ∗ g = h f g ⇐⇒ ( f ∗ g)(x) = h(x) f (x) g(x) .

Therefore wewill fix our attention in non-local algebras of functions for which theexpression of the product is given by an integral kernel depending on three variables:

( f ∗ g)(x) =∫E

K (x, y, z) f (y)g(z) dy dz .

Of course, in order to this product be associative the values of the kernels are notarbitrary but they should satisfy the following relation

∫E

K (x, y, z) K (z, t, u) dz =∫E

K (x, z, u) K (z, y, t) dz .

In fact,

[( f ∗ g) ∗ h](x) =∫

K (x, y, z)( f ∗ g)(y)h(z) dy dz

=∫

K (x, y, z) K (y, t, u) f (t) g(u) h(z) dy dz dt du ,

which, with the change of variables y → z, z → u, t → y, u → t , becomes

[( f ∗ g) ∗ h](x) =∫

K (x, z, u) K (z, y, t) f (y) g(t) h(u) dy dz dt du ,


while

[ f ∗ (g ∗ h)](x) =∫

K (x, y, z) f (y)(g ∗ h)(z) dy dz

=∫

K (x, y, z) K (z, t, u) f (y) g(t) h(u) dy dz dt du ,

and therefore the mentioned relation follows.This new product will be in general non commutative unless the following con-

dition be satisfied:

K (x, y, z) = K (x, z, y) .

A way of constructing an associative algebra structure is by transporting thestructure from an associative algebra bymeans of an invertible map. This mechanismhas been very often used in the formulation of Quantum Mechanics in phase space.The idea is to associate to a selfadjont operator A in a Hilbert space H a functionin the phase space, f A, which is usually called the symbol of the operator A, thiscorrespondence being one-to-one. This allows us to define a new composition lawin the space of such functions by

fA ∗ fB = f AB .

These twisted products can depend on a parameter, as indicated before. For in-stance, a remarkable example is the one giving raise to the so-called Moyal quanti-zation, which is defined on functions of a phase space as follows:

( f ∗ g) = exp [−λ D] ( f g) ,

where D( f g) is given by the rule

D( f g) = ∂ f

∂qi

∂g

∂ p j− ∂g

∂qi

∂ f

∂ p j= { f, g} .

Here λ is a parameter which in physical applications is related to the Planckconstant �.

In the most general case, a deformation of the associative and commutative point-wise product of functions will be given by

f ∗λ g = f g + λ F1( f, g)+ λ2F2( f, g)+ · · · .

The new deformed product is non-commutative, and the lowest degree term inthe difference of the deformed product defines a skew-symmetric bilinear function,

700 10 Appendices

{ f, g} = limλ→0

1

λ[ f ∗λ g − g ∗λ f ] = F1( f, g)− F1(g, f ) .

The new composition law is associative if the functions Fi satisfy some propertieswhich will be later established in full generality. For instance,

f1 F1( f2, f3)+ F1( f1, f2 f3) = F1( f1 f2, f3)+ F1( f1, f2) f3 .

In case in which this deformed algebra is non-commutative, we can construct thecorresponding Lie algebra and take the limit when λ goes to zero, obtaining in thisway a Lie algebra. It can also be shown that the bracket we have introduced satisfiesthe relation

{ f, g1 g2} = g1 { f, g2} + g2 { f, g1} ,

for any triple of functions, i.e., our bracket satisfies the properties of a Poisson bracket.See the next section after the digression for more details. As a consequence we havearrived in this way to a new structure which will play a relevant rôle as characterizingthe possible deformations of the algebra of functions. Next section is a digression ofa mathematical nature which can be skipped in a first lecture.

Appendix G: Differential Operators

G.1. Local Differential Operators

IfU is an open subset ofRn and Dk denotes the differential operator Dk = −i ∂/∂xk ,

we define, for every multi-index α = (α1, . . . , αn), αk ∈ N, k = 1, . . . , n: Dα =Dα11 · · · Dαn

n .

Definition 10.69 A differential operator P of order r is a linear map on the space ofcomplex-valued functions on U of the form Pu =∑

|α|≤r aα Dαu, u ∈ C∞(U ;C),whose coefficients aα ∈ C∞(U ;C), and where |α| denotes |α| = α1+· · ·+αr . Thesymbol of P is the function on U ×R

n given by p(x, ξ) =∑|α|≤r aα(x) ξα , where

ξα ≡ ξα11 · · · ξαn

n . The principal symbol of P is the function on U × Rn defined by

pr (x, ξ) =∑|α|=r aα(x) ξα .

Notice that if P is a differential operator of order r , then

(e−i t f Pei t f

)u =

r∑k=0

tr−k Pku ,

with Pk being a differential operator of order k which does not depend on t . Forinstance, P0 is just multiplication by the function p0(x) = P(x, ∂ f/∂x). In fact,from

Appendix G: Differential Operators 701

(e−i t f Dαei t f

)u =

(e−i t f D1ei t f

)α1 · · ·(

e−i t f Dnei t f)αn

u

and(e−i t f Dkei t f

)u = Dku + t ∂ f

∂xk, we see that

(e−i t f Dαei t f

)u =

(D1 + t

∂ f

∂x1

)α1

· · ·(

Dn + t∂ f

∂xn

)αn

u .

Consequently,(e−i t f Dαei t f

)u takes the above mentioned form and the coeffi-

cient of tr is the product (∂ f/∂x1)α1 · · · (∂ f/∂xn)αn . In particular, we can write(

e−i ξ ·x Pei ξ ·x) u = p(x, ξ) u.As an immediate consequence, if P and Q are differential operators with principal

symbols pr (x, ξ) and qs(x, ξ), then the principal symbol of P Q is the productpr (x, ξ) qs(x, ξ).

Let us observe that if u denotes the Fourier transform of the function u, i.e.

u(x) = 1

(2π)n

∫eix ·ξ u(ψ) dξ ,

then

Pu(x) = 1

(2π)n

∫eix ·ξ p(x, ξ )u(ψ) dξ ,

whenever the integral is meaningful.Note however that we are using the explicit form of the differential operators Dk

which depend very much of the choice for the coordinates or, in other terms, whenseen as generators of translations, they depend on the choice of a linear structureon R

n . There are however alternative ways of defining differential operators on Rn

which do not depend on such choices.Our first approach consists of considering the associative algebra of functions in

Rn together with the Lie algebra X(Rn) of vector fields in R

n . Then we can endowthe linear space X(Rn)⊕ C∞(Rn) with a Lie algebra structure by defining

[(X1, f1), (X2, f2)] = ([X1, X2], X1( f2)− X2( f1)) ,

usually called the ‘holomorph’ of C∞(Rn) and then, the enveloping algebra of suchLie algebra is the associative algebra of differential operators in R

n , that is thequotient algebra of the associative algebra generated by elements (X, f ) quotientedby the bilateral ideal generated by the relations (X, f )⊗ (Y, g)− (Y, g)⊗ (X, f )−[(X, f ), (Y, g)] for all (X, f ), (Y, g).

The differential operators of order zero are the only differential operators thatare not zero on constant functions and correspond to functions. The set D(Rn) of

702 10 Appendices

differential operators can be written asD(Rn) = C∞(Rn)⊕Dc(Rn), whereDc(R

n)

denotes the set of differential operators that are zero on constant functions.The preceding algebraic characterization allows for a definition of differential

operators on any subamnifold M ⊂ Rn . It would be enough to restrict ourselves to

consider only vector fields which are tangent to M instead of X(Rn), to use F(M)

instead of C∞(Rn) and consider the holomorph of F(M). The corresponding en-veloping algebra provides us the algebra of differential operators in the submanifoldM . In other words, consideringF(M) as an Abelian Lie algebra,D1(M) is the alge-bra of derivations X(M) and D(M) becomes what is known in the literature as the‘holomorph’ of F(M). In this way the algebra of differential operators becomes theenveloping algebra of the holomorph of F(M).

There is still another intrinsic way of defining the concept of differential operator.The idea again consists in replacing R

n by its algebra of differentiable functionsand the observation that the differential operators Dk are linear maps satisfying thefollowing property (see [Al91, Gr04]):

[Dk,m f

] = m Dk( f ) ,

where m f is the multiplication operation by f , i.e., the operation of degree zerom f (g) = f g, with f, g ∈ C∞(U ). It follows that if P is a differential operator ofdegree r of the form indicated, then

[P,m f ] =∑|σ |≤k

aσ

[Dk,m f

],

is of degree at most r−1. Iterating the previous formula for a set of r+1 functionsf0, · · · , fr ∈ C∞(U ), one finds that

[. . . [[P,m f0 ],m f1], . . . ,m fr ] = 0 ,

i.e., differential operators of degree no greater than r are such that for r+1 functionsf j ∈ C∞(U ) they satisfy the preceding identity. In particular the operators m f

are differential operators of order zero and the usual vector fields are differentialoperators of order one.

Then we may consider the algebra F(M) of differentiable functions in M asdifferential operators of order zero and then the differential operators P of degreeno greater than r > 0 are R-linear maps from F(M) into itself such that

[· · · [P,m f0 ],m f1 ], · · · ,m fr ] = 0 .

Usually one adds the locallity requirement that P does not increase the support, i.e.,supp(P f ) ⊂ supp( f ), ∀ f ∈ F(M).

The set of differential operators of degree not greater than r is both a left F(M)-module and a right F(M)-module and both actions commute, therefore, it is a bi-


module. We denote Dr (M) the set of differential operators of order at most r . Notethat Dp(M) ⊂ Dq(M) when p ≤ q, and that for any couple of nonnegative integernumbers,

Dp(M) ◦Dq(M) ⊂ Dp+q(M) , [Dp(M),Dq ](M)] ⊂ Dp+q−1(M) . (10.295)

This shows that D1(M) is a Lie subalgebra, while D0(M) is a subalgebra, andthe following sequence of Lie algebras is exact:

0→ D0(M)→ D1(M)→ DerF(M)→ 0

i.e.,D1(M) is a Lie algebra extension ofD1c(M) = X(M) = DerF(M) byD0(M).If we denote mx the ideal of functions vanishing on x , then differential operators

of degre not greater than k are alternatively characterized by the following property:P ∈ Dk(M) is a R-linear map P : F(M) → F(M) such that the image of mk+1

xlies in mx . In fact, given P ∈ Dk(M), then for any pair of functions f ∈ mx andg ∈ mk

x ,

P( f g) = f P(g)+ [P,m f ](g) ,

and on the right-hand side both terms are in mx , therefore P(mk+1x ) ⊂ mx . Con-

versely, asssume that P is a linear map such that P(mk+1x ) ⊂ mx , and consider first

the case k = 0. Then, for any g ∈ F(M) and x ∈ M , using the identity

(P − P(1))g = P(g − g(x))+ P(g(x))− P(1)(g − g(x))− P(1)g(x),

and taking into account that P(g(x)) = P(g(x) 1) = g(x)P(1) = P(1) g(x), itreduces to:

(P − P(1))g = P(g − g(x))− P(1)(g − g(x)) ,

and consequently (P− P(1))g ∈ mx , ∀x ∈ M , i.e., P = P(1). If k is different fromzero, the condition P(mk+1

x ) ⊂ mx says that if f ∈ F(M) and g ∈ mkx , then we

define f = f − f (x), and we can write

[P,m f ]g = P( f g)− f P(g) = P( f g)− f P(g) ,

and both terms on the right-hand side are in mx .The relations (10.295) also show that the set of differential operators on M , de-

noted as D(M), can be given the structure of a graded associative algebra and it isalso a bi-module over F .

704 10 Appendices

Differential Operators on Vector Bundles

Given two vector bundles over the same base, πa : Ea → M , a = 1, 2, the corre-sponding spaces of sections E1 and E2 are both F(M)-modules. The concept of dif-ferential operator is given by recurrence. AR-linear map P : �(E1)→ �(E2) is saidto be a differential operator of order zero. Therefore D0(E1, E2) = Hom(E1, E2).If k ≥ 1, we say that such a P is a differential operator of order no greater than kif for any function f ∈ F(M) the commutator [P,m f ] is a differential operator oforder no greater than k − 1, where m f denotes as usual the map of multiplying asection by the function f .

The set of differential operators of order ≤ k is a F(M)-module, to be denotedDk(E1, E2) and similarly, D(E1, E2) =⋃

k Dk(E1, E2).When E1 and E2 are the trivial bundle πa : M × R → M , a = 1, 2, then E1 =

E2 = F(M) and we recover the spaces D we mentioned before.There are other equivalent definitions of differential operators. For instance, P ∈

Dk(E1, E2), for k > 0, if and only if P is a linear map P : �(E1)→ �(E2), and forevery k + 1 functions fi ∈ F(M), i = 1, . . . , k + 1, such that fi (x) = 0, and everysection s : M → E1, we have P( f1 · · · fk+1s)(x) = 0. Or in another equivalentform, if for every function f such that f (x) = 0, and every section s : M → E1,we have that P( f k+1s)(x) = 0.

Example 10.8 1. Let M be a differentiable manifold and consider the vector bun-dles E1 = �r (T ∗M), E2 = �r+1(T ∗M). Then �(E1) is the set of r -forms inM while �(E2) is the set of (r + 1)-forms. The exterior dfferential acting onr -forms dr is a differential operator of order 1.

Note that given multi-indexes I = (i1, . . . , ir ), with 1 ≤ i1 < . . . < ir ≤dim M , and J = ( j1, . . . , jr+1) with 1 ≤ j1 < · · · < jr+1 ≤ n, then theoperator dr is given by

d

(∑I

f I dxi1 ∧ · · · ∧ dxir

)= i

∑J

(−1)k(Dk f j1··· jk ··· jr+1)dx j1∧· · ·∧dx jr+1 .

Consider now E1 = E2 = �•(T ∗M), then the exterior differential d is differ-ential operator of order 1.

2. Let be M be a differentiable manifold and E1 = �r (T ∗M) (with r > 1),E2 = �r−1(T ∗M) and X ∈ X(M) an arbitrary but fixedvector field.Contractionwith such vector field defines a differential operator iX of order 0 such that

iX(

fi1···ir dxi1 ∧ · · · ∧ dxir

) =∑k

(−1)k+1 fi1···ir dxi1 ∧ · · · ∧ dxk ∧ · · · ∧ dxir .

As before, we can define now a differential operator iX of order 0 acting onsections of the exterior bundle �•(T ∗M).


3. In the same vein, the Lie derivative LX is a differential operator of order 1 givenby Cartan’s formula: LXα = iX dα + diXα.

4. Pauli and Dirac operator.In non-relativistic mechanics the description of Stern-Gerlach experiment re-quires wave functions which have values in C

2 to take into account the spin de-grees of freedom. From our general point of view, thesewave functions, spinorialfunctions, may be considered as sections of a vector bundle with typical fibreC2. Thus the Hamiltonian operator and all other observables should be operators

acting on vector-valued wave functions to be able to extract expectation valuesout of measurements performed while the system is in these states.A convenient way to take into account the electron spin, is to consider the Hamil-ton operator represented by the following operator, called the Pauli Hamiltonian:

H = 1

2m(σ · P)2 +U (x, y, z) σ0 ,

where σ0, σ1, σ2, σ3, are Pauli matrices (Eq. 10.68).

The vector-valued operator P given by

P = p+ e

cA,

with p the standard momentum, and A being the vector potential of the externalelectromagnetic field B in which the electron is moving.

We thus have:

σ · p = �

i

⎛⎜⎝

∂

∂x3

∂

∂x1− i

∂

∂x2∂

∂x1+ i

∂

∂x2− ∂

∂x3

⎞⎟⎠ , σ · A = A1σ1 + A2σ2 + A3σ3.

These operators turn out to be finite-dimensional matrices whose entries aredifferential operators. They act on sections of vector bundles or on the tensorproduct of theHilbert space of complex-valued square integrable functions timesthe finite-dimensional Hilbert space describing the ‘internal structure’, in ourcase the spin degrees of freedom.Similarly, for the relativistically invariant Dirac equation one replaces σ ·p withthe quantity

γμ pμ = γ 0 p0 + γ 1 p1 + γ 2 p2 + γ 3 p3 ,

where (p0,p) is the four-momentum and (γ 0, γ 1, γ 2, γ 3) is a four-vector whosecomponents are matrices. These matrices (γ 0, γ ) transform like a vector, whilethe four-momentum transforms like a covector and then the total ‘scalar product’is Poincaré invariant. Here the γ -matrices may be represented as

706 10 Appendices

γ 0 =(1 00 1

), γ i =

(0 σ i

−σ i 0

), i = 1, 2, 3.

and the Dirac equation acquires the form

(γ 0 ∂

∂xo+ γ 1 ∂

∂x1+ γ 2 ∂

∂x2+ γ 3 ∂

∂x3

)ψ = imψ .

Remark 10.7 The matrices γμ are generators of a Clifford algebra and have thefollowing structure constants:

(γ 0)2 = 1 , (γ j )2 = −1 , j = 1, 2, 3

γ 0γ j + γ jγ 0 = 0 , γ jγ k + γ kγ j = 0 , j �= k .

Any product of a finite number of matrices may be expressed in the form

a0 1+∑

j

a j γj +

∑j<k

a jk γ jγ k +∑

j<k<l

a jkl γjγ kγ l + a0123 γ

0γ 1γ 2γ 3 .

By considering the left- and right-multiplication by functions onR4 we can construct

differential operators with coefficients given by the matrices generated by the γ ’s.

G.2. The Codifferential and the Laplace-Beltrami Operator

The Hodge-Star Operator

As a preliminary, we would like to extend the notion of the scalar product from1-forms to forms of higher rank. So, let us consider a manifold M with a (pseudo-)Riemannian metric g whose volume form is given by �.

Definition 10.70 The scalar product of two p-forms: α = 1p!αi1...i p dxi1∧ . . .∧dxi p

and: β = 1p!βi1···i p dxi1 ∧ · · · ∧ dxi p is given by

(α, β) = 1

p!αi1...i pβi1...i p (10.296)

where,βi1...i p = gi1 j1 . . . gi p jpβ j1··· jp (10.297)

In analogy with Eq. (10.297), the totally contravariant components of the volume-form can be defined as:


�i1,...,in = gi1 j1 . . . gin jn� j1,..., jn . (10.298)

Let β be a p-form. Then there exists a unique (n − p)-form ∗β such that,

α ∧ ∗β = (α, β)�, ∀α ∈ �p(M). (10.299)

The existence of ∗β follows from the explicit construction,

∗ β = 1

(n − p)! (∗β) j1··· jn−p dx j1 ∧ · · · ∧ dx jn−p , (10.300)

where,

(∗β) j1··· jn−p =1

p!�i1···i p, j1··· jn−pβi1···i p . (10.301)

The uniqueness follows in turn from the fact, whose proof is immediate, that if αis a p-form and γ an (n − p)-form, then, α ∧ γ = 0 ∀α implies γ = 0.

Definition 10.71 With the conventions before, the Hodge star operator is the linearisomorphism,

∗ : �p(M)→ �n−p(M) (10.302)

that associates with every p-form β the (n − p)-form ∗β.Proposition 10.72 The Hodge star operator satisfies the properties,

1. α ∧ (∗β) = β ∧ (∗α).2. (∗α, ∗β) = sgn(G)(α, β).3. ∗ ◦ ∗ = (−)p(n−p)sgn(G)Id�p on p–forms.4. ∗� = sgn (G).

where G = det(gi j ).

Exercise 10.9 The proof is left as an exercise on tensor algebra.

Example 10.10 Let us consider M = R3 with the standard Euclidean metric. As

(−1)p(n−p) = 1 and G = 1 then � = dx1 ∧ dx2 ∧ dx3 and:

1. ∗ f = f �2. If α = αi dxi , then ∗α = 1

2εi jk αi dx j ∧ dxk .3. If ω = 1

2ωi j dxi ∧ dx j , then ∗ω = 12εi jk ωi j dxk .

4. ∗(h dx1 ∧ dx2 ∧ dx3) = h.

708 10 Appendices

The Laplace-Beltrami Operator

The star operator and the exterior derivative can be combined to define a new differ-ential operator:

Definition 10.73 The codifferential δ is the differential operator whose action onp-forms is given by,

δ = (−1)n(p+1)+1sgn(G) ∗ d∗ (10.303)

Remark 10.8 The codifferential maps then p-forms into (p − 1)-forms, thus it is alinear map of degree −1 with respect to the natural grading of the exterior algebra�•(M) of forms.

Moreover δ is a differential operator of order 1 as it is easy to check. However,at variance with the exterior differential d, it is not a derivation on forms. Indeed, ascan be checked easily, δ does not obey the Leibnitz rule.

Proposition 10.74 The codifferential δ operator satisfies the following properties:

1. δ f = 0 on functions.2. δ2 = δ ◦ δ = 0.3. ∗δ d = d δ ∗; ∗ d δ = δ d ∗.4. ∗ δ ω = (−1)n+p+1d ∗ ω; ω ∈ �p.5. d ∗ δ = δ ∗ d = 0.6. δ ∗ ω = (−)n+p+1 ∗ dω; ω ∈ �p.

The proof of all of the above properties is elementary, and is left as an exercise.

The exterior differential d is ‘natural’with respect to diffeomorphisms, i.e.:φ∗d =dφ∗ for any φ ∈ Diff (M), a similar discussion was not done for the codifferential.We can pose now the same problem for the codifferential. Let’s use here a morecomplete notation, i.e. let’s denote by δg the codifferential associated with a givenmetric g, and let’s also denote with 〈·, ·〉g the global scalar product associated withthe same metric. Then:

Proposition 10.75 If φ ∈ Diff (M), then, φ∗δg = δφ∗gφ∗.

Indeed, it is easy to show that:

φ∗(⟨δgα, β

⟩g) = ⟨

φ∗δgα, φ∗β⟩φ∗g ,

but, on the other hand,

φ∗(⟨δgα, β

⟩g) = φ∗(〈α, dβ〉g) =

⟨φ∗α, dφ∗β

⟩φ∗g =

⟨δφ∗gφ

∗α, φ∗β⟩φ∗g .

Equating the right-hand side ’s we obtain the desired result.


It follows from the previous proposition that the codifferential δ is ‘natural’ onlywith respect to the diffeomorphisms that leave the metric invariant, i.e., such that:φ∗g = g. These are the isometries, and hence the codifferential is ‘natural’ withrespect to isometries.

Example 10.11 If F = 12 Fi j dxi ∧ dx j is a 2-form, then it is easy to prove that,

δF = (δF)i dxi (10.304)

where,

(δF)i = − 1√|G|gi j∂

∂xk

√|G|F jk (10.305)

A further differential operator that, as we shall shortly, is a convenient general-ization to forms of arbitrary rank of the more familiar Laplacian can be constructedby combining the actions of d and δ, and precisely:

Definition 10.76 The Hodge operator is the second-order differential operator� : �p → �p defined by,

� = (d + δ)2 ≡ dδ + δd = d ∗ d ∗ + ∗ d ∗ d. (10.306)

Restricted to functions, the Hodge operator becomes the familiar Laplace-Beltrami operator that, in local coordinates takes the explicit form:

� f = − 1√|G|∂

∂xi

√|G|gi j ∂

∂x jf. (10.307)

Vector Analysis in the Presence of a (Pseudo-)Riemannian Metric

Let g be a (pseudo-) Riemannian metric on M . Then the gradient of a functionf ∈ F(M) is the vector field,

grad( f ) = ∇ f = g−1(d f ) (10.308)

or, explicitly, grad( f ) = (grad( f ))i ∂∂xi , with (grad( f ))i = gi j ∂ f

∂x j .Let now � be the volume-form associated with g (actually we could do the same

with any other volume-form) and let X be a vector field.We recall that the divergenceof the vector field X is the unique function div (X) (or ∇ · X ) such that:

LX� = div(X) � (10.309)

710 10 Appendices

and that if � is the volume-form associated with an Euclidean metric in Rn , div(X)

becomes the familiar expression,

div(X) =∑

i

(∂Xi/∂xi ). (10.310)

Denoting now by αX the 1-form associated with X bymeans of g, i.e. αX = g(X),it is a simple exercise to show that,

iX� = ∗αX (10.311)

and we have furthermore:

divX = −δαX . (10.312)

Indeed, using the Cartan identity we get:

div (X)� = d (iX�) = d ∗ αX (10.313)

andEq. (10.312) follows then by taking the ‘star’ of both sides and usingEq. (10.303).

Exercise 10.12 Prove that, in local coordinates,

div (X) = 1√G

∂

∂xi

(√G Xi

)

and that, introducing the Christoffel symbols,

�ijk =

1

2gil

{∂gl j

∂xk+ ∂glk

∂x j− ∂g jk

∂xl

}

one has,

div X = ∂Xi

∂xi+ �i

ik Xk .

Let’s calculate now the function div(grad( f )). As it is obvious that αgrad( f ) = d f .Then we obtain,

− div(grad( f )) = δd f = � f, (10.314)

where � is the Laplace-Beltrami operator Eq. (10.307).

Example 10.13 (i) In R3 with the Euclidean metric,

� f = −∇2 f (10.315)


with ∇2 the usual Laplacian: ∇2 = ∑i ∂

2/∂(xi )2. The minus sign in the aboveequationmakes it clear that the Laplace-Beltrami operator is a positive (semi)definiteelliptic operator in this case.

(ii) In R4 with the Lorentz metric,

� f = −�2 f (10.316)

with �2 the D’Alambertian (or wave operator),

�2 = gi j ∂2

∂xi∂x j= ∂2

∂(x0)2−

3∑i=1

∂2

∂(xi )2(10.317)

Let’s specialize now the previous discussion to M a linear space of dimension 3.If X is a vector field, ∗dαX will be a 1-form, and because (−1)p(n−p) = 1. Then therotational of a vector field X is the vector field curl X defined by,

curl X = g−1(∗dαX ).

Exercise 10.14 Prove that, in coordinates,

∗ dαX = 1

2

√|G|εi jkgjlgkm

{∂Xl

∂xm− ∂Xm

∂xl

}dxi (10.318)

and that,

curl X = √|G|((

∂X2

∂x1 −∂X1

∂x2

)∂

∂x3 + cyclic permutations

)(10.319)

It is easy to check that:

αcurl X = g (curl X) = ∗dαX ,

and hence,

div(curl X) = −δ ∗ dαX ,

as well as,

g−1(δ ∗ d f ) = curl(grad( f )).

The familiar identities: ‘div · curl = 0’ and ‘curl · grad = 0’ of elementary vectoranalysis in R

3 all follow then from δ ∗ d = 0 (ultimately, from d2 = 0).Let now X have a vanishing curl. As both g and the star operator are isomorphisms,

712 10 Appendices

curl X = 0⇐⇒ dαX = 0,

and, byPoincare‘’s Lemma (which holds globally in the present context)αX = d f forsome f ∈ F(E). Similarly: div X = 0 implies δαX = 0 andhence,d∗αX = 0.Againby Poincare‘’s Lemma: ∗αX = dαY for some Y ∈ X(E) and hence, αX = ∗dαY . Itfollows then that the familiar theorems of vector analysis,

curl X = 0⇐⇒ X = grad( f )

and,

div X = 0⇐⇒ X = curl Y

are both consequences of the Poincare’ Lemma.

Exercise 10.15 Observing that α f X = f αX and using, d( f g) = g d f + f dg, anddα f X = f dαX + d f ∧ αX , prove that:

grad( f g) = f grad(g)+ g grad( f ),

and

curl( f X) = f curl X + grad( f )× X,

where the cross-product of any two vector fields X and Y , X × Y , is defined via,

αX×Y = ∗(αX ∧ αY ),

or,

X × Y = g−1(∗(αX ∧ αY )).

References

[Ja79] Jacobson, N.: Lie Algebras, No. 10. Dover Publications, New York (1979)[La85] E.A. Lacomba, A.B. Bugdor.: Circuits, Chain Complexes, and Symplectic Geometry. H.

Poincaré (1985)[Whit44] Whitney, H.: The self intersections of a smooth n-manifold in 2n-space. Ann. Math. 45,

220–246 (1944)[Ab78] Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin, Reading

(1978)[Ab88] Abraham R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis and Applications, 2nd

edn. Springer, New York (1988)[Ni55] Nijenhuis, A.: Jacobi-type identities for bilinear differential concomitants of certain ten-

sor fields I, II. Indag. Math. 17, 390–397 (1955); ibid. Indag. Math. 17, 397–403 (1955)

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[Ni64] Nijenhuis, A., Richardson, R.W.: Cohomology and deformations of algebraic structures.Bull. Amer. Math. Soc. 70, 406–411 (1964)

[Ni66] Nijenhuis, A., Richardson, R.W.: Cohomology and deformations in graded Lie algebras.Bull. Amer. Math. Soc. 72, 1–29 (1966)

[ML75] MacLane, S.: Homology. Graduate Texts of Math, vol. 114, p. 975. Springer, New York(1975)

[Mo64] Moore, C.: Extensions and low dimensional cohomology theory of locally compactgroups, I, II. Trans. Amer. Math. Soc. 113, 4063 (1964); ibid. Trans. Am. Math. Soc.136, 64–86 (1966)

[Mac57] Mackey, G.W.: Les ensembles Borliens et les extensions des groupes. J. Math. PuresAppl. 36, 171–178 (1957)

[Mac76] Mackey, G.W.: Induced representations of groups and quantum mechanics. Pubbl. dellaClasse de Scienze della Scuola Normale Superiore di Pisa (Published jointly with Ben-jamin Inc. and Editore Boringhiori) (1976)

[Ba54] Bargmann, V.: On unitary ray representations of continuous groups. Ann.Math. 59, 1–46(1954)

[Ho77] Houard, J.C.: On invariance groups and Lagrangian theories. J. Math. Phys. 18, 502–516(1977)

[Ch48] Chevalley, C., Eilenberg, S.: Cohomology theory of Lie groups and algebras. Trans.Amer. Math. Soc. 63, 85–124 (1948)

[Ho80] Houard, J.C.: An integral formula for cocycles of Lie groups. Ann. Inst. H. Poincaré32A, 221–247 (1980)

[G64] Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. 79, 59–103(1964)

[GH46] Hochschild, G.: On the cohomology theory for associative algebras. Ann. Math. 47,568–79 (1946)

[CI88] Cariñena, J.F., Ibort, L.A.: Noncanonical groups of transformations, anomalies and co-homology. J. Math. Phys. 29, 541–545 (1988)

[CGM00] Cariñena, J.F., Grabowski, J.,Marmo,G.: Quantumbi-Hamiltonian systems. Int. J.Mod.Phys. A 15, 4797–4810 (2000)

[RR84] Rubio, R.: Algèbres asociatives locales sur l’espace de sections d’un fibré en droites.C.R. Acad. Sci. Paris 299, 699–701 (1984)

[Al91] Alekseevskij, D.V., Vinogradov, A.M., Lycagin, V.V.E.: Geometry I: Basic ideas andconcepts of differential geometry. In: Gamkrelidze, R.V. (ed.). Springer (1991)

[Gr04] Grabowski, J., Poncin, N.: Automorphisms of quantum and classical Poisson algebras.Compos. Math. 140, 511–527 (2004); ibid., Lie algebraic characterization of manifolds.Cent. Eur. J. Math. 2, 811–825 (2004)

Index

SymbolsC∗-algebra, 467

imaginary elements, 467observables, 467pure state, 469real elements, 467state, 467vector states, 469

GL(E), 69, 615GL(n,C), 18, 19, 434GL(n,R), 86, 571H W (3), 222Hn(G,M), 681H•

C (g,M), 685H•dR(G,M), 683

O(2, 1), 28O(3), 28P(H), 443SL(2,R), 492SO(2, 1), 498SO(3,R), 594SO(p, q), 38SU (n), 19, 38U (H), 456CPn , 444D(M), 523D(Rn), 701D1

1 (H), 445F(X), 138Jl (E), 434Jl (E, �), 435JY , 139L↑

+, 358Aut(G), 612Diff (E), 87, 90End(E), 615Spec(R), 614

Sp(2n,R), 282Sp(ω), 282Spec

R(F), 141

1-formCartan, 341

2-form, 101Lagrangian, 342, 343

AAdmissible triple, 418Affine differentiable space, 142Algebra, 615

associative, 616Clifford, 705graded, 617Lie

graded, 617Poisson, 211tensor, 626

Almost complex structure, 435Anti-holomorphic, 431Antilinear, 424Atlas, 633

BBasis, 615Bi-unitary, 481Bimodule, 614Bracket

Jordan, 450Poisson, 450

Burgersequation, 493

© Springer Science+Business Media Dordrecht 2015J.F. Cariñena et al., Geometry from Dynamics, Classical and Quantum,DOI 10.1007/978-94-017-9220-2

715

716 Index

CCalogero-Moser

equations, 40Canonical coordinates, 297Canonical linear basis, 280Canonical momenta, 22Cauchy-Riemann operators, 432Cauchy-Schwartz inequality, 622Change of coordinates

holomorphic, 433linear, 71

Chart, 632compatible, 633domain, 632

Classical Hall conductivity, 12Coboundaries, 681Coccyges, 681Cochains, 681Cohomology

Bargmann-Mostow, 682Chevalley, 685de Rham, 683Eilenberg-MacLane, 681Hochschild, 686local, 682Mackey-Moore, 681

Coisotropic, 278Commutator, 69, 451, 616

graded, 617Compatible

symplectic, 415Compatible pair, 418Complete cotangent lift, 299Completely integrable, 3Complexification, 75Composition of independent motions, 7Contraction, 105Cotangent bundle, 98, 638, 642Covariant derivative, 160Cross sections, 97Curve

differentiable, 634Curves

tangent, 634

DDeformed

harmonic oscillator, 531Density states, 443, 445Derivation, 91, 616

inner, 616Differentiable space, 142

Differential equationBernoulli, 580first-order, 2

homogeneous, 3inhomogeneous, 5

second-order, 2Differential equations

second-order, 14Differential operator, 522, 700

principal symbol, 700symbol, 700

Dilationvector field, 98

Dimension, 615Dirac equation, 705Divergence, 105Dynamics

Lagrangian, 341Dyson’s formula, 81

EEasy tensorialization principle, 108Ehrenfest picture, 442Energy function, 345Equation

Heisenberg, 444Neumann, 443Schrödingier, 440

Equations of motionLorentz, 8

Euler differential operator, 98Events, 260Expectation values, 442Exterior differential, 101Exterior product, 100

FFactorizable system, 410Faraday tensor, 47Feynman’s problem, 247Field, 614Finite level

quantum, 420Flow, 2, 67Folium, 469Fréchet space, 140Free fall, 7Free motion, 304Frequency

cyclotron, 9Larmor, 9

Index 717

proper, 4Function

group, 497linear, 82

GGef’fand transform, 141Generating function, 283Generic position, 428Geodesic spray, 304Germs, 634GNS construction, 463Gradient

vector field, 456Group, 69, 328, 349, 351, 352, 354–358,

380, 611Abelian, 611affine, 570, 590cyclic, 19diffeomorphisms, 87epimorphism, 612Euclidean, 127Galilei, 126general linear, 615general linear group, 18Heisenberg-Weyl, 221homomorphism, 612isomorphism, 612Lorentz, 358monomorphism, 612orthogonal, 28paths, 570Poincaré, 261pseudorthogonal, 28quotient, 612rotation, 226, 495special unitary, 19sub-, 612transformations, 87

Group actionleft, 583

Group representation, 148

HHamilton equations, 22Hamilton’s equations, 297Hamiltonian

function, 22Pauli, 704

Harmonicoscillator, 524

Harmonic oscillator, 16, 18, 20, 27, 29, 174,324, 325, 347, 352–358, 376, 379,380, 429, 479, 510, 519, 531, 554

1D, 201isotropic, 4

Heisenberg equation, 474Heisenberg picture, 440Hermitean structure, 416

Fubini-Study, 454Hessian matrix, 22Hilbert space, 441Holomorph, 523, 701Holomorphic, 431Holomorphic map, 433Homomorphism

module, 615ring, 613

Hopf vibration, 20, 556

IIdeal, 613

bilateral, 613Gelfand, 468left, 613maximal, 613right, 613

IdentityJacobi, 616Leibnitz, 616

Impure state, 447Indicial functions, 497Inner product, 416Invariant

J , 417Inverse Hermitean problem, 419Inverse problem

Lagrangian, 21Inverse symplectic problem, 419Isotropic, 278

harmonic oscillator, 496Isotropy

group, 420

JJacobi identity, 91, 211Jordan bracket, 451Jordan-Schwinger map, 470

KKähler manifold, 436Kählerian function, 438

718 Index

Killingvector field, 455

Kinetic energy, 303

LLagrangian, 21, 278, 279, 341, 343, 346–

353, 355–358, 374–383, 496,500, 503, 504, 653

regular, 22, 342singular, 342

Lagrangian description, 347Leibniz identity, 211Leibniz’s rule, 91Level surface

regular, 630Levi-Civita, 340Lie algebra, 19, 69, 211, 240, 535, 616, 695Lie algebra cohomology, 286Lie algebras, 695Lie derivative, 106Lie-Poisson

Poisson, 308Linear, 321, 322, 325, 328, 336, 337, 345,

354–357, 377, 382space, 64symmetries, 18symmetry, 72system, 65

Linear mapsymplectic, 279

Linear operators, 472Linear spaces

topological, 619Liouville

vector field, 98, 156Local one-parameter

group, 94Lorentz group, 261

proper othocronous, 358

MManifold

complex, 435Matrix

nilpotent, 3symplectic, 23

Maximallysuperintegrable, 551

Mechanical systems, 303Minkowski space-time, 260Module

free, 615

left, 614right, 614sub-, 614

NNatural tensorialization, 165

OObservables, 439Operator

Hodge, 709Laplace-Beltrami, 709

PParallelizable

manifold, 308Phase space, 22Poincaré group, 260Poisson, 346, 381–383

algebra, 210bracket, 22, 210structure, 193, 216tensors, 211, 215

Poisson bracket, 23, 238, 295, 437, 472Poisson structure, 222Poisson tensor

localizable, 368Potential energy, 303Projective Hilbert space, 445Pure states, 444Purification, 447

QQuantum, 84, 472, 474–476, 479, 485, 528,

529, 697, 699

RRank, 214Realification, 74Realization

symplectic, 306Reciprocal function

group, 497Regular submanifold, 140Regular value, 630Reparametrized

harmonic oscillator, 34Representation

linear, 595Resolvent, 81

Index 719

Riccati transformation, 492Riemannian manifold, 304Ring, 612

commutative, 612division, 614sub-, 612unit element, 612

SSchrödinger picture, 440Second order

vector fields, 321, 495Second-order differential equations, 321Semidirect product, 263Space

linear, 615tangent, 634vector, 615

Spectrum, 141maximal, 614

Stablelinear, 420

States, 2quantum, 441

Strong Whitney topology, 140Structure constants, 616Subgroup

normal, 612Submanifold, 115

regular, 633Subspace

Lagrangian, 287linear, 615

Superintegrable, 551Superposition rule, 580

nonlinear, 574Supremum norm, 139Symmetry, 12, 186

infinitesimal, 195linear, 186

Symplectic, 208, 209, 275, 289, 305, 346,429, 665

action, 497algebra, 286conjugation, 287form, 286, 382inverse problem, 382isomorphism, 279, 282leaf, 236linear space, 273polar, 276rank, 287

realization, 497reduction, 498structures, 193, 410, 472subspace, 278transformation, 287transvections, 287

Symplectomorphism, 279, 282System

linear, 71

TTame, 411Tangent bundle, 97, 323, 324, 326, 344, 375,

380, 382, 497, 498, 507, 550, 678Tangent bundle structure, 323Time-ordered exponential, 81Trace topology, 139Transformation

Newtonian, 14point, 14

TransformationsNewtonoid, 14

Transition functions, 632Type II generating function, 283

UUnitary

group, 420

VVariables

action-angle, 3Vector

cyclic, 468tangent, 634

Vector bundle, 214, 321, 322, 549, 550Vector field, 90, 91, 321–323, 326–332, 336,

337, 340, 342, 343, 345, 346, 348,349, 351–356, 374, 375, 377, 378,380–382, 555

Euler, 491Hamiltonian, 212linear, 185, 186, 192, 429phase, 453

Velocitydrift, 11

WWave functions, 440

Geometry from Dynamics, Classical and Quantum

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