Contact Geometry and Non-linear Differential …...61 H. Groemer Geometric Applications of Fourier...

Contact Geometry and Nonlinear Differential Equations

Methods from contact and symplectic geometry can be used to solve highly non-trivialnon-linear partial and ordinary differential equations without resorting to approximatenumerical methods or algebraic computing software. This book explains how it’s done.It combines the clarity and accessibility of an advanced textbook with the completenessof an encyclopedia. The basic ideas that Lie and Cartan developed at the end of thenineteenth century to transform solving a differential equation into a problem ingeometry or algebra are here reworked in a novel and modern way. Differentialequations are considered as a part of contact and symplectic geometry, so that all themachinery of Hodge–de Rham calculus can be applied. In this way a wide class ofequations can be tackled, including quasi-linear equations, Monge–Ampère equations(which play an important role in modern theoretical physics and meteorology).

The main features of the book are geometric transparency, clear and almostimmediate applications to interesting problems, and exact solutions clarifying howapproximate numerical solutions can be better obtained. The types of problemconsidered range from the classical (e.g., Lie’s classification probelm) to the analysisof laser beams or the dynamics of cyclones. The authors balance rigor with the need tosolve problems, so it will serve as a reference and as a user’s guide.

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Editorial BoardP. Flajolet M. Ismail E. LutwakVolume 101

Contact Geometry and Nonlinear Differential Equations

All the titles listed below can be obtained from good booksellers or from Cambridge University Press.

For a complete series listing visit http://www.cambridge.org/uk/series/

60 J. Krajicek Bounded Arithmetic, Propositional Logic, and Complexity Theory61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory63 A. C. Thompson Minkowski Geometry64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications65 K. Engel Sperner Theory66 D. Cvetkovic, P. Rowlinson and S. Simic Eigenspaces of Graphs67 F. Bergeron, G. Labelle and P. Leroux Combinatorial Species and Tree-Like Structures68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2nd edn70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry71 G. E. Andrews, R. Askey and R. Roy Special Functions72 R. Ticciati Quantum Field Theory for Mathematicians73 M. Stern Semimodular Lattices74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II76 A. A. Ivanov Geometry of Sporadic Groups I77 A. Schinzel Polymomials with Special Regard to Reducibility78 H. Lenz, T. Beth, and D. Jungnickel Design Theory II, 2nd edn79 T. Palmer Banach Algebras and the General Theory of *-Algebras II80 O. Stormark Lie’s Structural Approach to PDE Systems81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets83 C. Foias, O. Manley, R. Rosa and R. Temam Navier–Stokes Equations and Turbulence84 B. Polster and G. Steinke Geometries on Surfaces85 R. B. Paris and D. Kaminski Asymptotics and Mellin–Barnes Integrals86 R. McEliece The Theory of Information and Coding, 2nd edn87 B. Magurn Algebraic Introduction to K-Theory88 T. Mora Solving Polynomial Equation Systems 189 K. Bichteler Stochastic Integration with Jumps90 M. Lothaire Algebraic Combinatorics on Words91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II92 P. McMullen and E. Schulte Abstract Regular Polytopes93 G. Gierz et al. Continuous Lattices and Domains94 S. Finch Mathematical Constants95 Y. Jabri The Mountain Pass Theorem96 G. Gasper and M. Rahman Basic Hypergeometric Series, 2nd edn97 M. C. Pedicchio and W. Tholen (eds.) Categorical Foundations98 M. E. H. Ismail Classical and Quantum Orthogonal Polynomials in One Variable99 T. Mora Solving Polynomial Equation Systems II

100 E. Olivieri and M. Eulália Vares Large Deviations and Metastability101 A. Kushner, V. Lychagin and V. Rubtsov Contact Geometry and Nonlinear Differential Equations102 L.W. Beineke, R. J. Wilson, P. J. Cameron. (eds.) Topics in Algebraic Graph Theory103 O. Staffans Well-Posed Linear Systems104 J. M. Lewis, S. Lakshmivarahan and S. Dhall Dynamic Data Assimilation105 M. Lothaire Applied Combinatorics on Words

http://www.cambridge.org/uk/series/

Contact Geometry and Non-linearDifferential Equations

ALEXEI KUSHNER, VALENTIN LYCHAGIN ANDVLADIMIR RUBTSOV

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 2RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521824767

© Cambridge University Press 2007

This publication is in copyright. Subject to statutory exception and to the provisions of relevantcollective licensing agreements, no reproduction of any part may take place without the written

permission of Cambridge University Press

First published 2007

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

ISBN-13 978-0-521-82476-7 hardbackISBN-10 0-521-82476-1 hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls forexternal or third-party internet websites referred to in this publication, and does not guarantee

that any content on such websites is, or will remain, accurate or appropriate.

http://www.cambridge.org

http://www.cambridge.org/9780521824767

Contents

Preface xiii

Part I Symmetries and Integrals 1

1 Distributions 31.1 Distributions and integral manifolds 3

1.1.1 Distributions 31.1.2 Morphisms of distributions 41.1.3 Integral manifolds 5

1.2 Symmetries of distributions 111.3 Characteristic and shuffling symmetries 151.4 Curvature of a distribution 181.5 Flat distributions and the Frobenius theorem 201.6 Complex distributions on real manifolds 231.7 The Lie–Bianchi theorem 24

1.7.1 The Maurer–Cartan equations 241.7.2 Distributions with a commutative symmetry

algebra 271.7.3 Lie–Bianchi theorem 30

2 Ordinary differential equations 322.1 Symmetries of ODEs 32

2.1.1 Generating functions 322.1.2 Lie algebra structure on generating functions 372.1.3 Commutative symmetry algebra 38

2.2 Non-linear second-order ODEs 402.2.1 Equation y′′ = y′ + F(y) 432.2.2 Integration 462.2.3 Non-linear third-order equations 48

vi Contents

2.3 Linear differential equations and linear symmetries 502.3.1 The variation of constants method 502.3.2 Linear symmetries 51

2.4 Linear symmetries of self-adjoint operators 542.5 Schrödinger operators 56

2.5.1 Integrable potentials 582.5.2 Spectral problems for KdV potentials 652.5.3 Lagrange integrals 73

3 Model differential equations and theLie superposition principle 763.1 Symmetry reduction 76

3.1.1 Reductions by symmetry ideals 763.1.2 Reductions by symmetry subalgebras 77

3.2 Model differential equations 783.2.1 One-dimensional model equations 803.2.2 Riccati equations 82

3.3 Model equations: the series Ak , Dk , Ck 833.3.1 Series Ak 833.3.2 Series Dk 863.3.3 Series Ck 87

3.4 The Lie superposition principle 893.4.1 Bianchi equations 92

3.5 AP-structures and their invariants 943.5.1 Decomposition of the de Rham complex 943.5.2 Classical almost product structures 963.5.3 Almost complex structures 983.5.4 AP-structures on five-dimensional manifolds 98

Part II Symplectic Algebra 101

4 Linear algebra of symplectic vector spaces 1034.1 Symplectic vector spaces 103

4.1.1 Bilinear skew-symmetric forms on vector spaces 1034.1.2 Symplectic structures on vector spaces 1044.1.3 Canonical bases and coordinates 107

4.2 Symplectic transformations 1084.2.1 Matrix representation of symplectic

transformations 110

Contents vii

4.3 Lagrangian subspaces 1134.3.1 Symplectic and Kähler spaces 117

5 Exterior algebra on symplectic vector spaces 1195.1 Operators ⊥ and � 1195.2 Effective forms and the Hodge–Lepage theorem 125

5.2.1 sl2-method 132

6 A symplectic classification of exterior 2-forms indimension 4 1356.1 Pfaffian 1356.2 Normal forms 1376.3 Jacobi planes 142

6.3.1 Classification of Jacobi planes 1436.3.2 Operators associated with Jacobi planes 145

7 Symplectic classification of exterior 2-forms 1477.1 Pfaffians and linear operators associated with

2-forms 1477.2 Symplectic classification of 2-forms with distinct real

characteristic numbers 1497.3 Symplectic classification of 2-forms with distinct

complex characteristic numbers 1527.4 Symplectic classification of 2-forms with multiple

characteristic numbers 1547.5 Symplectic classification of effective 2-forms in

dimension 6 160

8 Classification of exterior 3-forms on a six-dimensionalsymplectic space 1628.1 A symplectic invariant of effective 3-forms 162

8.1.1 The case of trivial invariants 1658.1.2 The case of non-trivial invariants 1678.1.3 Hitchin’s results on the geometry of

3-forms 1738.2 The stabilizers of orbits and their prolongations 175

8.2.1 Stabilizers 1758.2.2 Prolongations 178

viii Contents

Part III Monge–Ampère Equations 181

9 Symplectic manifolds 1839.1 Symplectic structures 183

9.1.1 The cotangent bundle and the standardsymplectic structure 184

9.1.2 Kähler manifolds 1869.1.3 Orbits and homogeneous symplectic spaces 187

9.2 Vector fields on symplectic manifolds 1899.2.1 Poisson bracket and Hamiltonian vector fields 1899.2.2 Canonical coordinates 191

9.3 Submanifolds of symplectic manifolds 1929.3.1 Presymplectic manifolds 1929.3.2 Lagrangian submanifolds 1949.3.3 Involutive submanifolds 1979.3.4 Lagrangian polarizations 198

10 Contact manifolds 20110.1 Contact structures 201

10.1.1 Examples 20210.2 Contact transformations and contact vector fields 208

10.2.1 Examples 20910.2.2 Contact vector fields 215

10.3 Darboux theorem 21910.4 A local description of contact transformations 221

10.4.1 Generating functions of Lagrangiansubmanifolds 221

10.4.2 A description of contact transformations in R3 222

11 Monge–Ampère equations 22411.1 Monge–Ampère operators 22411.2 Effective differential forms 22611.3 Calculus on �∗(C∗) 23011.4 The Euler operator 23311.5 Solutions 23611.6 Monge–Ampère equations of divergent type 241

12 Symmetries and contact transformations of Monge–Ampèreequations 24312.1 Contact transformations 243

Contents ix

12.2 Lie equations for contact symmetries 25112.3 Reduction 25612.4 Examples 259

12.4.1 The boundary layer equation 25912.4.2 The thermal conductivity equation 26112.4.3 The Petrovsky–Kolmogorov–Piskunov

equation 26212.4.4 The Von Karman equation 264

12.5 Symmetries of the reduction 26712.6 Monge–Ampère equations in symplectic geometry 270

13 Conservation laws 27313.1 Definition and examples 27313.2 Calculus for conservation laws 27413.3 Symmetries and conservations laws 27913.4 Shock waves and the Hugoniot–Rankine

condition 28013.4.1 Shock Waves for ODEs 28013.4.2 Discontinuous solutions 28113.4.3 Shock waves 283

13.5 Calculus of variations and the Monge–Ampèreequation 28513.5.1 The Euler operator 28513.5.2 Symmetries and conservation laws in

variational problems 28613.5.3 Classical variational problems 287

13.6 Effective cohomology and the Euler operator 288

14 Monge–Ampère equations on two-dimensionalmanifolds and geometric structures 29414.1 Non-holonomic geometric structures associated with

Monge–Ampère equations 29514.1.1 Non-holonomic structures on contact

manifolds 29514.1.2 Non-holonomic fields of endomorphisms

on generated by Monge–Ampèreequations 295

14.1.3 Non-degenerate equations 29814.1.4 Parabolic equations 302

x Contents

14.2 Intermediate integrals 30414.2.1 Classical and non-holonomic intermediate

integrals 30414.2.2 Cauchy problem and non-holonomic

intermediate integrals 30714.3 Symplectic Monge–Ampère equations 308

14.3.1 A field of endomorphisms Aω on T∗M 30814.3.2 Non-degenerate symplectic equations 31014.3.3 Symplectic parabolic equations 31214.3.4 Intermediate integrals 313

14.4 Cauchy problem for hyperbolic Monge–Ampèreequations 31314.4.1 Constructive methods for integration of

Cauchy problem 314

15 Systems of first-order partial differential equations ontwo-dimensional manifolds 31815.1 Non-linear differential operators of first order on

two-dimensional manifolds 31915.2 Jacobi equations 32115.3 Symmetries of Jacobi equations 32815.4 Geometric structures associated with

Jacobi’s equations 33015.5 Conservation laws of Jacobi equations 33215.6 Cauchy problem for hyperbolic Jacobi equations 334

Part IV Applications 337

16 Non-linear acoustics 33916.1 Symmetries and conservation laws of the KZ equation 340

16.1.1 KZ equation and its contact symmetries 34016.1.2 The structure of the symmetry algebra 34216.1.3 Classification of one-dimensional subalgebras of

sl(2, R) 34516.1.4 Classification of symmetries 34716.1.5 Conservation laws 348

16.2 Singularities of solutions of the KZ equation 34916.2.1 Caustics 34916.2.2 Contact shock waves 351

Contents xi

17 Non-linear thermal conductivity 35617.1 Symmetries of the TC equation 356

17.1.1 TC equation 35617.1.2 Group classification of TC equation 357

17.2 Invariant solutions 363

18 Meteorology applications 37118.1 Shallow water theory and balanced dynamics 37218.2 A geometric approach to semi-geostrophic theory 37418.3 Hyper-Kähler structure and Monge–Ampère

operators 37618.4 Monge–Ampère operators with constant

coefficients and plane balanced models 380

Part V Classification of Monge–Ampèreequations 383

19 Classification of symplectic MAOs on two-dimensionalmanifolds 38519.1 e-Structures 38619.2 Classification of non-degenerate Monge–Ampère

operators 38819.2.1 Differential invariants of non-degenerate

operators 38819.2.2 Hyperbolic operators 39219.2.3 Elliptic operators 401

19.3 Classification of degenerate Monge–Ampèreoperators 40619.3.1 Non-linear mixed-type operators 40619.3.2 Linear mixed-type operators 416

20 Classification of symplectic MAEs on two-dimensionalmanifolds 42220.1 Monge–Ampère equations with constant

coefficients 42220.1.1 Hyperbolic equations 42320.1.2 Elliptic equations 42520.1.3 Parabolic equations 426

20.2 Non-degenerate quasilinear equations 42820.3 Intermediate integrals and classification 429

xii Contents

20.4 Classification of generic Monge–Ampère equations 43020.4.1 Monge–Ampère equations and e-structures 43020.4.2 Normal forms of mixed-type equations 436

20.5 Applications 44020.5.1 The Born–Infeld equation 44020.5.2 Gas-dynamic equations 44220.5.3 Two-dimensional stationary irrotational

isentropic flow of a gas 445

21 Contact classification of MAEs on two-dimensionalmanifolds 44721.1 Classes Hk,l 44721.2 Invariants of non-degenerate Monge–Ampère equations 454

21.2.1 Tensor invariants 45421.2.2 Absolute and relative invariants 456

21.3 The problem of contact linearization 45921.4 The problem of equivalence for non-degenerate

equations 46421.4.1 e-Structure for non-degenerate equations 46421.4.2 Functional invariants 470

22 Symplectic classification of MAEs on three-dimensionalmanifolds 47222.1 Jets of submanifolds and differential equations on

submanifolds 47322.2 Prolongations of contact and symplectic manifolds and

overdetermined Monge–Ampère equations 47622.2.1 Prolongations of symplectic manifolds 47622.2.2 Prolongations of contact manifolds 479

22.3 Differential equations for symplectic equivalence 482

References 487

Index 493

Preface

xiii

xiv Preface

The aim of this book is to introduce the reader to a geometric study of partialdifferential equations of second order.

We begin the book with the most classical subject: the geometry of ordinarydifferential equations, or more general, differential equations of finite type. Themain item here is the various notions of symmetry and their use in solving a givendifferential equation. In Chapter 1 we discuss the distributions, integrability andsymmetries. In a form appropriate to our aims, we remind the reader of the mainnotions of the geometry of distributions: complete integrability, curvature, integ-ral manifolds and symmetries. The Frobenius integrability theorem is presentedin its geometric form: as a flatness condition for the distribution.

The main result of this chapter is the famous Lie–Bianchi theorem whichgives a condition and an constructive algorithm for integrability in quadrat-ures of a distribution in terms of a Lie algebra of the shuffling symmetries.The theorem clearly explains the etymology of the expression “solvable Liealgebra.”

In Chapter 2 we apply these results to explicit integration of scalar ordinarydifferential equations. We consider some standard examples of differential equa-tions integrable in quadratures but treat them in quite non-standard geometricway to demonstrate the advantage of the language and the method of symmet-ries. Even in the case of linear differential equations one is able to find some newand interesting results by systematically exploiting the notion of symmetries.The most instructive illustration of this methodology is the application of thelinear symmetries of (skew) self-adjoint linear operators. The space of linearsymmetries admits in this case the structure of a Lie superalgebra. For example,the even part of the linear symmetries for the Schrödinger operator L = ∂2+Wis isomorphic to the Lie algebra sl2, and the generating functions of the lin-ear symmetries satisfy the third-order differential equation. The correspondingthird-order operator is the second symmetric power of the Schrödinger one. Thisoperator is also known as a second Gelfand–Dikii Hamiltonian operator, whichtransforms the functional space of the potentials W under appropriate boundaryconditions into the infinite-dimensional Poisson algebra known as the Virasoroalgebra. We use this operator to obtain a description of integrable potentialsW such that the solutions of the Schrödinger equation Lu = 0 can be obtainedin quadratures. The operator is also used to find symmetries of the eigenvalueproblem for the Schrödinger operator. We show that if the potential W satis-fies the KdV equation, or one of their higher analogs, then the eigenvalues andeigenfunctions can be found by quadratures.

In Chapter 3 we illustrate the potency of the geometric approach to thesymmetries developing two constructions: a symmetry reduction and Lie’ssuperposition principle.

Preface xv

The first construction is very natural: given an ideal τ of the Lie algebra g

of shuffling symmetries of a completely integrable distribution, we decomposethe integrability problem into two steps: integrability of a completely integrablewith symmetry algebra τ , and then the new one with the symmetry factoralgebra g/τ .

Taking τ to be the radical of g we reduce by quadratures the integration ofthe initial distribution to a distribution with a semi-simple or simple symmetryalgebra. The last distributions correspond to ordinary differential equationswhich we call model equations. We give a description of the model equa-tions which correspond to the classical simple Lie algebras. One can see theadvantage of using the model system from Lie’s superposition principle. Theprinciple provides us with all the solutions of the model differential equa-tion when we know some finite number of solutions (a fundamental system ofsolutions) and a (in general-non-linear) superposition law. Thus, for the case ofthe three-dimensional Lie algebra sl2, the model is the Riccati equation and thesuperposition rule is given by the cross-ratio of four points.

Part II of the book is devoted to symplectic algebra. Here we collect necessaryinformation associated with the existence of a symplectic structure on a basicvector space. We had decided to gather together here all the main results notonly for the sake of completeness and to make the book self-contained but alsobecause of the conceptual importance of the symplectic structure for Monge–Ampère differential equations.

Historically, the appearance of a symplectic structure in the geometricstudies of differential equation has traditionally been attributed to Huygenspapers in geometric optics (though, strictly speaking, he had used merely acontact structure – an odd-dimensional cousin of the symplectic structure).The importance of symplectic geometry was recognized by J. Lagrange, G.Monge, A. Legendre and especially by Sophus Lie. E. Cartan and his Belgianstudent T. Lepage had used the symplectic machinery to study the geometry ofMonge–Ampère equations at the beginning of the twentieth century. It is curi-ous to note that T. Lepage had introduced a symplectic analog of Hodge theorybefore the appearance the “very” Hodge decomposition theorem on Riemannianmanifolds.

The necessity of symplectic and contact geometry in mechanics is wellknown. For Monge–Ampère differential equations one should go further anduse differential forms in the middle dimension.

The algebra of exterior forms on a symplectic vector spaces has some inter-esting specific features. In Chapter 5 we study as sl2- structure given by a couple

xvi Preface

of “rising” � and “lowering” ⊥ operators on the exterior forms:

�ω = ω ∧�, ⊥ ω = ıX�ω

and by its commutator.Here � is the given symplectic 2-form and ıX� is the contraction (the “inner

product” ) with the symplectically dual bivector X�.Their commutator acts on k–forms by a multiplication:

ω→ (n− k)ω.

The form ω annihilated by⊥ is called a “primitive” or effective k-form. Theseforms are extremely important – they correspond to Monge–Ampère operators.The arguments of the sl2-representation theory give the Hodge – Lepage decom-position theorem – the main result of this chapter – which states that anyexterior k-form ω on the symplectic vector space V is a sum of the formsωi ∧�i, i = 0, . . . , where ωi are effective forms uniquely determined by ω.

The classification problems for differential equations and operators have theirtrace in linear algebra – this is a classification of effective forms with respect tothe symplectic group. Chapter 6 deals with the easiest classifications problemsin dimension 4. In the next chapter we give a symplectic classification of exterior2-forms in arbitrary dimensions.

In Chapter 8 we classify effective 3-forms in six-dimensional symplecticspace with respect to a natural action of the symplectic group Sp3 . The problemhas a long history. Being in the spirit of the classical questions of the theoryof geometric invariants, this problem was well known within a classification ofspinors of dimension 12 and 14 (see [40], [90]) for the case when the base fieldis algebraically closed. Their methods do not work for the real classification.The first classification was obtained in our papers [74, 77] where the list ofnormal forms had a gap that was later filled by B. Banos [4, 5].

Part IV is devoted to the Monge–Ampère equations and to the related objects:Monge–Ampère operators and partial differential equation (PDE) systems ontwo-dimensional manifolds.

Chapters 9 and 10 contain some necessary information about symplectic andcontact manifolds.

The application of the algebraic machinery of Chapter 5 gives a descriptionof the Monge–Ampère equations and Monge – Ampère operators. The initialpoint of our approach is the following observation: to any differential k-formω ∈ �k(J1M), where J1M is the space of 1-jet functions on a manifold M, weattach a non-linear second-order differential operator�ω : C∞(M)→ �k(M),

Preface xvii

acting as

�ω(h) = j1(h)∗(ω),

where j1(h) : M → J1M is the 1-jet prolongation of a function h ∈ C∞(M).We see that the first advantage of this approach is a reduction of the order of

the jet spaces: we use a simpler space J1M instead of the space J2M where theMonge–Ampère equations should lie ad hoc, being second-order partial differ-ential equations. The space J1M has the Cartan distribution which in this case isnothing but the aforementioned contact structure which impacts fascinatinglyon the treatment of second-order differential operators and equations.

We should stress that our definition does not cover all non-linear second-order differential equations but only a certain subclass of them. This subclass israther wide and contains all linear, quasi-linear and Monge–Ampère equations.We call the operators �ω with ω ∈ �n(M), where n = dim M, Monge–Ampèreoperators. The following observation justifies this definition: being written in alocal canonical contact coordinates on J1M the operators�ω have the same typeof non-linearity as the Monge – Ampère operators. Namely, the non-linearityinvolves the determinant of the Hesse matrix and its minors.

The correspondence ω→ �ω is not one-to-one: this map has a huge kernel.If we denote the canonical contact 1-form on J1M by ω0, then the kernel isgenerated by the forms α ∧ ω0 + β ∧ dω0.

It is not hard to check that these forms produce an ideal C in the exterioralgebra �∗(J1M) which we call Cartan ideal, and the quotient �∗(J1M)/C bythis ideal is isomorphic to the effective exterior forms �∗ε (J1M) which we haddiscussed above. Hence, the effective exterior forms uniquely define Monge –Ampère operators and we can apply all of the machinery of contact/symplecticgeometry to a study of these operators and the related non-linear differen-tial equations. For example, from the geometrical point of view, solutions ofdifferential equations corresponding to �ω are nothing but the Legendre sub-manifolds L in J1M which are integral with respect to the form ω, that is,ω|L = 0. It is also much easier to apply the contact transformations to differ-ential forms than to the differential operators, so one can define (infinitesimal)symmetries of the Monge–Ampère operators and Monge–Ampère equations byusing the induced action of the contact diffeomorphisms (respectively contactvector fields) on the effective differential forms.

In Chapter 11 we introduce and discuss some operators acting on the effect-ive forms and (by correspondence) on the Monge – Ampère operators. First ofall the de Rham operator induces a complex on the algebra of effective forms.The cohomology of the complex coincides with the de Rham cohomology of

xviii Preface

the base M up to dimension n − 1, where n = dim M. They are trivial indimensions greater than n, and only in dimension n do we have an essentialdifference with the cohomology of M. This relates to conservation laws andan Euler operator E . By a conservation law we mean an (n − 1)-differentialform θ on J1M, such that dθ |L = 0 for all solutions L. One can check that thisis possible if and only if dθ = gωmod C for some function g ∈ C∞(J1M).We call such a function a generating function of the conservation law. Thereis one-to-one correspondence between generating functions and conservationlaws considered up to the trivial ones, and a function g is a generating functionif and only if E (gω) = 0. We study conservation laws in Chapter 13. We show arelation between contact symmetries and conservation laws for Monge–Ampèreequations of divergent type that generalize the classical Noeter theorem in vari-ational calculus. Conservation laws can be used in different directions and herewe discuss their application to the classical problem of “sewing” of two solu-tions by a border of codimension 1. This leads us to the contact analog of theclassical Hugoniot–Rankin conditions which are used further for description ofshock waves and discontinuous solutions. The end of this chapter is devoted toan application of the developed approach to variational problems. We show thatthe Euler operator is exactly the operator in the corresponding Euler–Lagrangeequations. The chapter closes with a description of non-holonomic filtrationin the exterior algebra of J1M and with an interpretation of the Euler oper-ator as a connecting differential in the spectral sequence associated with thisfiltration.

Chapter 14 deals with the special case when the base manifold M is two-dimensional. This situation has a lot of complementary geometric and algebraicstructures which come into play. The structures resulte from the naturallydefined non-holonomic field of endomorphism on J1M, that is, a field of operat-ors defined on the Cartan distribution only. This leads to a non-holonomic almostcomplex structure, for elliptic differential equations, and to a non-holonomicalmost product structure for the case of hyperbolic equations. For the parabolicone we get a non-holonomic almost tangent structure.

The theory becomes more enlightened in the case when J1M is replaced bythe cotangent bundle T∗M. We call the corresponding Monge–Ampère equationsymplectic and their geometry is defined by the corresponding structure on thephase space. Thus, for example, elliptic equations define an almost complexstructure on T∗M. The liaison between the geometric structures and equationswill be used profoundly in subsequent chapters to establish and to clarify theclassification and equivalence problems.

Part IV of the book has a somewhat specific feature – on one hand it isso important and voluminous that it could be chosen as the foundation for

Preface xix

a separate book. On the other hand, we had decided to include this part asan illustration how the proposed approach can be effectively used in practicefor many different type of equations, coming from practically all branchesof the natural sciences. We were clearly unable to cover all possible applica-tions but we had focused on the examples which were elaborated by ourselvesduring long-time contacts with physicists, engineers, biologists, etc. Most ofthe examples were developed up to numerical results and had figured as auseful part of some joint technical and scientific projects which were under-taken at the Applied Mathematical Department of Moscow Technical Universityof Civil Constructions during the period 1978–1990 under the direction ofone of the authors (VL) and with a strong participation of two others. Someof the examples in this part were developed at Astrakhan State Universityunder the guidance of one of the authors (AK) in a collaboration with theBiology Department of Moscow State University between 1999 and 2003.Another important collaboration which we acknowledge in this part is thelong-time cooperation of third author (VR) with applied mathematicians andmeteorologists from the Meteorological Office (Bracknell, UK) and ReadingUniversity.

Chapter 15 is devoted to a study of the Khokhlov–Zabolotskaya (KZ) equa-tion. We knew about this equation in the mid-1970s from contacts with thetheoretical acoustics group of R. Khokhlov (Physics Department of MoscowState University). The equation describes the propagation of three-dimensionalsound beams in a non-linear medium. We treat this equation in its full three-dimensional version. It worth mentioning that the two-dimensional version ofthis equation is a hierarchy member of the famous dispersionless Kadomtsev–Petviashvili integrable system. The numerous applications and physicallyrelevant versions of this dispersionless hierarchy are beyond the scope of ourbook (for statements and references see [78]).

We describe symmetries, conservation laws and exact solutions of the KZequation. We discuss singularities of the solutions, Hugoniot–Rankin conditionsand shock acoustic waves. Using this information we give a mathematicalexplanation to an experimentally verified phenomena of self-diffraction andperiodic oscillation of sound beams (which is completely similar to the beha-vior of beams) and give some explicit formulas for the parameters of thisbehavior.

A version of the Kolmogorov–Petrovsky–Piskunov equation with a non-linear diffusion coefficient is the subject of our study in Chapter 16. Thisequation has a lot of interest in biology, ecology, and heat and mass trans-fer theory. We compute the Lie algebra of its symmetries and show how to usethem to construct invariant solutions.

xx Preface

In Chapter 17 we collect all the applications of geometric studies of Monge–Ampère equations in theoretical meteorology where Monge–Ampère likeoperators had appeared some time ago in so-called semi-geostrophic models.They constitute an important class of models which are very useful in numer-ical weather prediction. This chapter is based on some on-going research byone of the authors with I. Roulstone (Surrey University, UK). We give a shortaccount of the geometric study of balanced rotational models, which mathemat-ically means a very special case of the Navier–Stokes system with the presenceof Coriolis-like forces. An important aero- and hydrodynamical notion asso-ciated with these models is a potential vorticity. The conservation law of thisquantity (under some mild restrictions) gives a non-linear differential equa-tion which is easily represented in a form of the symplectic Monge–Ampèreequation. The geometric structures related to this symplectic Monge–Ampèreequation define and sometimes (and in turn are defined by) some nearly balancedtwo-dimensional model.

The second main motive of this book is a contact equivalence problem fordifferential equations. The first and important algebraic step to this problem wasmade in the previous parts of the book when we had discussed the classificationand equivalence for (effective) exterior forms.

Part V of the book contains the contact classification results on Monge–Ampère equations (in analytic and smooth categories) which can be obtainedby our geometric approach and which have the most complete form when thebase manifolds are two- or three-dimensional.

The first case (dim M = 2) can be attributed as a classical Sophus Lieproblem. This problem was raised in the S. Lie article [66] and in our lan-guage it may be reformulated in the following way: to find the equivalenceclasses of second-order (non-linear) differential equations with respect to the(local) group of contact diffeomorphisms. Lie himself had stated some theorems(or without proofs or with some indications/hints of them) which can be con-sidered as an attempt to give answers to the problem in some special cases. Oneof his main results is a statement about (quasi-)linearizability of any analyticMonge–Ampère equation. He had also considered the Monge–Ampère equa-tion with constant coefficients and the Monge–Ampère equation in the presenceof so-called intermediate integrals.

The essential inroad to the classical Lie problem was given by the FrenchSchool and mainly by G. Darboux and E. Goursat who gave a classification of thetwo-dimensional hyperbolic Monge–Ampère equation under some restrictions.We should mention that the geometric version of Goursat’s results was given byT. Morimoto who had used the language of G-structures and whose approachdiffers cardinally from ours (see [83]).

Preface xxi

Chapters 18–21 contain a modern version of S. Lie’s results (which to ourknowledge were never completely proven until our papers in the early 1980s)and more general results on the symplectic and contact classification (A. Kush-ner, B. Kruglikov and D. Tunitskii), based on e-structures naturally defined byMonge–Ampère equations of the general type. Concerning the general equival-ence problem we outline classification results for Monge–Ampère equations inthe general case. We apply these results for classification and normal forms ofsymplectic Monge–Ampère equations on three-dimensional manifolds.

You will discover a family of sympathic cats which decorate the main text.Each cat has his own personal name and we hope that it will not be an enigmaticproblem to our cleverminded and brilliant readers to understand why one oranother cat appears in its proper place place in the text. A list of pictures ofthese cats with their names appear below.

Welcome Eureka Mentor Thinking Lazy Terminator

In conclusion we wish to thank our friends, colleagues and students for theirhelp and support during the preparation of this book. In particular, we wish tothank Marat Djamaletdinov for his beautiful pictures which, we believe, shouldcaptivate readers.

PART I

Symmetries and Integrals

1

Distributions

1.1 Distributions and integral manifolds

1.1.1 Distributions

Let M be an (n + m)-dimensional smooth manifold, and let τ : TM → Mbe the tangent bundle. By a distribution P on M one means a smooth fieldP : a ∈ M −→ Pa = P(a) ⊂ TaM of m-dimensional subspaces of the tangentspaces. The number m is called a dimension of the distribution, m = dim P,and n is called a codimension of P, n = codim P.

There are different ways to say that P is a smooth field, and to define adistribution. We give the more important ones.

1. As subbundlesLet EP = ⋃

a∈MP(a) ⊂ TM. Then P is a distribution if and only if τP :EP → M is a subbundle of the tangent bundle.

2. By local basesWe say that a vector field X ∈ D(M) belongs (or is tangent) to P on a subsetN ⊂ M, if Xa ∈ P(a) for all a ∈ N . Then smoothness of P means that thereare local bases for P consisting of vector fields that belong to P.

In other words, for any point a ∈ M there exists a neighborhood O of aand m vector fields X1, . . . , Xm that belong to P on O and such that vectorsX1,b, . . . , Xm,b form a basis of Pb at any b ∈ O. Note, that the conditionX belongs to P means that the vector field X is a section of the bundleτP : EP → M.

We denote by D(P) the set of all vector fields that belong to P. It is clearthat D(P) is a module over the algebra C∞(M) of smooth functions on M:

X , Y ∈ D(P) =⇒ X + Y ∈ D(P),

f ∈ C∞(M), X ∈ D(P) =⇒ f X ∈ D(P).

3

4 Distributions

In the case when D(P) admits a global basis, consisting, say, of vector fieldsX1, . . . , Xm we write P = F〈X1, . . . , Xm〉.

3. By equationsLet Ann(P(a)) ⊂ T∗a M be the annihilator of P(a) ⊂ TaM, that is

Ann(P(a)) = {ωa ∈ T∗a M|ωa = 0 on P(a)}.In other words, Ann(P(a)) contains all linear equations for P(a), that is allcovectors ωa vanishing on vectors from P(a). Note that dim Ann(P(a))= n.

We say that a differential 1-form ω ∈ �1(M) annihilates P on a subsetN ⊂ M if and only if ωa ∈ Ann(P(a)) for all a ∈ N . We denote byAnn(P) the set of all differential forms on M that annihilate P and by�1(M)

the C∞(M)-module of differential 1-forms on M. Obviously, Ann(P) is amodule over C∞(M):

α,β ∈ Ann(P) =⇒ α + β ∈ Ann(P),

f ∈ C∞(M), α ∈ Ann(P) =⇒ f α ∈ Ann(P).

In this terms the smoothness of P means that locally P can be defined by ndifferential 1-forms; that is, for any point a ∈ M there exists a neighborhoodO of a and n differential 1-formsω1, . . . ,ωn that annihilate P on O and suchthat covectors ω1,b, . . . ,ωn,b form a basis of Ann(P(b)) at any b ∈ O. In thecase when Ann(P) admits a global basis, say, ω1, . . . ,ωn we write

P = F〈ω1, . . . ,ωn〉.

For the distribution P = F〈X1, . . . , Xm〉 one can define its derivatives. Thedistribution P(1) on M which is generated by the vector fields X1, . . . , Xm andby all possible sorts of commutators [Xi, Xj] (i < j; i, j = 1, . . . , m) is calledthe first derivative of P, i.e.,

P(1) = F〈X1, . . . , Xm, [X1, X2], . . . , [X1, Xm], . . . , [Xm−1, Xm]〉.

Analogously one can define the higher derivatives: P(k+1) def= (P(k))(1)

(k > 1).

1.1.2 Morphisms of distributions

Let F : N → M be a smooth map and let P be a distribution on M. Thendifferentials

F∗,b : TbN → TaM

1.1 Distributions and integral manifolds 5

where a = F(b), define a family F∗(P) of vector spaces on N by

F∗(P)(b) = F−1∗ (P(a))

for all b ∈ N .Dimension dim F∗(P)(b) is equal to dim ker F∗,b+dim(Im F∗,b∩P(a)) and

therefore varies in the general case.Note that ifω1,a, . . . ,ωn,a is a basis in Ann(P(a)) then F∗(ω1,a), . . . , F∗(ωn,a)

generates Ann(F∗(P)(b)).We say that a map F is P-regular if the dimension function b ∈ N −→

dim F∗(P)(b) is locally constant.For the case of P-regular maps F∗(P) is a distribution on N . We call this

distribution the image of P under F.The following three cases have great importance in applications.

1. F is a diffeomorphism, then the image F∗(P) is well defined for anydistribution P.

2. F is a surjection, or a smooth bundle. Then the image F∗(P) is well definedfor any distribution P also.

3. F is an embedding. Thus N is a submanifold of M. Then regularity F meansthat the intersection TbN ∩ P(b) has a constant dimension for all b ∈ N . Inthis case we call F∗(P) the restriction of P on N .

1.1.3 Integral manifolds

Let P be a distribution on M. A submanifold i : N ↪→ M is said to be integralfor P if the restriction of P to N is equal to the tangent bundle, that is,

TaN ⊂ P(a) (1.1)

for any point a ∈ N .This definition implies that the dimension of an integral manifold cannot

exceed the dimension of a distribution.If we define distributions in terms of differential 1-forms, say, locally P =

F〈ω1, . . . ,ωn〉 then condition (1.1) takes the form

ω1|N = 0, . . . ,ωn|N = 0

of the so-called Pfaff system.An integral manifold N is called a maximal integral manifold if for any point

a ∈ N one can find a neighborhood O of a such that there is no integral mani-fold N ′ such that N ′ ⊃ N ∩O.

6 Distributions

From this it is clear that the dimension of maximal integral manifolds doesnot exceed the dimension of the distribution. The distributions which haveintegral manifolds of the dimension equal to the dimension of the distribu-tion are the simpler ones. They are called completely integrable distributions,or CIDs.

A smooth function H ∈ C∞(M) is called first integral for P if dH ∈ Ann(P).As we will see later on a distribution P, is completely integrable if and only if

it has (locally) n = codim P first functional independent integrals H1, . . . , Hn;that is, (locally) P = F〈dH1, . . . , dHn〉.

Note that any distribution has at least one-dimensional integral manifolds(integral curves). Indeed, any integral curve of a vector field X ∈ D(P) is anintegral curve of the distribution P. To find this curve one should solve someordinary differential equations (ODEs). This observation has a general nature;namely, the problem of finding integral manifolds of a distribution implies asolution of some differential equation, and vice versa, the problem of findingsolutions of differential equations is equivalent to finding integral manifolds ofsome distributions.

Let us look at a few examples.

Example 1.1.1 The simplest non-trivial distribution is a one-dimensional dis-tribution on the plane M = R2. Let x, y be coordinates on M, and let P = F〈ω〉where ω = a(x, y) dx + b(x, y) dy, and a2 + b2 �= 0. Let N ⊂M be an integralcurve. Assume, for example, that x can be chosen as a (local) coordinate on N.Then N is a graph of a function h(x),

N = {(x, h(x)), x ∈ R}

and the Pfaff equation ω|N = 0 takes the form of the first-order differentialequation

a(x, h(x))+ b(x, y(x))y′(x) = 0.

Note also that P = F〈X〉 where

X = b(x, y)∂

∂x− a(x, y)

∂

∂y

and therefore to find integral curves of the distribution one should solve thesystem of differential equations:

x = b(x, y), y = −a(x, y).


The existence theorem shows that these equations have smooth solutions andtherefore the distribution has integral manifolds of dimension 1. Thus P is a CID.

We observe from this example that integration of a first-order ODE, say,

y′ = F(x, y)

is equivalent to finding integral curves of the distribution P = F〈ω〉 where

ω = dy − F(x, y)dx.

In this case we also have

P = F〈D〉

where

D = ∂

∂x+ F(x, y)

∂

∂y.

The next example generalizes this observation for ODEs of arbitrary order.

Example 1.1.2 (Cartan distribution) Let M = Rk+1. Denote the coordinatesin M by x, p0, p1, . . . , pk and given a function F(x, p0, . . . , pk−1) consider thefollowing differential 1-forms:

ω0 = dp0 − p1 dx,

ω1 = dp1 − p2 dx,

......

......

ωk−2 = dpk−2 − pk−1 dx,

ωk−1 = dpk−1 − F(x, p0, . . . , pk−1) dx

and the distribution P = F〈ω0, . . . ,ωk−1〉. This is the one-dimensionaldistribution, called the Cartan distribution.

This distribution can also be described by a single vector field D, P = F〈D〉,where

D = ∂

∂x+ p1

∂

∂p0+ p2

∂

∂p1+ · · · + pk−1

∂

∂pk−2+ F(x, p0, . . . , pk−1)

∂

∂pk−1.

8 Distributions

If N is an integral curve of the distribution then x can be chosen as acoordinate on N , and therefore

N = {(x, h0(x), h1(x), . . . , hk−1(x)), x ∈ R}.

Conditions

ω0|N = 0, . . . , ωk−2|N = 0

imply that

h1 = h′0, h2 = h′1, . . . , hk−1 = h′k−2

or that

N ={(

x, h(x), h′(x), . . . , h(k−1)(x))

, x ∈ R}

for some function h(x).The last equation ωk−1|N = 0 gives us an ordinary differential equation

h(k)(x) = F(

x, h(x), h′(x), . . . , h(k−1)(x))

.

The existence theorem shows us once more that the integral curves do exist, andtherefore the Cartan distribution is a CID.

Note that three is the lowest number of dimensions where one can encountera non-CID.

Example 1.1.3 (Contact distribution, see Figure 1.1) Let M = R2n+1 andlet P = F〈ω〉 where

ω = du−n∑

i=1

pi dqi

in the coordinates (q1, . . . , qn, u, p1, . . . , pn) on R2n+1.Then P is a 2n-dimensional distribution, but there are no 2n-dimensional

integral manifolds of P. Let us assume, for example, that n = 1, and thatN is a two-dimensional integral manifold such that q and p, say, are (local)coordinates on N . Then N given by u = h(q, p) for some function h(q, p) andω|N = 0 imply contradictory differential equations for h:

hq = p, hp = 0.


uq

p

Figure 1.1. The contact distribution in R3.

On the other hand, every smooth function f (q1, . . . , qn) determines ann-dimensional submanifold

Lf ={

u = f (q), p1 = ∂f

∂q1, . . . , pn = ∂f

∂qn

}which is integral because of

ω|Lf = 0.

Note that this distribution can also be defined by 2n vector fields

X1 = ∂

∂q1+ p1

∂

∂u, . . . , Xn = ∂

∂qn+ pn

∂

∂u,

Y1 = ∂

∂p1, . . . , Yn = ∂

∂pn.

This example introduces the special case of a contact distribution, which isof great importance throughout this book.

Example 1.1.4 (Oricycle distribution) Let M = R × R+ × S1 be a three-dimensional manifold. The following differential form

ω = (1− cosφ) dx + sin φ dy− y dφ

determines the so-called oricycle distribution on M.Here x, y > 0 are coordinates on R× R+ and φ is the angle on S1.Integral curves of this distribution are oricycles.

Example 1.1.5 Let M be a Möbius strip (see Figure 1.2). Define a one-dimensional distribution P on M where P(a) is the line perpendicular to the

10 Distributions

Figure 1.2. The distribution on the Möbius strip.

central circle of the strip. Both modules D(P) and Ann(P) associated with thedistribution are not free because the fibering of the Möbius strip over its centralcircle is non-trivial. But both modules become free as soon we cut the bundlein such a way that it becomes trivial. Integral curves of this distribution are thefibres of the bundle M → S1.

Example 1.1.6 (Overdetermined system of PDEs) Consider the followingoverdetermined system of partial differential equations (PDEs):

φx = A(x, y,φ,ψ),

φy = B(x, y,φ,ψ),

ψx = C(x, y,φ,ψ),

ψy = D(x, y,φ,ψ),

with respect to functions φ(x, y) and ψ(x, y).Define two differential 1-forms

ω1 = du− A(x, y, u, v) dx − B(x, y, u, v) dy

and

ω2 = dv − C(x, y, u, v) dx − D(x, y, u, v) dy

on the space R4 with coordinates x, y, u, v.Then the pair of functions (φ(x, y),ψ(x, y)) is a solution of the system if and

only if the surface

= {u = φ(x, y), v = ψ(x, y)} ⊂ R4

is an integral manifold of the distribution F〈ω1,ω2〉.

1.2 Symmetries of distributions 11

1.2 Symmetries of distributions

A symmetry of a distribution is a transformation of the manifold M that mapsthe distribution into itself.

In other words, a diffeomorphism F : M → M is a symmetry of adistribution P if

F∗(Pa) = PF(a) (1.2)

for all a ∈ M.Assume that P = F〈ω1, . . . ,ωn〉, then (1.2) means that the differential 1-

forms F∗(ω1), . . . , F∗(ωn) determine the same distribution P and therefore canbe expressed in terms of basis forms ω1, . . . ,ωn.

That is

F∗(ω1) = a11ω1 + · · · + a1nωn,

......

...... (1.3)

F∗(ωn) = an1ω1 + · · · + annωn

for some non-degenerate matrix ‖aij‖ with aij ∈ C∞(M).On the other hand, the last conditions could be written without additional

functions aij:

F∗(ω1) ∧ ω1 ∧ · · · ∧ ωn = 0, . . . , F∗(ωn) ∧ ω1 ∧ · · · ∧ ωn = 0. (1.4)

Example 1.2.1 The Legendre transformation

F : (q, u, p) → ( p, u− qp,−q)

is a symmetry of the contact distribution.Indeed, in this case

F∗(ω) = d(u− pq)+ q dp = ω.

Example 1.2.2 The transformation

G : (q, u, p) → ( p, λqp− λu, λq),

where λ �= 0 is some constant, is a symmetry of the contact distribution. Really,G∗(ω) = −λω.

A vector field X on the manifold M is called an infinitesimal symmetry, orsimply a symmetry, of the distribution P if the flow At along the vector field X

12 Distributions

consist of symmetries of P, i.e.,

(At)∗(P(a)) = P(Ata)

for all points a ∈ M and t.Denote by Sym(P) the set of all infinitesimal symmetries of the distribution P.

Theorem 1.2.1 The following conditions are equivalent:

(a) X ∈ Sym(P),(b) ∀Y ∈ D(P) =⇒ [X , Y ] ∈ D(P),(c) ∀ω ∈ Ann(P) =⇒ LX(ω) ∈ Ann(P).

Proof

(a)⇒(b) Suppose that P is an m-dimensional distribution generated by vec-tor fields X1, . . . , Xm or by n differential 1-forms ω1, . . . ,ωn. LetX ∈ Sym(P) and let {At}t∈I be the one-parametric transformationgroup of X . Then the transformations At preserve the distribution P.Therefore,

(At)∗(Xi) =m∑

j=1

Aij(t)Xj, (1.5)

where Aij(t) ∈ C∞(M) are functions that depend smoothly on theparameter t. Differentiating at t = 0 and using the definition of theLie derivative we obtain:

[Xi, X] = LX(Xi) =m∑

j=1

aijXj

with aijdef= (d/dt)Aij(t)|t=0. Hence, [D(P), X] ⊂ D(P).

(b)⇒(c) Let ω ∈ Ann(P). Then using LX(ω) = ιX(dω) + d(ιXω) and theinfinitesimal Stokes’ formula one obtains:

LX(ω)(Xi)=dω(X , Xi)+ Xi(ω(X)) = X(ω(Xi))−ω([X, Xi])=0

for all i = 1, . . . , m. Therefore, LX(ω) ∈ Ann(P). Here ιX is anoperation of interior multiplication by the vector field X:

(ιXθ)(Y1, . . . , Yk−1)def= θ(X, Y1, . . . , Yk−1)

for any k-form θ . We also denote this operation by X�.

1.2 Symmetries of distributions 13

(c)⇒(a) Let us consider the following differential (n+ 1)-forms:

�j(t)def= A∗t (ωj) ∧ ω1 ∧ · · · ∧ ωn, (1.6)

which depend smoothly on the parameter t. We have �j(0) = 0,since A0 = id . Now we are going to prove that �j(t) ≡ 0 forall t ∈ I . This will imply that A∗t (ωj) are linear combination ofω1, . . . ,ωn for all t and that X is a symmetry of the distribution P.We find

d

dt�j(t) = A∗t (LX(ωj)) ∧ ω1 ∧ · · · ∧ ωn.

From (b) it follows that

LX(ωj) = n∑

i=1

bjiωi

( j = 1, . . . , n) for some functions bji ∈ C∞(M). Moreover,

A∗t(LX(ωj)) = n∑

i=1

A∗t(bji)

A∗t (ωi)

and, therefore,

d

dt�j(t) =

n∑i=1

A∗t (bji)�i(t). (1.7)

Equation (1.7) shows that the vector consisting of (n + 1)-forms�1(t), . . . ,�n(t) is a solution of the linear homogeneous system ofordinary differential equations with zero initial conditions. Hence,it must be equal to zero identically. �

Corollary 1.2.1 Sym(P) is a Lie R-algebra with respect to the commutator ofthe vector fields.

Proof We shall prove the following statements:

1. X , Y ∈ Sym(P) =⇒ X + Y ∈ Sym(P);2. X ∈ Sym(P), λ ∈ R =⇒ λX ∈ Sym(P);3. X , Y ∈ Sym(P) =⇒ [X , Y ] ∈ Sym(P).

14 Distributions

All of these statements follow from (c) and the properties of the Lie derivative:LX+Y = LX + LY , LλX = λLX , L[X,Y ] = [LX , LY ]. �

Example 1.2.3 The vector field

X = a∂

∂q+ b

∂

∂u,

where a, b ∈ R, is an infinitesimal symmetry of the contact distribution on R3.Indeed, the one-parameter group corresponding to the vector field is

At : (q, u, p) −→ (q + at, u+ bt, p).

Therefore, for ω = du− p dq, we have

(At)∗(ω) = ω.

Example 1.2.4 Consider a distribution P on Rn generated by differential1-forms

ωj = ωj1(x) dx1 + · · · + ωjn(x) dxn

where j = 1, . . . , n and ωj1, . . . ,ωjn are smooth functions.If all of these functions do not depend on the variable x1, then the vector field

∂/∂x1 is a symmetry of the distribution P because

L∂/∂x1(ωj) = 0.

Example 1.2.5 Let P be a one-codimensional distribution generated by a differ-ential 1-formω. A vector field X is an infinitesimal symmetry of this distributionif and only if the Lie derivative along X of the form ω is proportional to ω. Itmeans that LX(ω) = fω for some smooth function f , or LX(ω) ∧ ω = 0.

Example 1.2.6 The vector fields

X = ∂

∂x, Y = y

∂

∂y+ sin φ

∂

∂ϕ

are symmetries of the oricycle distribution.

Example 1.2.7 Let us find infinitesimal symmetries of the contact distributionin R3. We shall consider the more general case later on.

1.3 Characteristic and shuffling symmetries 15

Let

X = a(q, u, p)∂

∂q+ b(q, u, p)

∂

∂u+ c(q, u, p)

∂

∂p

be an infinitesimal symmetry of the distribution. Then the Lie derivative

LX(du− p dq) = db− c dq − p da

should vanish when du = p dq mod(ω).This gives two differential equations on the components of X:

p∂a

∂p− ∂b

∂p= 0,

∂b

∂q− p

∂a

∂q− c+ p

(∂b

∂u− p

∂a

∂u

)= 0.

Let us put

f (q, u, p) = b(q, u, p)− p a(q, u, p).

Then,

a = − ∂f

∂p, b = f − p

∂f

∂p, c = ∂f

∂q+ p

∂f

∂u,

and any infinitesimal symmetry of the contact distribution in R3 is determinedby a smooth function f and has the following form:

Xf = − ∂f

∂p

∂

∂q+(

f − p∂f

∂p

)∂

∂u+(∂f

∂q+ p

∂f

∂u

)∂

∂p.

Note that ω(Xf ) = f .

1.3 Characteristic and shuffling symmetries

Let X be a symmetry of a distribution P and let At be the flow along X. ThenAt(N) is an integral manifold of P for any integral manifold N . This reflects themain property of symmetries: one-parameter transformation groups generatedby symmetries act on the set of integral manifolds.

16 Distributions

There is, however, a very distinguished class of symmetries that trans-form each maximal integral manifold into itself. They are called characteristicsymmetries.

Namely, if an (infinitesimal) symmetry X belongs to the distribution P, thenit is called a characteristic symmetry.

Denote by Char(P) the set of all characteristic symmetries

Char(P)def= Sym(P) ∩ D(P).

Lemma 1.3.1 Let X be a characteristic symmetry of a distribution P and let Nbe a maximal integral manifold of P. Then X is tangent to N .

Proof Denote by At the corresponding one-parameter transformation groupgenerated by X , and assume that X is not tangent to X at some point a ∈ N .Then there is a neighborhood O of a such that

N ′ =⋃

−ε<t<ε

At(N ∩O)

is a smooth submanifold of M (for sufficiently small ε).Moreover, N ′ ⊃ N ∩O and N ′ is an integral manifold. To see this note that

tangent spaces to N ′ are sums of tangent spaces to At(N ∩ O) and the one-dimensional subspace generated by X , but both of them belong to P. Hencewith N ′ integral, N ′ should belong to N and X should be tangent to N . �

Theorem 1.3.11. Char(P) is an ideal of the Lie algebra Sym(P). That is, Char(P) is a linear

subspace of Sym(P) and

X ∈ Sym(P), Y ∈ Char(P) =⇒ [X, Y ] ∈ Char(P).

2. Char(P) is a C∞(M)-module:

f ∈ C∞(M), Y ∈ Char(P) =⇒ f Y ∈ Char(P).

1.4 Characteristic and shuffling symmetries 17

Proof

1. It is clear that [X, Y ] ∈ Sym(P). Applying the formula [LX , ιY ] = ι[X,Y ] to adifferential 1-form ω ∈ Ann(P) we obtain

ω([X, Y ]) = ι[X,Y ](ω) = LX(ιYω)− ιY (LX(ω)) = 0.

Because ιYω = 0 and LX(ω) ∈ Ann(P).2. We have [f Y , X] = f [Y , X]−X( f )Y ∈ D(P) for any vector field X ∈ D(P).

Thus, f Y ∈ Sym(P), but f Y ∈ D(P) too. Therefore, f X ∈ Char(P).

�

Since Char(P) is an ideal of the Lie algebra Sym(P), we can define thequotient Lie algebra

Shuf(P)def= Sym(P)/Char(P).

Elements of Shuf(P) are called shuffling symmetries of P.The geometrical sense of shuffling symmetries can be explained as follows.Let us denote by Sol(P) the set of all maximal integral manifolds of P.

Then any symmetry X ∈ Sym(P) generates a flow on Sol(P), and, as wehave seen, the characteristic symmetries generate trivial flows. Moreover, if forsome symmetries X, Y ∈ Sym(P) the difference X − Y ∈ Char(P), then thecorresponding flows on Sol(P) are the same.

In other words, classes X mod Char(P)mix or “shuffle” the set of all maximalmanifolds like a player shuffles a pack of cards.

Example 1.3.1 Let the distribution P on the puncturing plane R2\{0} withcoordinates x, y be given by ω = x dx + y dy. Maximal integral manifolds ofthis distribution are circles x2 + y2 = const. The vector field

X = −y∂

∂x+ x

∂

∂y

generates characteristic symmetries as a module over C∞(R2 \ {0}).The class of symmetry

Y = x∂

∂x+ y

∂

∂y

is a non-trivial shuffling symmetry of P (see Figure 1.3).

Example 1.3.2 The contact distribution in R3 has no characteristic symmet-ries. We shall see later on that this is true for any contact distribution. Indeed,let X = Xf be a characteristic symmetry. Then ω(X) = 0. But ω

(Xf) = f , and

therefore f = 0 and X = 0.

18 Distributions

Figure 1.3. The radial symmetry.

1.4 Curvature of a distribution

The commutator of vector fields

X , Y −→ [X , Y ]

is the skew-symmetric and R-bilinear map.This is not C∞(M)-bilinear because

[ fX, gY ] = fg[X , Y ] + f X(g)Y − gY( f )X.

Let us introduce a bundle νP normal to a distribution P as a factor-bundle TM/P,that is

νP,x = TxM/Px

for any x ∈ M, and restrict the commutator map on D(P).The above formula shows that map

�P : D(P)× D(P)→ (νP),

�P : X ∈ D(P), Y ∈ D(P) −→ [X, Y ]mod D(P)

is C∞(M)-bilinear, and therefore gives a skew-symmetric 2-form on thedistribution with values in the normal bundle.

1.4 Curvature of a distribution 19

We call �P a curvature of the distribution P.The value of �P at a point x ∈ M can be computed as follows:

�P(Xx , Yx) = [X , Y ]mod Px

where vector fields X , Y are any extensions of vectors Xx , Yx ∈ TxM on M.Note that Ann(Px) annihilates Px and

ν∗x = Ann(Px), P∗x = T∗M/Ann(Px).

Therefore, if ω1, . . . ,ωn is a (local) free basis for Ann(P) and Xω1 , . . . , Xωn isthe dual basis in νP then we obtain for the curvature form

�P(X , Y) = −n∑

i=1

dωi(X , Y) Xωi .

Indeed, if X, Y ∈ D(P) and θ ∈ Ann(P), then by definition

〈θ ,�P(X, Y)〉 = 〈θ , [X , Y ]〉 = X(θ(Y))− Y(θ(X))− dθ(X, Y) = −dθ(X, Y),

and therefore the ith coordinate of �P(X , Y) with respect to the basisXω1 , . . . , Xωn is equal to dωi(X , Y).

We shall write

�P = −n∑

i=1

dωi ⊗ Xωi . (1.8)

We say that a distribution P is flat, or, as we shall see later on, completelyintegrable, if �P = 0.

Example 1.4.1 All one-dimensional distributions are flat.

Example 1.4.2 If Ann(P) admits a (local) basis of closed 1-forms then (1.8)shows that P is flat.

Example 1.4.3 The contact distribution on R3 given by the 1-form ω =du− p dq has the following curvature:

� = dp ∧ dq⊗ ∂

∂u.

Example 1.4.4 The proposed construction of the curvature forms for distribu-tions is very similar to the construction of the curvature of connections. Namely,let π : M → B be a bundle over a base manifold B. Recall that vectors tan-gent to the fibres of the projection π are called vertical, and the subspace of

20 Distributions

TxM consisting of all vertical vectors at the point is called vertical and denotedby Tv

x :

Tvx = Tx(π

−1(b))

where b = π(x).A connection in the bundle is given by a distribution P, dim P = dim B, such

that subspaces Px and Tvx are transversal at each point x ∈ M. In other words,

TxM = Tvx ⊕ Px .

Vectors of Px are called horizontal with respect to connection.In this case the normal bundle νP can be identified with the vertical vectors

bundle and the curvature form �P defines a 2-form

�P : Px ∧ Px → Tvx

or, due to isomorphism π∗ : Px → TbB, where b = π(x) ∈ B, a 2-form

�P : TbB ∧ TbB → Tvx .

which is a curvature form of the connection.

1.5 Flat distributions and the Frobenius theorem

A connection in Example (1.4.4) is called flat if the curvature 2-form vanishes.In a similar way, we say that a distribution P on a manifold is flat if the curvature2-form vanishes.

The condition of flatness can be reformulated in terms of vector fields ordifferential 1-forms generating the distribution.

Lemma 1.5.1 The following conditions are equivalent:

(a) P = F〈X1, . . . , Xm〉 = F〈ω1, . . . ,ωn〉 is flat.(b) There are smooth functions cl

ij such that

[Xi, Xj] =∑

l

clijXl

for all i, j = 1, . . . , m.

1.5 Flat distributions and the Frobenius theorem 21

(c) There are differential 1-forms αij such that

dωi =∑

j

αij ∧ ωj

for all i = 1, . . . , n.(d)

dωi ∧ ω1 ∧ · · · ∧ ωn = 0

for all i = 1, . . . , n.

Proof

(a)⇐⇒(b) Indeed, �P = 0 if and only if the commutator [X, Y ] of any twovector fields X , Y ∈ D(P) belongs to D(P).

(a)⇐⇒(c) Condition (c) means that all dωi vanish on P, and due to (1.8) thisis equivalent to (a).

(c)⇐⇒(d) Both conditions state that dωi equals zero on ω1= · · · =ωn= 0.

�

Lemma 1.5.2 Let P be a distribution satisfying one of the above conditions.Suppose that a vector field X belongs to P and At is the one-parameter groupof shifts along X. Then the differential 1-forms A∗t (ω1), . . . , A∗t (ωn) vanish onP for all t and therefore they are linear combinations of the forms ω1, . . . ,ωn.

Proof Consider the differential (n+ 1)-forms

�j(t)def= A∗t (ωj) ∧ ω1 ∧ · · · ∧ ωn,

where j = 1, . . . , n.Then

d�j

dt= d

dtA∗t (ωj) ∧ ω1 ∧ · · · ∧ ωn = A∗t (LX(ωj)) ∧ ω1 ∧ · · · ∧ ωn

and

LX(ωj) = ιX(dωj)+ d(ωj(X)) = ιX(dωj) = αj1(X)ω1 + · · · + αjn(X)ωn.

Therefore,d�j

dt= �j1(t)�1 + · · · +�jn(t)�n, (1.9)

where �js(t) = A∗t (αjs(X)).

22 Distributions

In other words, the forms �1(t), . . . ,�n(t) satisfy the linear ordinary differ-ential equations (1.9). Since �j(0) = 0 for all j = 1, . . . , n, the uniquenesstheorem implies that � j(t) ≡ 0 for all j = 1, . . . , n and all t. �

Lemma 1.5.3 Let P be a distribution satisfying one of the above conditions,and let L be an integral manifold of the distribution P. Let X be a vector fieldsuch that X belongs to P and X is not tangent to L. Then for sufficiently small t

Ldef= ∪tAt(L)

is an integral manifold of P too.

Proof From the previous lemma it follows that

ωj|At(L) = A∗t (ωj)|L = 0.

Therefore the submanifolds At(L) are also integral manifolds of the distributionP. For sufficiently small t the set L = ∪tAt(L) is a smooth submanifold of Mand tangent vectors to L are sums of vectors tangent to At(L) for some t andvectors proportional to X . Both of them belong to P. �

We say, that a distribution P is completely integrable if for any point a ∈ Mthere exists an integral manifold N of P such that a ∈ N and dim N = dim P.

Theorem 1.5.1 (Frobenius) A distribution P is completely integrable if andonly if P is flat. Moreover, if P is completely integrable, and if N1 and N2 aremaximal integral submanifolds of P passing through a point a ∈ N1∩N2. ThenN1 = N2 in some neighborhood of a.

Proof The previous lemma shows how to construct locally the submanifoldN by integrating vector fields X ∈ D(P), and this N is a locally unique ifdim N = dim P.

It is also shows that any vector field X ∈ D(P) should be tangent to N , andtherefore [X, Y ] ∈ D(P) if X , Y ∈ D(P). �

Corollary 1.5.1 The distribution P is completely integrable if and only if P andits first derivative coincide: P(1) = P.

Example 1.5.1 (Distributions of codimension one) A distribution of codimen-sion one can be defined locally by one differential 1-form ω. This distributionis completely integrable if and only if

ω ∧ dω = 0.

1.6 Complex distributions on real manifolds 23

For example, for the distribution F〈ω〉 in R3, where

ω = A(x, y, z)dx + B(x, y, z, )dy+ C(x, y, z)dz

we get

ω ∧ dω=[

A

(∂B

∂z− ∂C

∂y

)−B

(∂A

∂z− ∂C

∂x

)+ C

(∂B

∂x− ∂A

∂y

)]dx ∧ dy ∧ dz

and therefore, this distribution is completely integrable if and only if

A

(∂B

∂z− ∂C

∂y

)− B

(∂A

∂z− ∂C

∂x

)+ C

(∂B

∂x− ∂A

∂y

)= 0.

Example 1.5.2 The oricycle distribution is completely integrable because of

dω = sin φ dφ ∧ dx + (1+ cosφ) dφ ∧ dy

and ω ∧ dω = 0.

Example 1.5.3 For distribution from Example 1.1.6 conditions of completeintegrability are simply conditions of integrability of the corresponding systemof differential equations. The Frobenius theorem requires that dωi∧ω1∧ω2 = 0for i = 1, 2, or

∂A

∂y− ∂B

∂x+ B

∂A

∂u+ D

∂A

∂v− A

∂B

∂u− C

∂B

∂v= 0,

∂C

∂y− ∂D

∂x− B

∂C

∂u− D

∂C

∂v+ A

∂D

∂u+ C

∂D

∂v= 0.

1.6 Complex distributions on real manifolds

Let M be a real (n+m)-dimensional smooth manifold, and let τ : TMC → Mbe the complexification of a tangent bundle. By a complex distribution P on Mone means a smooth field P : a ∈ M −→ Pa = P(a) ⊂ TaMC of complexsubspaces (dimC = m) of the complexification TaMC.

The number m is called a (complex) dimension of the distribution, m =dimC P, and n is called a (complex) codimension of P, n = codimC P.

Similar to the real case, complex distributions can be described by complexvector fields or by complex-valued differential 1-forms: P = F 〈X1, . . . , Xm〉,

24 Distributions

where Xi ∈ D(M)C, or P = F 〈ω1, . . . ,ωn〉, where ωi ∈ �1(M)C, andD(M)C and �1(M)C are complexification of the modules D(M) and �1(M)

respectively:

D(M)Cdef= {X + ιY |X , Y ∈ D(M)},

�1(M)Cdef= {α + ιβ|α,β ∈ �1(M)}.

They are modules over the algebra C∞(M, C) of smooth complex-valuedfunctions on M.

We denote by D(P) the C∞(M, C)-module of complex vector fields on Mwhich belong to P and by Ann P the annihilator of P, that is

Ann(P(a))def= {ωa ∈ T∗a MC

∣∣ωa = 0 on P(a)}.Let P be a complex distribution on M. A submanifold i : N ↪→ M is said to

be integral for P if

TaNC ⊂ P(a)

for any point a ∈ N .If distribution P is defined by complex-valued differential 1-forms, say, P =

F 〈ω1, . . . ,ωn〉, then i : N ↪→ M is integral if and only if

ω1|N = 0, . . . , ωn|N = 0.

A smooth function H is called first integral for P if dH ∈ Ann(P).A complex distribution P, is called completely integrable if it has (loc-

ally) n = codimC P functional independent first integrals H1, . . . , Hn; thatis, (locally) P = 〈dH1, . . . , dHn〉.

A complex distribution P is involutive if [X , Y ] ∈ D(P) for any X, Y ∈ D(P).

Theorem 1.6.1 (Frobenius–Nirenberg [35]) Let P be a complex involutivedistribution such that the distribution P + P is an involutive distribution anddimC

(P ∩ P

) = const. Then P is a completely integrable distribution.

1.7 The Lie–Bianchi theorem

1.7.1 The Maurer–Cartan equations

Let P be an n-codimensional completely integrable distribution generated bythe differential 1-forms ω1, . . . ,ωn and let g be a Lie subalgebra of Shuf(P).

1.7 The Lie–Bianchi theorem 25

We say that g is transversal to the distribution P if dim g = n, and values ofsymmetries generate the factor TaM/Pa at any point a ∈ M.

Denote by X ∈ Sym(P) a representative of a shuffling symmetry X ∈Shuf(P). It is clear that the value θ(X)

def= θ(X) is correctly defined for anyθ ∈ Ann(P).

Let X1, . . . , Xn be a basis of the Lie algebra g. Then the transversalitycondition for the algebra g means that the matrix

Wdef= ∣∣∣∣ωi(Xj)

∣∣∣∣i,j=1,...,n (1.10)

is non-degenerate at any point.Let us choose another basis ω1, . . . ,ωn in the module Ann(P) such that

ωi(Xj) = δij (i, j = 1, . . . , n) . (1.11)

To find this basis, it is enough to set

∥∥∥∥∥∥∥ω1...ωn

∥∥∥∥∥∥∥ def= W−1

∥∥∥∥∥∥∥ω1...ωn

∥∥∥∥∥∥∥ . (1.12)

Suppose now that for the forms ω1, . . . ,ωn condition (1.11) holds, and letcl

ij ∈ R be structure constants of the Lie algebra g (i, j, l = 1, . . . , m), i.e.,

[Xi, Xj

] = n∑l=1

clijXl (1.13)

for all i, j = 1, . . . , n.

Theorem 1.7.1 Let P be a completely integrable distribution which is defined bydifferential 1-formsω1, . . . ,ωn that satisfy (1.11) and (1.13). Then the followingformulae

dωl +∑i<j

clijωi ∧ ωj = 0, (1.14)

hold for all l = 1, . . . , n.

We say that (1.14) is the Maurer–Cartan equation by analogy with theMaurer–Cartan equation in Lie group theory.

26 Distributions

Proof From the Frobenius theorem it follows that

dωl =n∑

s=1

γls ∧ ωs,

where γls are some differential 1-forms.Under the conditions specified, these 1-forms belong to Ann(P), i.e., γls

vanish on the vectors of the distribution P.To prove this, we use the formula:

LXj(ωl) = d(ωl(Xj))+ Xj� dωl = Xj� dωl.

Since Xj ∈ Sym(P), then LXj(ωl) ∈ Ann(P) for all j, l = 1, . . . , n, and

LXj(ωl) = Xj� dωl =

n∑s=1

γls(Xj)ωj − γlj ∈ Ann(P),

which implies that γlj ∈ Ann(P).Therefore,

γlj =n∑

i=1

alijωi

for some smooth functions alij ∈ C∞(M), and

dωl = 2∑i<j

alijωi ∧ ωj,

and

dωl(Xi, Xj) = 2alij.

On the other hand,

dωl(Xi, Xj) = Xi(ωl(Xj))− Xj(ωl(Xj))− ωl([Xi, Xj])

= −ωl([Xi, Xj]) = −ωl

(n∑

k=i

ckijXk

)= −cl

ij.

Therefore,

2alij = −cl

ij


and

dωl = −∑i<j

clijωi ∧ ωj. �

1.7.2 Distributions with a commutative symmetry algebra

Now let g be a commutative symmetry Lie algebra which is transversal toa completely integrable distribution P. Then the above discussion gives thefollowing algorithm for finding first integrals.

Namely, let X1, . . . , Xn be a basis of g and let the distribution P be generatedby the differential 1-forms ω1, . . . ,ωn. Then,

�P(X , Y) = −n∑

i=1

dωi(X , Y) Xωi . (1.15)

Step 1. Form the matrix

W =

∥∥∥∥∥∥∥ω1(X1) · · · ω1(Xn)

......

...ωn(X1) · · · ωn(Xn)

∥∥∥∥∥∥∥and find W−1.

Step 2. Change the basis in Ann(P) for∥∥∥∥∥∥∥ω1...ωn

∥∥∥∥∥∥∥ = W−1

∥∥∥∥∥∥∥ω1...ωn

∥∥∥∥∥∥∥ .

Step 3. Differential 1-forms ω1, . . . , ωn are closed due to the Maurer–Cartanequations. Therefore, a complete set of first integrals of the distributioncan be found by quadratures:

Hj(a) =∫

ωj, (1.16)

if the base manifold M is, for example, connected and simplyconnected.

Here is a path from the fixed point a0 to a point a ∈ M.

28 Distributions

We shall use formula (1.16) for all manifolds and consider H1, . . . , Hn asmultivalued first integrals for P.

Example 1.7.1 (Distributions of codimension one) Let P = F〈ω〉 be acompletely integrable codimension one distribution and X be a transversalsymmetry of P. Then the differential 1-form

ω = 1

ω(X)ω

is closed and

H =∫

ω

ω(X)

is a first integral of P.

Example 1.7.2 Consider the oricycle distribution. It has the infinitesimalsymmetry X = ∂/∂x and W = ω(X) = 1− cosφ. Therefore,

ω = dx + sin φ

1− cosφdy− y

1− cosφdφ

is closed.Then,

H(x, y, z) =∫ x

0a(

t, 0,π

2

)dt +

∫ y

0b(

x, t,π

2

)dt +

∫ φ

π/2c(x, y, t) dt

= x + y cotφ

2

for the following path in the space (x, y,φ):(0, 0,

π

2

)→(

x, 0,π

2

)→(

x, y,π

2

)→ (x, y,φ).

Therefore,

H(x, y,φ) = x + y cotφ

2

is a first integral for the oricycle distribution.

Example 1.7.3 (Integrating factor) Consider a first-order differentialequation

y′ = F(x, y). (1.17)


The corresponding 1-form is

ω = dy − F(x, y) dx.

Suppose that X = a(x, y) ∂∂x + b(x, y) ∂

∂y is a transversal symmetry of the distri-bution, that is the symmetry of the equation. Then, W = b(x, y)−F(x, y)a(x, y)and the differential 1-form

ω = dy − F dx

b(x, y)− F(x, y)a(x, y)

is closed, and the function

H =∫

ω

is a first integral of the equation. The function W−1 is called an integratingfactor for the equation.

Example 1.7.4 (Homogeneous equations) Consider a homogeneous differ-ential equation

y′ = �(x, y) (1.18)

where �(λx, λy) = �(x, y) for all x, y and λ �= 0.This equation is invariant under the scale transformation x −→ λx, y −→

λy, and therefore the vector field

X = x∂

∂x+ y

∂

∂y

is a symmetry. In this case W = ω(X) = y − x�(x, y) and the form

ω = dy −�(x, y) dx

y − x �(x, y)

is closed and the function H = ∫ ω is a first integral of the equation.

30 Distributions

Example 1.7.5 (Generalized homogeneous equations) Let us consider ageneralization of (1.18):

y′ = �(x, y) (1.19)

where �(λax, λby

) = λb−a�(x, y) for some real numbers a, b. This equa-tion has the scale transformations symmetry group x −→ λax, y −→ λby.Therefore,

X = ax∂

∂x+ by

∂

∂y

is a symmetry and

W−1 = 1

by− ax �(x, y)

is an integrating factor.

1.7.3 Lie–Bianchi theorem

Let us now analyze the case of general symmetry Lie algebra.Denote by g(1) = [g, g] the commutator of g; that is, the ideal of g gen-

erated by all commutators [X , Y ] for X , Y ∈ g. Suppose that g(1) �= g andl = codimg g(1).

Let us choose a basis X1, . . . , Xl, . . . , Xn in g such that

X1, . . . , Xl /∈ g(1) but Xl+1, . . . , Xn ∈ g(1).

Then the Maurer–Cartan equations imply that

dωi = 0, i = 1, . . . , l.

Let

H1 =∫

ω1, . . . , Hl =∫

ωl

be the corresponding first integrals.Then submanifolds

Mc = {H1 = c1, . . . , Hl = cl},


where c = (c1, . . . , cl) ∈ Rl, are invariant with respect to the commutator g(1)

because

Xi(Hj) = dHj(Xi) = ωj(Xi) = 0

for i = l + 1, . . . , n and j = 1, . . . , l.Denote by Pc the restriction of distribution P on Mc. Since Hi are first integ-

rals, Pc is a completely integrable distribution of the same dimension n andcodimension dim g(1).

Therefore, g(1) is a transversal symmetry algebra for Pc for any c.We can apply the same procedure for the distribution Pc if the next

commutator

g(2) = [g(1), g(1)]

does not coincide with g(1), etc.Define by induction the derived subalgebras g(s+1) = [g(s), g(s)] and recall

that a Lie algebra g is said to be solvable if there is a natural number r such that

g(r) = 0.

For a solvable Lie algebra g one has the descending sequence of Liesubalgebras

g = g(0) ⊃ g(1) ⊃ · · · ⊃ g(r−1) ⊃ g(r) = 0,

and the above procedure shows that we can find the complete sequence of firstintegrals by integration of closed 1-forms only, or quadratures.

In other words we get the following result.

Theorem 1.7.2 (Lie–Bianchi) Let P be a completely integrable distributionand let g ⊂ Shuf(P) be a solvable symmetry Lie algebra transversal to P,dim g = codim P. Then P is integrable by quadratures.

Note that the above proof of this theorem provides us with an algorithm forfinding first integrals by quadratures.

2

Ordinary differential equations

2.1 Symmetries of ODEs

2.1.1 Generating functions

Let us consider an ordinary differential equation (ODE) of (k + 1)-th orderwhich is resolved with respect to the highest derivative:

y(k+1) = F(x, y, y′, . . . , y(k)). (2.1)

As we have seen this equation determines a one-dimensional distribution P onthe manifold of k-jets Jk = JkR with coordinates x, p0

def= y, p1, . . . , pk . Thedistribution is generated by the vector field

D = ∂

∂x+ p1

∂

∂p0+ · · · + pk

∂

∂pk−1+ F

∂

∂pk,

or by the Cartan differential 1-forms

ω1 = dp0 − p1 dx, . . . , ωk = dpk−1 − pk dx, ωk+1 = dpk − F dx.

Moreover, the vector field D is a characteristic symmetry of the distribution P.In order to describe shuffling symmetries we note that in the module Char(P)

the following relation

∂

∂x≡ −p1

∂

∂p0− · · · − pk

∂

∂pk−1− F

∂

∂pkmod Char(P)

holds.

32

2.1 Symmetries of ODEs 33

Therefore, any shuffling symmetry S ∈ Shuf(P) has a unique representativeof the form

S = α0(x, p0, p1, . . . , pk)∂

∂p0+ · · · + αk(x, p0, p1, . . . , pk)

∂

∂pk.

Moreover,

dpi ≡ pi+1 dx modωi+1,

and therefore,

df ≡ D(f ) dx mod Ann(P),

for any function f = f (x, p0, p1, . . . , pk).For Lie derivatives of the Cartan forms one obtains

LS(ωi) = (D(αi)− αi+1)dx mod Ann(P)

if i = 1, . . . , k.Therefore, S ∈ Sym(P) if

αi+1 = D(αi)

for all i = 0, . . . , k − 1.In other words, any shuffling symmetry can be represented in the form

S = φ∂

∂p0+D(φ)

∂

∂p1+ · · · +Dk(φ)

∂

∂pk, (2.2)

where φdef= α0 and Di def= D(Di−1), D0(φ)

def= φ.The function φ is called a generating function of the symmetry S and we

write Sφ instead of S.Note also that

φ = ω1(Sφ).

For the last Cartan form ωk+1 we have

LSφ (ωk+1) = (D(αk)− Sφ(F))dx mod Ann(P)

= (Dk+1(φ)− Sφ(F))dx mod Ann(P).

34 Ordinary differential equations

Therefore, Sφ is a shuffling symmetry of the ordinary differential equation ifand only if the generating function φ satisfies the following Lie equation:

Dk+1(φ)−k∑

i=0

∂F

∂piDi(φ) = 0. (2.3)

Let us denote by

�F : C∞(Rk+2) −→ C∞(Rk+2)

the following linear k-th order scalar differential operator:

�Fdef= Dk+1 −

k∑i=0

∂F

∂piDi

and call it linearization of (2.1).

Theorem 2.1.1 There exists the isomorphism

Shuf(P) ∼= ker�F

between solutions of the Lie equation and shuffling symmetries given by

ker�F � φ←→ Sφ ∈ Shuf(P).

Remark 2.1.1 To understand the geometrical meaning of the generating func-tions, let us consider a symmetry Sφ and let h(x) be a solution of the equation.Then the corresponding curve Lh = (p0 = h(x), p1 = h′(x), . . . , pk = h(k)(x))is an integral curve of the Cartan distribution. Let At be a one-parametricgroup of shifts along the trajectories of Sφ . Then locally and for small t we

have At(Lh) = Lht and ht = h+ t φ|Lh+ o(t), h(i)t = h+ Di(φ)

∣∣Lh

t+ o(t) fori = 1, . . . , k. In other words, the action of symmetry Sφ on a solution h corres-ponds to a transformation of the form ht = h + t φ|Lh

+ o(t) on functions, orsolutions of the evolutionary equation

∂h

∂t= φ

(x, h, . . . ,

∂kh

∂xk

).

Example 2.1.1 (The translation in x, see Figure 2.1) This symmetry cor-responds to transformations Tt : (x, p0, . . . , pk) −→ (x + t, p0, . . . , pk) and


Figure 2.1. The translation in x.

Figure 2.2. The translation in p0.

has generating function φ = −p1, and

Sφ = −p1∂

∂p0− · · · − −pk

∂

∂pk−1− F

∂

∂pk= ∂

∂x−D.

Equation (2.1) admits the symmetry if and only if

∂F

∂x= 0.

Example 2.1.2 (The translation in p0, see Figure 2.2) This symmetrycorresponds to transformations

Tt : (x, p0, . . . , pk) −→ (x, p0 + t, . . . , pk)

and has the generating function φ = 1 and

Sφ = ∂

∂p0.

Equation (2.1) admits the symmetry if and only if

∂F

∂p0= 0.


Figure 2.3. The scale symmetry.

Example 2.1.3 (Scale transformations, see Figure 2.3) This symmetrycorresponds to transformations

Tt : (x, p0, . . . , pi, . . . pk) −→ (eαtx, eβtp0, . . . , e(β−iα)tpi, . . . , e(β−kα)tpk)

and has the generating function

φ = αxp− βp0.

Example 2.1.4 (Translations in h(x)) Functions φ= h(x) generatetransformations

Tt : (x, p0, . . . , pi, . . . pk) −→ (x, p0+ h(x), . . . , pi+ h(i)(x), . . . , pk + h(k)(x)).

Example 2.1.5 (Point transformations) Functions φ = a(x, p0)p + b(x, p0)

generate so-called point transformations. They correspond to vector fields onR2 of the form

b(x, p0)∂

∂p0− a(x, p0)

∂

∂x.

Example 2.1.6 (Contact transformations) They correspond to the contactvector fields Xf on R3 and have generating functions of the form

φ = f (x, p0, p1).

Example 2.1.7 (KdV symmetries, see Figure 2.4) The picture below showssymmetries for the elliptic function differential equation

(y′)2 − 4y3 + a1y+ a0

generating by the Korteweg–de Vries (KdV) equation (see [68]).


2

1

–1

–1

–2

Figure 2.4. KdV symmetries.

2.1.2 Lie algebra structure on generating functions

We know that Shuf(P) is a Lie algebra for any distribution P. Because of thisfact and the isomorphism between solutions of the Lie equation (i.e., generatingfunctions) and symmetries we are able to introduce a Lie algebra structure ona space of solutions ker�F .

Namely, we define a bracket [φ,ψ] between two generating functions φ andψ as the generating function of the commutator

[Sφ , Sψ

], that is

S[φ,ψ]def= [Sφ , Sψ

].

We call this a Poisson–Lie bracket of functions φ,ψ ∈ ker�F . Obviously, thisbracket defines a Lie algebra structure on ker�F .

To find the bracket we remark that φ = ω1(Sφ), and therefore

[φ,ψ] = ω1(S[φ,ψ]) = S[φ,ψ](p0) = [Sφ , Sψ ](p0) = Sφ(ψ)− Sψ(φ)

or [φ,ψ] = Sφ(ψ)− Sψ(φ). (2.4)

In coordinates one finds

[φ,ψ] =k∑

i=0

(Di(φ)

∂ψ

∂pi−Di(ψ)

∂φ

∂pi

).


Theorem 2.1.2 The generating functions of symmetries (2.1) form a Lie algebrawith respect to the Poisson–Lie bracket.

2.1.3 Commutative symmetry algebra

We begin with the following observation. Let v1, . . . , vn be linear independentvector fields in a domain D ⊂ Rn, and let

[vi, vj] = 0

for all i, j = 1, . . . , n.Take independent functions f1, . . . , fn on D and define differential 1-forms

θ1, . . . , θn as

θ = W−1 df ,

where

θ =

∥∥∥∥∥∥∥∥∥θ1

θ2...θn

∥∥∥∥∥∥∥∥∥ , df =

∥∥∥∥∥∥∥∥∥df1df2...

dfn

∥∥∥∥∥∥∥∥∥ ,

and

W =

∥∥∥∥∥∥∥∥∥v1(f1) v2(f1) · · · vn(f1)v1(f2) v2(f2) · · · vn(f2)

...... · · · ...

v1(fn) v2(fn) · · · vn(fn)

∥∥∥∥∥∥∥∥∥ .

Then θ1, . . . , θn constitute the dual basis to v1, . . . , vn.

Lemma 2.1.1 dθi = 0 for all i = 1, . . . , n.

Proof We have θi(vj) = δij, and therefore

dθi(va, vb) = va(θi(vb))− vb(θi(va))− θi([va, vb]) = 0. �

We apply this result for integration of ordinary differential equations.Assume that one has (k+1) commuting linear independent shuffling symmet-

ries φ1, . . . ,φk+1 in a domain D = {pk+1=F(x, p0, p1, . . . , pk)}⊂Jk+1. Then[D, Xφi ] = 0. Indeed, by the definition of a symmetry [D, Xφi ] = λD for somefunction λ. Applying both sides to the coordinate function x we obtain λ = 0.


Therefore, the vector fields D, Xφ1 , . . . , Xφk+1 commute and are linearindependent at each point. To obtain first integrals we need the followingconstruction. Let us define a Cartan form ωf corresponding to a functionf ∈ C∞(Jk+1) as

ωf =k+1∑i=0

∂f

∂piωi,

where

ωi = dpi − pi+1 dx (0 ≤ i ≤ k)

andωk+1 = dpk+1 −D(F) dx.

Then,ωf (Xφ) = Xφ(f ) and ωf (D) = 0.

Theorem 2.1.3 Let φ1, . . . ,φk+1 be commuting shuffling symmetries for ordin-ary differential equation E = {pk+1 = F} ⊂ Jk+1, and let D ⊂ E be a domainwhere vector fields Xφ1 , . . . , Xφk+1 are linear independent. Let f1, . . . , fk+1 befunctions such that the functions x, f1, . . . , fk+1 are independent in D. Thendifferential 1-forms θ1, . . . , θk+1 defined by

θ = W−1ωf ,

where

θ =

∥∥∥∥∥∥∥∥∥θ1

θ2...

θk+1

∥∥∥∥∥∥∥∥∥ , ωf =

∥∥∥∥∥∥∥∥∥ωf1ωf2

...ωfk+1

∥∥∥∥∥∥∥∥∥ ,

and

W =

∥∥∥∥∥∥∥∥∥Xφ1(f1) Xφ2(f1) · · · Xφk+1(f1)Xφ1(f2) Xφ2(f2) · · · Xφk+1(f2)

...... · · · ...

Xφ1(fk+1) Xφ2(fk+1) · · · Xφk+1(fk+1)

∥∥∥∥∥∥∥∥∥ ,

are closed in D and θi(D) = 0 for all i = 1, . . . , k + 1.

Proof We havedx(D) = 1, ωfi(D) = 0,


and

dx(Xφi) = 0, ωfi(Xφj ) = Xφj (fi)

for all i, j = 1, . . . , k + 1.Therefore, the differential 1-forms

dx, θ1, . . . , θk+1

constitute a dual basis for

D, Xφ1 , . . . , Xφk+1

and the theorem follows from the above lemma. �

Theorem 2.1.4 Let φ,φ1, . . . ,φr be shuffling symmetries for an ordinary dif-ferential equation E ⊂ Jm, and let D ⊂ E be a domain where vector fieldsXφ1 , . . . , Xφr are linear independent. If a vector field Xφ is a linear combinationof the fields Xφ1 , . . . , Xφr then the shuffling symmetry φ is a linear combinationof φ1, . . . ,φr in the domain D:

φ = λ1φ1 + · · · + λrφr

where coefficients λ1, . . . , λr are first integrals for E .

Proof In the domain D we have

Xφ = λ1Xφ1 + · · · + λmXφm

for some functions λ1, . . . , λm. Then, φ = Xφ(p0) = λ1φ1 + · · · + λrφr .As we have seen [D, Xψ ] = 0, for all shuffling symmetries, therefore

[D, Xφ] = D(λ1)Xφ1 + · · · +D(λm)Xφm = 0

and D(λi) = 0 for all i = 1, . . . , m. �

2.2 Non-linear second-order ODEs

In this section we shall apply the Lie–Bianchi theorem and the aboveconstructions for ODEs of second order. Let

y′′ = F(x, y, y′)

2.2 Non-linear second-order ODEs 41

be such an equation, and letφ(x, p0, p1) andψ(x, p0, p1) be generating functionsof symmetries of the equation.

Here

D = ∂

∂x+ p1

∂

∂p0+ F

∂

∂p1,

ω1 = dp0 − p1 dx, ω2 = dp1 − F dx,

and the shuffling symmetry corresponding to the generating function φ hasthe form

Sφ = φ∂

∂p0+D(φ)

∂

∂p1.

The Poisson–Lie bracket has the form

[φ,ψ] = φ∂ψ

∂p0− ψ

∂φ

∂p0+D(φ)

∂ψ

∂p1−D(ψ)

∂φ

∂p1.

Assume that φ and ψ generates a two-dimensional Lie algebra. It is known thattwo-dimensional Lie algebras are either commutative or solvable and one canfind a new basis (we will use the same notation φ and ψ for the basis) in the Liealgebra, such that [φ,ψ] = cψ , where c = 0 for a commutative Lie algebraand c = 1 for a solvable Lie algebra.

Note also that

ω1(Sφ) = φ, ω2(Sφ) = D(φ).

Therefore matrix (20.14) takes the form of the Wandermond matrix

W =∥∥∥∥∥ φ ψ

D(φ) D(ψ)

∥∥∥∥∥and ∥∥∥∥∥ω1

ω2

∥∥∥∥∥ = W−1

∥∥∥∥∥ω1

ω2

∥∥∥∥∥ = 1

|W |

∥∥∥∥∥D(ψ)ω1 − ψω2

−D(φ)ω1 + φω2

∥∥∥∥∥or

ω1 = D(ψ)ω1 − ψω2

φD(ψ)− ψD(φ),

ω2 = − D(φ)ω1 − φω2

φD(ψ)− ψD(φ).

Here |W | is the determinant of W .


It follows from the Maurer–Cartan equations that ω1 is closed and therestrictions of ω2 on first integrals of ω1 are closed also.

Example 2.2.1 Let us consider the following differential equation y′′ =x(y − xy′)4. This equation shall appear later on when we will find solutionsof the Monge–Ampère equation (see Section 12.5). The vector field D has theform

D = ∂

∂x+ p1

∂

∂p0+ x(p0 − p1x)4

∂

∂p1.

Let us find partial solutions of the Lie equation which are linear in p1.These are so-called point symmetries of the equation. Substituting a func-tion φ(x, y, p) = α(x, p0)p1 + β(x, p0) in the Lie equation we find two linearindependent solutions:

φ = p0 + xp1, ψ = x

and

[φ,ψ] = −2ψ .

In the new basis

ω1= 1

|W |((2p21−xp0(p0−xp1)

4)dx−(2p1+x2(p0−xp1)4)dp0+(p0+xp1)dp1),

ω2= 1

|W |(

x2−p1(p0−xp1)2

(p0−xp1)2dx+dp0−xdp1

),

where |W | = (p0 − xp1)(1 − x3(p0 − xp1)3), the first differential 1-form is

closed, and indeed

ω1 = d

(ln

|p0 − xp1|3√|1− x3(p0 − xp1)3|

).

Therefore we get the first integral

H(x, p0, p1) = |p0 − xp1|3√|1− x3(p0 − xp1)3|

.

The restrictions of the 1-form ω1 on Ma = {H = a} are

ω2|Ma =(

1

x2 3√

x3 + a3− x2p0

)dx + 1

xdp0.


Integrating these forms, we find general solutions of the equation:

y = −x∫

dx

x2 3√

x3 + a3+ bx

=3√(x3 + a3)2

a3− x3

2F1(23 , 1

3 ; 53 ;−x3/a3)

2a4+ bx

where a and b are constants and 2F1 is the hypergeometric function.

2.2.1 Equation y′′ = y′ + F(y)

In this section we find (see [22]) all differential equations of the form y′′ =y′ + F(y), which have a 2-dimensional Lie algebra of point symmetries andthen solve these equations by quadratures.

First of all, the independent variable x does not enter explicitly into theequation, hence translation along the x axis is a symmetry. This symmetry hasthe generating function φ = p1.

The problem is to determine when the equation has an additional symmetrywith a generating function of the form ψ = α(x, p0)p1 + β(x, p0).

In what follows, we exclude the particular case of linear functions F(y).The Lie equation for the generating function ψ takes the following form:

∂2α

∂p20

p31+(

2∂2α

∂x∂p0+2

∂α

∂p0+∂2β

∂p20

)p2

1+(

3∂α

∂p0F+∂α

∂x+ ∂2α

∂x2+ 2

∂2β

∂x∂p0

)p1

+(

2F∂α

∂x+F

∂β

∂p0+ ∂2β

∂x2− ∂β

∂x− F ′β

)= 0.

Taking coefficients in p1s we obtain the following system of equations:

∂2α

∂p20

= 0,

2∂2α

∂x∂p0+ 2

∂α

∂p0+ ∂2β

∂p20

= 0,

3F∂α

∂p0+ ∂α

∂x+ ∂2α

∂x2+ 2

∂2β

∂x∂p0= 0,

2F∂α

∂x+ F

∂β

∂p0+ ∂2β

∂x2− ∂β

∂x− F ′β = 0.

The first equation yields

α = γ p0 + δ,

where γ and δ are some functions of x.


Substituting this into the second equation, we obtain

β = − (γ + γ ′)p20 + εp0 + ζ ,

where ε and ζ are again functions of x.Then the third equation is reduced to

3γF = 3(γ + γ ′)p0 − δ′ − δ′′ − 2ε′.

Since the function F(y) was supposed to be non-linear, it follows that

γ = 0,

and

ε = (d − δ − δ′)/2

where d = const.Finally, we find

(εp0 + ζ )F ′ − ηF = θp0 + λ

with

η = 2δ′ + ε, θ = ε′′ − ε′.

This is an ordinary differential equation with respect to F(p0), where the variablex enters as a parameter.

Its general solution, under the assumption

ε �= 0, η �= 0, ε �= η,

is given by the following formula:

F = µ

(p0 + ζ

ε

)η/ε+ θ

ε − η

(p0 + ζ

ε

)+ θζ − ελ

εη

where µ is an arbitrary function of x.From all such functions, we have to choose those which depend only on p0

and are non-linear in p0. The former requirement holds if and only if all thefunctions

µ,ζ

ε,

η

ε,

θ

ε − η,

θζ − ελ

εη

are constants.


Denoting the constantsµ, ζ/ε, η/ε by a, b and c respectively, and taking intoaccount all the relations among functions under consideration, we arrive at thefollowing expressions:

ε = −k + 1

2a exp(kx), λ = (k2 − k)bε,

η = cε, θ = (k2 − k)ε, ζ = bε,

where

k = 1− c

3+ c.

Hence,

F(p0) = a(p0 + b)c − 2c+ 2

(c+ 3)2p0.

Now we consider the previously excluded cases. Either assumption η = 0 orη = ε results in linearity of the function F(p0).

In the case where ε = 0, we obtain a new series of functions

F(p0) = a exp(bp0)− 2

b, a, b ∈ R.

These computations can be summarized as follows.

Theorem 2.2.1 Non-linear differential equations of the form

y′′ = y′ + F(y)

that have a two-dimensional Lie algebra of point symmetries split in twoclasses:

1.

y′′ = y′ + a(y + b)c − 2c+ 2

(c+ 3)2(y + b),

where a, b, c ∈ R, c �= −3, 0, 1 and2.

y′′ = y′ + a exp(by)− 2

b,

where a, b ∈ R, b �= 0.


In the first case the symmetry Lie algebra is generated by

φ = p1, ψ =(

p1 − k + 1

2(p0 + b)

)exp(kx)

where

k = 1− c

3+ c,

and

[φ,ψ] = −kψ .

In the second case

φ = p1, ψ =(

p1 − 2

bp0

)exp(−x)

and

[φ,ψ] = ψ .

2.2.2 Integration

We will carry out the integration procedure for the equation of the first series(see Theorem 2.2.1) with a = 1, b = 0, c = −2. In this case we have thefollowing equation:

y′′ = y′ + y−2 + 2y. (2.5)

The corresponding Lie algebra of point symmetries is generated by twofunctions

φ = p1 and ψ = (p1 − 2p0) exp(3x)

and we have

[φ,ψ] = −3ψ .

We see that the first commutator subalgebra g(1) is generated by the functionψ . Hence, the basis φ,ψ is written in the order prescribed by the algorithm.


The Wronski matrix W is

W =∥∥∥∥∥ φ ψ

D(φ) D(ψ)

∥∥∥∥∥ =∥∥∥∥∥ p1 (p1 − 2p0) exp (3x)

p1 + p−20 + 2p0 (p1 + p−2

0 − 4p0) exp (3x)

∥∥∥∥∥Its determinant |W | is equal to T exp (3x), where

T = p21 − 4p0p1 + 2p−1

0 + 4p20.

The new basis of differential 1-forms is as follows:

ω1 = 1

T[(2p1 + p−2

0 − 4p0) dp0 − (2p0 − p1) dp1] − dx,

ω2 = 1

T[−exp(−3x)(p1 + p−2

0 + 2p0) dp0 + exp(−3x)p1 dp1].

The form ω1 is closed and its integral is

−1

2ln |T | − x.

Instead of this function it is more convenient to take

H1 = exp(2x)(p21 − 4p0p1 + 2p−1

0 + 4p20)

which is also an integral of the equation.

Now we integrate the differential 1-forms ω2 on the surfaces MC1

def= {H1 =C1 = const}.

On these surfaces

p1 = 2p0 ±√

C1 exp(−2x)− 2p−10

and

ω2=−exp(−3x)∓ 2p0 exp(−3x)√

C1 exp (−2x)− 2p−10

dx ± exp(−3x)√C1 exp(−2x)− 2p−1

0

dp0.


The integral of this differential form depends on the sign of the constant C1.For C1 > 0 we have

H2 =1

3exp(−3x)± y exp(−2x)

C1

√C1 − 2y−1 exp(2x)

±C−3/21 ln

∣∣∣∣∣√

C1 − 2y−1 exp(2x)+√C1√C1 − 2y−1 exp(2x)−√C1

∣∣∣∣∣+ C2,

and for C1 < 0

H2 =1

3exp(−3x)± y exp(−2x)

C1

√C1 − 2y−1 exp(2x)

∓(−C1)−3/2 arctan

√2

C1y−1 exp(2x)− 1+ C2.

2.2.3 Non-linear third-order equations

Here we consider ordinary differential equations of the following type:

y2y′′′ = ayy′y′′ + b(y′)3

where a, b ∈ R.Equations of this type have a three-dimensional solvable Lie algebra of

symmetries:

φ1 = p0, φ2 = p0 − xp1, φ3 = p1

with commutation relations

[φ1,φ2] = 0, [φ1,φ3] = 0, [φ2,φ3] = −φ3.

The corresponding transformations for these symmetries are

A1t : (x, p0, p1, p2) −→ (x, etp0, etp1, etp2),

A2t : (x, p0, p1, p2) −→ (etx, etp0, p1, e−tp2),

A3t : (x, p0, p1, p2) −→ (x + t, p0, p1, p2)

respectively.Hence, if we take the symmetry Lie algebra g = 〈Sφ1 , Sφ2 , Sφ3〉 generated

by these symmetries, then the commutator subalgebra g(1) is generated by Sφ3 ,and g(2) = 0.


Therefore the equation can be solved in two steps.First of all, we find the Wronski matrix

W =∥∥∥∥∥∥

p0 p0 − xp1 p1

p1 −xp2 p2

p2 −p2 − xF F

∥∥∥∥∥∥and the inverse matrix

W−1= 1

|W |

∥∥∥∥∥∥∥∥p2

2 −p0F − p1p2 p0p2

−p1F + p22 p0F − p1p2 p2

1 − p0p2

xp1F + p2(xp2 − p1) xp0F + 2p0p2 − xp1p2 xp21 − p0(p1 + xp2)

∥∥∥∥∥∥∥∥where |W | is the determinant of the matrix W and

F(x, p0, p1, p2) = ap0p1p2 + bp31

p20

, |W | = 2p0p22 − p2

1p2 − p1p0F.

Thus, the differential 1-forms ω1,ω2,ω3 are

|W |ω1 = p22 dp0 − (p0F + p1p2)dp1 + p0p2 dp2,

|W |ω2 = (p22 − p1F) dp0 + (p0F − p1p2) dp1 + (p2

1 − p0p2) dp2,

|W |ω3 = (p0p1F + p21p2 − 2p0p2

2) dx + (xp22 − p1p2 − xp1F) dp0

+ (xp0F + 2p0p2 − xp1p2) dp1 + (xp21 − xp0p2 − p0p1) dp2.

It follows from the Maurer–Cartan equations that

dω1 = 0,dω2 = 0,dω3 = ω2 ∧ ω3.

Hence, we find two integrals H1 and H2 such that dH1 = ω1, dH2 = ω2 andthen on the level surface

MC = {H1 = c1, H2 = c2}

we find the third integral.


2.3 Linear differential equations and linear symmetries

2.3.1 The variation of constants method

Let us consider a linear ordinary differential equation

y(k+1) + Ak(x)y(k) + · · · + A0(x)y = g(x), (2.6)

where A0(x), . . . , A0(x), g(x) are given smooth functions of x.Note that a function f (x) is a generating function for some symmetry of the

equation if and only if

�F(f ) = f (k+1) + Ak(x)f(k)(x)+ · · · + A0(x)f (x) = 0. (2.7)

Hence, an arbitrary solution of the corresponding homogeneous equation is asymmetry (a generating function of a symmetry, to be more precise) of theinitial equation.

The space of all solutions forms a (k+1)-dimensional commutative algebra,and if one knows its basis (i.e., a fundamental system of solutions of the homo-geneous equation), then the non-homogeneous equation can be integrated bymeans of quadratures.

Let f1(x), . . . , fk+1(x) be a fundamental system of solutions of the homogen-eous equation (2.7). Then,

Sf = f (x)∂

∂p0+ f ′(x) ∂

∂p1+ · · · + f (k)(x)

∂

∂pk,

and

[Sfi , Sfj ] = 0.

We have

ωi(Sf ) = f (i−1),

and

W =

∥∥∥∥∥∥∥∥∥∥f1 . . . fk+1

f ′1 · · · f ′k+1...

......

f (k)1 · · · f (k)k+1

∥∥∥∥∥∥∥∥∥∥is the ordinary Wronski matrix.

2.3 Linear differential equations and linear symmetries 51

The corresponding forms ω1, . . . ,ωk are closed and define a complete set ofintegrals,

Hj =∫

ωi.

2.3.2 Linear symmetries

A shuffling symmetry Sφ is called a linear symmetry if the generating functionφ is linear in p0, . . . , pk :

φ = b0(x)p0 + · · · + bk(x)pk .

With any linear symmetry we associate a linear operator

�φ = b0 + · · · + bk∂k

where ∂ = d/dx, and we rewrite the Lie equation for linear symmetries in termsof the algebra of linear differential operators. To this end we need the followinglemma.

Lemma 2.3.1 For any linear differential operators A = a0 + · · · + an∂n and

L = l0 + · · · + lk∂k + ∂k+1

there are differential operators CA and RA of order ≤ n − k − 1 and ≤ krespectively such that

A = CA ◦ L + RA.

Moreover, these operators CA and RA are uniquely determined by A.

Proof The result follows from the observation that

A = an∂l−n−1 ◦ L + A1

where the order of A1 is not greater than n− 1 and induction on n.To prove the uniqueness assume that A = C1 ◦ L+ R1 = C2 ◦ B+ R2. Then

(C2 − C1) ◦ L = R1 − R2. If C1 �= C2 then the order of (C2 − C1) ◦ L is notless then k + 1 and the order of R1 − R2 is not greater than k − 1. Therefore,C1 = C2 and R1 = R2. �


Remark 2.3.1 In a similar way one proves that there are operators cA and rA

such that

A = L ◦ cA + rA

and these operators are uniquely determined by A.

Lemma 2.3.2 Let A be a differential operator such that A(h) = 0 for anysolution h of equation L(h) = 0. Then A = C ◦ L for some differentialoperator C.

Proof Using the above lemma we obtain A = C◦L+R for some operator R oforder≤ k. Moreover, ker R ⊃ ker L by assumption. If R �= 0 then dim ker R ≤ kand dim ker L = k + 1. Therefore, R = 0. �

Theorem 2.3.1 A differential operator �φ = b0 + · · · + bk∂k corresponds to

a shuffling symmetry φ = b0p0 + · · · + bkpk of the linear differential equationL(h) = 0, where

L = A0 + · · · + Ak∂k + ∂k+1

if and only if there is a differential operator ∇φ of order k and such that

L ◦�φ = ∇φ ◦ L.

Moreover, the commutator [φ,ψ] of linear symmetries corresponds to theremainder R of division [�φ ,�ψ ] by L; that is,

R[�φ ,�ψ ] = �[φ,ψ].

Proof Consider a solution h and let ht be the 1-parameter family of solutions,obtained by shifting h along symmetry Sφ . Then, (see Remark 2.1.1), ht =h+ t�φ(h)+ o(t) and L(ht) = tL(�φ(h))+ o(t) and therefore, L ◦�φ(h) = 0for all solutions L(h) = 0. The existence of ∇φ now follows from the abovelemma.

To prove the last statement of the theorem we denote the above transformationby �t : h −→ ht = h+ t�φ(h)+ o(t).

Let �s be the transformation corresponding to a linear symmetry ψ . Then�s : h −→ hs = h + s�ψ(h)+ o(s), and the commutator [φ,ψ] correspondsto the st-term in the difference �t ◦�s −�s ◦�t . We have

�t ◦�s : h�s−→ h+ s�ψ(h)+ o(s)

�t−→ h+ s�ψ(h)+ t�φ(h)+ st�φ ◦�ψ(h)+ o(t)+ o(s)

2.4 Linear differential equations and linear symmetries 53

and, on the other hand,

�s ◦�t : h�t−→ h+ t�φ(h)+ o(t)

�s−→ h+ s�ψ(h)+ t�φ(h)+ st�ψ ◦�φ(h)+ o(t)+ o(s).

Therefore,

�t ◦�s −�s ◦�t : h −→ st[�φ ,�ψ ](h)+ o(t)+ o(s)

= stR[�φ ,�ψ ](h)+ o(t)+ o(s). �

Let us denote by G(L) a vector space of all differential operators � such that

L ◦� = ∇ ◦ L

for some (uniquely determined) differential operator ∇.First of all G(L) is an algebra with respect to the composition of operators,

and a Lie algebra with respect to the commutator of operators.Denote by Sym(L) the Lie algebra of linear symmetries of L.

Theorem 2.3.2 1. If � ∈ G(L) then R� ∈ Sym(L).2. The residue map

R : G(L)→ Sym(L),

R : � ∈ G(L) −→ R� ∈ Sym(L)

is a Lie algebra homomorphism.

Proof Indeed, from L ◦� = ∇ ◦ L we find

L ◦ (C� − c∇) ◦ L = r∇ ◦ L − L ◦ R�.

If C� �= c∇ then the operator L ◦ (C� − c∇) ◦ L has order greater than 2k + 2but the order of r∇ ◦ L − L ◦ R� does not exceed 2k + 1. Therefore, C� = c∇and r∇ ◦ L = L ◦ R�.

The second part of the theorem follows from Theorem 2.3.1. �


2.4 Linear symmetries of self-adjoint operators

Recall that the differential operator

Lt = (−1)k+1∂k+1 +k∑

i=0

(−1)i∂ i ◦ Ai

is said to be adjoint to the operator

L = ∂k+1 +k∑

i=0

Ai∂i.

The conjugation operation is an involution in the algebra of differentialoperators, that is

(At)t = A, (A ◦ B)t = Bt ◦ At .

A differential operator L is said to be self-adjoint if Lt = L and skew-adjointif Lt = −L.

Example 2.4.1

1. One has only skew-adjoint operators in order one. They have the form

2f (x)∂ + f ′(x).

2. In order two one has only self-adjoint operators. They have the form

f (x)∂2 + f ′(x)∂ + g(x).

3. In order three one has only skew-adjoint operators. They have the form

2f (x)∂3 + 3f ′(x)∂2 + (f ′′(x)+ 2g(x))∂ + g′(x).

Theorem 2.4.1 The correspondence �φ ←→ ∇ tφ establishes an isomorphism

between linear symmetries of the differential equation L(h) = 0 and linearsymmetries of the adjoint equation Lt(h) = 0.

Let us now assume that L is self-adjoint or skew-adjoint. Then if � ∈ G(L)so ∇ t does.

Using this involution we can decompose G(L) into the direct sum:

G(L) = G0(L)⊕G1(L)

2.5 Linear symmetries of self-adjoint operators 55

where

G0(L) ={�| L ◦� = −�t ◦ L

},

G1(L) ={�| L ◦� = �t ◦ L

}.

We will define Z2-parity ε(�) = 0 ∈ Z2 for� ∈ G0(L) and ε(�) = 1 ∈ Z2 for� ∈ G1(L), and will consider the above decomposition as Z2-grading on G(L).

Theorem 2.4.2 Let L be a self- or skew-adjoint differential operator.

1. Then the commutator of operators determines a Lie algebra structure onG(L), such that

[�a,�b] ∈ Ga+b(L)

if �a ∈ Ga(L), �b ∈ Gb(L), a, b ∈ Z2.2. Let Sym(L) be the Lie algebra of linear symmetries of operator L, and

let Syma(L) = R(Ga(L)) for a ∈ Z2. Then Sym(L) = Sym0(L) ⊕Sym1(L) and

[Syma(L), Symb(L)] ⊂ Syma+b(L).

Proof We have

L ◦�a = −(−1)a�ta ◦ L.

Therefore,

L ◦�a ◦�b = (−1)a+b�ta ◦�t

b ◦ L = (−1)a+b(�b ◦�a)t ◦ L

and

L ◦ [�a,�b] = −[�a,�b]t ◦ L.

The second statement of the theorem follows from Theorems 2.3.1 and 2.3.2.�

We call elements of Sym0(L) by even symmetries and elements of Sym1(L)by odd symmetries of the equation L(h) = 0.


2.5 Schrödinger operators

In this section we apply the above results to the Schrödinger operator of theform

L = ∂2 +W

where W = W(x) is a so-called potential.This operator is self-adjoint. Therefore, the algebra of linear symmetries is

Z2-graded.Let us begin with Sym0(L). If � = A0 + A1∂ ∈ Sym0(L) then

L ◦� = −�t ◦ L.We have �t = (A0 − A′1)− A1∂ and

L ◦� = A1∂3 + (A0 + 2A′1)∂2 + (A′′1 + 2A′0 +WA1)∂ +WA0,

�t ◦ L = −A1∂3 + (A0 − A′1)∂2 − A1W∂ − (A1W ′ + A′1W −WA0).

Therefore, � ∈ Sym0 (L) implies A0 = − 12 A′1, and the function z = A1 should

satisfy the following differential equation:

z′′′ + 4Wz′ + 2W ′z = 0. (2.8)

We denote the differential operator on the right-hand side of this equation by

L(2) = ∂3 + 4W∂ + 2W ′.

If � ∈ Sym1(L) then L ◦� = �t ◦ L and we obtain A1 = 0, A′0 = 0. Thus� ∈ Sym1(L) if and only if � proportional to the identity operator.

Finally, we obtain the following theorem.

Theorem 2.5.1 The Lie algebra of linear symmetries of the Schrödingeroperator has the following description:

1. Sym0(L) = {− 12 z′ + z∂|L(2)(z) = 0};

2. Sym1(L) = R · id;3. The Lie algebra Sym0(L) is isomorphic to the Lie algebra sl2(R).

Proof To prove this theorem we should establish isomorphism betweenSym0(L) and sl2(R). First of all we note that the solution space of the equationL(2)(z) = 0 is a Lie algebra with respect to the bracket

[z1, z2] = z1z′2 − z′1z2

induced by the bracket in Sym0(L).

2.5 Schrödinger operators 57

This is a three-dimensional Lie algebra. To find an explicit expression for thecommutator we take a fundamental system of solutions, say, z1, z2, z3 where

z1(0) = 1, z′1(0) = 0, z′′1(0) = 0;

z2(0) = 0, z′2(0) = 1, z′′2(0) = 0;

z3(0) = 0, z′3(0) = 0, z′′3(0) = 1.

Let z = [z1, z2]. Then,

z(0) = 1, z′(0) = 0, z′′(0) = −4W(0).

Therefore,

[z1, z2] = z1 − 4W(0)z3.

In a similar way we find

[z2, z3] = z3, [z1, z3] = z2

and isomorphism with sl2(R). �

The straightforward calculations give the following interpretation of thesymmetry operator L(2).

Proposition 2.5.1 The differential operator L(2) is the second symmetric powerof the Schrödinger operator L. That is, if y1 and y2 is a fundamental system ofsolutions for the Schrödinger equation L(y) = 0. Then L(2) is the differentialoperator of third order with a fundamental system of solutions z1 = 1

2 y21, z2 =

y1y2, z3 = 12 y2

2.

Corollary 2.5.1 Let y be a solution of a Schrödinger equation. Then

� = y2∂ − yy′

is a linear symmetry of the equation.

Remark 2.5.1 Any skew-adjoint operator of third order ∂3 + A∂ + B is thesymmetric power of a Schrödinger operator.

Remark 2.5.2 The ratio z = y1/y2 satisfies the Schwartzian equation

S[z] = 2W ,


where

S[z] = z(3)

z(1)− 3

2

(z(2)

z(1)

)2

is the Schwartzian of z. Therefore linear symmetry of the Schrödinger equationy′′+Wy = 0 generates a Möbius transform and a symmetry for the Schwartzianequation

2p3p1 − 3p22 − 2Wp2

1 = 0.

2.5.1 Integrable potentials

We say that a potential W is integrable if for the Schrödinger equation a linearsymmetry z∂ − z′/2 is known.

In this case the solution space of the Schrödinger equation is also knownbecause the Schrödinger equation has two commuting symmetries p0 and φz =zp− 1

2 z′p0.For example, a potential W is integrable if we know a non-zero solution or a

non-zero even symmetry of the equation.In this section we shall discuss a few methods of finding new integrable

potentials from the known ones.

IntegrationAssume that the Schrödinger equation

y′′ +W(x)y = 0 (2.9)

has the linear symmetry

φz = z(x)p1 − z′(x)2

p0,

where the function z = z(x) satisfies the following equation:

z′′′ + 4wz′ + 2w′z = 0. (2.10)

Note that symmetries p0 and φz commute and assuming that z given onecan find first integrals by quadratures. Namely, taking f1 = p0, f2 = p1 inTheorem 2.1.3 one can obtain two differential 1-forms ω1 andω2, and integralsH1 and H2.

The first integral H = H1 can be chosen to be quadratic in p0, p1:

H = 2zp21 − 2z′p0p1 + (z′′ + 2wz)p2

0.

We will rewrite this integral in the following way.


Let us note that (2.10) is defined by the skew-adjoint operator

L = d3

dx3+ 4w

d

dx+ 2w′

and the Lagrange formula (see Section 2.5.3) shows that

K(z) = 2z(z′′ + 2wz)− z′2

is a first integral for (2.10).We say that the symmetry φz is elliptic, hyperbolic or parabolic if K(z) > 0,

K(z) < 0 or K(z) = 0 respectively.Using the generating function of the symmetry we can rewrite the first integral

in the following form:

H = 2(φ2z + kp2

0)

z,

where 4k = K(z).Now taking f1 = H, f2 = p0 in Theorem 2.1.3 we find two differential

1-forms with

θ1 = dH

2H,

and the restriction θ of the second form θ2 on levels H = 2c equals

θ = dp0 − p1 dx

φz.

Let

α = φz√|z| , β = p0√|z| .

Then,

H = 2(α2 + kβ2) = 2c

and the restriction θ takes the following form:

θ = dβ

α− dx

z.

Integration of θ gives the following solutions of the Schrödinger equation (see[68]).


Theorem 2.5.2 Let φz be a linear symmetry of (2.9). Then solutions of theSchrödinger equation have the following form:

• for elliptic symmetry φz

y(x) =√ |cz(x)|

ksin

(√k∫

dx

z(x)

),

• for hyperbolic symmetry φz

y(x) =√ |cz(x)|

−ksinh

(√−k∫

dx

z(x)

),

• for parabolic symmetry φz

y(x) = √|cz(x)|∫

dx

z(x).

Here H = 2c.

Potentials with a common even symmetryLet us assume that we have a potential W0 with a non-zero linear symmetryEz0 = − 1

2 z′0 + z0∂ and consider (2.8) as an equation for W with given z0:

z′′′0 + 4Wz′0 + 2W ′z0 = 0.

Then W0 is a solution of this equation and if W has the same linearsymmetry, then

W = W0 + Cz−20 .

Proposition 2.5.2 Assume that the Schrödinger equation with potential W0 hasan even symmetry Ez0 , then the Schrödinger equation with potential W0+ cz−2

0has the same symmetry for all constants c.

Corollary 2.5.2 Assume that the Schrödinger equation with potential W0 hastwo solutions y1 and y2. Then the Schrödinger equation with potential

W0 + c

(a11y21 + 2a12y1y2 + a22y2

2)2

has the symmetry Ez with z = a11y21 + 2a12y1y2 + a22y2

2 and is thereforeintegrable.


Example 2.5.1

1. Take W0 = 0. Then the Schrödinger equations with potentials

W = c

(a11 + 2a12x + a22x2)2

are integrable in quadratures.2. Take W0 = ω2. Then the Schrödinger equations with potentials

W = ω2 + c

(a11 cos2 ωx + 2a12 cosωx sinωx + a22 sin2 ωx)2

are integrable in quadratures.3. Take W0 = −k2. Then the Schrödinger equations with potentials

W = −k2 + c

(a11ekx + 2a12 + a22e−kx)2

are integrable in quadratures.

sl2(R)-representationsAs we have seen Sym0(L) gives us a representation of the Lie algebra sl2 (R)using first-order differential operators Ez, or a representation by vector fields z∂on R where z runs over solutions of (2.8). On the other hand, any representationof the Lie algebra by vector fields on R corresponds to some Schrödingerequation. Indeed, if z∂ is a non-zero vector field from this representation, thensolving (2.8) with respect to W we find the Schrödinger equation with evensymmetry z∂ − 1

2 z′. This equation can be solved by quadratures and due to(2.5.1) we are able to find all symmetries and therefore the representation ofsl2(R).

Let us take W = 0. Then (2.8) gives the standard representation of sl2(R) byvector fields

∂ , x∂ , x2∂ .

Take Ez with z = a + bx + cx2. This is an even symmetry of the equationy′′ = 0. Therefore, from the previous section, Schrödinger equations withpotentials

W = c1

(a+ bx + cx2)2

are integrable in quadratures (see Figure 2.5).


Figure 2.5. The potential (1+ x + x2)−2.

W

x

Figure 2.6. The potential 1− (2+ sin 2x)−2.

Now take W = ω2. Then (2.8) gives a representation of sl2(R) by vectorfields

∂ , cos 2ωx ∂ , sin 2ωx ∂

and Ez with z = a+ b cos 2ωx + c sin 2ωx is an even symmetry of the harmonicoscillator y′′ + ω2y = 0.

Therefore, Schrödinger equations with potentials

W = ω2 + c1

(a+ b cos 2ωx + c sin 2ωx)2

are integrable in quadratures (see Figure 2.6).


Figure 2.7. The potential −1+ (1+ e2x)−2.

In the similar way, for the potential W = −k2 we have a representation ofsl2(R) by vector fields

∂ , e2kx∂ , e−2kx∂

and Schrödinger equations with potentials

W = −k2 + c1

(a+ be2kx + ce−2kx)2

are also integrable in quadratures (see Figure 2.7).

Relations with the Korteweg–de Vries equationAll previous constructions of integrable potentials have used relations betweenpotentials and even symmetries. Let us make this relation explicit. Namely, letus assume that the potential function W is a function of symmetry:

W = w(z).

In this case (2.8) takes the following form:

z′′′ + 4wz′ + 2w′z′z = 0.

We look at a particular case when this equation can be easily integrated.Namely, let us consider the case of linear dependency

W = a z + b

for some constants a �= 0 and b.


Figure 2.8. The Weierstrass potential.

Then we obtain the Korteweg–de Vries equation

z′′′ + 6azz′ + 4bz′ = 0

for traveling waves.Integrating, we find

z′′ + 3az2 + 4bz = c1.

Multiplying both sides by z′ and integrating once more we obtain

12 (z

′)2 + az3 + 2bz2 − c1z = c2

and

z = − 13√

2a℘(x, g2, g3)− b

3a

where ℘ (x, g2, g3) is the Weierstrass function.Thus Schrödinger equations with potentials

W = − 3

√a2

2℘(x, g2, g3)+ 2

3b

have even linear symmetries Ez with z = −(1/ 3√

2a)℘ (x, g2, g3) − b/3a andtherefore can be integrated by quadratures (see Figure 2.8).


Figure 2.9. The Weierstrass potential with symmetry.

It is also true for potentials (see Figure 2.9)

− 3

√a2

2℘(x, g2, g3)+ 2

3b+ c

(−(1/ 3√

2a)℘ (x, g2, g3)− b/3a)2.

2.5.2 Spectral problems for KdV potentials

In this section we consider potentials w(x) such that the correspondingeigenvalue problem

y′′ + w(x)y = λy (2.11)

possesses linear symmetries z(x, λ) which are polynomial in λ.Let

z(x, λ) = z0(x)λn + z1(x)λ

n−1 + · · · + zn−1(x)λ+ zn(x)

be a linear symmetry for (2.11).Then the Lie equation gives a polynomial (in λ) of degree n+ 1

z′′′(x, λ)+ 4(w(x)− λ) z′(x, λ)+ 2w′(x) z(x, λ) = 0


and a recursive set of equations on zk :

z′0 = 0,

z′k+1 = 14 L(2)(zk), k = 0, . . . , n− 1,

L(2)(zn) = 0.

Taking z0 = 1, we find inductively functions zk = zk(w) by

zk+1(w) = 1

4

∫L(2)(zk(w)) dx,

(k = 0, 1, . . .).The first functions are as follows:

z1(w) =w

2+ c1,

z2(w) =w′′ + 3w2

8+ c1

2w+ c2,

z3(w) =w(4)

32+ 5

16

(ww′′ + w′2

2+ w3

)+ c1

8(w′′ + 3w2)+ c2

2w+ c3,

z4(w) =w(6)

128+ 7w′w(3)

32+ 7(3w′′2 + 10w2w′′ + 10ww′2 + 5w4)

128

+ c1

32

(w(4) + 10(ww′′ + w′2

2+ w3

)+ c2

8(w′′ + 3w2)+ c3

2w+ c4.

The conditions L(2) (zn(w)) = 0 which can also be reformulated as z′n+1(w) = 0are called nth KdV stationary equations.

Below we list the first KdV equations:

zeroth KdV:

w′ = 0,

first KdV:

w′′′ + 6ww′ + 4c1w′ = 0,

second KdV:

w(5) + 10(ww′′′ + 2w′w′′ + 3w2w′)

+ 4c1(w′′ + 6ww′)+ 16c2w′ = 0.


We conclude that potentials w which satisfy the n-th stationary KdV equationpossess linear symmetry φSn with

Sn = λn +n∑

k=1

zkλn−k .

As we have seen, the function K = 2z(z′′ + 2(w−λ)z)− z′2 is the first integralof the Lie equation and therefore coefficients of the polynomials

Qn = 2Sn(S′′n + 2(w− λ)Sn)− S′2n

are first integrals for the n-th KdV equation.For example, for the classical (first) KdV equation, w′′′ +6ww′ +4c1w′ = 0,

one has

S1 = λ+ w

2+ c1

and

Q1 = −4λ3 − 8c1λ2 + q11λ+ q10,

where

q11 = w′′ + 3w2 + 4c1w− 4c21,

q10 = 2ww′′ − w′2 + 4w3

4+ c1(w

′′ + 4w2 + 4c1w)

are first integrals.Solving the KdV equation together with equations q11 = constant, q10 =

constant we obtain the first-order ODE for w:

w′2 = −2w3 − 4c1w2 + 2(q11 + 4c21)w+ 4(c1q11 − q10 + 4c3

1)

and solutions

w = −2℘(x + c, g2, g3)− 2c1

3

where the invariants of the Weierstrass elliptic function are equal to

g2 = 4c21 − 6c1

3, g3 = −152c3

1

27− 5q11c1

3+ 2q10.

For the second KdV equation,

w(5) + 10(ww′′′ + 2w′w′′ + 3w2w′)+ 4c1(w′′ + 6ww′)+ 16c2w′ = 0,


one has

S2 = λ2 +(w

2+ c1

)λ+ w′′ + 3w2

8+ c1

2w+ c2

and

Q2 = −4λ5 − 8c1λ4 − 4(c2

1 + 2c2)λ3 + q22λ

2 + q21λ+ q20,

where

q22 = 10w3 + 5w′2 + 10ww′′ + w(4)

10− 8c1c2 + (w′′ + 3w2)c1 + 4wc2,

q21 = 2ww(4) − 2w′w′′′ + w′′2 + 20w2w′′ + 15w4

16+ (w′′ + 3w2)c2

1

+ 4wc1c2 − 4c22 +

4w(4) + 12ww′′ + 4w′2 + 14w3

4c1 + w2c2,

q20=2w′′w(4)+6w2w(4)+4w(4w′′2−3w′w′′′)−w′′′2+12w′2w′′+60w3w′′+36w5

64

+4w3 − w′2 + 2ww′′

4c2

1 + (w′′ + 4w2)c1c2 + 4wc22

+ 12w4 + 13w2w′′ + w′′2 − w′w′′′ + ww(4)

8c1

+ 12w3 + 6w′2 + 10ww′′ + w(4)

4c2

are the first integrals for the second KdV.Using these integrals one can reduce the second KdV equation to the

following second-order ODE:

25w8 + 80w7c1 + 32w6(2c21 + 5c2)− 16w5(24c1c2 + 5Q22)

+ 32w4(−72c21c2 + 28c2

2 + 5Q21 − 9c1Q22)

+ 256(−8c21c2 + 4c2

2 + Q21 − c1Q22)2

+ 256w(8c1c2 + Q22)(8c21c2 − 4c2

2 − Q21 + c1Q22)

− 256w3(8c31c2 − c1(−4c2

2 + Q21))

− 256w3(c21Q22 + c2Q22)+ 64w2(32c3

2 + Q222 + 8c2(Q21 + c1Q22))


+ 76w5(w′)2 + 152w4c1(w′)2 + 64w3(c2

1 + 3c2)(w′)2

− 64w(4c22 + Q21)(w

′)2 − 64w2(6c1c2 + Q22)(w′)2

+ 128(8c31c2 − c1(20c2

2 + Q21)+ c21Q22 + 2(2Q20 − c2Q22))(w

′)2

− 20w2(w′)4 − 16wc1(w′)4 + 16(c2

1 − 4c2)(w′)4 + 80w3(w′)2w′′

+ 96w2c1(w′)2w′′ + 128wc2(w

′)2w′′ − 32(8c1c2 + Q22)(w′)2w′′

− 8(w′)4w′′ − 10w4(w′′)2 − 16w3c1(w′′)2 − 32w2c2(w

′′)2

+ 16w(8c1c2 + Q22)(w′′)2 − 32(−8c2

1c2 + 4c22 + Q21 − c1Q22)(w

′′)2

+ (20w+ 8c1)(w′)2(w′′)2 + (w′′)4 = 0.

Two cases when c = 0, and c = 0, q = 0 give us shorter ODEs:

c = 0 :

25w8 + 160w4Q21 + 256Q221 − 80w5Q22 − 256wQ21Q22 + 64w2Q2

22

+ 76w5(w′)2 + 512Q20(w′)2 − 64wQ21(w

′)2 − 64w2Q22(w′)2

− 20w2(w′)4 + 80w3(w′)2w′′ − 32Q22(w′)2w′′ − 8(w′)4w′′

− 10w4(w′′)2 − 32Q21(w′′)2 + 16wQ22(w

′′)2

+ 20w(w′)2(w′′)2 + (w′′)4 = 0.

c = 0, q = 0 :

25w8 + 76w5(w′)2 − 20w2(w′)4 + 80w3(w′)2w′′

− 8(w′)4w′′ − 10w4(w′′)2 + 20w(w′)2(w′′)2 + (w′′)4 = 0.

Note that these equations have a two-dimensional commutative symmetryLie algebra generated by translations and the first KdV, and therefore can besolved in quadratures.

Now one can apply Theorem 2.5.2 to spectral problems for the Schrödingerordinary differential equations in which potentials are solutions of n-th KdVequations. This gives complete and explicit solutions of the spectral problems.

We illustrate this method for potentials which satisfy the first KdV equation(this is the case of a special Lamé equation) and for the following boundaryproblem on an interval [a, b]:

y(a) = y(b) = 0.


Then Theorem 2.5.2 shows that smooth eigenfunctions

y(x) = 2

√|S1(x)|Q1(λ)

sin

(√Q1(λ)

2

∫ x

a

dτ

S1(τ )

)do exist if:

• S1 = λ+ w/2+ c1 �= 0 on the interval,• Q1 (λ) > 0, ∫ b

a

dτ

S1(τ )= 2πn√

Q1(λ)

for n ∈ Z.

Summarizing we obtain the following result.

Theorem 2.5.3 Let the potential w satisfies the classical KdV equation,

w′′′ + 6ww′ + 4c1w′ = 0,

then spectral values λ for the boundary problem (2.11) given by the formula

λ = ℘(α, g2, g3)− 2c1/3

where α are solutions of the equations

2(b− a)ζ(α)+ lnσ(b+ c− α)σ(a+ c+ α)

σ(a+ c− α)σ(b+ c+ α)= ι2πn (n ∈ Z, ι = √−1)

such that Q1(λ) > 0 and

λ > −c1 − 1

2min

a≤x≤bw(x)

or

λ < −c1 − 1

2min

a≤x≤bw(x).

Here

Q1(λ) = −4λ3 − 8c1λ2 + q11(w)λ+ q10(w),

and constants q11(w) and q10(w) are values of first integrals q10 and q11 on thesolution w. The function ζ(α) is the Weierstrass zeta function and σ(z) is theWeierstrass sigma function with invariants

g2 = 4c21 − 6c1

3, g3 = −152c3

1

27− 5q11(w)c1

3+ 2q10(w).


The eigenfunctions corresponding to the eigenvalue λ has the following form:

yλ(x) = 2

√|λ+ 1

2 w(x)+ c1|Q1(λ)

× sin

(√Q1(λ)

2

(2(x − a)ζ(α)+ ln

σ(x − α)σ(a+ α)

σ(x + α)σ(a− α)

)).

Proof Since

w = −2℘(x + c, g2, g3)− 2c1

3

we have

I =∫ b

a

dτ

λ+ 12 w+ c1

=∫ b

a

dτ

λ− ℘(τ + c, g2, g3)+ 2c1/3.

Let α be such that

℘(α, g2, g3) = λ+ 2c1/3.

Then,

I = −∫ b+c

a+c

dτ

℘ (τ , g2, g3)− ℘(α, g2, g3)

= − 1

℘′(α, g2, g3)

(2zξ(α)+ ln

σ(z − α)

σ(z + α)

) ∣∣∣∣b+c

a+c.

Because

℘′2(x, g2, g3) = 4℘3(x, g2, g3)− g2℘(x, g2, g3)− g3

and

℘′′(x, g2, g3) = 6℘2(x, g2, g3)− g2/2,

we obtain the following values of the first integrals q11 and q10:

q11(w) = w′′ + 3w2 + 4c1w− 4c21 = g2 − 16/3c2

1 = −2c1 − 4c21,

q10(w) = 2ww′′ − w′2 + 4w3

4+ c1(w

′′ + 4w2 + 4c1w)

= 2/3c1g2 + g3 − 32/27c31.


Note that for

℘(α, g2, g3) = λ+ 2c1/3

we have

℘′2(α, g2, g3) = 4℘3(α, g2, g3)− g2℘(α, g2, g3)− g3 = −Q1(λ).

Therefore

℘′(α, g2, g3) = ±√−Q1(λ)

and we find that

I = ± 1√−Q1(λ)

(2zξ(α)+ ln

σ(z − α)

σ(z + α)

) ∣∣∣∣b+c

a+c= 2πn√

Q1(λ).

So, we have

±.

((2zζ(α)+ ln

σ(z − α)

σ(z + α))

) ∣∣∣∣b+c

a+c= ι2πn,

where ζ(α) is the Weierstrass zeta function and σ(z) is the Weierstrass sigmafunction. Finally, we obtain the following equations for α:

2(b− a)ζ(α)+ lnσ(b+ c− α)σ(a+ c+ α)

σ(a+ c− α)σ(b+ c+ α)= ι2πn. �

In the similar way one finds the following result.

Theorem 2.5.4 Let a potential w satisfies the n-th KdV equation, then spectralvalues λ for the boundary problem (2.11) given by solutions of the equation∫ b

a

dτ

Sn(τ )= 2πm√

Qn(λ), m ∈ Z,

such that Qn(λ) > 0 and Sn(τ ) �= 0 on the interval [a, b].The eigenfunctions corresponding to the eigenvalue λ have the following

form:

yλ(x) = 2

√|Sn(x)|Qn(λ)

sin

(√Qn(λ)

2

∫ x

a

dτ

Sn(τ )

).


2.5.3 Lagrange integrals

Recall that a linear differential operator

L = ∂k+1 +k∑

i=0

Ai∂i

and its adjoint

Lt = (−1)k+1 +k∑

i=0

(−1)i∂ i ◦ Ai

are connected by the Lagrange formula

vL(u)− uLt(v) = ∂PL(u, v)

where

PL(u, v) =k+1∑i=1

i−1∑j=0

(−1)ju(i−j−1)∂ j(Aiv)

is the Lagrange form.In this section we shall use the Lagrange forms to obtain first integrals from

linear symmetries.

Example 2.5.2

1. For anti-self-adjoint operators L = 2A∂ + A′of first order the Lagrangeform is

PL(u, v) = 2Auv.

2. For self-adjoint operators L = A∂2+A′∂+B of second order the Lagrangeform is

PL(u, v) = A(u′v − uv′).

3. For anti-self-adjoint operators L = 2A∂3 + 3A′∂2 + (A′′ + 2B)∂ + B′ ofthird order the Lagrange form is

PL(u, v) = 2A(u′′v + uv′′ − u′v′)+ A′(u′v + uv′)+ 2Buv.


Let Sol(L) be the solution space of the homogeneous equation L(h) = 0, andlet L be self- or anti-self-adjoint operator. It follows from the Lagrange formulathat ∂PL(h1, h2) = 0 for any h1, h2 ∈ Sol(L), or PL(h1, h2) = const .

In other words,

PL : Sol(L)⊗ Sol(L)→ R,

PL : h1 ⊗ h2 −→ PL(h1, h2)

determines a bilinear form on the space of solutions.This form is symmetric if L is skew-adjoint, and skew-symmetric if L is

self-adjoint. We call it the Lagrange form.

Lemma 2.5.1 PL is a non-degenerate bilinear form.

Proof Let h0 ∈ Sol(L) belong to the kernel of the form, and h0 �= 0, say,h0(0) �= 0. Then PL(h0, h) = 0 for all h ∈ Sol(L). Take the solution h such thath(0) = 0, . . . , h(k−1)(0) = 0, h(k)(0) = 1. Then the Lagrange formula givesPL(h0, h) = h(0) �= 0. �

Summarizing we obtain the following theorem.

Theorem 2.5.5 The Lagrange form PL determines

(a) a symplectic structure on Sol(L) if L is self-adjoint, and(b) a pseudo-Euclidean structure if L is anti-self-adjoint.

Note that the space Sol(L) can be described, for example, by the initialconditions:

x0 = h(0), x1 = h′(0), . . . , xk = h(k)(0).

This gives an isomorphism between Sol(L) and Rk+1, and the Lagrange formulaallows one to compute PL explicitly.

Example 2.5.3 For the Schrödinger operator L = ∂2 +W we obtain

PL(x, y) = x1y0 − x0y1

or

PL(f , g) = f ′g− g′f

for f , g ∈ Sol(L).


Example 2.5.4 For the symmetric power L(2) = ∂3 + 4W∂ + 2W ′ we obtain

PL(2) (x, y) = x2y0 + y2x0 − x1y1 + 4W(0)x0y0

and the first integral

K(z) = PL(2) (z, z) = 2zz′′ − (z′)2 + 4Wz2

on the solution space, z ∈ Sol(L(2)).

Let � be an even symmetry of the equation, and let L be a self-adjointoperator, then L ◦� = −�t ◦L, or, in other words, L ◦� is a skew-self adjointoperator and L ◦ �(h) = 0 for all solutions h ∈ Sol(L). Therefore, PL◦� is asymmetric form on the space of solutions Sol(L). Similar reasoning proves thefollowing.

Theorem 2.5.6

1. Let L be a self-adjoint operator. Then the Lagrange form PL determines asymplectic structure on the solution space Sol(L), and(a) PL◦� ∈ �2(Sol(L)) is a skew-symmetric form on the space of solutions

Sol(L) for any � ∈ Sym0(L).(b) PL◦� ∈ S2(Sol(L)) is a symmetric form on the space of solutions Sol(L)

for any � ∈ Sym1(L).2. Let L be an anti-self-adjoint operator. Then the Lagrange form PL determines

a pseudo-Eucledean structure on the solution space Sol(L), and(a) PL◦� ∈ S2(Sol(L)) is a symmetric form on the space of solutions Sol(L)

for any � ∈ Sym0(L).(b) PL◦� ∈ S2(Sol(L)) is a skew-symmetric form on the space of solutions

Sol(L) for any � ∈ Sym1(L).

Example 2.5.5 Let L = ∂2+W be the Schrödinger operator . Then the identityoperator � = id ∈ Sym1(L) determines the symplectic structure on Sol(L).Moreover, as we have seen any solution y ∈ Sol(L) determines the symmetryEy = y2∂ − yy′ ∈ Sym0(L) and PL(2) (x

2, y2) = 2(x′y − y′x)2 for all x, y ∈Sol(L).

3

Model differential equations and theLie superposition principle

3.1 Symmetry reduction

3.1.1 Reductions by symmetry ideals

We begin with reduction of distributions by ideals of symmetry algebra.Let P be a completely integrable distribution on manifold M and let g be

a Lie algebra of shuffling symmetries. For any ideal j of g we define a newdistribution Pj as follows: the subspace Pj(a) ⊂ TaM is generated by vectorsof the former distribution P(a) and values of representatives of j at the point.

Theorem 3.1.1 The distribution Pj is completely integrable, and g/j is ashuffling symmetry algebra of Pj.

Proof The following two types of vector fields generate Pj:

(a) vector fields X ∈ D(P) that belong to the distribution P, and(b) representatives Z of shuffling symmetries Z ∈ j.

For all of them we have the following commutation relations:

X1 ∈ D(P), X2 ∈ D(P) =⇒ [X1, X2] ∈ D(P),

Z1 ∈ D(Pj), Z2 ∈ D(Pj) =⇒ [Z1, Z2] = [Z1, Z2] ∈ D(Pj),

Z1 ∈ D(Pj), X1 ∈ D(P) =⇒ [Z1, X1] ∈ D(P)

where the first relation is true because P is completely integrable, the secondone follows from the condition that j is closed with respect to the bracketand the third one follows from j ⊂Sym(P).

The last statement follows from the construction of Pj and the fact that j isan ideal of the Lie algebra g. �

76

3.1 Symmetry reduction 77

This theorem shows that integration of a completely integrable distributioncan be decomposed into two steps:

1. integration of the completely integrable distribution Pj with symmetry Liealgebra g/j, and

2. integration on integral manifolds Mc of the restrictions Pc with symmetryalgebra j.

First, applying this procedure to the radical r of algebra g we decompose theintegration problem into two parts: the integration of the distribution with thesemi-simple algebra g0 = g/r, and then the integration of the restrictions Pc

with the solvable symmetry algebra r.We have seen that the last step can be performed by quadratures. Moreover,

every semi-simple algebra Lie g0 is a direct sum of simple ones which areideals in g0. Thus, the above theorem together with the Lie–Bianchi the-orem reduces the integration problem to integrating completely integrabledistributions equipped with simple algebras of shuffling symmetries.

Before proceeding with the integration of this type of distribution we considera reduction procedure that gives us a way to reduce essentially dimensions.

3.1.2 Reductions by symmetry subalgebras

Together with actions of Lie groups we need actions of Lie algebras, where byan action of a Lie algebra g on a manifold N we mean an injective Lie algebrahomomorphism λ : g→D(N) of the Lie algebra g to the Lie algebra D(N) ofvector fields on N . We say that an action λ is transitive if values of vector fieldsλ(X), X ∈ g, at any point a ∈ N generate the whole tangent space TaN .

By ga = {X ∈ g |λ(X)a = 0}we denote the stability subalgebra of the pointa ∈ N .

Let P be a completely integrable distribution on M with a Lie algebra g

of shuffling symmetries of P and let N be a manifold with a transitive actionλ : g→D(N).

We will assume that dim g = codim P.Let us construct a new distribution P on the manifold M ×N as follows. For

any point x = (a, b) ∈ M × N denote by Px the subspace of Tx(M × N) =TaM ⊕ TbN generated by vectors in P(a) and, in addition, by vectors of theform Xa + λ(X)b for all X ∈ g.

Lemma 3.1.1 The distribution P is completely integrable, and codimP = dim g.

78 Model differential equations and the Lie superposition principle

Proof Vector fields in D(P) are linear combinations of the following vectorfields: X + λ(X) for X ∈ g, and Y ∈ D(P).

For them we have

[X1 + λ(X1), X1 + λ(X1)] = [X1, X2] + λ[X1, X2]mod D(P)

and

[X1 + λ(X1), Y ] = [X1, Y ] ∈ D(P). �

We say that a surjection

h : M → N

is an N-integral of P if the graph Lh = {(a, h(a))|∀a ∈ M} is an integralmanifold for the distribution P.

It is easy to see that any maximal integral manifold of P has the form ofN-integral.

Theorem 3.1.2 Let h be an N-integral of P and let Mb = h−1(b) ⊂ M, b ∈ N .Then

1. P(a) ⊂ Ta(Mb) for any a ∈ Mb,2. the restriction Pb of P on Mb is a completely integrable distribution with

shuffling symmetry algebra gb, and3. codim Pb = dim gb.

Proof It follows from the construction P that P(a) belongs to TaMb. Further,vector fields X, X ∈ g, tangent to Mb if λ(X)b = 0. Therefore, gb is a symmetryalgebra for Pb. The last statement of theorem follows from transitivity λ. �

3.2 Model differential equations

To find an integral h : M → N we need solutions of some ordinary differentialequations on the homogeneous space N . Let us fix points a ∈ M and b ∈ N . Tofind the value of h at x ∈ M providing h(a) = b, we take a path x(t) on M suchthat x(0) = a, x(1) = x. Then tangent vectors to (x(t), y(t) = h(x(t))) shouldbe in P(y(t)). To find this system of ODEs we consider the local descriptionof the problem. Let differential 1-forms ω1, . . . ,ωm be a local basis for P, letdifferential 1-forms α1, . . . ,αk be a local basis for the cotangent bundle T∗N ,and let X1, . . . , Xm be representatives of a basis X1, . . . , Xm in g. We also assumethat ωj(Xi) = δij.

3.2 Model differential equations 79

Then differential 1-forms

β1 = α1 − 〈α1, λ(X1)〉ω1 − · · · − 〈α1, λ(Xm)〉ωm,

......

......

...

βk = αk − 〈αk , λ(X1)〉ω1 − · · · − 〈αk , λ(Xm)〉ωm

determine P.Indeed, if Y ∈ D(P) then βi(Y) = 0 and ωj(Y) = 0, therefore αi(Y) = 0,

and

βi(Xj + λ(Xj)) = 〈αi, λ(Xj)〉 − 〈αi, λ(Xj)〉 = 0.

Therefore, the ODEs system for y(t) takes the form

〈α1, y(t)〉 = 〈α1, λ(X1)〉〈ω1, x(t)〉 − · · · − 〈α1, λ(Xm)〉〈ωm, x(t)〉,...

......

......

......

〈αk , y(t)〉 = 〈αk , λ(X1)〉〈ω1, x(t)〉 − · · · − 〈αk , λ(Xm)〉〈ωm, x(t)〉.

Let us denote 〈ωi, x(t)〉 by Ai(t). Then the above system gives

〈αi, y(t)− A1(t)λ(X1)− · · · − Am(t)λ(Xm)〉 = 0

for all i = 1, . . . , r, and therefore

y(t) = A1(t)λ(X1)+ · · · + Am(t)λ(Xm). (3.1)

In other words, to find the N-integral h one should solve (3.1).Differential equations of this form are extremely important to us. We call

them model differential equations .Formally these equations can be described as follows. Let N = G/H be a

homogeneous space and let λ : g → D(N) be the natural representation of theLie algebra of group G by vector fields on the homogeneous space. Let Xt be apath on the Lie algebra g, Xt ∈ g. Then the system of ODEs corresponding tothe vector field

∂

∂t+ λ(Xt) (3.2)

on R × N is called the model differential equations corresponding to modelN = G/H .


Theorem 3.2.1 Let N = G/H be a homogeneous space. A system of ODEsZ = ∂/∂t + Y on R× N has the form (3.2) for some path Xt ∈ g if and only ifthere is a path g(t) ∈ G, g(0) = e, on G such that any trajectory y(t) of Z canbe written in the following form:

y(t) = g(t)y(0). (3.3)

Proof Let g(t) be given. Then identifying g with TeG, where e is the unit ofG, we define the path

Xt = g∗(t)−1(g(t)) (3.4)

on g, and differentiating (3.3) we obtain y(t) = λ(Xt)y(t).Since y(0) is arbitrary the relation (3.3) gives all solutions of the model ODEs

with path (3.4).Conversely, assume that the path Xt is given. Then considering (3.4) as ODEs

for g(t) with initial conditions g(0) = e we get g(t). �

3.2.1 One-dimensional model equations

Model ODEs on one-dimensional homogeneous spaces correspond to com-monly used scalar valued integrals h : M → R.

We begin with a local description (at 0 ∈ R) of finite-dimensional Liealgebras which can be realized as transitive Lie algebras of vector fieldson R.

Let λ : g→D(R) be a transitive action, and let

λ(X) = fX(t)d

dt

for X ∈ g.Transitivity of the action means that there is X1 ∈ g such that fX1(0) �= 0,

and assume that the coordinate t is chosen so that

λ(X1) = d

dt.

Denote by ord(Y), Y ∈ g, the order of zero λ(Y) at 0 ∈ R, and let Z be thevector of maximal order k. Then the commutator [Y , Z] has order ≤ l + k − 1if ord(Y) = l. Therefore, l + k − 1 ≤ k, or l ≤ 1, and dim g ≤ 3.

3.2 Model differential equations 81

Suppose that dim g = 2. Then g = 〈X1, X2〉 where

λ(X1) = d

dt, λ(X2) = f (t)

d

dt.

The commutator

[λ(X1), λ(X2)] = λ([X1, X2]) = f ′(t) d

dt

is a linear combination of λ(X1) and λ(X2).Therefore,

f ′ = a+ bf

for some constants a, b ∈ R, and f = a+ Cebt , b �= 0.Therefore, we can change the coordinate t −→ ebt and the basis in g in such

a way that

λ(X1) = d

dt, λ(X2) = t

d

dt.

Let dim g = 3 and let X1, X2, X3 be a basis X1, X2, X3 in g such thatord(Xi) = i − 1.

Then as above we can assume that λ(X1) = d/dt, λ(X2) = td/dt and

λ(X3) = g(t)d

dt

where g(t) has a second-order zero at t = 0.Moreover, g′ and tg′ − g are linear combinations of 1, t, g. From this we

conclude that the third basis vector X3 can be chosen in such a way that

λ(X3) = t2 d

dt.

Summarizing we get the following.

Theorem 3.2.2 (Sophus Lie) Suppose that a Lie algebra λ : g→D(R) actstransitively on R. Then dim g ≤ 3 and there exists a local coordinate t in someneighborhood of 0 ∈ R and a basis 〈Xi〉 in g such that the representation λ hasthe following form:

1. dim g = 1:

λ(X1) = d

dt;


2. dim g = 2:

λ(X1) = d

dt, λ(X2) = t

d

dt;

3. dim g = 3:

λ(X1) = d

dt, λ(X2) = t

d

dt, λ(X3) = t2 d

dt.

Corollary 3.2.1 Model ODEs of dimension one have one of the followingforms:

1. y′ = a(t);2. y′ = a(t)+ b(t)y;3. y′ = a(t)+ b(t)y + c(t)y2.

3.2.2 Riccati equations

Here we discuss the geometry that produces the third model or Riccati equation.First of all, let us note that the one-dimensional realization of the three-

dimensional Lie algebra is a representation of the Lie algebra g = sl2(R) byvector fields.

Let G = SL2(R) be the Lie group of 2× 2 matrices with determinant equalto 1. This group acts on R2 in the natural way. Let us fix a vector v0 ∈ R2, sayv = (1, 0), and consider the following subgroup:

H = {g ∈ SL2(R)|gv0 = λv0 for some λ}

or

H ={∥∥∥∥a11 a12

0 a22

∥∥∥∥ ∈ SL2(R)

}.

The homogeneous space N = G/H isomorphic to the projective line RP1 by

g =∥∥∥∥a11 a12

a21 a22

∥∥∥∥ ∈ SL2(R) −→ [a11 : a21] ∈ RP1.

Under this identification the natural action of SL2(R) on N corresponds to theprojective transformations

[x1 : x2] −→ [a11x1 + a12x2 : a21x1 + a22x2].

3.3 Model equations: the series Ak , Dk , Ck 83

The differential of this action determines the representation λ of sl2(R). To findthis action consider the canonical basis

X+ =∥∥∥∥0 1

0 0

∥∥∥∥ , X− =∥∥∥∥0 0

1 0

∥∥∥∥ , H =∥∥∥∥1 0

0 −1

∥∥∥∥in sl2(R).

Then,

exp(tH) : [x1 : x2] −→ [etx1 : e−tx2],exp(tX+) : [x1 : x2] −→ [(x1 + tx2) : x2],exp(tX−) : [x1 : x2] −→ [x1 : (x2 + tx1)].

Let us identify R with the affine part of RP1 by x −→ [x : 1], then

exp(tH) : x −→ e2tx,

exp(tX+) : x −→ x + t,

exp(tX−) : x −→ x

1+ tx.

Therefore,

λ : H −→ 2x∂ ,

λ : X+ −→ ∂ ,

λ : X− −→ −x2∂ ,

and the corresponding model equation is the Riccati equation.

3.3 Model equations: the series Ak , Dk , Ck

In this and the next two sections we discuss model equations that correspondto the simple Lie algebras of the series Ak , Bk and Dk .

3.3.1 Series Ak

Let E be a vector space of dimension n+ k equipped with a volume form andlet Grn+k,k be the Grassmanian of k-dimensional subspaces in E.


The Lie group SLn+k(R) = SL(E) acts in the natural way on Grn+k,k:

g : L ⊂ E −→ g(L) ⊂ E

for all g ∈ SL(E), L ∈ Grn+k,k.This action is obviously transitive, and the stationary subgroup H = HL of

an element L ∈ Grn+k,k is

H = {g ∈ SL(E), g(L) = L}.Thus,

Grn+k,k = SLn+k(R)/H.

To find a coordinate description of the above action and therefore a modelequation of the series A we introduce the canonical coordinates into Grn+k,k.

Let us fix an element L0 ∈ Grn+k,k and let L⊥0 be a complementaryn-dimensional subspace:

E = L0 ⊕ L⊥0 .

Then all elements L ∈ Grn+k,k that belong to a sufficiently small neighborhoodof L0 are transversal to L⊥0 and can be represented by graphs of linear operatorsA : L0 → L⊥0 , in other words, we obtain local coordinates

A ∈ hom(L0, L⊥0 ) −→ LA = {x ⊕ Ax|x ∈ L0} ∈ Grn+k,k .

Let us write down the SLn+k(R)-action in these coordinates.To this end we represent the vectors x ∈ E as pairs

x =∥∥∥∥x1

x2

∥∥∥∥where x1 ∈ L0 and x2 ∈ L⊥0 , and operators g ∈ SL(E) as matrices

g =∥∥∥∥g11 g12

g21 g22

∥∥∥∥with g11 ∈ hom(L0, L0), g12 ∈ hom(L⊥0 , L0), g21 ∈ hom(L0, L⊥0 ), g22 ∈hom(L⊥0 , L⊥0 ), and elements of the stability subgroup HL0 are operators ofthe form ∥∥∥∥h11 h12

0 h22

∥∥∥∥ .


The action in these coordinates has the form

x =∥∥∥∥ x1

x2 = Wx1

∥∥∥∥ g−→∥∥∥∥g11x1 + g12Wx1

g21x1 + g22Wx1

∥∥∥∥ = ∥∥∥∥ x′1g(W)x′1

∥∥∥∥ ,

where

g : W −→ g(W) = (g21 + g22W) ◦ (g11 + g12W)−1.

Let us find the corresponding representation λ : sln+k(R) → D(Grn+k,k) ofthe Lie algebra sln+k(R) into vector fields on Grn+k,k. First of all, we note thatelements of sln+k(R) correspond to matrices

X =∥∥∥∥x11 x12

x21 x22

∥∥∥∥ ,

where

tr x11 + tr x22 = 0.

To find λ(X) let us denote by xij(t) the components of exp(tX):

exp(tX) =∥∥∥∥x11(t) x12(t)

x21(t) x22(t)

∥∥∥∥ .

Then,

dxij(t)

dt

∣∣∣∣t=0= xij, and xij(0) = δij.

We have

W(t) = exp(tX)(W) = (x21(t)+ x22(t)W) ◦ (x11(t)+ x12(t)W)−1

and therefore

λ(X) = (x21 + x22W −W(x11 + x12W))∂

∂W

where tr x11 + tr x22 = 0.The A-model differential equation therefore takes the form of the matrix

sl-Riccati equation

W = A21(t)+ A11(t)W −WA22(t)−WA12(t)W


with

tr A11 + tr A22 = 0.

Example 3.3.1 Take n = 1, k = 2. Then g = sl3(R) and Gr3,2 =RP2. Thecorresponding Riccati equation is a system of the following form:

x1 = a1(t)+ (2a11(t)+ a22(t))x1 + a12(t)x2 + Q1(x1, x2),

x2 = a2(t)+ a21(t)x1 + (a11(t)+ 2a22(t))x2 + Q2(x1, x2),

where Q1 and Q2 are quadrics in x1 and x2.

3.3.2 Series Dk

Let E be a Euclidean vector space of dimension n + k with a metric g, and letSO(g) = SO(n+ k) be the group of orthogonal transformations preserving theorientation. Because SO(n+k) ⊂ SL(n+k), the group G = SO(g) acts on theGrassmanian N = Grn+k,k and the action is transitive. To find the correspondingrepresentation of the Lie algebra son+k we use the same coordinate systems asabove with L⊥0 being the orthogonal complement of L0 ∈ Grn+k,k. In thesecoordinates the operator

X =∥∥∥∥x11 x12

x21 x22

∥∥∥∥belongs to son+k if and only if

xt11 + x11 = 0, x22 + xt

22 = 0, x12 + xt21 = 0.

Therefore

λ(X) = (x21 + x22W −W(x11 − xt21W))

∂

∂W

and the D-model differential equations are matrix so-Riccati equations ofthe form

W = A12(t)+ A11(t)W −WA22(t)+WAt12(t)W

where A11 and A22 are skew-symmetric matrices.


Example 3.3.2 Take n = 1, k = 2, g = so3(R) and Gr3,2 =RP2. Thecorresponding so-Riccati equation is the system of the following form:

x1 = a1(t)− a(t)x2 + a1(t)x21 + a1(t)a2(t)x1x2,

x2 = a2(t)+ a(t)x1 + a2(t)x22 + a1(t)a2(t)x1x2.

3.3.3 Series Ck

Let E be a symplectic space (see), of dimension 2n with a structure of the formω, and let G = Sp(E) = Sp(2n) be the group of symplectic transformations of(E,ω):

Sp(E) = {A ∈ GL(E),ω(Ax, Ay) = ω(x, y)}.

Owing to Darbouxus, theorem there exists a canonical basis e1, . . . , en,en+1, . . . , e2n in E such that

ω(x, y) = (x1yn+1 − xn+1y1)+ · · · + (xny2n − x2nyn).

In this basis the matrix

A =∥∥∥∥a11 a12

a21 a22

∥∥∥∥of an operator A, where aij are n× n matrices, belongs to Sp(E) if and only if

A�At = �

where

� =∥∥∥∥ 0 1−1 0

∥∥∥∥is the matrix of ω in the canonical basis.

Elements of the Lie algebra sp2n can therefore be represented by matrices

X =∥∥∥∥x11 x12

x21 x22

∥∥∥∥where

x11 + xt22 = 0, x12 = xt

12, x21 = xt21, (3.5)


or by matrices

X =∥∥∥∥∥x11 x12

x21 −xt11

∥∥∥∥∥where x12 and x21 are symmetric matrices.

Consider the action Sp(E) on Grassmanians Gr2n,k. This action is not trans-itive if 1 < k < 2n because, due to the Witt theorem, two k-dimensionalsubspaces L and L′ belong to the same Sp-orbit if and only if restrictions of thestructure form ω|L and ω|L′ have the same rank.

We consider model equations that correspond to the case of the Grassmanianof Lagrangian subspaces.

A subspace L ⊂ V is called Lagrangian if

1. dim L = n;2. L is an isotropic subspace, that is, ω(x, y) = 0 for all x, y ∈ L.

Denote by LGr2n the Grassmanian of Lagrangian subspaces. Then the actionof the Lie group G = Sp(E) on N = LGr2n is transitive.

To find the coordinate description of the corresponding model equations. Weshall use the same coordinate system as in series A with the extra requirementthat L⊥0 is a Lagrangian subspace too.

Then L⊥0 is isomorphic to the dual space L∗0 by the correspondence

X ∈ L⊥0 −→ X = ω(X , ·) ∈ L∗0 .

In this case a graph linear map W : L0 → L⊥0 = L∗0 determines a Lagrangiansubspace

LW = {x ⊕Wx|x ∈ L0}

if and only if W is self-adjoint

W = W∗.

Indeed, if x, y ∈ L0, then

ω(x +Wx, y +Wy) = ω(x, Wy)+ ω(Wx, y)

= −(Wy)(x)+ y(Wx) = (−Wy + W∗(y))(x) = 0.

3.4 The Lie superposition principle 89

Using (3.5) we obtain

λ(X) = (x21 + x11W −W(−xt11 + x12W))

∂

∂W

where x12 and x21 are symmetric.Finally, the corresponding matrix sp2n-Riccati equation has the form

W = A21(t)+ A11(t)W +WAt11(t)−WA12(t)W

where W and A12, A21 are symmetric n× n matrices.

3.4 The Lie superposition principle

Sophus Lie found that solutions of model differential equations on homogen-eous spaces possess some superposition rules.

Let us consider a homogeneous space N = G/H and a model system

∂

∂t+ λ(Xt), Xt ∈ g. (3.6)

As we have seen in (3.2.1) any such equation determines a path g(t), g(0) = e,on the group G such that any solution x(t) of the model equation can be obtainedusing the formula

x(t) = g(t)x(0). (3.7)

Therefore, instead of finding solutions of (3.6) we can try to find the path g(t)and then obtain all solutions by (3.7).

Remarkably g(t) can be found by using some solutions and the geometry ofthe homogeneous space.

Let us consider the diagonal action of the group Lie G on the direct productNk = N × · · · × N :

g : (a1, . . . , ak) −→ g(a1, . . . , ak) = (ga1, . . . , gak).

We say that the homogeneous space N is k-stiff if the set of points (a1, . . . , ak)

with trivial stationary subgroup is dense in Nk .Denote by Ga the stationary subgroup of a ∈ N . Then the stationary subgroup

of (a1, . . . , ak) ∈ Nk is equal to the intersection Ga1 ∩ Ga2 ∩ · · · ∩ Gak .


Let us call a point (a1, . . . , ak) ∈ Nk regular if

Ga1 ∩ Ga2 ∩ · · · ∩ Gak = e.

On orbits of regular points we obtain a function F(a1, . . . , ak ; b1, . . . , bk) ∈ Gsuch that

(b1, . . . , bk) = F(a1, . . . , ak ; b1, . . . , bk) · (a1, . . . , ak)

for all regular points (b1, . . . , bk) ∈ G · (a1, . . . , ak).A family y1(t), . . . , yk(t) of solutions of the model equation is called a

fundamental solution if (y1(0), . . . , yk(0)) is a regular point in Nk .The fundamental solution allows us to find the path g(t) as follows:

g(t) = F(y1(0), . . . , yk(0); y1(t), . . . , yk(t))

and solutions of the model equation are

x(t) = F(y1(0), . . . , yk(0); y1(t), . . . , yk(t)) · x(0). (3.8)

The function F is called the superposition function, and the way of obtainingsolutions by (3.8) is called the Lie superposition principle.

Theorem 3.4.1 (The Lie superposition principle) Let ∂/∂t+ λ(Xt), Xt ∈ g,be a model equation on a homogeneous space G/H. Let k be the stiff numberof G/H and F(a1, . . . , ak ; b1, . . . , bk) be the superposition function. Then forany fundamental solution y1(t), . . . , yk(t) (3.8) gives all solutions of the modelequation.

Example 3.4.1 Let us consider the case of one-dimensional homogeneousspaces (3.2.2). Then

1. If g = R, then k = 1, and F(x, y) = y− x.The fundamental solution is a solution of the equation y′ = a(t). Then thesuperposition principle gives

x(t) = x(0)+ (y(t)− y(0))

for any solution x(t).2. If dim g = 2, G is the “ax + b”-group of affine transformations of the line.

Therefore k = 2, and F(x1, x2; y1, y2) corresponds to the transformation

x −→ (x − x2)y1 − (x − x1)y2

x1 − x2.


The fundamental solution of the model equation y′ = a(t)y+b(t) is a pair ofsolutions (y1(t), y2(t)) such that y1(0) �= y2(0). The superposition principlegives

x(t) = (x(0)− y2(0))y1(t)− (x(0)− y1(0))y2(t)

y1(0)− y2(0).

3. If dim g = 3, G = SL2(R) is the group of projective transformations ofthe form

x −→ a11x + a12

a21x + a22.

It is known in projective geometry that every projective transformation isdetermined by the images of three different points. Therefore the stiffness ofRP1 is equal to 3. It is also known that the double ratio

x − x1

x − x2

x3 − x2

x3 − x1

is invariant in the projective transformations. Therefore,

y = F(x1, x2, x3; y1, y2, y3) · x

can be found from the equation

x − x1

x − x2

x3 − x2

x3 − x1= y − y1

y − y2

y3 − y2

y3 − y1.

The fundamental solution is a triple (y1(t), y2(t), y3(t)) of solutions of theRiccati equation y′ = a(t) + b(t)y + c(t)y2 such that y1(0), y2(0), y3(0)are distinct. In this case any solution x(t) can be found from the doubleratio:

x(0)− y1(0)

x(0)− y2(0)

y3(0)− y2(0)

y3(0)− y1(0)= x(t)− y1(t)

x(t)− y2(t)

y3(t)− y2(t)

y3(t)− y1(t).

4. Stiffness of RPn is equal to n+ 2.5. Stiffness of Gr2n,n for sl-Riccati equations is equal to 5.6. Stiffness of Gr2n,n for so-Riccati equations is equal to 4.7. Stiffness of LGr2n for sp-Riccati equations is equal to 4 if n ≥ 2.


3.4.1 Bianchi equations

Bianchi classified all primitive and symplectic actions of Lie algebras on theplane. They are:

1. dim g = 5, and

g =⟨∂

∂x1,∂

∂x2, x1

∂

∂x2, x2

∂

∂x1, x1

∂

∂x1− x2

∂

∂x2

⟩

is the Lie algebra of all affine symplectic transformations on R2.The corresponding model equations has the form

x1 = a1(t)+ a11(t)x1 + a12(t)x2,

x2 = a2(t)+ a21(t)x1 − a11(t)x2

and the stiffness equals to 3.2. dim g = 3, and

g =⟨∂

∂x1,∂

∂x2, x1

∂

∂x2− x2

∂

∂x1

⟩

is the Lie algebra of the Lie group of motions on R2.The corresponding model equations are

x1 = a1(t)+ a(t)x2,

x2 = a2(t)− a(t)x1

and the stiffness equals 2.3. dim g = 3, and

g =⟨x1

∂

∂x2− x2

∂

∂x1, (1+ x2

1 − x22)

∂

∂x1+ 2x1x2

∂

∂x2,

2x1x2∂

∂x1+ (1+ x2

2 − x21)

∂

∂x2

⟩is the Lie algebra of motions on the sphere equipped with the metric

g = dx21 + dx2

2

(1+ x21 + x2

2)2

.


The corresponding model equations are

x1 = −a1(t)x2 + a2(t)(1+ x21 − x2

2)+ 2a3(t)x1x2,

x2 = a1(t)x1 + 2a2(t)x1x2 + a3(t)(1+ x22 − x2

1)


g =⟨x1

∂

∂x2− x2

∂

∂x1, (1+ x2

2 − x21)

∂

∂x1− 2x1x2

∂

∂x2,

− 2x1x2∂

∂x1+ (1+ x2

1 − x22)

∂

∂x2

⟩is the Lie algebra of motions on the pseudosphere equipped with the metric

g = dx21 + dx2

2

(1− x21 − x2

2)2

.


x1 = −a1(t)x2 + a2(t)(1+ x22 − x2

1)− 2a3(t)x1x2,

x2 = a1(t)x1 − 2a2(t)x1x2 + a3(t)(1+ x21 − x2

2)


g =⟨∂

∂x2, x1

∂

∂x2− x2

∂

∂x1, 2x1x2

∂

∂x1+ (x−2

1 − x22)

∂

∂x2

⟩is the Lie algebra of motions on the sphere equipped with the metric

g = x−21 dx2

1 + x21 dx2

2.


x1 = a2(t)x2 + 2a3(t)x1x2,

x2 = a1(t)− a2(t)x2 + a3(t)(x−21 − x2

2)

and the stiffness equals 2.


3.5 AP-structures and their invariants

3.5.1 Decomposition of the de Rham complex

A real AP-structure on a smooth real manifold M is an ordered set P of rdistributions P1, . . . , Pr , P = (P1, . . . , Pr), such that at each point a ∈ M thetangent space TaM splits into the direct sum

TaM =r⊕

i=1

Pi(a).

A complex AP-structure on a smooth real manifold M is an ordered setP of r complex distributions P1, . . . , Pr , P = (P1, . . . , Pr), such that at eachpoint a ∈ M the complexification of the tangent space TaM splits into thedirect sum

TaMC =r⊕

i=1

Pi(a).

Let us consider a real AP-structure on M. The vector space �s(T∗a M) ofexterior s-forms on TaM falls into the direct sum

�s(T∗a M) =⊕|k|=s

�k(T∗a M),

where k is a multi-index, k =(k1, . . . , kr), ki ∈ {0, 1, . . . , dim Pi}, |k| = k1 +· · · + kr , and

�k(T∗a M)def=

r⊗i=1

�ki(Pi,a).

Here

�ki(Pi,a)def= {α ∈ �ki(T∗a M)|X�α = 0,∀X ∈ P1,a ⊕ · · · ⊕ Pi,a ⊕ · · · ⊕ Pr,a}.

We obtain the following decomposition of the de Rham complex theC∞(M)-modules of differential s-forms �s(M) split into the direct sums

�s(M) =⊕|k|=s

�k(M), (3.9)

where

�k(M)def=

r⊗i=1

�ki(Pi)

3.5 AP-structures and their invariants 95

and

�ki(Pi)={α ∈ �ki(M)|X�α = 0, ∀X ∈ D(P1)⊕ · · · ⊕ D(Pi)⊕ · · · ⊕ D(Pr)

}.

The de Rham differential d splits into the following direct sum:

d =⊕|σ |=1

dσ ,

where σj ∈ Ijdef= {z ∈ Z||z| ≤ dim Pj} and

dσ : �k(M)→ �k+σ (M).

The following theorem allows one to construct tensor invariants of aAP-structure.

Theorem 3.5.1 If one of the components ti of a multi-index t is negative, thenthe operator dt is a C∞(M)-homomorphism.

Proof It is sufficient to prove that dt(hα) = hdt(α) for any function h ∈C∞(M). For α ∈ �k(M) we obtain

d(hα) =∑|σ |=1

dα(hα) =r∑

i=1

d1i(hα)+∑|t|=1

dt(hα),

where 1i = (0, . . . , 1i, . . . , 0) and the multi-indexes t have negative compon-ents. On the other hand,

d(hα) = dh ∧ α + hdα =(

r∑i=1

d1i h

)∧ α + h

∑|t|=1

dtα.

Therefore d1i h∧α /∈ �k+t(M) for each i = 1, . . . , r and one finds that dt(hα) =hdtα for any h ∈ C∞(M). �

In other words, if one of the components ti of the multi-index t is negative,then the operator dt is a tensor which acts from �k(M) to �k+t(M).

Suppose that dt is such a tensor. Then the difference between the number ofnon-negative components s and the number of negative components in t is 1.The tensor dt is a sum of the tensors of the type θ ⊗X , where θ is a differentials-form and X is the (s− 1)-vector field on M.


Recall that the tensor θ ⊗ X acts on a differential form α as

(θ ⊗ X)(α) = θ ∧ (X�α).

Remark 3.5.1 The constructed tensors dt are analogs of the Nijenhuis tensor[26]. Recall that the Nijenhuis tensor arises in a bi-graded complex (see [75]).

A tensor field TP on M that is associated with P is called a tensor invariantof P if F∗(TP ) = TF∗(P) for any diffeomorphism F : M → M. Here F∗(P) =(F∗(P1), . . . , F∗(Pr)) and a diffeomorphism F acts on a tensor β ⊗ X by thefollowing rule: F∗(β ⊗ X) = F∗(α)⊗ F−1∗ (X).

Constructed tensors dt are invariants with respect to diffeomorphisms of themanifolds M.

In the case of a complex AP-structure one can construct similar tensor invari-ants. In this case instead of the decomposition (3.9) one must consider thedecomposition

�s(M)C =⊕|k|=s

�k(M)C,

where k is a multi-index, k =(k1, . . . , kr), ki ∈ {0, 1, . . . , dimC Pi},

�k(M)Cdef=

r⊗i=1

�ki(Pi),

and

�ki(Pi)

= {α ∈ �ki(M)C | X�α = 0, ∀X ∈ D(P1)⊕ · · · ⊕ D(Pi)⊕ · · · ⊕ D(Pr)},

where

D(Pi) = {X ∈ D(M)C | Xa ∈ Pa, ∀a ∈ M}

is a C∞(M)C-module of complex vector fields from the distribution Pi

(i = 1, . . . , r).

3.5.2 Classical almost product structures

Recall that a field of endomorphisms (a tensor field of (1, 1)-type) A on a 2n-dimensional smooth manifold M is called a classical almost product structureif A2= 1.


If on a smooth manifold M an almost product structure is defined, at each pointa ∈ M the tangent space TaM splits into the direct sum of two eigensubspacesof A: TaM = V+(a)⊕ V−(a):

V±(a)def= {X ∈ TaM | AaX = ±X}.

In the other words, an almost product structure on a 2n-dimensional smoothmanifold M generates the pair of n-dimensional distributions V+ and V−on M.

Let us consider an almost product structure A on a four-dimensional smoothmanifold M. Then we obtain P =(P1, P2), where dim P1 = dim P1 = 2. Themodule �s(M) falls into the direct sum

�s(M) = ⊕p+q=s

�p,q(M),

where

�p,q(M)def= �p(P1)⊗�q(P2).

Here

�p(P1) = {α ∈ �p(M)|X�α = 0 for any vector field X ∈ D(P2)},�q(P2) = {α ∈ �q(M)|X�α = 0 for any vector field X ∈ D(P1)}.

Let us write these decompositions for each s:

�0(M) = C∞(M),

�1(M) = �1,0(M)⊕�0,1(M),

�2(M) = �2,0(M)⊕�1,1(M)⊕�0,2(M),

�3(M) = �2,1(M)⊕�1,2(M),

�4(M) = �2,2(M).

Moreover, we obtain the following decomposition of the exterior differentiald : �s(M)→ �s+1(M):

d = d1,0 ⊕ d0,1 ⊕ d2,−1 ⊕ d−1,2.

The components d2,−1 and d−1,2 in this sum are tensors.


Proposition 3.5.1

(1) The distribution P1 is completely integrable if and only if d2,−1 = 0.(2) The distribution P2 is completely integrable if and only if d−1,2 = 0.

Proof We prove the first statement only. Suppose that the distribution P1 isgenerated by a pair of 1-forms α,β ∈ �0,1(M),α ∧ β �= 0. This distributionis completely integrable if and only if dα = α ∧ γ1 + β ∧ δ1, and dβ =α ∧ γ2 + β ∧ δ2 for some 1-forms γi, δi ∈ �1(M), (i = 1, 2). This means thatdα, dβ /∈ �2,0(M). �

3.5.3 Almost complex structures

A field of real endomorphisms A on a 2n-dimensional smooth manifold M iscalled an almost complex structure if A2 = −1.

An almost product structure defines at each point a ∈ M a splitting of the com-plexification of the tangent space TaM in the direct sum of two eigensubspacesof A: TaMC = V+(a)⊕ V−(a):

V±(a)def= {X ∈ TaMC | AaX = ±ιX},

where ι = √−1. This means that an almost complex structure generates thepair of complex distributions V+ and V− on a real smooth manifold M.

Suppose that n = 2. We obtain the following decomposition of the de Rhamcomplex:

�s(M)C =⊕

p+q=s

�p,q(M),

d = d1,0 ⊕ d0,1 ⊕ d2,−1 ⊕ d−1,2

where �p,q(M) = �p(V+)⊗�q(V−) and di,j : �p,q(M)→ �p+i,q+k(M).Following the above results, d−1,2 and d2,−1 are tensor invariants of the

Monge – Ampère equation Eω. Since A is real, the tensors d−1,2 and d2,−1 arecomplex conjugated: d−1,2 = d2,−1.

3.5.4 AP-structures on five-dimensional manifolds

Let M be a five-dimensional smooth manifold and P = (P1, P2, P3) a realAP-structure. Here dim P1 = dim P3 = 2 and dim P2 = 1.


The module of differential s-forms on M falls into the direct sum

�s(M) =⊕

r+p+q=s

�p,r,q(M),

where

�p,r,q(M)def= �p(P1)⊗�r(P2)⊗�q(P3)

and

�p(P1) = {α ∈ �p(M)|X�α = 0 for any vector field X ∈ D(P2)⊕ D(P3)},�r(P2) = {α ∈ �r(M)|X�α = 0 for any vector field X ∈ D(P1)⊕ D(P3)},�q(P3) = {α ∈ �q(M)|X�α = 0 for any vector field X ∈ D(P1)⊕ D(P2)}.

Let us write these decompositions for each s:

�0(M) = C∞(M),

�1(M) = �1,0,0(M)⊕�0,1,0(M)⊕�0,0,1(M),

�2(M) = �2,0,0(M)⊕�1,1,0(M)⊕�1,0,1(M)⊕�0,1,1(M)⊕�0,0,2(M),

�3(M) = �2,1,0(M)⊕�2,0,1(M)⊕�1,1,1(M)⊕�1,0,2(M)⊕�0,1,2(M),

�4(M) = �2,1,1(M)⊕�2,0,2(M)⊕�1,1,2(M),

�5(M) = �2,1,2(M).

Moreover, we have the following decomposition of the exterior differential:

d =⊕

i+j+k=1i,k∈I1;j∈I2

di,j,k ,

where I1 = {−2,−1, 0, 1, 2}, I2 = {−1, 0, 1} and

di,j,k : �p,r,q(M)→ �p+i,r+j,q+k(M).

The following operators are tensors: d−1,1,1, d1,1,−1, d1,−1,1, d0,−1,2, d2,−1,0,d2,1,−2 and d−2,1,2.

These tensors will be used later on for classification of hyperbolic Monge –Ampère equations.


For the complex AP-structure P = (P1, P2, P3), dimC P1 = dimC P3 = 2,dimC P2 = 1 one has similar complex tensor invariants.

PART II

Symplectic Algebra

4

Linear algebra of symplectic vector spaces

4.1 Symplectic vector spaces

4.1.1 Bilinear skew-symmetric forms on vector spaces

Let V be a real finite-dimensional vector space and let � : V × V → R be abilinear skew-symmetric form on V .

Denote by

: V → V∗

a linear map from V to its dual space V∗ defined by the formula

(x) = ıx�,

in other words

〈 (x), y〉 = �(x, y)

for all x, y ∈ V .The dimension of the image of the map is called a rang of �, and the kernelof , which we denote by ker �, is called a kernel of �.

Note that

ker� = {x ∈ V |�(x, y) = 0, for all y ∈ V}.

The adjoint map ∗ is equal to − because of the skew-symmetry of �:

〈 ∗x, y〉 = 〈x, y〉 = −�(x, y) = −〈 x, y〉

and one could identify skew-symmetric 2-forms � with skew-adjoint maps : V → V∗.

103

104 Linear algebra of symplectic vector spaces

Let V0 = V/ ker� be the factor space. Then� determines a skew-symmetricform �0 on V0 in the natural way:

�0(x mod ker�, y mod ker�) = �(x, y)

and this 2-form is non-degenerate: ker �0 = 0.Moreover,

V∗0 = Ann ker� ⊂ V∗.

Therefore, we decomposed any skew-symmetric form in the composition

VΓ

Γ0

V*

V0*V0

where the vertical arrows are the natural projection and the embedding.

4.1.2 Symplectic structures on vector spaces

A non-degenerate skew-symmetric bilinear form � is called a symplectic formor a symplectic structure on the vector space V .

A vector space V equipped with a symplectic form � is called a symplecticvector space and is denoted by (V ,�).

Let (V1,�1) and (V2,�2) be symplectic vector spaces. Then V1⊕V2 has anobvious symplectic structure defined by

�(x1 ⊕ y1, x2 ⊕ y2) = �1(x1, x2)+�2(y1, y2),

where x1, x2 ∈ V1 and y1, y2 ∈ V2.We call this structure the direct sum of (V1,�1) and (V2,�2) and write

� = �1 ⊕�2.

Let (V ,�) be a symplectic space and let : V → V∗ be the correspondingmap. Then the operator −1 : V∗ → V = (V∗)∗ defines a symplectic structureon the dual space V∗. We denote this structure by (V∗,�∗) and call it the dualstructure.

4.1 Symplectic vector spaces 105

Note that

�∗(x∗, y∗) = �( −1x∗, −1y∗)

for all x∗, y∗ ∈ V∗.

Example 4.1.1 Let V be a two-dimensional space. Then the space �2(V∗) ofexterior 2-forms on V is one-dimensional, and therefore any non-zero exter-ior 2-form � determines a symplectic structure on V. Using the direct sumoperation we can construct a series of symplectic spaces:

V ⊕ · · · ⊕ V︸︷︷︸k times

equipped with the symplectic form

� = �1 ⊕ · · · ⊕�k

where �1, . . . ,�k are non-zero 2-forms on V.

Example 4.1.2 Let E be a vector space, and let

V = E ⊕ E∗.

Define an exterior 2-form on V as follows:

�(x ⊕ x∗, y⊕ y∗) = 〈x∗, y〉 − 〈y∗, x〉

for all x, y ∈ E, and x∗, y∗ ∈ E∗.Then, (E ⊕ E∗,�) is a symplectic space.

Let (V ,�) be a symplectic vector space. We say that two vectors x, y ∈ Vare skew-orthogonal if

�(x, y) = 0

and write x ⊥ y.Let W ⊂ V be a vector subspace and let

W⊥ = {x ∈ V |�(x, y) = 0 for all y ∈ W}

be the space of all vectors skew-orthogonal to W .We call W⊥ the skew-orthogonal of W .We collect some properties of the skew-orthogonality in the following.


Proposition 4.1.1 Let W , W1, W2 be subspaces of a symplectic vectorspace. Then

dim W⊥ = codim W ,

ker�W = W ∩W⊥,

(W⊥)⊥ = W ,

(W1 +W2)⊥ = W⊥

1 ∩W⊥2 ,

(W1 ∩W2)⊥ = W⊥

1 +W⊥2 ,

W1 ⊂ W2 =⇒ W⊥1 ⊃ W⊥

2 ,

where �W ∈ �2(W∗) is the restriction of the structure form � on W.

The proof of this proposition is straightforward.Note that �W is non-degenerate if and only if

W ∩W⊥ = 0.

We call the subspace W regular, or symplectic if �W is non-degenerate. Inthis case (W ,�W ) is also symplectic, and we get a splitting

V = W ⊕W⊥.

Moreover, W⊥ is also regular and therefore

(V ,�) = (W ,�W )⊕ (W⊥,�W⊥)

is a decomposition of the symplectic structure into a direct sum of two structures.Remark also that any symplectic space contains a regular plane W ,

dim W = 2, and thus, applying this remark and the above splitting to W⊥ wedecompose V into a direct sum of symplectic planes.

Summing up we get the following.

Theorem 4.1.1 Any symplectic space (V ,�) is a direct sum of symplecticplanes:

(V ,�) = (W1,�1)⊕ · · · ⊕ (Wn,�n)

where W1, . . . , Wn are symplectic planes in W, �i = �Wi , Wi ⊂ W⊥j for

i = 1, . . . , n, and j �= i.

4.1 Symplectic vector spaces 107

4.1.3 Canonical bases and coordinates

We begin with a two-dimensional symplectic space (W ,�). If e1, e2 is a basisin W and e∗1, e∗2 is the dual basis in W∗ then

� = λe∗1 ∧ e∗2

for some λ ∈ R, λ �= 0.The scale transformation e = λ−1e1, f = e2 gives a basis in W such that

� = e∗ ∧ f ∗.

We call this basis canonical.Note that a basis e′ = a11e + a12 f , f ′ = a21e + a22 f is canonical if and

only if det ‖aij‖ = 1.Now let us take an arbitrary symplectic vector space (V ,�). From decom-

position Theorem 4.1.1 we get canonical bases: (e 1, f1) in the plane W1, . . . ,and (e n, fn) in the plane Wn.

We call such a basis (e1, . . . , en, f1, . . . , fn) canonical.The basic property of the canonical basis is

�(ei, ej) = �( fi, fj) = 0 and �(ei, fj) = δij.

Therefore, in the basis

(e∗1, . . . , e∗n, f ∗1 , . . . , f ∗n )

dual to the canonical basis

(e1, . . . , en, f1, . . . , fn)

the structure 2-form � can be written as

� = e∗1 ∧ f ∗1 + · · · + e∗n ∧ f ∗n .

Let (x1, . . . , x2n) and (y1, . . . , y2n) be coordinates of vectors X and Y in thecanonical basis, then

�(X , Y) =n∑

i=1

(xiyi+n − xi+nyi).

One can define the symplectic structure by the above formula. This structure isusually called the standard symplectic structure on R2n.


Note that the corresponding linear operator : V → V∗ in the canonicalbasis acts as follows:

: ei −→ f ∗i ,

: fi −→ −e∗i .

4.2 Symplectic transformations

Let (V1,�1) and (V2,�2) be symplectic spaces. A linear operator A : V1 → V2

is called symplectic if

A∗(�2) = �1

or, in other words, if

�2(Ax, Ay) = �1(x, y)

for all x, y ∈ V1.Note that ker A = ker�1 = 0, and A is an isomorphism if dim V1 = dim V2.

Example 4.2.1 An embedding W ↪→ V is symplectic if and only if W is asymplectic subspace of V.

Example 4.2.2 The decomposition theorem shows that any two symplecticspaces (V1,�1) and (V2,�2) of the same dimension are symplecticallyisomorphic.

Symplectic isomorphisms A : V → V are called symplectic transformations.They form a group with respect to the composition. This group is called thesymplectic group of the symplectic vector space (V ,�) and we denote it bySp (V ,�).

The group of symplectic transformations of the standard symplectic structureis denoted by Sp (n) or Sp (n, R).

The group Sp (V ,�) is a Lie group, and the corresponding Lie algebrasp(V ,�) consists of the linear operators B : V → V that satisfy the followingrelation:

�(Bx, y)+�(x, By) = 0

for all x, y ∈ V .We call them Hamiltonian operators.Note that exp (tB) ∈ Sp (V ,�), t ∈ R, if B ∈ sp(V ,�).

4.2 Symplectic transformations 109

Proposition 4.2.1 1. Let B ∈ sp (V ,�), then

qB(x, y)def= �(Bx, y)

is a symmetric bilinear form on V.2. Let q : V × V → R be a symmetric bilinear form on V , then the linear

operator Bq : V → V given by

q (x, y)def= �(Bqx, y)

belongs to sp(V ,�).

Proof One has

qB(x, y) = �(Bx, y) = −�(x, By) = �(By, x) = qB(x, y)

and

�(Bqx, y) = q(x, y) = q(y, x) = �(Bqy, x) = −�(x, Bqy). �

Corollary 4.2.1 dim Sp(V ,�) = (dim V(dim V + 1))/2.

This proposition establishes an isomorphism between the Lie algebra ofHamiltonian operators and the space S2(V∗) of bilinear symmetric forms on V ,and therefore induces a Lie bracket on S2(V∗).

Namely, let q1, q2 ∈ S2(V∗) then the bracket

[q1, q2] ∈ S2(V∗)

corresponds to the commutator [Bq1 , Bq2 ].In other words,

[q1, q2](x, y) = �([Bq1 , Bq2 ]x, y) = q1(x, Bq2 y)− q2(x, Bq1 y).

Let α ∈ V∗ and let q = α2 ∈ S2(V∗) be the symmetric square of α:

α2(x, y)def= α(x)α(y).

The corresponding Hamiltonian operator Bq ∈ sp(V ,�) is called a symplectictransvection and is denoted by τα .


We have

α2(x, y) = α(x)α(y) = �( −1α, x)�( −1α, y)

or

τα(x) = α(x) α,

where α = −1α ∈ V .Note that

τ 2α = 0

and

ker τa = ker α, Im τa = Rα.

The corresponding symplectic transformations act as follows:

exp(tτα) = 1+ tτα ,

where t ∈ R.

Proposition 4.2.2 (E. Artin [3])

1. The symplectic transvections τα ,α ∈ V∗ generate the Lie algebra sp(V ,�),and

[τα , τβ ] = �(α, β)

2(τα+β − τα−β).

2. The symplectic transformations 1 + tτα (α ∈ V∗, t ∈ R) generate the Liegroup Sp(V ,�).

4.2.1 Matrix representation of symplectic transformations

The symplectic transformations of the standard symplectic structure in R2n arerepresented by matrices A ∈ GL(2n, R).

Let us write vectors X ∈ R2n as

X =∥∥∥∥X1

X2

∥∥∥∥ ,

4.2 Symplectic transformations 111

where

X1 =n∑

i=1

xiei and X2 =n∑

i=1

xi+n fi

and let

〈X , Y〉 =2n∑

i=1

xiyi.

Then,

�(x, y) =n∑

i=1

(xiyi+n − xi+nyi) = 〈Jx, y〉,

where

J =∥∥∥∥ 0 1−1 0

∥∥∥∥is the matrix of the standard symplectic structure.

Let

A =∥∥∥∥A11 A12

A21 A22

∥∥∥∥be a matrix of a symplectic transformation A : R2n → R2n.

Then the condition �(AX , AY) = �(X , Y) means

AtJA = J ,

or, in coordinates,

At21A11 − At

11A21 = 0,

At22A12 − At

12A22 = 0,

At11A22 − At

21A12 = 1.

Example 4.2.3 The matrices of the form

A =∥∥∥∥a 0

0 b

∥∥∥∥


are symplectic if and only if

atb = 1.

Example 4.2.4 The matrices of the form

A =∥∥∥∥1 a

0 1

∥∥∥∥are symplectic if and only if

at = a.

Hamiltonian operators B ∈ spn have matrices

B =∥∥∥∥B11 B12

B21 B22

∥∥∥∥that satisfy the following condition:

BtJ + JB = 0,

or

B12 − Bt12 = 0,

B21 − Bt21 = 0,

B11 + Bt22 = 0.

We call the matrices of such a type Hamiltonian.

Example 4.2.5 Matrices of the form

A =∥∥∥∥a 0

0 b

∥∥∥∥are Hamiltonian if and only if

at + b = 0.

Example 4.2.6 Matrices of the form

A =∥∥∥∥0 a

b 0

∥∥∥∥

4.3 Lagrangian subspaces 113

are Hamiltonian if and only if

at = a, bt = b.

4.3 Lagrangian subspaces

We have called a subspace W ⊂ V of a symplectic space (V ,�) regular if therestriction�W of the structure 2-form� on W is non-degenerate or symplectic.

It follows that dimensions of regular subspaces are even, and V = W ⊕W⊥for any regular W .

On the other hand, any subspace W of codim W = 1 can be defined by acovector α, that is, W = ker α, and the corresponding vector α = −1(α) ∈ Vbelongs to ker α, because of

α(α) = �(α, α) = 0.

Therefore, in this case W⊥ ⊂ W .In general, we say that a subspace W ⊂ V is involutive if the above inclusion

holds:

W⊥ ⊂ W .

Note that a subspace W is involutive if and only if

�W⊥ = 0.

A subspace E of V is said to be isotropic if �E = 0.In other words, the notions of involutive and isotropic subspaces are dual

to each other with respect to the operation W =⇒ W⊥ : W is isotropic, orinvolutive if and only if the orthogonal W⊥ is involutive or isotropic respectively.

Example 4.3.1 Any one-dimensional subspace is isotropic and any one-codimensional subspace is involutive.

A Lagrangian subspace is a maximal isotropic subspace; that is, if L ⊂ V isLagrangian and W ⊃ L is isotropic then W = L.

One could also define Lagrangian subspaces as minimal involutive subspaces.

Theorem 4.3.1

1. Lagrangian subspaces do exist.2. A subspace L is Lagrangian if and only if L⊥ = L.3. An isotropic subspace L ⊂ V is Lagrangian if and only if dim L = 1

2 dim V.


Proof As we have seen, any one-dimensional subspace is isotropic and V isnot. Therefore any chain W0 ⊂ W1 ⊂ · · · ⊂ Wk ⊂ · · · of isotropic subspaceshas a proper maximal element which corresponds to a Lagrangian subspace. Onthe other hand, if W is isotropic, then W⊥ ⊃ W and therefore dim V−dim W ≥dim W or dim W ≤ n. If W⊥ �= W then there is a non-zero vector x ∈ W⊥ suchthat x /∈ W .

Therefore, W ⊕Rx ⊃ W is isotropic too, and W is Lagrangian if and only ifW⊥ = W . �

Example 4.3.2 Let (e, f ) be a canonical basis of V. Then Lin(e1, . . . , en) andLin( f1, . . . , fn) are Lagrangian subspaces, but Lin(e1, f1, e3, . . . , en) is not.

Example 4.3.3 Let V = E ⊕ E∗ be the symplectic space, and let LA ⊂ V bea graph of a linear operator A : E → E∗. Then LA is Lagrangian if and only ifA is a self-adjoint operator.

Example 4.3.4 The subspaces (e1, . . . , ek , fk+1 , . . . , fn), 1≤ k≤ n areLagrangian.

Theorem 4.3.2 For any two Lagrangian subspaces L, L′ ⊂ V , and any linearisomorphism A0 : L → L′ there exists a symplectic linear transformationA : V → V such that the restriction A|L coincides with A0.

Proof Let us consider a filtration

L = Ln ⊃ Ln−1 ⊃ · · · ⊃ L1 ⊃ 0

of the Lagrangian subspace L by subspaces Li, where dim Li = i, and letL′i = A0(Li) be the corresponding filtration of L′:

L′ = L′n ⊃ L′n−1 ⊃ · · · ⊃ L′1 ⊃ 0.

Then the corresponding orthogonal L⊥i together with Li gives a filtration of V :

V = 0⊥ ⊃ L⊥1 ⊃ · · · ⊃ L⊥n−1 ⊃ L = Ln ⊃ Ln−1 ⊃ · · · ⊃ L1 ⊃ 0.

We obtain a similar filtration with L′:

V = 0⊥ ⊃ L′⊥1 ⊃ · · · ⊃ L′⊥n−1 ⊃ L′ = L′n ⊃ L′n−1 ⊃ · · · ⊃ L′1 ⊃ 0.

To construct A we shall use an induction in n − i. To find the symplecticextension A0 on L⊥n−1 we pick a non-zero vector en ∈ L such that en /∈ Ln−1

and a vector fn ∈ L⊥n−1 but fn /∈ L, and such that �(en, fn) = 1. Then the


image of vector fn under operator A shall be a vector f ′n ∈ L′⊥n−1 such that�(Aen, f ′n) = �(en, fn) = 1. This vector is defined up to vectors from L′. Letus pick one of them. This choice gives us an extension of A0 on L⊥n−1 such thatA : L⊥n−1 → L′⊥n−1 sends �L′⊥n−1

to �L⊥n−1. A similar procedure shows that one

can extend A from L⊥i to L⊥i−1 in such a way that A transforms �L′⊥i−1to �L⊥i−1

if it does for �L′⊥ito �L⊥i

. For i = 0 we find the required A. �

Corollary 4.3.1 For any two isotropic subspaces W and W ′, dim W =dim W ′, of a symplectic space V and any linear isomorphism A0 : W → W ′there exists a symplectic linear transformation A : V → V such that therestriction A|W coincides with A0.

Proof It is enough to embed W and W ′ into Lagrangian subspaces L and L′and consider any extension of A0 to a linear isomorphism L → L′. �

Theorem 4.3.3 (Witt) Let W and W ′ be subspaces of a symplectic vector space(V ,�), and let A0 : W → W ′ be a linear isomorphism such that

A∗0(�W ′) = �W .

Then A0 can be extended to a symplectic isomorphism A, such that therestriction A|W coincides with A0.

Proof Let W0 = W ∩W⊥ and let W1 be a complementary subspace of W0 inW ; that is,

W = W0 ⊕W1.

Let

W ′ = W ′0 ⊕W ′

1

be the corresponding decomposition of W ′ with W ′1 = A0(W1).

The 2-form �|W1 is non-degenerate. Therefore, W1 and W⊥1 are regular,

V = W1 ⊕W⊥1

and A0 : W1 → W ′1 is a symplectic isomorphism.

Moreover, W0 ⊂ W⊥1 and W ′

0 ⊂ W ′⊥1 are isotropic subspaces, and due to

the above theorem there is an extension of A0 : W0 → W ′0 to symplectic


isomorphism A1 : W⊥1 → W ′⊥

1 . Then the symplectic isomorphism

A0 ⊕ A1 : W1 ⊕W⊥1 → W ′

1 ⊕W ′⊥1

is the required extension. �

Theorem 4.3.4 Let W ⊂ V be a subspace of a symplectic space (V ,�). Then,

1. W contains a Lagrangian subspace if and only if W is involutive;2. if W is involutive and L ⊂ V is a Lagrangian subspace such that L ⊂ W,

then L ⊃ W⊥.

Proof Indeed, let L be Lagrangian and let L ⊂ W . Then L⊥ ⊃ W⊥ and

W ⊃ L = L⊥ ⊃ W⊥.

On the other hand, let W be involutive. Then W⊥ is isotropic, and thereforethere exists a Lagrangian subspace L such that L ⊃ W⊥ and L = L⊥ ⊂(W⊥)⊥ = W . �

Let us now consider the decomposition

(V ,�) = (�1,�1)⊕ · · · ⊕ (�n,�n)

into a direct sum of symplectic planes, �i = �|�i , and let

πi : V → �i

be the corresponding projectors.Note that all embeddings

�i1 ⊕ · · · ⊕�ik ↪→ V

are symplectic.Let L ⊂ V be a Lagrangian subspace. Then images Li = πi(L) �= 0 for any

i = 1, . . . , n.Indeed, if, say, π1(L) = 0, then L ⊂ �2 ⊕ · · · ⊕ �n is isotropic and

dim L = n > n− 1.This observation gives us a method to choose coordinates on Lagrangian

subspaces.


Theorem 4.3.5 Let (e1, . . . , en, f1, . . . , fn) be a canonical basis of V, L be aLagrangian subspace. Then there is a subset I ⊂ {1, . . . , n}, #I = k, such thatlinear functions

e∗i1 , . . . , e∗ik , is ∈ I,

f ∗j1 , . . . , f ∗jn−k, jl /∈ I

are coordinates on L.

4.3.1 Symplectic and Kähler spaces

Recall that a Hermitian space is a complex vector space (V , I) equipped witha positive Hermitian form h.

Let

g(x, y) = Re h(x, y)

be the real part and

�(x, y) = Im h(x, y)

be the imaginary part of h.Then

h(x, y) = g(x, y)+ ı�(x, y),

ı = √−1, and the Hermitian properties of h:

h(x, y) = h(y, x),

h(Ix, y) = ıh(x, y)

implies that:

1. g(x, y) is a Euclidian structure on V ;2. �(x, y) is a symplectic structure on V ; and3. g(x, y) = �(Ix, y) or I ∈ sp(V ,�).

We can reformulate the definition of Hermitian space in terms of thesymplectic structure (V ,�) equipped with a complex structure I : V → V .


To do this we say that a complex structure I : V → V and a symplecticstructure are compatible if:

1. I ∈ sp(V ,�); and2. �(Ix, x) > 0 if x �= 0.

A triple (V ,�, I)with compatible symplectic and complex structures is calleda Kähler space.

If (V ,�, I) is a Kähler space then the form

h(x, y) = g(x, y)+ ı�(x, y)

determines a Hermitian space structure on V .

Example 4.3.5 If the Hermitian form h has the form

h =n∑

k=1

zkzk

in coordinates zk = pk + ıqk, then the symplectic form that corresponds to theimaginary part of h has the form

� =n∑

k=1

pk ∧ qk .

Let us discuss briefly relations between symplectic structures and compatiblecomplex structures.

First of all, let us note that for any unitary transformation A : V → V thecondition h(Ax, Ay) = h(x, y) implies g(Ax, Ay) = g(x, y) and �(Ax, Ay) =�(x, y), and therefore the unitary group U(V , h) belongs to the intersectionof the symplectic group Sp(V ,�) and the orthogonal group O(V , g), and therelation g(x, y) = �(Ix, y) shows that

U(V , h) = Sp(V ,�) ∩ O(V , g).

Note also that Hermitian orthogonality h(x, y) = 0 implies both the Euclidiang(x, y) = 0 and skew-orthogonality �(x, y) = 0.

On the other hand, if W is an isotropic subspace and e1, . . . , ek is an orthonor-mal basis in W with respect to g, then these vectors are orthonormal with respectto h, and therefore give us an orthonormal C-basis in the C-vector space W⊕ IW .If W is a Lagrangian subspace then any g-orthonormal basis e1, . . . , en in Wgives an h-orthonormal basis in V . In other words, Lagrangian subspaces arereal enveloping spaces of h-orthogonal C-bases in V .

5

Exterior algebra on symplectic vector spaces

Let (V ,�) be a symplectic vector space, dim V = 2n, and let �∗(V∗) bethe exterior algebra of forms on V . The structure form � generates an idealIC = � ∧�∗(V∗).

We call it the Cartan ideal of the symplectic space. The corresponding factoralgebra

�ε(V∗) = �∗(V∗)�IC

is called the algebra of effective forms.This algebra has great importance for the partial differential equation (PDE)

theory, and in this chapter we give a detailed description of effective forms assome special exterior forms on the symplectic space V .

5.1 Operators ⊥ and �We extend the isomorphism : V → V∗,

(X)def= ιX (�),

to isomorphisms

k : �k(V)→ �k(V∗)

between k-vectors and k-forms on V as follows:

k(X1 ∧ · · · ∧ Xk)def= (X1) ∧ · · · ∧ (Xk),

119

120 Exterior algebra on symplectic vector spaces

and denote by Xα ∈ �k(V) the k-vector that corresponds to a k-formα ∈ �k(V∗):

Xαdef= −1

k (α).

Example 5.1.1 Let k = 1 and α = ∑ni=1(aie∗i + bi f ∗i ) in a canonical basis

(e1, . . . , en, f1, . . . , fn). Then,

Xα =n∑

i=1

(biei − aifi).

Example 5.1.2 For the structure 2-form � =∑ni=1 e∗i ∧ f ∗i we obtain

X� =n∑

i=1

ei ∧ fi.

Define two operators on the algebra of exterior forms �∗(V∗).The first operator

� : �k(V∗)→ �k+2(V∗) (5.1)

is the operator of exterior multiplication by the structure 2-form �:

�(ω) def= ω ∧�.

The second operator

⊥ :�i(V∗)→ �i−2(V∗) (5.2)

is an operator of inner multiplication by the bivector X�:

⊥(ω) def= ιX� (ω).

Lemma 5.1.1 Let

�k = 1

k!� ∧ · · · ∧�︸︷︷︸k times

.

Then

⊥(�k) = (n− k + 1)�k−1.

5.1 Operators ⊥ and � 121

Proof Let e1, . . . , en, f1, . . . , fn be a canonical basis of the vector space V .Then,

ιei(�) = f ∗i , ιfi(�) = −e∗i

and

ιei(�k) = f ∗i ∧�k−1, ιfi(�k) = −e∗i ∧�k−1.

Therefore

ιei∧fi(�k) = ιfi(ιei(�k)) = ιfi(f∗i ∧�k−1)

= �k−1 − f ∗i ∧ ιfi(�k−1)

= �k−1 + f ∗i ∧ e∗i ∧�k−2

and

⊥ (�k) =n∑

i=1

ιei∧fi(�k) =n∑

i=1

(�k−1 + f i ∧ ei ∧�k−2)

= n�k−1 −� ∧�k−2

= n�k−1 − (k − 1)�k−1 = (n− k + 1)�k−1. �

Lemma 5.1.2 If θ ∈ V∗ and ω ∈ �k(V∗), then

⊥ (θ ∧ ω) = θ ∧ (⊥ ω)− ιXθ (ω).

Proof If e1, . . . , en, f1, . . . , fn is a canonical basis of V , then

⊥(θ ∧ ω) = ιX�(θ ∧ ω) =n∑

i=1

ιfi(ιei(θ ∧ ω))

=n∑

i=1

ιfi(ιei(θ)ω − θ ∧ ιei(ω))

=n∑

i=1

(θ(ei)ιfi(ω)− θ( fi)ιei(ω)+ θ ∧ ιei∧fi(ω))

=n∑

i=1

(θ(ei)ιfi(ω)− θ( fi)ιei(ω))+ θ ∧ ⊥(ω).


Note that

θ =n∑

i=1

(θ(ei)e∗i + θ( fi)f

∗i )

and

Xθ =n∑

i=1

(θ( fi)ei − θ(ei)fi).

Therefore, the last formula reads as

⊥(θ ∧ ω) = −ιXθ (ω)+ θ ∧ (⊥(ω)). �

Lemma 5.1.3 For any k-form ω ∈ �k(V) we have

[⊥,�](ω) = (n− k)ω.

Proof The proof follows by induction on k.If k = 1, then ⊥ (ω) = 0, and due to Lemma 5.1.1, we have

[⊥,�](ω) = ⊥(�(ω)) = ιX�(� ∧ ω)

= ιX�(�)ω = ⊥(�1)ω

= (n− 1)ω.

Suppose that for any ω ∈ �k−1(V) we have

[⊥,�](ω) = (n− (k − 1))ω.

Note that any k-form can be represented as a sum of forms θ ∧ ω whereθ is an exterior 1-form, and ω is an exterior (k − 1)-form. For these formswe have

[⊥,�](θ ∧ ω) = ⊥(� ∧ θ ∧ ω)−�(⊥(θ ∧ ω))

= ⊥(θ ∧� ∧ ω)−�(θ ∧ ⊥(ω))+�(ιXθ (ω))

= ⊥(θ ∧ �(ω))−� ∧ θ ∧ ⊥(ω)+� ∧ ιXθ (ω)

= θ ∧ ⊥�(ω)− ιXθ (� ∧ ω)− θ ∧ �⊥(ω)+� ∧ ιXθ (ω)

= θ ∧ [⊥,�](ω)− ιXθ (�) ∧ ω −� ∧ ιXθ ω +� ∧ ιXθ (ω)

= θ ∧ [⊥,�](ω)− ιXθ (�) ∧ ω

= (n− (k − 1))θ ∧ ω − θ ∧ ω = (n− k)θ ∧ ω. �

5.1 Operators ⊥ and � 123

Let

�kdef= 1

k!�k

and

⊥kdef= 1

k!⊥k .

Lemma 5.1.4 Let ω ∈ �k(V), then

[⊥,�s](ω) = (n− k − s+ 1)�s−1(ω),

[⊥s,�](ω) = (n− k + s− 1)⊥s−1(ω)

and

�s⊥s(ω) = (−1)s(n− k + s− 1)(n− k + s− 2) · · · (n− k)ω,

if �ω = 0,

⊥s�s(ω) = (n− k − s+ 1)(n− k − s+ 2) · · · (n− k)ω,

if ⊥ω = 0.

Proof The proof is induction on s. We prove only the first formula; the secondone can be proved in the similar way.

Suppose that

[⊥,�s−1](ω) = (n− k − s+ 2)�s−2(ω)

and

[⊥s−1,�](ω) = (n− k + s− 2)⊥s−2(ω).

Then,

[⊥,�s](ω) = ⊥�s(ω)−�s⊥(ω)

= 1

s(⊥�s−1(�(ω))−�s−1(�⊥(ω)))

= 1

s(�s−1⊥(�(ω))+ [⊥,�s−1](�(ω))−�s−1(�⊥(ω)))


= 1

s(�s−1([⊥,�s−1](ω))+ (n− (k + 2)− s+ 2)�s−2(�(ω)))

= 1

s((n− k)�s−1(ω)+ (s− 1)(n− k − s)�s−1(ω))

= 1

s(n− k + (s− 1)(n− k)− (s− 1)s)�s−1(ω)

= (n− k − s+ 1)�s−1(ω).

For the last two formulas we have: if s = 1, then it follows from the previousformula, and for s+ 1 we have

�s+1⊥s+1(ω) = �s�⊥s+1(ω) = �s([�,⊥s+1](ω)−⊥s+1�(ω))= −(n− k + s)�s⊥s(ω)

= (−1)s+1(n− k + s)(n− k + s− 1) · · · (n− k)ω.

The last formula can be proved in a similar way. �

Lemma 5.1.5

1. The operators

� : �k(V)→ �k+2(V)

are injective if k ≤ n− 1.2. The operators

⊥ :�k(V)→ �k−2(V)

are injective if k ≥ n+ 1.

Proof Suppose k ≤ n− 1 and �(ω) = 0 for some form ω ∈ �k(V).Then,

�n+1⊥n+1(ω) = (−1)n+1(2n− k)(2n− k − 1) · · · (n− k)ω.

Since k ≤ n− 1 then ⊥n+1(ω) = 0. Therefore,

(2n− k)(2n− k − 1) · · · (n− k)ω = 0

and ω = 0.

5.2 Effective forms and the Hodge–Lepage theorem 125

To prove the second part of the statement suppose that k ≥ n + 1 and⊥(ω) = 0 for some form ω ∈ �k(V).

Then,

⊥n�n(ω) = (1− k)(2− k) · · · (n− k)ω.

Since k ≥ n+ 1, we see that �n(ω) = 0,

(1− k)(2− k) · · · (n− k)ω = 0.

and ω = 0. �

5.2 Effective forms and the Hodge–Lepage theorem

A form ω ∈ �k(V∗) is called effective if

⊥ (ω) = 0.

We denote by �kε(V

∗) the space of all effective k-forms on V . Note thatnon-trivial effective k-forms exist only if k ≤ n.

Theorem 5.2.1 (Hodge–Lepage) Any k-form ω ∈ �k(V) can be decom-posed as

ω = ω0 +�(ω1)+�2(ω2)+ · · ·

where ωi ∈ �k−2i(V∗) are uniquely determined effective forms.

Proof The proof is by induction on k. The case of k = 1 is trivial.Assume that the proposition holds for forms of degree less then k.Let ω ∈ �k(V∗). Then ⊥ (ω) ∈ �k−2(V∗) and by the inductive hypothesis,

⊥ (ω) = α0 +�(α1)+�2(α2)+ · · · ,

where α0,α1,α2, . . . are uniquely determined effective forms.Hence, if we set

ω = x0 +�(x1)+�2(x2)+ · · · ,

then under the assumption that x0, x1, . . . are effective, we obtain

⊥ (ω) = (n− k + 2)x1 + (n− k + 3)�(x2)+ · · · .


Consequently, if we take

x1 = 1

n− k + 2α0, x2 = 1

n− k + 3α1, . . . ,

then

⊥ (ω −�(x1)−�2(x2)− · · · ) = 0,

i.e.,

ω0 = ω −�(x1)−�2(x2)− · · ·

is an effective form. �

Suppose ω ∈ �k(V∗) and ω =∑s≥0�s(ωs) is the Hodge–Lepage decom-position. Straightforward calculations lead us to the following expressions:

⊥r (ω) =∑s≥r

Crn−k+s+r�s−r(ωs),

(�r◦ ⊥r)(ω) =∑s≥r

Crn−k+s+rCr

s�s(ωs)

and to the following expression

ω0 =(∑

s≥0

(−1)s1

s+ 1�s ⊥s

)(ω)

for the effective part of ω.

Theorem 5.2.2 The operators

⊥k :�n+k(V∗)→ �n−k(V∗)

and

�k : �n−k(V∗)→ �n+k(V∗),

k = 1, . . . , n− 1, are isomorphisms and for effective forms ω we have

⊥k ◦�k(ω) = ω.


Proof Let ω ∈ �n−k(V∗) and �k(ω) = 0.Then from the Hodge–Lepage decomposition we obtain

⊥k ◦�k(ω) = ω0 + (C1k+1)

2�(ω1)+ · · · .

From the uniqueness of the Hodge–Lepage decomposition we haveω0 = ω,ω1 = 0, . . . . �

Example 5.2.1 Let (V ,�)be a symplectic space, dim V = 4, and let e1, e2, f1, f2be a canonical basis.

Then,

� = e∗1 ∧ f ∗1 + e∗2 ∧ f ∗2

and

X� = e1 ∧ f1 + e2 ∧ f2.

Let

ω = a1e1 + a2e2 + b1f 1 + b2f 2 ∈ �1(V∗),

be a 1-form. Note that �1(V∗) = �1ε(V

∗),

�(ω) = a1e∗1 ∧ e∗2 ∧ f ∗2 − a2e∗1 ∧ e∗2 ∧ f ∗2 − b1e∗2 ∧ f ∗1 ∧ f ∗2 + b2e∗1 ∧ f ∗1 ∧ f ∗2

and � : �1(V∗)→ �3(V∗) is an isomorphism.If

ω = Ae∗1 ∧ e∗2 + Be∗1 ∧ f ∗1 + Ce∗1 ∧ f ∗2 + De∗2 ∧ f ∗1 + Ee∗2 ∧ f ∗2+ Ff ∗1 ∧ f ∗2 ∈ �2(V∗),

is a 2-form, then

⊥ (ω) = B+ E

and

�(ω) = (B+ E)e1 ∧ f 1 ∧ e2 ∧ f 2 = B+ E

2�2.


Hence, �2ε(V

∗) is the space of 2-forms such that

B+ E = 0

and

ker� = ker ⊥ .

Theorem 5.2.3 The formω ∈ �n−k(V) is effective if and only if�k+1(ω) = 0.

Proof First of all, note that for the form ω ∈ �n−k(V) we have

⊥ �k+1(ω) = �k+1 ⊥ (ω).

Therefore, if ⊥ (ω) = 0, then ⊥ �k+1(ω) = 0 and �k+1(ω) = 0. Conversely,if �k+1(ω) = 0, then �k+1 ⊥ (ω) =⊥ �k+1(ω) = 0. So, ⊥ (ω) = 0. �

Theorem 5.2.4 Let ω1,ω2 ∈ �n−kε (V∗), 0≤k≤n, be effective (n − k) forms,

and let the form ω2 vanish on every isotropic subspace L ⊂ V, dim L = n− k,such that ω1|L = 0.

Then ω2 = λω1 for some λ ∈ R.

Proof The proof follows by induction on 12 dim V . The case n = 1 is trivial.

Let V be a symplectic space of dimension 2n. Choose a pair of covectorsα,α+ ∈ V∗ such that ιXα (α

+) = 1. Then the restriction �′ of the form � to thesubspace

V ′ = ker α ∩ ker α+

defines a symplectic structure. Denote by �′ and ⊥′ the correspondingoperators. Since

V = V ′ ⊕ RXα ⊕ RXα+

we can identify forms on V ′ with forms on V that are degenerated onRXα ⊕ RXα+ .

An arbitrary form β ∈ �s(V∗) can be decomposed as

β = β0 + α ∧ β1 + α+ ∧ β2 + α+ ∧ α ∧ β3


where β0,β1,β2 and β3 are uniquely determined forms on V ′ and therefore

ια(βi) = ια+(βi) = 0.

Let us now assume that β is an effective form. From

X� = Xα+ ∧ Xα + X�′

we obtain

0 = ⊥β = ⊥′(β0)+ α ∧ ⊥′(β1)+ α+ ∧ ⊥′(β2)+ α+ ∧ α ∧ ⊥′(β3)+ β3.

Thus, β1,β2 and β3 are effective forms, and

⊥′ (β0) = −β3.

Let

β0 = β0 +�′(x1)+ (�′)2(x2)+ · · ·

be the Hodge–Lepage decomposition. Then

⊥′ (β0) = (n− s+ 1)x1 + 2(n− s+ 2)�′(x2)+ · · ·

and hence

x1 = − 1

n− s+ 1β3, x2 = 0, . . . .

Finally, we obtain the following decomposition of effective forms:

β = β0 + α ∧ β1 + α+ ∧ β2 + α+ ∧ α ∧ β3 − 1

n− s+ 1�′(β3)

in which all the βi are effective.In particular, for effective forms ωi ∈ �n−k(V∗), i = 1, 2, we find

ωi = ω0i + α ∧ ω1i + α+ ∧ ω2i + α+ ∧ α ∧ ω3i − 1

k + 1�′(ω3i).

Let us now go back to the conditions of the theorem. We construct an isotropicsubspace annihilating the form ωi in the following way. We choose arbitraryvectors Z1, . . . , Zn−k−2 ∈ V ′ which are linear independent and are in involution:

�′(Zi, Zj) = 0,


and add to them vectors u+Xα , v+Xα+ , u, v ∈ V ′, in such a way that vectors

u+ Xα , v + Xα+ , Z1, . . . , Zn−k−2

are in involution also. To do this it is enough to require that

�′(Zi, u) = �′(Zi, v) = 0 and �′(u, v) = 1.

Elementary computations show that

(u+ Xα) ∧ (v + Xα+) ∧ Z�ωi

= u ∧ v ∧ Z�ω0i + u ∧ Z�ω1i + v ∧ Z�ω2i − k + 2

k + 1Z�ω3i,

where Z = Z1∧· · ·∧Zn−k−2. Therefore, for a fixed vector v the linear equation

u� (v ∧ Z�ω02 + Z�ω12) = k + 2

k + 1Z�ω32 − v ∧ Z�ω22

on u is satisfied wheneveru� (V�Z�ω01 + Z�ω11) = k + 2

k + 1Z�ω31 − v�Z�ω21,

u� (v��′) = −1, u� (Zi��′) = 0.

Hence, the forms{γ0 = v ∧ Z�ω02 + Z�ω12, γ1 = v ∧ Z�ω01 + Z�ω11

γ2 = ′(v), γi+2 = ′(Zi) (i = 1, . . . , n− k − 2)

must be linear dependent for any choice of vectors v, Z1, . . . , Zn−k−2 ininvolution. Therefore,

γ0 ∧ γ1 ∧ γ2 ∧� = 0,

where � = γ3 ∧ · · · ∧ γn−k . We consider the last equation as an equation for v:

(v ∧ Z�ω02) ∧ (v ∧ Z�ω01) ∧ ′(v)�+ (v ∧ Z�ω02) ∧ (Z�ω11)

+ (Z�ω12) ∧ (v ∧ Z�ω01) ∧ ′(v) ∧�+ (Z�ω12) ∧ (Z ∧ ω11)

×∧ ′(v) ∧� = 0. (5.3)


Let us fix a set Z1, . . . , Zn−k−2 of vectors in involution. The equation shouldhold for all vectors v for which

�′(Zi, v) = 0, i = 1, . . . , n− k − 2.

Let us now assume that ω11 �= 0 and Z�ω11 �= 0. Replace v by tv in (5.3)and differentiate with respect to t at t = 0. We obtain the following equation:

(Z�ω12) ∧ (Z�ω11) ∧ ′(v) ∧� = 0,

for all v,�′(v, Zi) = 0. In other words, the restrictions of Z�ω12 and Z�ω11

on ∩ ker ′(Zi) for any involutive set Z1, . . . , Zn−k−2 lead to linear dependentforms. In particular, this gives us the fact that ω12 vanishes on all isotropicsubspaces that annihilate ω11. Hence,

ω12 = λω11

by induction.We now consider terms of second order in v in (5.3). To do this we replace v

by tv, and differentiate twice with respect to t at t = 0.We obtain

(v ∧ Z�ω02 − λv ∧ Z�ω01) ∧ (Z�ω11) ∧ ′(v) ∧� = 0

and thus on ∩ ker ′(Zi) the forms v ∧ Z� (ω02 − λω01) and ′(v) are lineardependent. Therefore,ω02−λω01 vanishes on all isotropic subspaces, and sinceω02 − λω01 is effective, we find that

ω02 − λω01 = 0.

Similarly, replacing v by u, we prove that ω12 = λ′ω21.If ω01 �= 0, λ �= λ′, then for any involutive set Z1, . . . , Zn−k−2, we obtain

Z� (ω32 − λω31) = 0.

Since ω32 − λω31 is effective we obtain ω32 = λω31 and ω2 = λω1. �

Corollary 5.2.1 (Lepage’s theorem) A k-form ω ∈ �k(V∗) belongs to theimage of � if and only if ω|L = 0 for any Lagrangian subspace L ⊂ V.

Corollary 5.2.2 Letω ∈ �k(V∗) be an effective k-form, and let θ ∈ �s(V∗) bean s-form, s < k, such that θ |L = 0 for any isotropic k-dimensional subspaceW ⊂ V on with ω|L = 0. Then θ ∈ Im�.


5.2.1 sl2-method

In this section we will discuss a more conceptual approach to the Hodge–Lepagedecomposition. This approach is based on the following observation.

Let us consider the following three operators acting in the exterior algebra�∗(V∗) of a symplectic vector space (V ,�): operators ⊥,� and

I : �∗(V∗)→ �∗(V∗),

I : ω ∈ �k(V∗) −→ (n− k)ω ∈ �k(V∗).

As we have seen

[⊥,�] = I.

Moreover,

[I,⊥] = 2⊥,

[I,�] = −2�.

In other words, these three operators give a representation of the Lie algebrasl2 in the exterior algebra of the symplectic vector space. We will use thisrepresentation in order to obtain the Hodge–Lepage decomposition.

sl2-representationsWe remind the reader of some standard facts from the representation theory forthe Lie algebra sl2.

Let sl2 be the Lie algebra of 2× 2 matrices with zero trace. This is a three-dimensional simple Lie algebra of rank 1.

We choose the Chevalley–Cartan basis E+, E−, H in sl2, where

E+ =∥∥∥∥0 1

0 0

∥∥∥∥ , E− =∥∥∥∥0 0

1 0

∥∥∥∥ , H =∥∥∥∥1 0

0 −1

∥∥∥∥ .

The commutation relations takes the form

[E+, E−] = H, [H, E+] = 2E+, [H, E−] = −2E−.

Let ρ : sl2 → End W be a representation of the Lie algebra. A vector v issaid to be of height λ if

Hv = λv.


It is easy to see that the vector E+v is of height (λ + 2) and the vector E−v isof height (λ− 2) if the vector v has height λ.

Assume that the vector space W is finite dimensional, then beginning withan eigenvector H and repeatedly applying operator E+ we should arrive at aneigenvector of H that belongs to ker E+. Such vectors are called primitive .

Let v ∈ W be a primitive vector of height λ, and let

vn = En−v

for n ≥ 0 and v−1 = 0.Then straightforward computations show that

Hvn = (λ− 2n)vn,

E+vn = (λ− n+ 1)vn−1,

E−vn = (n+ 1)vn+1.

The first relation shows that vectors vn are linear independent if they arenon-zero vectors. Therefore, vn+1 = 0 for some n, and assume that vn �= 0.Then the second relation gives λ = n. In other words, the heights of primitivevectors are natural numbers.

These relations also show how to construct an irreducible representation inany dimension n.

Namely, let us denote by Wn an (n+1)-dimensional vector space with a basis(v0, v1, . . . , vn) and sl2 action given by the above formulas. It is clear that Wn

is irreducible, and any sl2-representation is a sum of modules of Wn-type.To obtain such a decomposition one should remark that the subspace ker E+

is invariant with respect to the operator H. As we have seen, the spectrum ofthe restriction of H on ker E+ consist, of natural numbers. Therefore,

ker E+ = ⊕iZni ,

where Zni is a subspace of eigenvectors with eigenvalue ni ∈ N.Each non-zero vector v ∈ Zni gives an irreducible

Wni = {v = v0, v1 = E−v, . . . , vni = Eni−v}

sl2-representation of dimension ni + 1, and W is a direct sum of Wni withmultiplicities dim Zni .


Effective forms and primitive vectorsLet us apply the description of sl2-representations to the representation givenby operators ⊥,� and I.

As we have seen, the operator ⊥ corresponds to E+, the operator �correspond to E− and I corresponds to H.

Therefore, primitive vectors are exactly effective forms. They correspondsto non-negative eigenvalues of I and therefore belong to �k(V∗) with k ≤ n.

Any non-zero effective formω ∈ �kε(V

∗) has height n−k and thus generatesan irreducible sl2-module (ω,�(ω), . . . ,�n−k(ω)) of dimension n− k + 1.

Note that we obtain the relation

�n−k+1(ω) = 0

once more.Effective forms in the middle dimension ω ∈ �n

ε(V∗) have height 0.

Therefore they generate the trivial sl2-modules.Finally note, that the Hodge–Lepage decomposition corresponds to the

decomposition of the sl2-module �∗(V∗) into a sum of irreducibles.

6

A symplectic classification of exterior 2-formsin dimension 4

6.1 Pfaffian

Let (V ,�) be a four-dimensional symplectic vector space with a structure2-form �, and let ω ∈ �2(V∗) be a 2-form.

Define a linear operator Aω : V → V by the following condition:

X�ω = AωX�� (6.1)

for any vector X ∈ V .

Proposition 6.1.1 Operator Aω is uniquely determined by ω, and

1. Aω is symmetric with respect to �, i.e.,

�(AωX , Y) = �(X , AωY)

for any X , Y ∈ V ;2. vectors X and AωX are skew-orthogonal, i.e.,

�(AωX , X) = 0

for any vector X ∈ V;3. the map A : �2(V)→ End(V),ω −→ Aω, is R-linear, i.e.,

Ah1ω1+h2ω2 = h1Aω1 + h2Aω2

for any ω1,ω2 ∈ �2(V) and any h1, h2 ∈ R;4. let L ⊂ V be a Lagrangian subspace such that ω|L = 0. Then L is invariant

under Aω:

AωL = L.

135

136 A symplectic classification of exterior 2-forms in dimension 4

Proof We prove the last property only.Let L be the Lagrangian subspace. Then, L = L⊥. We have �(AωX, X) =

0 for each vector X ∈ V . If X ∈ L, then AX ∈ L⊥ = L. Therefore L is aninvariant space of Aω. �

For any 2-form ω ∈ �2(V∗) the 4-form ω2 = ω ∧ ω is proportional tothe volume form �2 = � ∧ �, and we define the Pfaffian Pf(ω) to be thecoefficient of proportionality:

ω2 = Pf(ω)�2,

Lemma 6.1.1 Let ω be an effective 2-form on V. Then,

A2ω + Pf(ω) = 0. (6.2)

Proof The effectivity condition for 2-form in four-dimensional symplecticspace takes the form

ω ∧� = 0

and we obtain

0 = (AωX ∧ X)�(ω∧�)= X�((AωX�ω) ∧�+ ω∧(AωX��))= ω(AωX, X)�− (AωX�ω)∧(X��)+ (X�ω)∧(AωX��)+�(AωX, X)ω

= (A2ωX��)∧(X��)+ (AωX��)∧(AωX��) = (A2

ωX��)∧(X��)

for any vector X ∈ V .This means that 1-forms A2

ωX� � and X� � are linearly dependent and since� is non-degenerate, we find that the vectors A2

ωX and X are linearly dependentfor all vectors X . Therefore A2

ω + α = 0 for some α ∈ R. Now we have

(X ∧ AωX)� (� ∧�) = −2X�ω ∧ X��

and

(X ∧ AωX)� (ω ∧ ω) = 2αX�ω ∧ X��.

Therefore

(α − Pf(ω))(X�ω ∧ X��) = 0

for all vectors X . Hence α = Pf(ω) or X�ω ∧ X�� = 0 for all vectors X.

6.2 Normal forms 137

In the last case all vectors are linear dependent and therefore Aω = λ, andω = λ�, that impossible. �

Corollary 6.1.1 det Aω = (Pf(ω))2.

Let us give a coordinate representation for Pf(ω) and Aω.Suppose that e1, e2, f1, f2 is a canonical basis in V . Then 2-forms

e∗1 ∧ e∗2, e∗1 ∧ f ∗1 − e∗2 ∧ f ∗2 ,

e∗1 ∧ f ∗2 , e∗2 ∧ f ∗1 , f ∗1 ∧ f ∗2

constitute a basis in the space of effective 2-forms on V .Let

ω = Ee∗1 ∧ e∗2 + B(e∗1 ∧ f ∗1 − e∗2 ∧ f ∗2 )+ Ce∗1 ∧ f ∗2 − Ae∗2 ∧ f ∗1 + Df ∗1 ∧ f ∗2 .

Then

ω2 = 2(AC − B2 − DE) e∗1 ∧ f ∗1 ∧ e∗2 ∧ f ∗2 = (AC − B2 − DE)�2

and

Pf(ω) = AC − B2 − DE.

Simple and straightforward calculations show that

Aω =

∥∥∥∥∥∥∥∥∥∥B 0 −A −D

0 B E C

C D −B 0

−E −A 0 −B

∥∥∥∥∥∥∥∥∥∥in the canonical basis.

6.2 Normal forms

In this section we consider the problem of symplectic equivalence for effectiveforms.

We say that effective 2-forms ω1 and ω2 on V are equivalent if there existsa symplectic isomorphism � ∈ Sp(V ,�) such that

�∗(ω1) = ω2.


Proposition 6.2.1 The Pfaffian is an invariant of the symplectic group action

Pf(�∗(ω)) = Pf(ω)

for all � ∈ Sp(V ,�).

Proof Indeed,

Pf(�∗(ω))� ∧� = �∗(ω) ∧�∗(ω) = �∗(ω ∧ ω) = �∗(Pf(ω)� ∧�)

= Pf(ω)�∗(�) ∧�∗(�) = Pf(ω)�2. �

Moreover

Pf(tω) = t2 Pf(ω)

for t ∈ R.Therefore the Pfaffian divides the space of effective forms into three parts

corresponding to sign(Pf(ω)).We say that an effective 2-form ω �= 0 on a four-dimensional symplectic

space is

1. hyperbolic, if Pf(ω) < 0,2. elliptic, if Pf(ω) > 0, and3. parabolic, if Pf(ω) = 0.

Theorem 6.2.1 Let ω ∈ �2ε(V) be an effective form on four-dimensional

symplectic space.Then there is a canonical basis (e1, e2, f1, f2) in V such that

1.

ω = √Pf(ω)(e∗1 ∧ f ∗2 − e∗2 ∧ f ∗1 )

if ω is elliptic,2.

ω = √−Pf(ω)(e∗1 ∧ f ∗1 − e∗2 ∧ f ∗2 )

if ω is hyperbolic, and3.

ω = e∗1 ∧ f ∗2

if ω is parabolic.


Proof Let us consider first the hyperbolic case and let

λ = √−Pf(ω).

Then the forms ω ± λ� are decomposable because

(ω ± λ�) ∧ (ω ± λ�) = ω ∧ ω + λ2� ∧� = 0.

Therefore,

ω + λ� = γ1 ∧ γ2,

ω − λ� = γ3 ∧ γ4

for some 1-forms γ1, γ2, γ3, γ4, and

� = 1

2λ(γ1 ∧ γ2 − γ3 ∧ γ4),

ω = 1

2(γ1 ∧ γ2 + γ3 ∧ γ4).

Therefore the basis dual to

e∗1 =1

2λγ1, e∗2 = −

1

2λγ3, f ∗1 = γ2, f ∗2 = γ4

is a canonical basis. In this basis we have

� = e∗1 ∧ f ∗1 + e∗2 ∧ f ∗2 ,

ω = λ(e∗1 ∧ f ∗1 − e∗2 ∧ f ∗2 ).

In the elliptic case we put

ω0 = ω√Pf(ω)

.

Then the operator

I = Aω0

defines a complex structure on V and a complex-valued 2-form

�(x, y) = �(x, y)− ıω0(x, y)


on the complex space V :

�(Ix, y) = �(x, Iy) = ı�(x, y).

In other words, (V ,�) is a complex symplectic plane. Let e1, f1 be a canonicalbasis for �, and let e2 = Ie1, f2 = −If1. Then,

�(e1, f1) = 1 =⇒ �(e1, f1) = 1, ω0(e1, f1) = 0,

�(e2, f1) = ı�(e1, f1) = ı =⇒ �(e2, f1) = 0, ω0(e2, f1) = −1,

�(e2, f2) = �(e1, f1) = 1 =⇒ �(e2, f2) = 1, ω0(e2, f2) = 0,

�(e1, f2) = −ı�(e1, f1) = −ı =⇒ �(e1, f2) = 0, ω0(e1, f2) = 1,

�(e1, e2) = ı�(e1, e1) = 0 =⇒ �(e1, e2) = 0, ω0(e1, e2) = 0,

�(f1, f2) = −ı�(f1, f1) = 0 =⇒ �(f1, f2) = 0, ω0(f1, f2) = 0.

Therefore e1, e2, f1, f2 is a canonical basis in (V ,�) andω has the required formin the basis.

If ω �= 0 is parabolic, then

ω ∧ ω = 0 =⇒ ω = α ∧ β

for some linear independent 1-forms α,β ∈ V∗.The condition ⊥ ω = 0 gives

α(Xβ) = β(Xα) = �(Xα , Xβ) = 0.

In other words, the plane

�ω = ker α ∩ ker β

corresponding to ω has a basis Xα , Xβ and is Lagrangian. �

Remark 6.2.1 Note that in hyperbolic and elliptic cases for normalized2-forms

ω = 1√|Pf (ω)|ω

we have

A2ω + sign(Pf(ω)) = 0.


This means that elliptic forms define complex structures on V and hyperbolicforms give a splitting of V into direct sums of two symplectic planes. Parabolicforms correspond to Lagrangian planes in V.

We say that a Lagrangian plane L ⊂ V is an ω-plane if

ω|L = 0.

The following theorem gives a description of ω-planes.

Theorem 6.2.2

1. Elliptic forms. Let ω be an elliptic 2-form. Then L is an ω-plane if andonly if

Aω(L) = L.

2. Hyperbolic forms. Let ω be an hyperbolic 2-form, and let

V = V+ ⊕ V−

be the decomposition of V into sum of symplectic planes

V± = ker(Aω ∓ λ)

where λ = √−Pf(ω).Then L is an ω-plane if and only if

dim(L ∩ V±) = 1

and

L = L ∩ V+ ⊕ L ∩ V−.

3. Parabolic forms. Let ω be a parabolic 2-form and let

Lω = Im Aω

be the corresponding Lagrangian plane. Then L is an ω-plane if and only if

dim(L ∩ Lω) = 1.

Proof Elliptic forms. For any non-zero vector X ∈ V we have

�(X , AωX) = −ω(X , X) = 0


and

ω(X , AωX) = �(AωX , AωX) = 0.

Therefore, the plane L = span(X , AωX) is the ω-plane.Let L be an ω-plane, and let X , Y ∈ L be a basis. Then

�(AωX , X) = ω(X , X) = 0

and

�(AωX , Y) = ω(X , Y) = 0.

Therefore,

AωX ∈ L⊥ = L.

Hyperbolic forms. As we have seen in the proof of the classification theorem

V = V+ ⊕ V−

splits into a sum of two regular planes and

Aω|V± = ±λ.

Let L be an ω-plane. Then the above reasons show that

Aω(L) = L.

Aω|L is not scalar operator because V± are regular, but L is Lagrangian.Therefore Aω|L has two distinct eigenvalues and L = L ∩ V+ ⊕ L ∩ V−.

Parabolic forms. It follows from the representation

Lω = ker α ∩ ker β

if ω = α ∧ β. �

6.3 Jacobi planes

By Jacobi planes we mean two-dimensional subspaces � ⊂ �2(V∗) in thespace of 2-forms on a four-dimensional vector space V over R.

Such planes arise in the geometrical approach to systems of non-linear partialdifferential equations on two-dimensional manifolds.

6.3 Jacobi planes 143

6.3.1 Classification of Jacobi planes

We classify Jacobi planes with respect to the group GL(V) of lineartransformations of V .

Let � ⊂ �2(V∗) be a two-dimensional plane and let us fix a volume 4-formµ ∈ �4(V∗).

One can define a symmetric bilinear form q on �:

q : �×� � (α,β) −→ q(α,β) ∈ R

where

α ∧ β = q(α,β)µ.

By analogy with effective 2-forms on four-dimensional symplectic spaces wedistinguish Jacobi planes by restrictions of q on �.

We say that a Jacobi plane � is

1. elliptic, if q|� is a non-degenerate definite quadratic form,2. hyperbolic, if q|� is a non-degenerate sign indefinite quadratic form,3. parabolic, if q|� is a degenerate non-zero quadratic form, and4. Euler, if q|� = 0.

Let us take a basis ω1,ω2 in � and let

Q =∥∥∥∥q(ω1,ω1) q(ω1,ω2)

q(ω2,ω1) q(ω2,ω2)

∥∥∥∥be the matrix of q, and

ε(�) = sign det Q.

The following result follows directly from the structure of quadratic forms onthe plane.

Proposition 6.3.1 Let � ⊂ �2(V∗) be a Jacobi plane. Then

1. � is elliptic⇐⇒ ε(�) = 1, and there exists a basis ω1,ω2 in �, such that

ω1 ∧ ω2 = 0, ω1 ∧ ω1 = ω2 ∧ ω2 �= 0;

2. � is hyperbolic ⇐⇒ ε(�) = −1, and there exists a basis ω1,ω2 on �,such that

ω1 ∧ ω2 = 0, ω1 ∧ ω1 = −ω2 ∧ ω2 �= 0;


3. � is parabolic⇐⇒ ε(�) = 0, and there exists a basisω1,ω2 on�, such that

ω1 ∧ ω2 = 0, ω1 ∧ ω1 �= 0, ω2 ∧ ω2 = 0;

4. � is Euler⇐⇒ q|� = 0, and there exists a basis ω1,ω2 on �, such that

ω1 ∧ ω2 = ω1 ∧ ω1 = ω2 ∧ ω2 = 0.

Note that ω1,ω2 for elliptic, hyperbolic and parabolic cases are orthogonalwith respect to q and the 2-form ω1 determines a symplectic structure on V ,and the 2-form ω2 is effective. Therefore, we can use the above classificationof effective forms to obtain the following theorem.

Theorem 6.3.1 Let � ⊂ �2(V∗) be a Jacobi plane, and let ω1,ω2 be theorthogonal basis in �. Then there is a basis e1, e2, e3, e4 in V such that ω1,ω2

take one of the following normal forms.

1. Elliptic Jacobi planes:

ω1 = e∗1 ∧ f ∗1 + e∗2 ∧ f ∗2 ,

ω2 = e∗1 ∧ f ∗2 − e∗2 ∧ f ∗1 .

2. Hyperbolic Jacobi planes:

ω1 = e∗1 ∧ f ∗1 + e∗2 ∧ f ∗2 ,

ω2 = e∗1 ∧ f ∗1 − e∗2 ∧ f ∗2 .

3. Parabolic Jacobi planes:

ω1 = e∗1 ∧ f ∗1 + e∗2 ∧ f ∗2 ,

ω2 = e∗1 ∧ f ∗2 .

4. Euler planes:

ω1 = e∗1 ∧ f ∗1 ,

ω2 = e∗1 ∧ f ∗2 .

Corollary 6.3.1 There are four orbits of the group GL(V) on the set of Jacobiplanes: elliptic, hyperbolic, parabolic and Euler planes.

6.3 Jacobi planes 145

6.3.2 Operators associated with Jacobi planes

Let bdef= (ω1,ω2) be an oriented orthogonal basis of �, where ω1 is a non-

degenerate 2-form. We can consider ω2 = � as a symplectic structure on Vand ω1 = ω as an effective 2-form.

Consider the linear operator Ab = Aω : V → V , where

X�ω2 = AbX�ω1. (6.3)

As we have seen Ab satisfies the equation

A2b + Pf(ω) = 0

and Pf(ω) = ε(�) in our case.Let � be an elliptic or a hyperbolic plane, then transformations between

oriented orthogonal bases, say (ω1,ω2) and (ω′1,ω′2), are given by elliptic orhyperbolic similitudes.

That is,

ω1 = (t cos s)ω′1 − (t sin s)ω′2,

ω2 = (t sin s)ω′1 + (t cos s)ω′2

for elliptic planes and

ω1 = (t cosh s)ω′1 + (t sinh s)ω′2,

ω2 = (t sinh s)ω′1 + (t cosh s)ω′2

for hyperbolic planes.Here s ∈ R, t ∈ R � {0}.

Proposition 6.3.2 The operator Ab does not depend on the choice of orientedorthogonal basis in elliptic Jacobi planes and on the orthogonal basis for thehyperbolic ones.

Proof We prove the proposition for the elliptic case only.Let Ab′ corresponds to the orthogonal basis b′ = (ω′1,ω′2). We show that

Ab = Ab′ .Indeed, the definition of Ab, X�ω2 = AbX�ω1 reads as

X�(sin s ω′1 + cos s ω′2) = AbX�(cos s ω′1 − sin s ω′2)


or

(−sin s X + cos s AbX)�ω′1 = (cos s X + sin s AbX)�ω′2.

Therefore

Ab′ = (−sin s+ cos s Ab)(cos s+ sin s Ab)−1 = Ab.

The proof for the hyperbolic case goes in the same way, with an additionalremark: bases (ω1,ω2) and (ω2,ω1) define the same operator. �

Since the operator Ab does not depend on the choice of (oriented) orthogonalbasis in �, we will denote the operator Ab by A�. Thus, any (oriented) ellipticor hyperbolic Jacobi plane � generates a complex structure in the elliptic caseor a product structure in the hyperbolic case on V .

Moreover, operators A� inherit all the properties of the operators Aω.For example, if � is hyperbolic, the space V splits into a direct sum oftwo-dimensional planes V = V− ⊕ V+ and A�|V± = ±1.

7

Symplectic classification of exterior 2-forms

In this chapter we classify exterior 2-forms on symplectic spaces with respectto the symplectic group.

7.1 Pfaffians and linear operators associated with 2-forms

We have introduced the Pfaffian of 2-forms on symplectic four-dimensionalspaces. The same idea can be used to find this notion for symplectic spaces ofarbitrary dimension.

Namely, we note that the space �2n(V∗)dimV = 2n has dimension one andthe volume n-form is fixed by �n. Then for all ω ∈ �2(V∗) there is a uniquescalar λω such that

ωn = λω�n.

This scalar is called a Pfaffian of the form ω and is denoted by Pf(ω):

ωn = Pf(ω)�n. (7.1)

The Pfaffian is obviously an invariant of the Sp(V)-action, i.e., if S ∈Sp(V ,�), then

Pf(S∗(ω)) = Pf(ω).

In fact,

S∗(ωn) = S∗(Pf(ω)�n) = Pf(ω)S∗(�n) = Pf(ω) �n,

and

S∗(ωn) = (S∗(ω))n = Pf(S∗(ω))�n.

147

148 Symplectic classification of exterior 2-forms

We also define a characteristic polynomial of an exterior 2-form ω ∈�2(V∗), as

Pω(λ) = Pf(ω − λ�),

λ ∈ R.The polynomial Pω(λ) is an invariant of the Sp(V ,�)-action:

Pω(λ) = PS∗(ω)(λ). (7.2)

Therefore the coefficients I0(ω), . . . , In−1(ω)of the characteristic polynomial

Pω(λ) = I0(ω)− λI1(ω)+ · · · + (−1)n−1In−1(ω)λn−1 + (−1)nλn

are invariants of the symplectic group Sp(V ,�).The roots of the polynomial Pω are called characteristic numbers of the

2-form ω. We denote by char ω ⊂ C the set of all characteristic numbers of ω.Let λ ∈ R be a characteristic number ofω, then (ω−λ�)n = 0 and therefore

ω − λ� is the degenerate 2-form.Denote by K(λ) ⊂ V the kernel of this form. This is an even dimensional

subspace and ω − λ� is non-degenerate on the factor-space V�K(λ).If λ ∈ C is a complex characteristic number of ω then the above holds for

the complexifications of ω and V .Let us denote by Aω : V → V the linear operator defined by the 2-form ω:

ιXω = ιAωX�

for all X ∈ V .

Lemma 7.1.1

1. The operator Aω is symmetric with respect to the structure form �, i.e.

�(AωX , Y) = �(X, AωY)

for all X , Y ∈ V.2. For any vector X ∈ V the vectors X and AωX are skew-orthogonal:

�(AωX , X) = 0.3.

AS∗(ω) = S−1 ◦ Aω ◦ S

for all S ∈ Sp(V ,�).

7.2 2-forms: distinct real characteristic numbers 149

4. λ is a characteristic number for ω if and only if λ is an eigenvalueof Aω.

Proof Indeed, for X and Y ∈ V we have

�(X, AωY) = −�(AωY , X) = −ω(Y , X) = ω(X, Y) = �(AωX, Y)

and

�(AωX , X) = ω(X , X) = 0.

In addition,

ιXS∗(ω)(Y) = ω(SX , SY)

= �(AωSX , SY) = �(S−1AωSX, Y)

= ιS−1AωS(X)(�)(Y).

Assume that λ ∈ R is a real characteristic number for ω and let X ∈ K(λ) bea non-zero vector. Then

(ω − λ�)(X , Y) = 0

for all Y ∈ V .But

(ω − λ�)(X , Y) = �((Aω − λ)X, Y).

Therefore, (Aω − λ)X = 0. �

Remark 7.1.1 It follows that all eigenvalues of Aω have even multiplicities,and, as we will see below, (see also, [3]) :

(Pf(ω))2 = det(Aω).

7.2 Symplectic classification of 2-forms with distinct realcharacteristic numbers

Proposition 7.2.1 Let λ1 and λ2 be distinct real characteristic numbers of a2-form ω. Then K(λ1) and K(λ2) are skew-orthogonal with respect to �:

�(X , Y) = 0,


as well as with respect to ω,

ω(X , Y) = 0

for all X ∈ K(λ1), Y ∈ K(λ2).Moreover,

K(λ1) ∩ K(λ2) = 0.

Proof We have

ω(X , Y) = �(AωX , Y) = λ1�(X, Y)

and

ω(X , Y) = �(X , AωY) = λ2�(X, Y).

Therefore, (λ1 − λ2)�(X , Y) = 0 and �(X , Y) = 0. This also implies thatω(X, Y) = 0.

Subspaces K(λ1) and K(λ2) have zero intersection because they are eigensubspaces of the operator Aω corresponding to distinct eigenvalues. �

Let us assume now that characteristic numbers of the 2-form ω are real anddistinct, say, λ1, . . . , λn, where dim V = 2n.

Then

V = K(λ1)⊕ · · · ⊕ K(λn)

is a skew-orthogonal decomposition symplectic space V into the direct sum ofplanes. These planes are symplectic too because the structure form � has nokernel.

The restrictions ωi = ω|K(λi) are proportional to the restrictions �i =�|K(λi), and

ωi(X , Y) = �i(AωX , Y) = λι�i(X, Y)

for all X, Y ∈ K(λi).Therefore,

ωi = λi�i

for all i = 1, . . . , n and

ω = λ1�1 ⊕ · · · ⊕ λn�n.

7.3 2-forms: distinct real characteristic numbers 151


Theorem 7.2.1 Let ω ∈ �2(V∗) be a 2-form on the symplectic vector space(V ,�). Assume that all characteristic numbers λ1, . . . , λn of ω are distinct andreal. Then

1. the symplectic vector space V is the direct sum of the symplectic planesK(λi), λi ∈ char ω:

V = K(λ1)⊕ · · · ⊕ K(λn);

2. all planes K(λi) and K(λj), i �= j, are skew-orthogonal with respect to bothforms ω and �;

3. the form ω is the direct sum of plane 2-forms λi�i, where �i = �|K(λi) arethe restrictions of the structure form � on the planes

ω = λ1�1 ⊕ · · · ⊕ λn�n

and

� = �1 ⊕ · · · ⊕�n.

Corollary 7.2.1 Let ω ∈ �2(V∗) be a 2-form on the symplectic vector space(V ,�) having all characteristic numbers λ1, . . . , λn ofω distinct and real. Thenthere is a canonical basis e1, . . . , en, f1, . . . , fn in the symplectic vector space Vsuch that

ω = λ1e∗1 ∧ f ∗1 + · · · + λne∗n ∧ f ∗n .

Corollary 7.2.2

Pf(ω) = λ1 · · · λn.

Corollary 7.2.3 Two 2-forms ω ∈ �2(V∗) and ω′ ∈ �2(V∗) on the symplecticvector space (V ,�)with distinct and real characteristic numbers are equivalentwith respect to Sp(V ,�) if and only if their characteristic numbers, or equallythe values of invariants I0, . . . , In−1, are coincident:

I0(ω) = I0(ω′), . . . , In−1(ω) = In−1(ω

′).


7.3 Symplectic classification of 2-forms with distinct complexcharacteristic numbers

In this section we consider the case of complex characteristic numbers.Let us denote by VC the complexification of the symplectic vector space V ,

and let �C and ωC be the complexifications of forms � and ω respectively.Obviously, all of the above results are valid for 2-forms ωC in the complexsymplectic space (VC,�C). Assume now that the complex characteristic num-bers λ1, . . . , λn of ωC are distinct. Then one has the direct decomposition ofthe symplectic space

VC = K(λ1)⊕ · · · ⊕ K(λn)

and forms

ωC = λ1�1 ⊕ · · · ⊕ λn�n,

�C = �1 ⊕ · · · ⊕�n

where �i = �C|K(λi),ωi = ωC|K(λi).For v = x+ ιy ∈ VC (x, y ∈ V) denote its complex conjugate by v = x− ιy ∈

VC. Then elements of V (real vectors) are fixed vectors for the conjugation:

v ∈ V ⇐⇒ v = v.

Note also that, for example, ωC(v, w) = ωC(v, w), and therefore,

K(λ) = K(λ).

Hence, for the characteristic number λ with non-trivial imaginary part, λ �= λ,the subspace

K(λ)⊕ K(λ)

is a four-dimensional (over C) symplectic vector space with respect to therestriction �λ ⊕�λ.

The real part

K(λ, λ) = (K(λ)⊕ K(λ)) ∩ V

is a four-dimensional (over R) symplectic vector space with structure form�λ,λ = �|K(λ,λ).

7.3 2-forms: distinct complex characteristic numbers 153

This observation produces the decomposition

V =⊕

Im λ=0

K(λ)⊕⊕

Im λ�=0

K(λ, λ)

of the symplectic vector space into the direct sum of symplectic planes K(λ)

and symplectic four-dimensional spaces K(λ, λ).They are pairwise skew-orthogonal with respect to � and ω, and

� =⊕

λ∈char ω,Im λ=0

�λ ⊕⊕

λ∈char ω,Im λ�=0

�λ,λ,

ω =⊕


ωλ ⊕⊕


ωλ,λ,

where ωλ,λ = ω|K(λ,λ).

Note that on the symplectic four-dimensional space K(λ, λ) we have

I0(ωλ,λ) = λλ = |λ|2, I1(ωλ,λ) = 2 Re λ.

Therefore the 2-form

θλ,λ = ωλ,λ − Re λ �λ,λ

is an effective form of elliptic type with

Pf(θλ,λ) = (Im λ)2.

It follows from the above classification of 2-forms on four-dimensional vectorspaces that in a canonical basis

ωλ,λ = Re λ (e∗1 ∧ f ∗1 + e∗2 ∧ f ∗2 )+ Im λ (e∗1 ∧ f ∗2 − e∗2 ∧ f ∗1 ).

Finally, we obtain the following theorem.

Theorem 7.3.1 Let ω ∈ �2(V∗) be a 2-form on the symplectic vector space(V ,�) with district characteristic numbers λ1, . . . , λn. Then,

1. the symplectic vector space V is the direct sum of the symplectic planesK(λ) corresponding to real characteristic numbers λ and four-dimensionalsymplectic spaces K(λ, λ) corresponding to the complex ones:

V =⊕


K(λ)⊕⊕


K(λ, λ);


2. all of these subspaces are pairwise skew-orthogonal with respect to formsω and �, and

ω =⊕


λ�λ ⊕⊕


ωλ,λ,

� =⊕


�λ ⊕⊕


�λ,λ,

where

ωλ,λ = Re λ �λ,λ + Im λ θλ,λ

and θλ,λ is an elliptic effective form on the four-dimensional symplectic space

(K(λ, λ),�λ,λ) with Pf(θλ,λ) = (Im λ)2.

Corollary 7.3.1 Let ω ∈ �2(V∗) be a 2-form on the symplectic vector space(V ,�) with distinct characteristic numbers λ1, . . . , λn. Then there is a canon-ical basis e1, . . . , ek , f1, . . . , fk , ek+1, ek+1, fk , fk+1, . . . in the symplectic vectorspace V such that

ω =∑

Im λi=0

λie∗1 ∧ f ∗1

+∑

Im λi �=0

(Re λi (e∗i ∧ f ∗i + e∗i ∧ f ∗i )+ Im λi (e

∗i ∧ f ∗i − e∗i ∧ f ∗i )).

Corollary 7.3.2 Pf(ω) = λ1 · · · λn.

Corollary 7.3.3 Two 2-forms ω ∈ �2(V∗) and ω′ ∈ �2(V∗) on the symplecticvector space (V ,�) with distinct characteristic numbers are equivalent withrespect to Sp(V ,�) if and only if their characteristic numbers, or equally thevalues of invariants I0, . . . , In−1, coincide:

I0(ω) = I0(ω′), . . . , In−1(ω) = In−1(ω

′).

7.4 Symplectic classification of 2-forms with multiplecharacteristic numbers

Let

Ki(λ) = ker(Aω − λ)i

7.4 2-forms: multiple characteristic numbers 155

and

K∗(λ) =⋃i≥1

Ki(λ)

be subspaces of vectors associated with a characteristic number λ.

Proposition 7.4.1 K∗(λ1) and K∗(λ2) are skew-orthogonal with respect to ω

and � if λ1 �= λ2.

Proof The proof is based on the following formula:

�((Aω − λ)X , Y)−�(X , (Aω − ν)Y) = (ν − λ)�(X, Y). (7.3)

It follows immediately from this formula that K1(λ1) and K1(λ2) are skew-orthogonal if λ1 �= λ2.

Assume that we have proved that Ki(λ1) and Kj(λ2) are skew-orthogonalunder the condition λ1 �= λ2 for all i and j such that i + j ≤ k − 1. Let us takeX ∈ Ki(λ1) and Y ∈ Kj(λ2), with i + j = k.

Then,

�((Aω − λ1)X , Y) = 0 and �(X , (Aω − λ2)Y) = 0

by the induction hypothesis. Therefore, it follows from (7.3) that �(X, Y) = 0.In the same way one obtains ω-skew orthogonality from the similar formula

ω((Aω − λ)X , Y)− ω(X , (Aω − ν)Y) = (ν − λ)ω(X, Y)

which follows from (7.3). �

Theorem 7.4.1 Let ω ∈ �2(V∗) be a 2-form on the symplectic vector space(V ,�). Then,

V =⊕

λ∈char ω

K∗(λ)

is the decomposition of V into the direct sum of ω- and �-skew orthogonalsymplectic subspaces.

We shall investigate the nilpotent case only.

Proposition 7.4.2 Let ω ∈ �2(V∗) be a 2-form on the symplectic vector space(V ,�) such that the operator Aω is a nilpotent, Ak

ω = 0, but Ak−1ω �= 0. Let


X ∈ V be a vector of height k, Ak−1ω X �= 0. Then,

1. the subspace generated by the vectors X , AωX, . . . , Ak−1ω X is isotropic;

2. there exists a symplectic 2k-dimensional subspace W ⊂ V such that X ∈ Wand AωW ⊂ W;

3. the vector X can be included in a basis

X = Xk , Xk−1 = AωXk , . . . , X1 = Ak−1ω Xk ,

Yk , Yk−1 = AωYk , . . . , Y1 = Ak−1ω Yk

of W in such a way that

�(Xi, Xj) = �(Yi, Yj) = 0

and

�(Xi, Yj) = δi+j, k+1;

4. the skew-orthogonal subspace W⊥ is an Aω-invariant;5. there is a canonical basic e1, . . . , ek , f1, . . . , fk such that the 2-form ω|W can

be written in the basis as

ω|W =k−1∑i=1

e∗i ∧ f ∗i+1. (7.4)

Proof First of all let us prove that

�(AiωX , X) = 0

for all i = 1, 2, . . . .Indeed, if i is even, say i = 2l, then

�(AiωX , X) = �(A2l

ω X , X) = �(AlωX, Al

ωX) = 0,

and if i is odd, say i = 2l + 1, then

�(AiωX, X) = �(A2l+1

ω X , X) = �(AωAlωX , Al

ωX) = ω(AlωX, Al

ωX) = 0.

Let Z be a such that

�(X1, Z) = 1.


We obtain

1 = �(X1, Z) = �(Ak−1ω X , Z) = �(X, Ak−1

ω Z).

Therefore, Ak−1ω Z �= 0 and Z is a vector of height k. We define vectors

Zi = Ak−iω Z

for i = 0, 1, . . . , k − 1. Let us take Y1 = Z1 and use induction to construct thevectors Yj such that

Aω(Yj) = Yj−1 (7.5)

for all j = 2, . . . , k, and

�(Xk , Yj) = 0.

To obtain Y2 we change Z2 for

Y2 = Z2 − aZ1

for some a ∈ R. One has

Aω(Z2 − aZ1) = Aω(Z2) = Y1

and

�(Xk , Z2 − aZ1) = �(Xk , Z2 − aZ1) = �(Xk , Z2)− a�(Xk , Z1)

= �(Xk , Z2)− a�(Xk , Ak−1ω Zk) = �(Xk , Z2)− a�(X1, Zk)

= �(Xk , Z2)− a.

Therefore, the vector

Y2 = Z2 −�(Xk , Z2) Z1

satisfies the above requirement.Assume that we have constructed the vectors Y2, . . . , Yj−1 such that (7.5) is

satisfied and �(Xk , Yi) = 0 for i = 2, . . . , j − 1.


To find Yj let us consider the following vector:

v = Zj − a1Y1 − · · · − aj−1Yj−1.

We have

Aωv = Zj−1 − a2Y1 − · · · − aj−1Yj−2.

Let

Yj−1 = Zj−1 + b1Y1 + b2Y2 + · · · + bj−2Yj−2.

If we take

a2 + b1 = 0, . . . , aj−1 + bj−2 = 0

and arbitrary a1, then

Aωv = Yj−1.

Finally, the condition �(Xk , v) = 0 defines a1 because �(Xk , Y1) = 1. �

Theorem 7.4.2 Let ω ∈ �2(V∗) be a 2-form on a symplectic vector space(V ,�) such that operator Aω is nilpotent. Then there is a decomposition of thesymplectic vector space V into the direct sum of symplectic subspaces

V =1/2 dim ker Aω⊕

j=1

Wj

that are ω-skew-orthogonal (or Aω-invariant), and the restrictions ω|Wj can bewritten in normal forms (7.4) for all j = 1, . . . , 1/2 dim ker Aω.

Summarizing all of these results we obtain the following.

Theorem 7.4.3 Let ω ∈ �2(V∗) be a 2-form on the symplectic vector space(V ,�). Then there exists a decomposition of V into a direct sum of symplecticsubspaces V = ⊕λ∈char ω V(λ) in such a way that:

1. V(λ) are �- and ω-skew orthogonal;2. V(λ) are symplectic planes K(λ) for distinct real characteristic numbers

and four-dimensional symplectic subspaces K(λ, λ) for complex distinct realcharacteristic numbers;


3. if λ is a multiple characteristic number then there is an extra decompositionof V(λ) into a direct sum of symplectic subspaces

V(λ) =1/2 dim ker(Aω−λ)⊕

j=1

Wj(λ)

which are skew-orthogonal with respect to forms ω and �;4. in all three cases there are canonical basises (e, f ) such that the restrictions

ω have one of the following normal forms:•

λe∗1 ∧ f ∗1

if λ is a real and simple characteristic number,•

Re λ (e∗1 ∧ f ∗1 + e∗2 ∧ f ∗2 )+ Im λ (e∗1 ∧ f ∗2 − e∗2 ∧ f ∗1 )

if λ is a complex and simple characteristic number,•

λ

k∑i=1

e∗i ∧ f ∗i +k−1∑i=1

e∗i ∧ f ∗i+1

if λ is a real characteristic number of multiplicity k.•

Re λ2k∑

i=1

e∗i ∧ f ∗i + Im λ

k∑i=1

(e∗i ∧ f ∗i+k − e∗i+k ∧ f ∗i )

+k−1∑i=1

(e∗i ∧ f ∗i+1 + e∗k+i ∧ f ∗k+i+1)

if λ is a complex characteristic number of multiplicity k.

Proof For any real and distinct complex characteristic numbers the theoremfollows from the above results and the formula �(AωX, Y) = ω(X, Y). Theproof for complex multiple characteristic numbers is as follows. Let λ ∈ Cbe a complex multiple characteristic number, then after complexification weobtain V(λ) = V(λ) ⊂ VC and Wj(λ) can be chosen in such a way that


Wj(λ) = Wj(λ). Moreover, if (Xk , . . . , X1, Yk , . . . , Y1) is the canonical basisin Wj(λ), then (Xk , . . . , X1, Yk , . . . , Y1) is a canonical basis in Wj(λ). Then thebasis

e1 = X1 + X1

2, . . . , ek = Xk + Xk

2,

ek+1 = X1 − X1

2ı, . . . , e2k = Xk − Xk

2ı,

f1 = Y1 + Y1

2, . . . , fk = Yk + Yk

2,

fk+1 = Y1 − Y1

2ı, . . . , f2k = Yk − Yk

2ı

is the required basis for ω on (Wj(λ)⊕Wj(λ)) ∩ V . �

Corollary 7.4.1 Two 2-forms ω and ω′ on a symplectic vector space V areSp-equivalent if and only if the corresponding linear operators Aω and Aω′ areGL-equivalent.

Corollary 7.4.2 det Aω = (Pf ω)2.

7.5 Symplectic classification of effective 2-forms indimension 6

Let V be a symplectic vector space of dimension 6, and let ω be an effective2-form on V . Then ω ∧ �2 = 0 and I2(ω) = 0. Therefore, the characteristicequation for this case takes the following form:

Pf(ω − λ�) = −λ3 − I1(ω)λ+ I0(ω) = 0.

There are four possible cases for the characteristic numbers λ1, λ2, λ3.

1. Three real and distinct characteristic numbers with

λ1 + λ2 + λ3 = 0.

2. Three real characteristic numbers with one real number, say λ1, ofmultiplicity 2:

λ1 = λ2 = λ, λ3 = −2λ.

7.5 2-forms in dimension 6 161

3. One real number, say λ1, and two complex characteristic numbers:

λ1 = 2λ, λ2 = −λ+ ıµ, λ3 = −λ− ıµ.

4. One real characteristic number of multiplicity 3:

λ1 = λ2 = λ3 = 0.

Applying the classification theorem for these four cases we obtain thefollowing theorem.

Theorem 7.5.1 Let ω ∈ �2ε(V

∗6 ) be an effective 2-form on a symplectic real

space V of dimension 6. Then there is a canonical basis e1, e2, e3, f1, f2, f3 suchthat ω can be written in one of the following normal forms:

1.

λ1e∗1 ∧ f ∗1 + λ2e∗2 ∧ f ∗2 + λ3e∗3 ∧ f ∗3

with λ1 + λ2 + λ3 = 0;

2.

2λe∗1 ∧ f ∗1 − λ(e∗2 ∧ f ∗2 + e∗3 ∧ f ∗3 )+ µ(e∗3 ∧ f ∗2 − e∗2 ∧ f ∗3 );

3.

2λe∗1 ∧ f ∗1 − λ(e∗2 ∧ f ∗2 + e∗3 ∧ f ∗3 )+ e∗2 ∧ f ∗3 ;

4.

e∗1 ∧ f ∗2 + e∗2 ∧ f ∗3 .

8

Classification of exterior 3-forms ona six-dimensional symplectic space

This chapter is devoted to a classification of effective 3-forms in six-dimensionalsymplectic space under the natural Sp-action. The question has a surprisinglylong history. Being in the spirit of the classical problems of geometric invari-ants theory, the solution of this problem over C was found in [40, 90]. Thecase of real symplectic spaces which has great importance in classification ofMonge–Ampère equations was investigated in [77] and completed in [5].

8.1 A symplectic invariant of effective 3-forms

Let V be a six-dimensional symplectic vector space and let ω ∈ �3ε(V) be an

effective 3-form.We associate with ω the following quadratic form qω ∈ S2(V∗):

qω(X) = −1

2⊥2ω

2X ,

where

ωXdef= ιXω ∈ �2(V∗).

This form is clearly Sp-invariant:

qS∗(ω) = S∗(qω),

for all S ∈ Sp(V).Indeed, we have (S∗ω)X = S∗(ωS(X)) for any operators S, and

S∗ ◦ ⊥=⊥ ◦ S∗

162

8.1 A symplectic invariant of effective 3-forms 163

if S ∈ Sp (V). Therefore,

qS∗(ω)(X) = −1

2⊥2(S

∗(ω))2X = −1

2⊥2S∗(ωS(X))

2 = −1

2⊥2(ωS(X))

2

= qω(SX) = S∗(qω)(X).

Consider now the characteristic polynomial of ωX :

(ωX − λ�)3 = ω3X − 3λω2

X ∧�+ 3λ2ωX ∧�2 − λ3�3.

We have ω3X = 0 because ωX is a degenerated 2-form, X ∈ ker ωX , and ωX ∧

�2 = 0 because ωX is an effective form. Moreover,

ω2X ∧� = −1

3qω(X)�

3,

because

⊥3 (ω2X ∧�) =⊥3 ◦�(ω2

X) = [⊥3,�](ω2X)= ⊥2 (ω

2X) = −2qω(X)

and

⊥3 (�3) = 6.

Therefore,

Pf(ωX − λ�) = −λ3 + qω(X) λ

and the characteristic numbers of ωX are

λ1 = 0, λ2 =√

qω(X), λ3 = −√

qω(X).

Below we give another interpretation of the invariant qω.To this end we choose two vectors X and Y in V in such a way that�(X, Y)= 1

and denote by W a symplectic subspace which is skew-orthogonal to the sym-plectic plane 〈X, Y〉. Let �′ be the restriction of � on W , and �′ and ⊥′ be thecorresponding operators.

Let X ∈ V∗ be the covector dual to the vector X ∈ V , X = ιX�. The 2-formω can be uniquely decomposed as

ω = ω0 ∧ X ∧ Y + ω1 ∧ X + ω2 ∧ Y + ω3,

164 Classification of exterior 3-forms on a 6D

with ω0 ∈ W∗,ω1,ω2 ∈ �2(W∗),ω3 ∈ �3(W∗). Using the decompositionV = W ⊕ 〈X, Y〉 and ⊥=⊥′ +X ∧ Y we obtain

0 =⊥ (ω) = ω0+ ⊥′ (ω1) ∧ X+ ⊥′ (ω2) ∧ Y+ ⊥′ (ω3).

Therefore,

⊥′ (ω1) =⊥′ (ω2) = 0

and

ω0+ ⊥′ (ω3) = 0.

In other words, 2-formsω1 andω2 are effective, butω0 andω3 satisfy the aboverelation.

Note also that

⊥′ (�′ω0 + ω3) = ω0+ ⊥′ (ω3) = 0

and ⊥′ :�3(W∗)→ W∗ is an isomorphism. Therefore,

ω3 = −�′ω0,

and finally we obtain the decomposition

ω = ω0 ∧ (X ∧ Y −�′)+ ω1 ∧ X + ω2 ∧ Y

where all forms ω0,ω1 and ω2 are effective on W .The invariant qω provides us with some information about this decomposi-

tion. Namely, we find

ωX = −ω0 ∧ X − ω2

and

qω(X) = −1

2⊥′2(ω2 ∧ ω2) = −Pf(ω2).

Thus, if we denote by qω(X , Y) the symmetric bilinear form associated withqω, then

qω(X , Y) = −1

2⊥′2(ω1 ∧ ω2).

Therefore, we arrive at the following result.


Proposition 8.1.1 The skew-orthogonal decomposition V = W ⊕〈X, Y〉 givesrise to the decomposition of an effective 3-form,

ω = ω0 ∧ (X ∧ Y −�′)+ ω1 ∧ X + ω2 ∧ Y

with effective forms ω1,ω2 and ω0 on W.Moreover,

qω(X) = −Pf(ω2)

and

qω(X , Y) = −1

2⊥′2(ω1 ∧ ω2).

8.1.1 The case of trivial invariants

In this section we investigate a structure of effective 3-forms with qω = 0.

Lemma 8.1.1 Let ω ∈ �3ε(V) satisfy ω ∧ ωX = 0 for all vectors X ∈ V. Then

there exists a non-zero vector v such that ωv = 0.

Proof Let ω ∧ ωX = 0 for all vectors X . Then also

ωX ∧ ωX = 0,

for all vectors X.Note that this relation implies that ωX+Y ∧ ωX+Y = 0 and therefore

ωX ∧ ωY = 0

for all vectors X and Y .We shall consider two cases:

1. there is a pair of vectors X and Y such that �(X , Y) = 1 and Y ∈ ker ωX ;2. for all non-zero vectors X ∈ V , �(X , Y) = 0 for all Y ∈ ker ωX .

In the latter case one has

ωX ∧ X = 0

for all vectors X.


Let us now take vectors X and Z such that �(X, Z) = 1. It follows from thedecomposition ω = ω0 ∧ (X ∧ Z −�′)+ ω1 ∧ X + ω2 ∧ Z that

ωX = −ω0 ∧ X − ω2

and

ωX ∧ X = −ω2 ∧ X = 0

if and only if ω2 = 0. For the same reasons we have ω1 = 0 and finally

ω = ω0 ∧ (X ∧ Z −�′).

Taking v = Xω0 ∈ W we obtain

ωv = ω0 ∧ ω0 = 0.

Now let us consider the first case, and let X, Y be the pair of vectors suchthat �(X, Y) = 1 and Y ∈ ker ωX . Then for the symplectic decompositionV = W ⊕ 〈X, Y〉 we obtain

ω = ω1 ∧ X + ω2 ∧ Y

with effective 2-formsω1 andω2. It follows fromωY∧ωY = 0 andωY∧ωY = 0that ω1 ∧ω1 = 0, and ω2 ∧ω2 = 0. Moreover, the relation ωX ∧ωY = 0 givesω1 ∧ ω2 = 0.

We shall assume that both ω1 = ωY and ω2 = −ωX are non-zero forms. Inthe other case the lemma is proved. Since both 2-forms ω1 and ω2 are effectivethen the corresponding planes ker ω1 and ker ω2 are Lagrangian, and the relationω1∧ω2 = 0 implies that these Lagrangian planes have a non-trivial intersection.Let v ∈ ker ω1 ∩ ker ω2 be a non-zero vector. Then

ωv = ω1v ∧ X + ω2v ∧ Y = 0. �

Theorem 8.1.1 Let ω ∈ �3ε(V) be a non-zero effective 3-form on a

six-dimensional symplectic vector space having the trivial invariant qω, νi. Then


there exists a canonical basis e1, e2, e3, f1, f2, f3 such that

ω = e∗1 ∧ e∗2 ∧ e∗3.

Proof As we have seen, qω = 0 implies

� ∧ ωX ∧ ωX = 0,

and therefore

� ∧ ωX ∧ ωY = 0

for all vectors X and Y .Note also that the ω ∈ �3

ε(V) means ω ∧� = 0, and

� ∧ ωX = −�X ∧ ω

for all vectors X.Therefore,

�X ∧ ω ∧ ωY = 0,

for all X, Y ∈ V and the relation ω ∧ ωY = 0 holds for all Y ∈ V .Applying the above lemma, we find a non-zero vector X such that ωX = 0.Then let Y ∈ V be such that �(X , Y) = 1. In the symplectic decomposition

W ⊕ 〈X, Y〉 we obtain

ω = ω1 ∧ X ,

where ω1 is an effective 2-form on W .Moreover, ω1 is a parabolic effective 2-form on W , since qω(Y) = 0.From the classification of effective 2-forms on four-dimensional symplectic

spaces we conclude that there is a canonical basis (e1, e2, f1, f2) in W such that

ω1 = e∗1 ∧ e∗2. �

8.1.2 The case of non-trivial invariants

In this section we shall assume that qω �= 0.

Lemma 8.1.2 For any non-zero vector X such that qω(X) �= 0 the subspaceker ωX is a symplectic plane.


Proof One has dim ker ωX = 2 because ωX is a degenerate 2-form withω2

X �= 0. Assume now that ker ωx is an isotropic plane and pick a vectorZ ∈ ker ωX such that X ∧ Z �= 0. Let Y be a vector such that �(X, Y) = 1 and�(X, Z) = 0.

Let us take the decomposition ω = ω0 ∧ (X ∧ Y −�′)+ ω1 ∧ X + ω2 ∧ Yand compute ωZ = 0.

We obtain

ωZ = ω0(Z)(X ∧ Y −�′)+ ω0 ∧ Z + (ω1)Z ∧ X + (ω2)Z ∧ Y = 0

and therefore

(ω1)Z = (ω2)Z = 0,

ω0(Z) = 0, ω0 ∧ Z = 0.

However, since qω(X) = −Pf(ω2) �= 0, the 2-form ω2 is non-degenerate onW and then Z = 0. �

Proposition 8.1.2 Let ω ∈ �3ε(V) be an effective 3-form and let X, Y ∈ ker ωX

be a canonical basis, such that qω(X) �= 0 and qω(X, Y) = 0. Then in thedecomposition V = W ⊕ 〈X , Y〉, where 〈X , Y〉 def= span(X, Y), the form ω canbe written as

ω = ω1 ∧ X + ω2 ∧ Y ,

where ω1 and ω2 are effective 2-forms on the symplectic space (W ,�′) andω1 ∧ ω2 = 0.

Proof The decomposition shall take the above form because

ω0 = (ω)XY = 0.

As we have seen both formsω1 andω2 are effective and qω(X) = −Pf(ω2) �= 0.The property ω1 ∧ ω2 = 0 follows from qω(X, Y) = 0 in the follow-

ing way. As we have seen qω(X , Y)=− 12⊥′2(ω1 ∧ ω2) and hence ⊥2(ω1 ∧

ω2)=−2qω(X, Y)= 0. This is equivalent to the fact that

�◦ ⊥2 (ω1 ∧ ω2) =⊥2 ◦�(ω1 ∧ ω2) = 0

because � and ⊥2 commute on �4(V) for six-dimensional symplectic vectorspaces.


Furthermore, ⊥2: �6(V)→ �4(V) is a monomorphism and therefore

�(ω1 ∧ ω2) = � ∧ ω1 ∧ ω2 = 0.

Substituting the vector X into � ∧ ω1 ∧ ω2 we obtain

X ∧ ω1 ∧ ω2 = 0

and therefore

ω1 ∧ ω2 = 0. �

Theorem 8.1.2 Any non-zero effective 3-form on a six-dimensional symplecticvector space V can be written in a canonical basis e1, e2, e3, f1, f2, f3 in one ofthe following forms.

1.

ω = e∗1 ∧ e∗2 ∧ e∗3 + γ f ∗1 ∧ f ∗2 ∧ f ∗3

with the invariant

qω = γ

2(e∗1 f ∗1 + e∗2 f ∗2 + e∗3 f ∗3 )

and γ �= 0.2.

ω = f ∗1 ∧ e∗2 ∧ e∗3 + f ∗2 ∧ e∗1 ∧ e∗3 ++f ∗3 ∧ e∗1 ∧ e∗2 + ν2f ∗1 ∧ f ∗2 ∧ f ∗3

with the invariant

qω = (e∗1)2 − (e∗2)2 + (e∗3)2 + ν2((f ∗1 )2 − (f ∗2 )2 + (f ∗3 )2)

and ν �= 0.


3.

ω = f ∗1 ∧ e∗2 ∧ e∗3 − f ∗2 ∧ e∗1 ∧ e∗3 + f ∗3 ∧ e∗1 ∧ e∗2 − ν2f ∗1 ∧ f ∗2 ∧ f ∗3

with the invariant

qω = −(e∗1)2 − (e∗2)2 − (e∗3)2 − ν2((f ∗1 )2 + (f ∗2 )2 + (f ∗3 )2)

and ν �= 04.

ω = f ∗1 ∧ e∗2 ∧ e∗3 + f ∗2 ∧ e∗1 ∧ e∗3 + f ∗3 ∧ e∗1 ∧ e∗2

with the invariant

qω = (e∗1)2 − (e∗2)2 + (e∗3)2.

5.

ω = f ∗1 ∧ e∗2 ∧ e∗3 − f ∗2 ∧ e∗1 ∧ e∗3 + f ∗3 ∧ e∗1 ∧ e∗2

with the invariant

qω = −(e∗1)2 − (e∗2)2 − (e∗3)2.

6.

ω = f ∗3 ∧ e∗1 ∧ e∗2 + f ∗2 ∧ e∗1 ∧ e∗3

with the invariant

qω = (e∗1)2.

7.

ω = f ∗3 ∧ e∗1 ∧ e∗2 − f ∗2 ∧ e∗1 ∧ e∗3

with the invariant

qω = −(e∗1)2.


8.

ω = e∗1 ∧ e∗2 ∧ e∗3

with the invariant

qω = 0.

Proof We shall consider the case of non-trivial invariants only. Let us take apair of vectors X, Y as in the above proposition and use the decomposition

ω = ω1 ∧ X + ω2 ∧ Y .

Then the problem of finding normal forms for ω shall be reduced to a studyof 2-forms �′,ω1,ω2 on the symplectic four-dimensional space W .

The forms ω1,ω2 are effective and can be hyperbolic, elliptic, parabolic andω2 can be trivial. A careful study of all cases allows us to obtain all possibleorbits. This study is a little bit long and tiresome, so we will only discuss thedetails of one case.

Namely, the case when ω2 and � are elliptic in the symplectic vectorspace (W ,ω1). Then there exists a canonical basis e1, e2, f1, f2 in (W ,ω1)

such that

ω2 = λ(e∗1 ∧ f ∗2 + f ∗1 ∧ e∗2)

for some λ �= 0.From the relations �′ ∧ ω1 = �′ ∧ ω2 = 0, we conclude that

�′ = p f ∗1 ∧ f ∗2 + qe∗1 ∧ e∗2 + r(e∗1 ∧ f ∗2 + e∗2 ∧ f ∗1 )+ s(e∗1 ∧ f ∗1 − e∗2 ∧ f ∗2 )

where pq + r2 + s2 < 0, since �′ is elliptic.In particular, q cannot be equal to zero.Let At and Bt be transformations of W that depend on the real parameter t

and in the basis e1, e2, f1, f2 has the following forms:

At =

∥∥∥∥∥∥∥∥1 0 0 t0 1 t 00 0 1 00 0 0 1

∥∥∥∥∥∥∥∥ , Bt =

∥∥∥∥∥∥∥∥1 0 t 00 1 0 −t0 0 1 00 0 0 1

∥∥∥∥∥∥∥∥ .


Then At and Bt preserve ω1,ω2 and act on � in the following way:

At : (p, q, r, s)At−→ (p− qt2 − 2st, q, r, s+ qt),

Bt : (p, q, r, s)Bt−→ (p− qt2 + 2rt, q, r, r − qt).

If we apply the transformation Bu and then the transformation Av withu = −r/q and v = −s/q, we obtain the following 2-forms:

ω1 = x1 ∧ y1 + x2 ∧ y2,

ω2 = λ(x1 ∧ y2 − x2 ∧ y1),

�′ = px1 ∧ x2 + qy1 ∧ y2,

where λ �= 0 and pq < 0.After applying the following transformation

F =

∥∥∥∥∥∥∥∥et 0 0 00 et 0 00 0 e−t 00 0 0 e−t

∥∥∥∥∥∥∥∥ ,

with e4t = −q/p > 0, we obtain the following expression for �′:

�′ = µ(x1 ∧ x2 − y1 ∧ y2)

and F does not change ω1 and ω2.In the canonical basis

e′1 = X, e′2 = µe1, e′3 = µf1, f ′1 = Y , f ′2 = e2, f ′3 = −f2

of (V ,�) one obtains

ω = 1

µ2f ′∗1 ∧ f ′∗2 ∧ f ′∗3 − f ′∗1 ∧ e′∗2 ∧ e′∗3 +

λ

µe′∗1 ∧ f ′∗2 ∧ e′∗3 +

λ

µe′∗1 ∧ e′∗2 ∧ f ′∗3

and in the canonical basis e1, e2, e3, f1, f2, f3, where

e1 = µe′1, e2 = µνf ′2, e3 = νf ′3,

f1 = 1

µf ′1, f2 = 1

µνf ′2, f3 = −1

νe′3,


with µ/λ = εν2, ε = ±1, we have

ω = e∗2 ∧ f ∗1 ∧ f ∗3 − e∗3 ∧ f ∗1 ∧ f ∗2 − εe∗1 ∧ f ∗2 ∧ f ∗3 + εν2e∗1 ∧ e∗2 ∧ e∗3. �

8.1.3 Hitchin’s results on the geometry of 3-forms

Here we describe some results of N. Hitchin on the geometry of 3-forms in six-dimensional vector spaces and their relations to the above symplectic geometryof 3-forms.

Let V be a six-dimensional real vector space and let θ be a (fixed) volumeform on V . With a given 3-form ω ∈ �3(V∗) we associate a linear operatorHω : V → V by

Hω(X)�θ def= ıX(ω) ∧ ω.

The Hitchin Pfaffian of a 3-form ω ∈ �3(V∗) is

h(ω)def= 1

6Tr H2

ω.

If h(ω) is non-zero, then we say that ω is non-degenerate.

Proposition 8.1.3 (N. Hitchin [38]) Let ω ∈ �3(V∗) be non-degenerate.Then,

1. H2ω = h(ω);

2. h(ω) > 0 if and only ifω = α+β, where α and β are decomposable 3-formson V. Furthermore, if we suppose that

α ∧ β

θ> 0,

then α and β are unique and

2α = ω + |h(ω)|−3/2(Hω)∗(ω)

2β = ω − |h(ω)|−3/2(Hω)∗(ω);

3. h(ω) < 0 if and only if ω = α + α, where α is a complex decomposable3-form on V. Furthermore, if we suppose that

α ∧ α

ιθ> 0,


then α is a unique:

2α = ω + ι|h(ω)|−3/2(Hω)∗(ω).

Remark 8.1.1 Let e1, . . . , e6 be a base of V and form the volume

θ = e∗1 ∧ · · · ∧ e∗6,

then,

1. h(ω) > 0 if and only if ω is in the GL(V)-orbit of

e∗1 ∧ e∗2 ∧ e∗3 + e∗4 ∧ e∗5 ∧ e∗6;

2. h(ω) < 0 if and only if ω is in the GL(V)-orbit of

(e∗1 + ιe∗4) ∧ (e∗2 + ιe∗5) ∧ (e∗3 + ιe∗6)+ (e∗1 − ιe∗4) ∧ (e∗2 − ιe∗5) ∧ (e∗1 − ιe∗6).

That is why the GL(V)-action on �3(V∗) has two open orbits separated bythe quartic hypersurface h = 0.

Let ω be a non-degenerate 3-form on V .The dual (by N. Hitchin) form ω is defined as follows:

if ω = α + β, then ωdef= α − β,

if ω = α + α, then ωdef= ι(α − α).

Hitchin also noted that the space �3(V∗) is a symplectic vector space withrespect to the skew-form � ∈ �2(�3(V∗)):

�(ω,ω′) def= ω ∧ ω′

θ.

Proposition 8.1.4 (N. Hitchin [38]) The action of SL(V , θ) on �3(V∗)enabled with the symplectic structure � is a Hamiltonian with the momentmap H : �3(V∗)→ sl6(V).

Suppose now that V is a symplectic vector space with the symplectic form� and we fix the volume form in such a way that θ = −16�3.

The space of all effective forms is a symplectic subspace in (�3(V∗),�). Itfollows immediately from Hodge–Lepage decomposition.

8.2 The stabilizers of orbits and their prolongations 175

Lemma 8.1.3 The action of Sp(V) on the symplectic linear variety(�3

ε(V∗),�) is Hamiltonian with momentum map

H : �3ε(V

∗)→ sp(V).

It is a natural question to compare this invariant of the Sp(V)-action with ourquadratic invariant qω ∈ S2(V∗) = sp(V).

We have the following proposition.

Proposition 8.1.5 (B. Banos [4]) Let ω be an effective 3-form on V. Then forall X ∈ V

qω(X) = �(HωX , X).

Hence, the invariant quadratic form qω is a moment map of the Hamiltonianaction of Sp(V) on �3

ε(V∗)):

qω : �3ε(V

∗)→ S2(V∗),

and the operator Hω is the dual to qω with respect to the symplectic form �.

8.2 The stabilizers of orbits and their prolongations

8.2.1 Stabilizers

In this section we describe the Lie algebras for stabilizers of effective 3-forms.Let ω be an effective 3-form on the six-dimensional symplectic vector space(V ,�).

We denote by

gωdef= {A ∈ sp(V ,�) | ω(AX , Y)+ ω(X , AY) = 0, ∀X, Y ∈ V}

the stabilizer of ω.In order to find stabilizers we shall use the normal forms and the canonical

bases for effective 3-forms.In these bases elements of gω will have the matrix form∥∥∥∥A B

C −At

∥∥∥∥with Bt = B, Ct = C.

A straightforward computation (using Maple) shows the following.


Theorem 8.2.1

1. Let ω = e∗1 ∧ e∗2 ∧ e∗3 + γ f ∗1 ∧ f ∗2 ∧ f ∗3 , then gω = sl(3, R), and elements ofgω have the following matrix form:

gω ={∥∥∥∥B 0

0 −Bt

∥∥∥∥}with B ∈ sl(3, R).

2. Let ω = f ∗1 ∧ e∗2 ∧ e∗3 + f ∗2 ∧ e∗1 ∧ e∗3 + f ∗3 ∧ e∗1 ∧ e∗2 + ν2f ∗1 ∧ f ∗2 ∧ f ∗3 , thengω = su(2, 1), and elements of gω have the following matrix form:

gω =

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

0 α β λ1 ξ1 ξ2

α 0 γ ξ1 λ2 ξ3

−β γ 0 ξ2 ξ3 λ3

−ν2λ1 ν2ξ1 −ν2ξ2 0 −α β

ν2ξ1 −ν2λ2 ν2ξ3 −α 0 −γ−ν2ξ2 ν2ξ3 −ν2λ3 −β −γ 0

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

.

3. Let ω = f ∗1 ∧ e∗2 ∧ e∗3 − f ∗2 ∧ e∗1 ∧ e∗3 + f ∗3 ∧ e∗1 ∧ e∗2 − ν2f ∗1 ∧ f ∗2 ∧ f ∗3 , thengω = su(3), and elements of gω have the following matrix form:

gω =

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

0 α β λ1 ξ1 ξ2

−α 0 γ ξ1 λ2 ξ3

−β −γ 0 ξ2 ξ3 λ3

−ν2λ1 −ν2ξ1 −ν2ξ2 0 α β

−ν2ξ1 −ν2λ2 −ν2ξ3 −α 0 γ

−ν2ξ2 −ν2ξ3 −ν2λ3 −β −γ 0

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

where λ1 + λ2 + λ3 = 0.

4. If ω = f ∗1 ∧ e∗2 ∧ e∗3+ f ∗2 ∧ e∗1 ∧ e∗3+ f ∗3 ∧ e∗1 ∧ e∗2, then the stabilizer algebrais the semidirect product gω = so(2, 1) × H2(2, 1), where H2(2, 1) is thespace of pseudo-harmonic symmetric 2-tensors and elements of gω, have thefollowing matrix form:

gω =

∥∥∥∥∥∥∥∥∥∥∥∥∥

0 α β λ1 ξ1 ξ2

α 0 γ ξ1 λ2 ξ3

−β γ 0 ξ2 ξ3 λ3

0 0 0 0 −α β

0 0 0 −α 0 −γ0 0 0 −β −γ 0

∥∥∥∥∥∥∥∥∥∥∥∥∥

with λ1 − λ2 + λ3 = 0.


5. Letω = f ∗1 ∧e∗2∧e∗3− f ∗2 ∧e∗1∧e∗3+ f ∗3 ∧e∗1∧e∗2. Then the stabilizer algebrais the semidirect product gω = so(3) � H2(3), where H2(3) is the spaceof harmonic symmetric 2-tensors, and elements of gω have the followingmatrix form:

gω =

∥∥∥∥∥∥∥∥∥∥∥∥∥

0 α β λ1 ξ1 ξ2

−α 0 γ ξ1 λ2 ξ3

−β −γ 0 ξ2 ξ3 λ3

0 0 0 0 α β

0 0 0 −α 0 γ

0 0 0 −β −γ 0

∥∥∥∥∥∥∥∥∥∥∥∥∥

with λ1 + λ2 + λ3 = 0.

6. Let ω = f ∗3 ∧ e∗1 ∧ e∗2+ f ∗2 ∧ e∗1 ∧ e∗3. Then elements of the stabilizer algebrahave the following matrix form:

gω =

∥∥∥∥∥∥∥∥∥∥∥∥∥

0 0 0 0 0 0λ1 µ ν 0 α1 α2

λ2 −ν µ 0 α2 −α1

γ1 γ2 γ3 0 −λ1 −λ2

γ2 γ4 γ5 0 −µ ν

γ3 γ5 −γ4 0 −ν −ν

∥∥∥∥∥∥∥∥∥∥∥∥∥

.

7. Let ω = f ∗3 ∧ e∗1 ∧ e∗2− f ∗2 ∧ e∗1 ∧ e∗3. Then elements of the stabilizer algebrahave the following matrix form:

gω =

∥∥∥∥∥∥∥∥∥∥∥∥∥

0 0 0 0 0 0λ1 µ ν 0 α1 α2

λ2 ν µ 0 α2 α1

γ1 γ2 γ3 0 −λ1 −λ2

γ2 γ4 γ5 0 −µ −νγ3 γ5 −γ4 0 −ν −ν

∥∥∥∥∥∥∥∥∥∥∥∥∥

.

8. Let ω = e∗1 ∧ e∗2 ∧ e∗3. Then the stabilizer algebra is the semidirect productgω = sl(3, R) � S2(3), where S2(3) is the space of symmetric 2-tensorson three-dimensional vector space, and elements of gω have the followingmatrix form:

gω ={∥∥∥∥A 0

B −A

∥∥∥∥ , A ∈ sl(3, R), Bt = B

}.


8.2.2 Prolongations

Recall that the prolongation of a linear subspace g of S2(V∗) is a subspaceg(1) of S3(V∗) such that

θ ∈ g(1) ⇐⇒ v(θ) ∈ g

for all v ∈ V .Here we denoted by v(θ) ∈ S2(V∗) the derivative of θ ∈ S3(V∗) along a

vector v ∈ V .We identify the Lie algebra gω with a subspace in S2(V∗) by taking

Hamiltonians A ∈ gω −→ hA ∈ S2(V∗), where hA(x, y) = �(Ax, y).

Theorem 8.2.2 Let ω be an effective 3-form on a six-dimensional symplecticvector space. Then

g(1)ω = 0

for all ω that are equivalent to one of the normal forms (1)–(5) in Theorem8.2.1.

Proof To illustrate the proof we consider the first normal form only. If thematrix

A =∥∥∥∥B 0

0 −Bt

∥∥∥∥ ∈ gω,

then the corresponding quadric form is

hA = Be∗f ∗.

Assume that θ ∈ S3(V∗) belongs to g(1)ω and let

θ = α f ∗3 + β f ∗2e∗ + γ f ∗e∗2 + δ e∗3.

Then e(θ) = β f ∗2 + 2γ f ∗e∗ + 3δ e∗2 ∈ gω implies β = 0, δ = 0, andf (θ) = 3α f ∗2 + 2β f ∗e∗ + γ e∗2 ∈ gω implies α = 0, γ = 0. Therefore,θ = 0. �

PART III

Monge–Ampère Equations

9

Symplectic manifolds

9.1 Symplectic structures

A smooth manifold M is called symplectic, if there is a non-degenerate closeddifferential 2-form defined on M.

That is, there is a differential 2-form � ∈ �2(M) such that d� = 0, and�a ∈ �2(T∗a M) is a symplectic structure on TaM for any point a ∈ M.

A smooth map ϕ : (M1,�1)→ (M2,�2) of symplectic manifolds is calledsymplectic if

ϕ∗(�2) = �1.

Symplectic diffeomorphisms are also called symplectomorphisms, and, in thecase where M1 = M2, symplectic or canonical transformations.

Theorem 9.1.1 (Darboux) Let (M1,�1) and (M2,�2) be symplectic manifoldsof the same dimension. Then for any points a ∈ M1 and b ∈ M2 there areneighborhoods O1 � a, O2 � b and a diffeomorphism ϕ : O1 → O2 such thatϕ(a) = b and ϕ∗(�2) = �1 in O1.

Proof Without loss of generality one can assume that two symplectic forms,say�0 and�1, are given in a neighborhood of a point a ∈ M, where in addition�0,a = �1,a. Let us take a path �t = (1− t)�0 + t�1 connecting �0 and �1,and find a path φt in local diffeomorphisms of M, such that

φ∗t (�t) = �0 (9.1)

and φ0 = 1, φt(a) = a.Note that forms �t are symplectic in a neighborhood of a, and

d�t

dt= �1 −�0 = dθ

in a smaller neighborhood.

183

184 Symplectic manifolds

Denote by Xt the vector field corresponding to φt , that is φ∗t+ε = φ∗t (1 +εXt + o(ε)). Then (9.1) is equivalent to

dφ∗t (�t)

dt= 0

or

LXt (�t)+ d�t

dt= 0.

The last equation can be rewritten as

dıXt�t + dθ = 0.

Therefore the vector field should be choosen such, that

ıXt�t + θ = dS

for some function S.Let S be a function such that θa = daS, then we obtain a uniquie vector field

Xt satisfying the equation and Xt,a = 0. �

Corollary 9.1.1 Let (M,�) be a 2n-dimensional symplectic manifold. Then forany point a ∈ M there are local canonical coordinates (q1, . . . , qn, p1, . . . , pn)

such that qi(a) = pi(a) = 0 (i = 1, . . . , n) and � has the followingcanonical form:

ω = dq1 ∧ dp1 + · · · + dqn ∧ dpn.

9.1.1 The cotangent bundle and the standard symplectic structure

Let τ ∗ : T∗B → B be the cotangent bundle of a manifold B. A differential1-form ρ on T∗B defined by

ρa(Xa)def= a(τ∗Xa),

for any covector a ∈ T∗B and any vector Xa ∈ Ta(T∗B), is called universal ora Liouville form.

If q1, . . . , qn are local coordinates in B and (q, p) = (q1, . . . , qn, p1, . . . , pn)

are the induced local coordinates on T∗B, then

ρ =n∑

i=1

pi dqi.

9.1 Symplectic structures 185

The 2-form

� = −dρ,

or in local coordinates (q, p)

� =n∑

i=1

dqi ∧ dpi,

is closed and non-degenerate, and therefore defines a symplectic structureon T∗M.

For B = Rn this 2-form� defines the so-called standard symplectic structureon the arithmetic space R2n = T∗Rn.

One can use the standard structure as a model. Thus, due to the Darboux the-orem, for any symplectic manifold M, and any a ∈ M, there are local coordinates(x1, . . . , xn, y1, . . . , yn) called canonical such that

� =n∑

i=1

dxi ∧ dyi

in some neighborhood of a.

Example 9.1.1 Any diffeomorphism f : B → B can be lifted to automorphismf (1) : T∗B → T∗B of the cotangent bundle as

f (1)|T∗a B = ( f ∗)(−1) : T∗a B → T∗f (a)B

and because of the property

( f (1))∗ρ = ρ

this automorphism is a symplectic transformation.

Example 9.1.2 Any smooth function h ∈ C∞(B) determines a shift

Sh : T∗B → T∗B

by

Sh : (x, λ) −→ (x, λ+ dxh)

for all x ∈ B, λ ∈ T∗x B.


Then

S∗h(ρ) = ρ + dτ ∗(h)

and therefore Sh is symplectic.

9.1.2 Kähler manifolds

A Kähler manifold is a complex manifold equipped with a compatiblesymplectic structure.

That is, at each point a ∈ M of the Kähler manifold M the tangent spaceTaM has the complex structure Ia : TaM → TaM, induced by the complexstructure on M, and the symplectic structure �a, induced by the symplecticform � ∈ �2(M), and they are compatible in the sense that

ga(x, y) = �a(Iax, y)

defines a Riemannian structure on M.Locally Kähler structure has a potential, called the Kähler potential, i.e. a

smooth function K , such that

� = ι

2∂∂K = ι

2

∑s, t

∂2K

∂zs∂ ztdzs ∧ dzt .

The potential functions are strictly plurisubharmonic in the sense that thematrix ∥∥∥∥ ∂2K

∂zl∂ zs

∥∥∥∥is positive definite in any local complex coordinates.

Moreover, any such function defines (locally) Kähler structure.The advantage of using Kähler manifolds is based on the fact that complex

submanifolds of Kähler manifolds are also Kähler.

Example 9.1.3 A complex vector space Cn is Kähler with

� = ι

2

n∑k=1

dzk ∧ dzk .

Therefore every complex submanifold of Cn is Kähler.

9.1 Symplectic structures 187

Example 9.1.4 The potential

K = log(1+ |z|2)

defines Kähler structure by the Fubini–Study form

� = ι

2∂∂K

on Cn.Let CPn be the complex projective space and one use the standard affine

charts

[z0 : · · · : zn] →(

z0

zk, . . . ,

zk−1

zk,

zk+1

zk, . . . ,

zn

zk

)then the Fubini–Study forms defines a Kähler structure on CPn.

Example 9.1.5 Smooth projective manifolds are Kähler.

9.1.3 Orbits and homogeneous symplectic spaces

Let (M,�) be a symplectic manifold and let G be a Lie group that acts on themanifold

G×M → M,

g : x −→ gx

by symplectomorphisms

g∗(�) = �,

for all g ∈ G.The symplectic space is called homogeneous if the action is transitive.We recall the construction of homogeneous symplectic spaces by

A. Kirillov [47].Any Lie group G acts on itself by conjugations:

Adg : G → G,

Adg : x −→ gxg−1.


Let us identify the Lie algebra g of G with the tangent space TeG at unite ∈ G, and let

Ad∗ : G → GL(g)

be the adjoint representation given by the differentials of conjugations:

Ad∗(g) = (Adg)∗,e : TeG → TeG.

The corresponding Lie algebra representation is the adjoint representation

ad : g→End g,

adX : Y −→ [X , Y ].

Denote by Ad∗ and ad∗ the corresponding dual and coadjoint representations:

Ad∗ : G → GL(g∗),

Ad∗(g) = (Adg−1)∗,e : T∗e G → T∗e G,

and

ad∗ : g→End g∗,

〈ad∗X α, Y〉 = 〈α, [Y , X]〉

where X, Y ∈ g, α ∈ g∗. Then any vector X ∈ g generates a vector field X ong∗ corresponding to the one-parameter group exp(t ad∗X) : g∗→ g∗. The valueXα of X at the point α ∈ g∗ can be identified with a vector in g∗ and

〈Xα , Y〉 = 〈α, [Y , X]〉.

Let us define

�α(Xα , Yα) = 〈α, [Y , X]〉.

One can easily check that� is determined a closed and non-degenerated 2-formon orbits of coadjoint representation.

9.2 Vector fields on symplectic manifolds 189

This structure is called the Konstant–Kirillov symplectic structure.

Example 9.1.6 Let G = SU(n+ 1) be the special unitary group. Then the Liealgebra g = su(n+1) is the Lie algebra of skew-Hermitian traceless (n+1)×(n+ 1) matrices. The Killing form

κ(X , Y) = Re(tr(XtY))

identifiesgwithg∗ by the isomorphism X → κ(X , ·)and the (co)adjoint action isthe conjugation g(X) = gXg−1. Therefore, eigenvalues describe the orbits. LetX0 be a traceless Hermitian matrix with eigenvalues λ1 = · · · = λn = 1/n, andλn+1 = −1. Then the Konstant–Kirillov form defines the symplectic structureon the orbit MX0 = SU(n+ 1)/U(n) ' CPn.

The Konstant–Kirillov structures describe practically all homogeneoussymplectic spaces.

Theorem 9.1.2 (Kostant–Souriau) Let G be a Lie group and g be the Liealgebra of G. Then for Lie algebras with trivial first and second cohomologygroups,

H1(g) = H2(g) = 0,

there is, up to covering, a one-to-one correspondence between homogeneoussymplectic spaces with structure group G and coadjoint orbits.

9.2 Vector fields on symplectic manifolds

9.2.1 Poisson bracket and Hamiltonian vector fields

Let (M,�) be a symplectic manifold and X ∈ D(M) be a vector field on M. Wesay that X is canonical if the flow At corresponding to X consists of symplecticdiffeomorphisms.

Proposition 9.2.1 The following conditions are equivalent:

(a) X is a canonical vector field;(b) LX(�) = 0;(c) ιX(�) is closed.


Proof

(a)⇐⇒ (b) If X is canonical and At is the corresponding flow, then

A∗t (�) = �

and therefore

LX(�) = d

dt

∣∣∣∣t=0

A∗t (�) = 0.

On the other hand, if LX(�) = 0, then

d

dt

∣∣∣∣t=s

A∗t (�) = A∗s (LX�) = 0

and therefore

A∗t (�) = A∗0(�) = �.

(b)⇐⇒ (c) It follows from the formula for the Lie derivative:

LX(�) = (ιX ◦ d + d ◦ ιX)(�) = d(ιX�). �

Let α ∈ �1(M) be a differential 1-form and let Xα ∈ D(M) be the vectorfield corresponding to α under isomorphism

: D(M)→ �1(M),

: X −→ ιX(�).

That is,

Xα�� = α.

Moreover,

LXα (�) = d(Xα��) = dα.

Define the Poisson bracket [α,β] of differential 1-forms α and β as follows:

X[α,β] = [Xα , Xβ ].

9.2 Vector fields on symplectic manifolds 191

Theorem 9.2.1 The Poisson bracket determines a Lie algebra structure on�1(M) and

[α,β] = Xα�dβ − Xβ�dα + d�(Xβ , Xα).

Proof Indeed,

ι[Xα ,Xβ ](�) = [LXα , ιXβ ](�) = LXα (β)− Xβ� dα

= Xα�dβ + d(β(Xα))− Xβ� dα. �

We have seen that canonical vector fields correspond to closed 1-forms, andthe above formula shows that the Poisson bracket of closed 1-forms is an exact1-form.

Canonical vector fields that correspond to exact forms are called Hamiltonian.If α = df then we will denote the Hamiltonian vector field Xα by Xf and call

the function f the Hamiltonian of Xf .The above formula for the Poisson bracket gives us

[df , dg] = d�(Xg, Xf ).

The function �(Xg, Xf ) is called a Poisson bracket of functions f and g and isdenoted by

{ f , g} def= �(Xg, Xf ).

In this notation the formula for Poisson brackets reads as

[Xf , Xg] = X{ f ,g}.

Note also that

{ f , g} = �(Xg, Xf ) = dg(Xf ) = Xf (g).

The Poisson bracket defines a Lie algebra structure on the space of functionsC∞(M).

9.2.2 Canonical coordinates

Let (x1, . . . , xn, y1, . . . , yn) be canonical coordinates for �, and let

α = a1(x, y) dx1 + · · · + an(x, y) dxn + a1(x, y) dy1 + · · · + an(x, y) dyn

in these coordinates.


Then,

Xα = a1(x, y)∂

∂x1+ · · · + an(x, y)

∂

∂xn− a1(x, y)

∂

∂y1− · · · − an(x, y)

∂

∂yn

and

Xf = ∂f

∂y1

∂

∂x1+ · · · + ∂f

∂yn

∂

∂xn− ∂f

∂x1

∂

∂y1− · · · − ∂f

∂xn

∂

∂yn.

The Poisson bracket in the canonical coordinates has the form

{ f , g} =(∂f

∂y1

∂g

∂x1− ∂f

∂x1

∂g

∂y1

)+ · · · +

(∂f

∂yn

∂g

∂xn− ∂f

∂xn

∂g

∂yn

).

Example 9.2.1 Let M = T2 = R2/Z2 be the two-dimensional torus with thecoordinates x and y, and let � = dx ∧ dy be the symplectic structure on T2.Then

Xα = a(x, y)∂

∂x− a(x, y)

∂

∂y

corresponds to α = a(x, y) dx + a(x, y) dy.Differential forms c1 dx+ c2 dy with constant coefficients are closed but not

exact. Therefore, the vector fields

c2∂

∂x− c1

∂

∂y

are canonical, but not Hamiltonian.

9.3 Submanifolds of symplectic manifolds

9.3.1 Presymplectic manifolds

Let M be a manifold and let ω be a differential 2-form on M.At each point a ∈ M one has a subspace

ker ωa = {X ∈ TaM| X�ωa = 0}.

Assume that a −→ dim ker ωa is a locally constant function, that is, the 2-formhas a constant rank.

9.3 Submanifolds of symplectic manifolds 193

Then the distribution

ker ω : a ∈ M −→ ker ωa

is called the characteristic distribution of ω.

Lemma 9.3.1 The characteristic distribution ker ω is completely integrable if

dω = 0.

Proof Let vector fields X and Y belong to ker ω. Then

LXω = ιX dω + dιX ω = 0

and

ι[X,Y ]ω = [LX , ιY ](ω) = 0. �

A presymplectic structure on the manifold M is a closed differential 2-formω ∈ �2(M) of constant rank.

For presymplectic structures ker ω is completely integrable. Let us denoteby S = M/ ker ω the set of all maximal integral manifolds of the distribution.Assume that there is a smooth manifold structure on S such that the projectionπ : M → S is a smooth bundle. In this case vertical vector fields v, that isvector fields tangent to fibres of π , are characteristics and therefore satisfy theproperty

ιvω = 0, Lvω = 0,

and thus the 2-form ω defines a 2-form ω on the base S as

ωs(Xs, Ys) = ωa(Xa, Ya)

where s ∈ S, a ∈ M,π(a) = s, and Xa, Ya ∈ TaM are arbitrary lifts ofXs, Ys ∈ TsS:

π∗(Xa) = Xs, π∗(Ya) = Ys.

Moreover, the form ω is closed and non-degenerate also.


Indeed, the non-degeneracy of ω follows from the construction and if X, Yand Z are vector fields on S and X , Y and Z are their lifts on M, then

dω(X, Y , Z) = X(ω(Y , Z))− Y(ω(X , Z))+ Z(ω(X, Y))

− ω([X , Y ], Z)+ ω([X, Z], Y)− ω([Y , Z], X)

= X(ω(Y , Z))− Y(ω(X , Z))+ Z(ω(X, Y))

− ω([X , Y ], Z)+ ω([X, Z], Y)− ω([Y , Z], X)

= dω(X , Y , Z) = 0.

The symplectic manifold (S, ω) is called reduction of the presymplecticmanifold (M,ω).

One particular case of the reduction is very important for applications.Namely, let us assume that ⊂ M is a section of π ; that is, a submanifoldsuch that π : → S is a diffeomorphism, then the symplectic manifold (S, ω)is equivalent to the symplectic manifold ( ,ω| ).

9.3.2 Lagrangian submanifolds

We say that a submanifold L of a symplectic manifold (M,�) is Lagrangian ifthe tangent space TaL is a Lagrangian subspace of the symplectic vector space(TaM,�a) at every point a ∈ L.

Example 9.3.1 Let (T∗B,� = dρ) be the phase space with the standardsymplectic structure. Any differential 1-form α ∈ �1(B) can be viewed as thesection

Sα : B → T∗B,

Sα : x ∈ B −→ αx ∈ T∗x B

of the cotangent bundle τ ∗ : T∗B → B.Moreover, by the definition of ρ, one gets

S∗α(ρ) = α.

Namely this property defines ρ uniquely and for this reason ρ is called theuniversal 1-form.

Denote by Bα = Sα(B) ⊂ T∗B the graph of α. Then the universality propertyreads as

ρ|Bα = α,

and Bα is a Lagrangian submanifold if and only if α is closed.


If this form is exact, α = df , for some function f ∈ C∞(B), then we denoteBdf by Lf and call f the generating function of Lf .

Note that a Lagrangian submanifold L ⊂ T∗B has the form Bα or Lf if andonly if: (1) τ ∗ : L → B is a diffeomorphism and (2) ρ|L is closed or exactrespectively.

In the canonical coordinates (q1, . . . , qn, p1, . . . , pn) in T∗B the Lagrangiansubmanifolds Lf have the form

p = ∂f

∂q.

Example 9.3.2 Let B0 ⊂ B be a submanifold. Define B0 ⊂ T∗B as follows:

B0 =⋃

x∈B0

Ann TxB0.

Then ρ|B0= 0 and dim B0 = dim B. Therefore B0 is Lagrangian.

Let L ⊂ M be a Lagrangian submanifold, and let (x1, . . . , xn, y1, . . . , yn) becanonical coordinates in a neighborhood O of a ∈ L.

If O is sufficiently small, there is a subset I = {i1, . . . , ik} ⊂ {1, . . . , n}such that xI = (xi1 , . . . , xik ) and yIc = (yj1 , . . . , yjn−k ), where js /∈ I are localcoordinates in L ∩O.

Let us write the structure form � as follows:

� = dx ∧ dy = dxI ∧ dyI + d(−yIc) ∧ dxIc .

Assume that L∩O is simply connected. Then due to the above example L ∩O =Lf for some function

f = f (xI , yIc)

and

yI = ∂f

∂xI,

xIc = − ∂f

∂yIc

on L.The function f (xI , yIc) is called the generating function for the Lagrangian

manifold L.


The above description of Lagrangian submanifolds can be used for a localdescription of canonical transformations. This is based on the followingobservation.

Let φ : (M1,�1)→ (M2,�2) be a diffeomorphism, and let Gφ ⊂ M1 ×M2

be the graph of φ. Define a differential 2-form on M = M1 ×M2 as

� = π∗1�1 − π∗2�2,

where πi : M → Mi, i = 1, 2 are the projections.Then � defines the symplectic structure on M and

�|Gφ = �1 − φ∗(�2).

Proposition 9.3.1 A diffeomorphism φ : M1 → M2 is symplectic if and only ifthe graph Gφ is a Lagrangian submanifold.

The local description of Lagrangian submanifolds gives us the followingdescription of symplectic diffeomorphisms in terms of generating functions.

Let (x, y) be canonical coordinates on M1, (u, v) be canonical coordinates onM2 and let

φ : (x, y)→ (u = U(x, y), v = V(x, y))

be a local symplectic diffeomorphism.Then

� = dx ∧ dy − du ∧ dv

and therefore canonical coordinates on Gφ can be taken as follows:

xI , uJ , yIc , vJc

where #I + #J = n.Let

f = f (xI , uJ , yIc , vJc)

be the generating function for Gφ . Then the equations

yI = ∂f

∂xI, xIc = − ∂f

∂yIc,

vJ = − ∂f

∂uJ, uJc = ∂f

∂vJc

determine φ.


To find φ one should solve this system with respect to u = U(x, y) andv = V(x, y).

Example 9.3.3 Let M = (T∗B, dρ), and let (p, q) be the canonical coordin-ates, ρ = p dq. Taking f = f (q, q′) we obtain the following symplectictransformations:

p = ∂f

∂q, p′ = − ∂f

∂q′.

General symplectic diffeomorphisms can be obtained from generating functionsof the form

f = f (qI , pIc , q′J , p′Jc)

as follows:

pI = ∂f

∂qI, qIc = − ∂f

∂pIc,

p′J =∂f

∂q′J, q′Jc = − ∂f

∂p′Jc.

9.3.3 Involutive submanifolds

A submanifold E of a symplectic manifold (M,�) is said to be involutive ifthe tangent space TaE is an involutive subspace of the symplectic vector space(TaM,�a) at every point a ∈ L.

Note that the restriction �E = �|E of the structure form has constant rankbecause

dim ker�E = codim E,

and therefore an involutive submanifold E determines a presymplectic structure(E,�E).

The characteristic distribution of �E coincides with the distribution of skew-orthogonal complements:

charE : a ∈ E −→ (ker�E)(a) = (TaE)⊥

and we obtain the following property of involutive submanifolds.


Theorem 9.3.1 Let E be an involutive submanifold of a symplectic manifold(M,�). Then,

1. the characteristic distribution charE is completely integrable, and2. any Lagrangian submanifold L ⊂ M such that L ⊂ E consists of leaves of

charE; that is,

TaL ⊃ charE (a)

for any a ∈ L.Moreover, if π : E → E/ charE is a smooth bundle then π(L) is aLagrangian submanifold of the symplectic manifold E/ charE , and for anyLagrangian submanifold L′ ⊂ E/ charE the preimage L = π−1(L′) ⊂ E isa Lagrangian submanifold of M.

Example 9.3.4 Any submanifold E of codimension 1 is involutive. Let E =f−1(0) for some function f ∈ C∞(M), and da f does not vanish at pointsa ∈ E. Then charE generates by the Hamiltonian vector field Xf , and thereforeany Lagrangian submanifold L ⊂ E is invariant with respect to Xf .

9.3.4 Lagrangian polarizations

A Lagrangian polarization on a symplectic manifold (M,�) is a completelyintegrable distribution P such that Pa ⊂ TaM is a Lagrangian subspace for eacha ∈ M.

Example 9.3.5 (Standard polarizations) Let (M = T∗B, dρ) be the sym-plectic manifold. Then fibres of the projection τ ∗ : T∗B → B are Lagrangianbecause ρ = 0 on the fibres, and therefore define a Lagrangian polarization onT∗B.

Theorem 9.3.2 Let P be a Lagrangian polarization. Then

1. Xf ∈ D(P),2. { f , g} = 0,

for any first integrals f and g of the distribution.

Proof Let f be a first integral of P. Then ker da f is an involutive subspace andPa ⊂ ker daf . Therefore, (ker da f )⊥ ⊂ P⊥a = Pa. But (ker da f )⊥ = R · Xf ,a

and therefore

Xf ,a ∈ Pa

for all a ∈ M.


If g is a first integral too, then { f , g} = Xf (g) = dg(Xf ) = 0 becauseXf ∈ D(P) and dg = 0 on P. �

The following theorem shows that Lagrangian polarizations are locallyequivalent.

Theorem 9.3.3 Let P and P′ be Lagrangian polarizations on a connected sym-plectic manifold (M,�). Then for any points a, a′ in M there are neighborhoodsO � a and O′ � a′ and a local symplectomorphism φ : O → O′ such thatφ(a) = a′ and φ∗(P) = P′.

Proof To prove this theorem we show a method of finding canonical coordin-ates in some neighborhood of a such that P will coincide with a standardpolarization. Let dim M = 2n, and let us take n independent first integrals,say f1, . . . , fn, for P in a neighborhood of a. The corresponding Hamiltonianvector fields Xf1 , . . . , Xfn commute due to the above theorem. Therefore, we canfind a function g1 such that

Xf1(g1) = 1, Xf2(g1) = 0, . . . , Xfn(g1) = 0

in a neighborhood of a.Then Hamiltonian vector fields Xf1 , . . . , Xfn , Xg1 commute because

[Xfi , Xg1 ] = X{fi ,g1}

and { fi, g1} = constant .Note also that df1, . . . , dfn, dg1 are linear independent because dg1(Xf1) = 1.In the same way we can find a function g2 such that

Xf1(g2) = 0, Xf2(g2) = 1, Xf3(g2) = 0, . . . , Xfn(g2) = 0, Xg1(g2) = 0

in a neighborhood of a.For the same reasons Hamiltonian vector fields Xf1 , . . . , Xfn , Xg1 , Xg2

commute and df1, . . . , dfn, dg1, dg2 are linear independent.Finally, we construct n functions g1, . . . , gn such that df1, . . . , dfn,

dg1, . . . , dgn are linear independent and

{fi, gj} = δij but {gi, gj} = 0.

These 2n functions provide local canonical coordinates for � and integralmanifolds of P are

f1 = c1, . . . , fn = cn. �


Remark 9.3.1 The above method is not applicable in practice. It is much bet-ter to use the method of Section 2.1.3. Namely, let us choose functions h1, . . . , hn

in a domain D such that functions f1, . . . , fn, h1, . . . , hn are independent in D,and form the matrix

W =

∥∥∥∥∥∥∥∥∥{f1, h1} {f2, h1} · · · {fn, h1}{f1, h2} {f2, h2} · · · {fn, h2}

......

......

{f1, hn} {f2, hn} · · · {fn, hn}

∥∥∥∥∥∥∥∥∥ .

Then differential 1-forms θ1, . . . , θn, where

θ = W−1 dh,

and

θ =

∥∥∥∥∥∥∥∥∥θ1

θ2...θn

∥∥∥∥∥∥∥∥∥ , dh =

∥∥∥∥∥∥∥∥∥dh1

dh2...

dhn

∥∥∥∥∥∥∥∥∥ ,

are closed on submanifolds Mc = {f1 = c1, . . . , fn = cn} ∩ D. If theforms are exact, say θk = dgk on Mc, k = 1, 2, . . . , n, then 2n functionsf1, . . . , fn, g1, . . . , gn provide us with local canonical coordinates for � indomain D.

10

Contact manifolds

10.1 Contact structures

A contact structure on a manifold M is a distribution C of codimension 1 suchthat the curvature form �C is non-degenerate.

In other words, if C is locally defined by a differential 1-form ω, C = ker ω,then the restriction dω|C is non-degenerate.

A contact manifold is a manifold equipped with a contact structure C.A contact differential 1-form is a form ω such that the distribution ker ω

is contact. A strict contact manifold is a pair (M,ω) where ω is a contactform on M.

By definition any contact structure locally admits a contact form, and there-fore is strict. A global form ω exists if and only if the one-dimensional bundleTM/C has a non-zero section, i.e. is orientable.

At each point a ∈ M of a strict contact manifold one has the splitting

TaM = ker ωa ⊕ ker daω

and daω is a symplectic structure on Ca = ker ωa.Therefore, the dimension of C is even, say 2n, and dim M = 2n + 1 is

odd. Note that the restriction of (daω)n on Ca is a volume form, and therefore

ω ∧ (dω)n is a volume form on M.One could rephrase the definition of contact forms as follows: a differential

1-form ω is contact if and only if

ω ∧ (dω)n �= 0

i.e., ω ∧ (dω)n is a volume form.

201

202 Contact manifolds

Note that the Frobenius theorem gives the following integrability condition

ω ∧ dω = 0

for distribution ker ω.This shows that completely integrable and contact distributions provide two

extreme cases for distributions in codimension 1.

10.1.1 Examples

Standard contact structureLet M = R2n+1 be the arithmetic space with coordinates (x1, . . . , xn, y1, . . . ,yn, z) and let

ω = dz −n∑

i=1

yi dxi.

Then

ω ∧ (dω)n = dz ∧ (dx ∧ dy)n �= 0.

The distribution ker ω is called the standard contact structure.

The Cartan distributionRecall that 1-jet of function f ∈ C∞(M) at a point a ∈ M is a pair [f ]1a =(f (a), daf ).

Denote by J1M the space of all 1-jets. Then J1M = R×T∗M, and we definethe Cartan form on J1M as

ω = du− ρ,

where u : J1M = R× T∗M → R is the first projection and ρ is the universal1-form on T∗M.

If q1, . . . , qn are local coordinates on M and (q1, . . . , qn, p1, . . . , pn) are thecorresponding coordinates on T∗M then (q1, . . . , qn, p1, . . . , pn, u) are localcoordinates on J1M and

ω = du− p dq

in these coordinates.

10.1 Contact structures 203

Therefore ω is a contact form and

C = ker ω

is a contact distribution on J1M called the Cartan distribution.

The space of contact elementsA contact element at a point a ∈ M is a hyperplane Ha ⊂ TaM.

A hyperplane Ha can be defined by Ann Ha which is a one-dimensionalsubspace of T∗a M. Thus the set of all contact elements at point a ∈ M isthe projectivization PT∗a M of the cotangent space and the space of all contactelements is the projectivization of the cotangent bundle

π : PT∗M =⋃a∈M

PT∗a M → M.

Let α ∈ T∗a M, α �= 0, be a covector and let [α] ∈ PT∗a M be the one-dimensional subspace generated by α.

Define a hyperplane

C[α] ⊂ T[α](PT∗M)

to be a space of vectors X ∈ T[α](PT∗M) such that their projections π∗(X)belong to the contact element ker α, that is

C[α] = π−1∗ (ker α).

This distribution can be obtained in a different way.Let us consider the manifold of non-zero covectors T∗M\{0} and distribution

ker ρ on it. This distribution has characteristics; namely, if we define the vectorfield Xρ dual to ρ as above

Xρ�dρ = ρ

then Xρ generates the kernel of restriction dρ on ker ρ, and

Xρ�ρ = 0 and LXρ (ρ) = ρ.

The one-parameter group of shifts along Xρ corresponds to the standardaction of the multiplicative group R+ = {λ > 0}

λ ∈ R+ : (x, p) −→ (x, λp),

on T∗M \ {0}.


The orbit space ST∗M is called the spherization of the cotangent bundle.The distribution ker ρ defines a distribution C on ST∗M. Note, that the vector

field Xρ preserves this distribution and scales dρ, LXρ (dρ) = dρ. Moreover Xρ

generates the kernel of dρ, and therefore dρ defines a non-degenerate 2-formon the factor ker ρ�Xρ . This means that the distribution C on the spherizationis contact.

The projectivization PT∗M can be viewed as an orbit space of the multiplic-ative group R∗ = R \ {0} action on

λ ∈ R∗ : (x, p) −→ (x, λp),

where a ∈ M, p ∈ T∗a M \ {0}.As above the distribution ker ρ is invariant with respect to actions and orbits

correspond to the kernel of the restriction dρ on ker ρ. This means that thedistribution ker ρ can be pushed down and we obtain a contact distribution onPT∗M. The definition of the universal 1-form ρ shows that this distributioncoincides with the distribution C on the space of contact elements.

To find a local representation of this distribution we introduce the localhomogeneous coordinates (q1, . . . , qn; p1 : · · · : pn) on PT∗M, then C given by

n∑i=1

pi dqi = 0.

If, say p1 �= 0, then in the affine coordinates (q1, . . . , qn; p2/p1, . . . , pn/p1)

our distribution can be given by the 1-form

dq1 +n∑

i=2

pi

p1dqi = 0,

and it shows that C is contact.

Exact symplectic manifoldsThe above constructions can be used to construct more examples of contactmanifolds.

Assume that (M,�) is an exact symplectic manifold; that is, � = dβ forsome 1-form β, and βa �= 0 for all a ∈ M.

Let Xβ be the vector field corresponding to β : Xβ�� = β, and let us assumethat the space B of trajectories of Xβ is a smooth manifold and π : M → B is


a smooth bundle. Then the above discussion shows that the distribution ker βon M defines a contact distribution on B.

Example 10.1.1 Let M = R2n\{0}with coordinates (x1, . . . , xn, y1, . . . , yn)and

β =n∑

i=1

(xi dyi − yi dxi).

Then dβ = 2dx ∧ dy and

Xβ = 1

2

n∑i=1

(xi

∂

∂xi+ yi

∂

∂yi

).

The space of orbits can be identified with the sphere S2n−1

n∑i=1

(x2i + y2

i ) = 1

and restriction of β on the sphere gives a contact structure.

One-dimensional bundlesAt first, we briefly recall basic constructions on connections in a form suitablefor us.

Let π : E → B be a bundle. A connection in the bundle is a distribution P,dim P = dim B, transversal to fibres ofπ . In other words, if Ta(π) is the tangentspace to the fibre passing through a ∈ E, then a connection gives a splitting

TaE = Ta(π)⊕ Pa

at each point a ∈ E.Vectors of Ta(π) are said to be vertical and vectors of Pa are horizontal.As we have seen the curvature of distribution P defines a 2-form �P. Owing

to the splitting the normal space νP,a can be identified with the vertical spaceTa(π) and the curvature form can be identified with the 2-form on P with valuesin T(π). Under this identification our construction of the curvature coincideswith the standard one.

Namely, let us denote by X the horizontal lifting of the vector field X ∈ D(B)into E, i.e. Xa ∈ Pa and the differential π∗ : Pa → TbB sends Xa to Xb, whereb = π(a).

Then the commutator [X , Y ] of two liftings projects on [X, Y ] and therefore[X, Y ] − [X, Y ] is a vertical vector field on E coinciding with �P(X, Y).


Thus,

�P(X , Y) = [X , Y ] − [X, Y ]for all vector fields X , Y ∈ D(B).

Let At be a flow of X and let At be a flow of X. Then At give automorphismsof the bundle such that the following diagram

EAt

At

E

B

π π

B

commutes.If the bundle under consideration is equipped with an extra structure then the

reasonable class of connections should provide us with automorphisms At thatpreserves the structure.

For example, let π : E → B be a vector bundle. Then a connection P is saidto be linear if At are linear automorphisms for any vector field X on B.

In the last case the connection form�P of the linear connection reduces to anoperator-valued 2-form. Namely, the value �P,a(Xb, Yb), a ∈ π−1(b) dependson a in a linear way; that is,

�P,a(Xb, Yb) = �P,b(Xb, Yb)(a)

for some linear operator

�P,b(Xb, Yb) : π−1(b)→ π−1(b).

This gives us a 2-form �P ∈ �2(B)⊗End π on the base manifold B with valuesin endomorphisms of π .

Let π : E → B be a one-dimensional vector bundle, then �P ∈ �2(B) forany linear connection P, and �P is closed due to the Bianchi identity.

The case �P = 0 corresponds to flat connections and the case when �P isnon-degenerate is the second extreme class of connections. This correspondsto linear connections P in line bundles π : E → B over symplectic manifolds(B,�) such that �P = �. The corresponding horizontal distributions in E arecontact.

Example 10.1.2 Let π : B × R →B be the trivial line bundle. A linearconnection in π can be given by the differential 1-form θ ∈ �1(B).


Namely,

P = Pθ = ker(du− θ)

where u : B× R → R is the projection.In this case

�P = −dθ .

Therefore, if (B,−dθ) is an exact symplectic structure, then Pθ defines a contactstructure on B× R.

Example 10.1.3 The projection π : J1M→T∗M is the trivial line bundleover the exact symplectic manifold (T∗M, dρ). The above example gives us theCartan distribution on J1M.

Hermitian line bundlesLet π : E → B be a Hermitian complex line bundle, and let P be a Hermitianconnection. In this case values of the curvature �P ∈ �2(B) ⊗ End π belongto the space of skew-Hermitian operators, and because of dim π = 1, ι�P is areal-valued closed 2-form on M. This form can be obtained in a different andmore geometrical way.

Let h be an Hermitian form on π and let

Sx(π) = {v ∈ π−1(x)|hx(v, v) = 1}

be the circle of unit vectors at x ∈ M.Consider the subbundle

πS : S(E) =⋃x∈M

Sx(π)→ B

of the bundle π .This is a one-dimensional bundle over M. Fibres of πS are circles, the abelian

group S1 = {z ∈ C | |z| = 1} acts on S(E) and fibres π−1S (x) are orbits of the

action. Thus πS is a principal S1-bundle.Now let P be a linear connection on π , then P is Hermitian if and only if P

is tangent to S(E):

Pa ⊂ Ta(S(E))

for all a ∈ S(E).


Denote by PS the restriction of P on S(E). This is a connection in πS , andmoreover PS is invariant with respect to S1-actions. Therefore the curvatureform�PS is S1-invariant. Using the standard identification of the vertical vectorswith elements of the Lie algebra of S1, that is R, we obtain the closed 2-formι�P on the base manifold B.

This form is non-degenerate if and only if the distribution PS iscontact.

Note also that the symplectic structure ι�P just obtained is not arbitrary. Thecohomology class [ι�P] of the form should be integral:

[ι�P] ∈ H2(B, Z).

It is known that for any integral 2-cocycle θ there is a Hermitian bundle πand a Hermitian connection P such that θ = ι�P. Thus for any symplecticmanifold (M,�) with integral structure form there is a Hermitian line bundle πand corresponding contact manifold that produce (M,�). Boothby and Wang[10] have proved that all strict compact and regular contact structures can beobtained in this way.

10.2 Contact transformations and contact vector fields

Let (M, C) be a contact manifold. A diffeomorphism ϕ : M → M is called acontact transformation if

ϕ∗(C) = C.

If (M,ω) is a strict contact manifold then contact transformations arediffeomorphisms ϕ such that

ϕ∗(ω) = λϕω

for some smooth function λϕ ∈ C∞(M).If in addition

ϕ∗(ω) = ω

the contact diffeomorphism is called strict.

10.2 Contact transformations and contact vector fields 209

10.2.1 Examples

The standard contact manifoldsLet M = R2n+1 and let

ω = dz −n∑

i=1

yi dxi

be the standard contact structure C on M.Then the following transformations are contact.

Proposition 10.2.1

1. Translations:

Tc : (x1, . . . , xn, z, y1, . . . , yn) −→ (x1 + c1, . . . , xn + cn, z + c0, y1, . . . , yn),

where c = (c0, c1, . . . , cn) ∈ Rn+1.2. Scale transformations:

Sλ : (x1, . . . , xn, z, y1, . . . , yn)

−→ (eλ1 x1, . . . , eλn xn, eλ0 z, e(λ0−λ1)y1, . . . , e(λ0−λn)yn).

3. The Legendre transformation:

L : (x1, . . . , xn, z, y1, . . . , yn) →(

y1, . . . , yn, z −n∑

i=1

yixi,−x1, . . . ,−xn,

).

4. The Euler (or partial Legendre) transformation:

Lk : (x1, . . . xk , xk+1, . . . , xn, z, y1, . . . , yk , yk+1, . . . , yn)

−→(

y1, . . . , yk , xk+1, . . . , xn, z −k∑

i=1

yixi,−x1, . . . ,−xk , yk+1, . . . , yn

).

5. Shifts:

Sh : (x1, . . . , xn, z, y1, . . . , yn)

→(

x1, . . . , xn, z + h(x1, . . . , xn), y1 + ∂h

∂x1, . . . , yn + ∂h

∂xn

),

where h ∈ C∞(Rn).


The Cartan distribution1. Let φ : M → M be a diffeomorphism, and let p1(φ) : J1M → J1M be the

prolongation of φ on 1-jets:

p1(φ) : [h]1a −→ [h ◦ φ−1]1φ−1(a).

Then

p1(φ)∗(ω) = ω,

i.e. p1(φ) is a strict contact transformation.2. Letφ : T∗M → T∗M be a symplectomorphism. Thenφ∗(dρ) = dρ implies

thatφ∗(ρ)−ρ is a closed 1-form on T∗M. Assume that this form is exact, then

φ∗(ρ)− ρ = dS

for some function S. Define a lifting φ : J1M → J1M as follows

φ : (u, x) −→ (u− S(x),φ(x))

where x ∈ T∗M. Then,

φ∗(du− ρ) = d(u− S)− φ∗(ρ) = du− ρ.

3. Any function h ∈ C∞(M) determines a strict contact transformation:

Sh : [f ]1a −→ [f + h]1a.

The space of contact elementsLet φ : M → M be a diffeomorphism. Define a prolongation φ : PT∗M →PT∗M on contact elements as follows:

φ : [α] −→ [(φ−1)∗(α)]where [α] is a class of non-zero covector α ∈ T∗M.

Then φ is a projectivization of the symplectic transformation φ(1) : T∗M →T∗M, and because (φ(1))∗(ρ) = ρ, φ is a contact diffeomorphism. Thesecontact diffeomorphisms are called point transformations.

Exact symplectic manifoldsLet (M, dβ) be an exact symplectic manifold and let us assume that the corres-ponding contact manifold B exists. Then any strict symplectic transformationφ : M → M, i.e. φ∗(β) = β induces a contact transformation B.


Contact transformations of hypersurfaces in classicdifferential geometry

In this section we discuss the classical coordinate approach to contacttransformations.

We are going to investigate different types of transformation of hypersur-faces in Rn+1. The fist type is given by diffeomorphisms, or so-called pointtransformations.

Let ϕ : Rn+1 → Rn+1 be a diffeomorphism

ϕ : (x, u) = (x1, . . . , xn, u) −→ (X , U) = (φ(x, u),ψ(x, u)),

and let

Sh = {u = h(x)}

be a graph of a smooth function.Assume that the hypersurface ϕ(Sh) is a graph of a new function, say hϕ :

ϕ(Sh) = Shϕ .

This function is a solution U = hϕ(X) of equations

X = φ(x, h(x)), U = ψ(x, h(x)). (10.1)

Obviously images ϕ(S) and ϕ(S′) are tangent at ϕ(a) if S and S′ were tangentat a ∈ S ∩ S′. Therefore, ϕ induces a transformation on contact elements (i.e.,tangent planes to hypersurfaces).

It is easy to find the corresponding transformation of contact elements;namely, a tangent plane to Sh given by the equation

du− ∂h

∂xdx = 0.

To find tangent planes to ϕ(Sh) we differentiate hϕ and obtain

dφ

dx

∂hϕ

∂X= dψ

dx

where, for example,

dψ

dx= ∂ψ

∂x+ ∂ψ

∂u

∂h

∂x.


Therefore,

∂hϕ

∂X=(

dφ

dx

)−1 dψ

dx. (10.2)

Denoting coordinates of contact elements by p = (p1, . . . , pn) we obtain thefollowing transformation:

(x, u, p) −→(φ(x, u),ψ(x, u),

(∂φ

∂x+ ∂φ

∂up

)−1 (∂ψ

∂x+ ∂ψ

∂up

)).

This transformation sends contact elements to contact elements. Moreoverit is easy to check that this transformation preserves zeros of the contact formω = du− p dx.

Suppose now that transformation (10.1) in addition depends on contactelements; that is,

X = φ

(x, h(x),

∂h

∂x

),

U = ψ

(x, h(x),

∂h

∂x

).

Then the corresponding transformation of contact elements will be(∂φ

∂x+ ∂φ

∂u

∂h

∂x+ ∂2h

∂x2

∂φ

∂p

)∂hϕ

∂X=(∂ψ

∂x+ ∂ψ

∂u

∂h

∂x+ ∂2h

∂x2

∂ψ

∂p

)and therefore it includes the second derivatives of h.

To get transformation on contact elements only we need some extra require-ments on φ and ψ , such that the above equation on ∂hϕ/∂X gives us a solutionindependent on ∂2h/∂x2. This gives us immediately the conditions:

∂φ

∂p

∂hϕ

∂X= ∂ψ

∂p,(

∂φ

∂x+ ∂φ

∂u

∂h

∂x

)∂hϕ

∂X=(∂ψ

∂x+ ∂ψ

∂u

∂h

∂x

)and finally (

∂φ

∂x+ ∂φ

∂u

∂h

∂x

)(∂φ

∂p

)−1∂ψ

∂p=(∂ψ

∂x+ ∂ψ

∂u

∂h

∂x

). (10.3)


The corresponding transformation has the form

(x, u, p) −→(φ(x, u, p),ψ(x, u, p),

(∂φ

∂p

)−1∂ψ

∂p

).

An alternative method of obtaining the conditions (10.3) is provided bydifferential forms.

As we have seen zeros of the differential 1-form ω = du − p dx describecontact elements. Let us try to extend the transformation

(x, u) −→ (φ(x, u, p),ψ(x, u, p))

to a transformation of the form

(x, u, p) −→ (φ(x, u, p),ψ(x, u, p), P(x, u, p))

in such a way that the form

dψ − P dφ

equals zero every time ω = 0.This gives us

dψ

dx− P

dφ

dx= 0,

∂ψ

∂p− P

∂φ

∂p= 0,

and therefore (10.3).

Example 10.2.1 (Pedal transformations, see Figures 10.1 and 10.2) Con-sider a smooth hypersurface L ⊂ Rn+1 and a fixed point P, say P = 0. Foran arbitrary point a of L we denote by S(a) the intersection of the tangenthyperplane TaL and the perpendicular from P to the tangent Ta. Obviously thistransformation depends on contact elements and one obtains:

S : (x, u) −→(− (u− px)

1+ p2p,

u− px

1+ p2

).

The corresponding contact transformation is called a pedal. It has the form

(x, u, p) −→(− (u− px)

1+ p2p,

u− px

1+ p2,

x + (2u− px)p

u(1− p2)− 2px

).


Figure 10.1. The pedal transformation

Figure 10.2. The pedal transformation of an ellipse.

Example 10.2.2 (Polar transformations) Let E be an ellipse in R2, and let abe a point in the exterior part of the ellipse. Let t1 and t2 be tangent lines to Epassing through a, say t1 = Ta1 E, t2 = Ta2 E. The line l(a) passing through thepoints a1 and a2 is called a polar of the point a. A polar transformation sends acontact element (a, p) to a point A in such a way that (1) A ∈ l(a), (2) a ∈ l(A)and (3) l(A) has slope p. In coordinates the polar transformation with respectto the ellipse x2/a2 + y2/b2 = 1 has the form

(x, u, p) −→(

a2p

px − u,

b2

u− px,−b2x

a2u

).

Example 10.2.3 (Dilatations, see Figure 10.3) Dilatations or optical trans-formations came from geometrical optics. Under this transformation the contactelement (x, u, p) moves in the orthogonal direction to p with velocity 1 during


-2 -1 1 2

-3

-2

-1

1

2

3

0.7

Figure 10.3. Dilitations of an ellipse.

time t, that is

(x, u) −→(

x − tp√1+ p2

, u+ t√1+ p2

)

and

(x, u, p) −→(

x − tp√1+ p2

, u+ t√1+ p2

, p

).

10.2.2 Contact vector fields

Contact vector fields and their generating functionsAn infinitesimal symmetry of a contact distribution is called a contact vectorfield.

For the strict contact structure this means that a vector field X is contact ifand only if

LX(ω) = λXω,

for some smooth function λX ∈ C∞(M).We can also re-write this condition without using the function λX :

LX(ω) ∧ ω = 0.


The tangent bundle of a strict contact manifold (M,ω) can be spit into adirect sum:

TaM = Ca ⊕ ker daω

because daω is a symplectic structure on Ca = ker ωa.Therefore, any vector field X on M can be decomposed into the sum

X = Xh + Xv,

where Xh ∈ D(C) is called horizontal and Xv ∈ D(ker dω) is the verticalcomponent of the vector field.

The distribution ker dω is one dimensional and the 1-form ω can be viewedas a coordinate on this distribution.

Let us denote by X1 the basis vector field in ker dω with coordinate 1; that is,

ιX1 dω = 0 and ιX1ω = 1.

Then

LX1(ω) = ιX1 dω + d(ιX1ω) = 0

and, therefore, X1 is a contact vector field.Below we show that any contact vector field X on a strict contact manifold is

uniquely determined by its vertical component Xv or by a function fX = ω(X).This function is called a generating function or a contact Hamiltonian of the

contact vector field X .

Theorem 10.2.1 Let (M,ω) be a strict contact manifold. Then for any functionf ∈ C∞(M) there exists a unique contact vector field Xf such that:

ω(Xf ) = f ,

Xf �dω = X1( f )ω − df ,

LXf (ω) = X1( f )ω.

Moreover,

Xf+g = Xf + Xg,

Xfg = f Xg + gXf − fgX1,

Xf (g)+ Xg( f ) = X1( f )g+ X1(g)f

for all f , g ∈ C∞(M).


The vector field X1 is called the Reeb vector field.

Proof Let X be a contact field on M and LXω = λXω. Then,

X = Xh + f X1

if

f = ω(X)

and

LX(ω) = X�dω + d(X�ω) = Xh�dω + df = λXω.

In other words,

Xh�dω = λXω − df . (10.4)

Since X1 ∈ ker dω, we have

ιX1(λXω − df ) = λX − X1( f ) = 0

or

λX = X1( f ).

Now the differential 1-form

X1( f )ω − df

vanishes on ker dω, and therefore can be viewed as a differential 1-form on thecontact distribution C. The restriction of dω on C is symplectic and therefore(10.4) uniquely determines the horizontal component Xh.

Let us denote the contact vector field X by Xf . It follows that

LXf (ω) = X1( f )ω

and

ιXf dω = X1( f )ω − df .


Consider now values of both 1-forms in the last equality on a contact vectorfield Xg. We obtain

dω(Xf , Xg) = X1( f )g− Xg( f ).

By the same reasoning

dω(Xg, Xf ) = X1(g)f − Xf (g)

and therefore

X1( f )g− Xg( f ) = Xf (g)− X1(g)f .

To finish the proof we denote by Y the following vector field:

Y = f Xg + gXf − fgX1.

Then,

ω(Y) = fg,

Y�dω = f ιXg dω − gιXf dω = ( fX1(g)+ gX1( f ))ω − f dg− g df

= X1( fg)ω − d( fg)

and therefore

LY (ω) = X1( fg)ω.

In other words, Y is a contact vector field with generating function fg, andY = Xfg. �

In canonical coordinates contact vector fields have the following form:

Xf = −n∑

i=1

∂f

∂pi

∂

∂qi+(

f −n∑

i=1

pi∂f

∂pi

)∂

∂u+

n∑i=1

∂f

∂qi

∂

∂pi

or

Xf = −n∑

i=1

∂f

∂pi

d

dqi+ f

∂

∂u+

n∑i=1

∂f

∂qi

∂

∂pi.

10.3 Darboux theorem 219

Lagrange bracketsThe commutator of two contact vector fields Xf and Xg with generating functionsf and g on a contact manifold M is again a contact vector field [Xf , Xg] = Xh

with a generating function h, which is defined by

h = ω([Xf , Xg])

and is called a Lagrange bracket of the functions f and g. We denote theLagrange bracket ω([Xf , Xg]) by [ f , g].Theorem 10.2.2 The Lagrange bracket defines a Lie algebra structure onC∞(M) and

[ f , g] = Xf (g)− X1( f )g.

Proof The fact that C∞(M) is a Lie algebra with respect to the Lagrangebracket is an immediate consequence of the definition. To verify the secondstatement one needs the standard formula:

ι[Xf ,Xg]ω = [LXf , ιXg ](ω) = Xf (g)− X1( f )g,

which completes the proof. �

Finally, we give expressions for the Lagrange bracket in canonical coordin-ates:

[ f , g] = f∂g

∂u− g

∂f

∂u+

n∑i=1

(df

dqi

∂g

∂pi− dg

dqi

∂f

∂pi

).

10.3 Darboux theorem

The following theorem claims that all contact manifolds are locally equivalent.

Theorem 10.3.1 (Darboux) Let ω and " be contact forms on a smooth man-ifold M. Then for any point a ∈ M there exists a local diffeomorphism ϕ suchthat ϕ(a) = a and ϕ∗(") = ω.

Proof Without loss of generality it can be assumed thatωa = "a and (dω)a =(d")a. Consider the one-parameter family of 1-forms

ωt = t" + (1− t)ω,


where t ∈ [0, 1]. Then ω0 = ω and ω1 = " . We will find a one-parameterfamily of local diffeomorphisms ϕt such that ϕt(a) = a,ϕ0 = id and

ϕ∗t (ωt) = ω0. (10.5)

Differentiating both sides of the last equality we obtain

d

dtϕ∗t (ωt) = ϕ∗t (LXt (ωt)+ ωt) = 0, (10.6)

where ωtdef= dωt/dt = ω−" and Xt is a one-parameter family of vector fields

defined in some neighborhood of a such that the translations along Xt fromt = 0 to t coincide with ϕt . From (10.6) it follows that

ιXt dωt + d(ωt(Xt)) = −ωt . (10.7)

We will find the vector field Xt in the form Xt = ftXt1 + Yt , where ωt(Yt) = 0

and Xt1 is the contact vector field such that ωt(Xt

1) = 1 for all t ∈ [0, 1]. Thenfrom (10.7) it follows that

ιYt dωt = −ωt − dft . (10.8)

Let ft be a one-parameter family of smooth functions on M such that in someneighborhood of the point a we have (−ωt − dft)(Xt

1) = 0, or

Xt1(ft) = ωt(Xt). (10.9)

Thus, choosing functions ft and the vector field Yt so that ft(a) = 0, ft satisfy(10.9) and Yt satisfies (10.8), we see that the vector fields Xt vanish at the pointa and the family ϕt satisfies condition (10.5). Therefore, ϕ∗1 (ω1) = ω0. �

Corollary 10.3.1 Suppose that M is a (2n + 1) – dimensional contact man-ifold with a contact form ω. Then one can find local canonical coordinates(q1, . . . , qn, u, p1, . . . , pn) in a neighborhood Oa of a point a ∈ M such thatqi(a) = pi(a) = u(a) = 0 (i = 1, . . . , n) and ω has the following canonicalform:

ω = du− p1 dq1 − · · · − pn dqn. (10.10)

10.4 A local description of contact transformations 221

10.4 A local description of contact transformations

10.4.1 Generating functions of Lagrangian submanifolds

We begin with a local description of integral manifolds of a contact structure.Let (R2n+1,ω = du−p dq) be the standard contact structure, and let L ⊂ R2n+1

be an n-dimensional integral manifold, ω|L = 0. Note that any smooth functionS(q) defines the integral manifold LS = {u = S(q), p = ∂S/∂q}, and moreoverany integral manifold where functions q1, . . . , qn are independent has this formwith function S = u|L .

For general integral manifolds a tangent space TaL ⊂ ker ωa is a Lagrangiansubspace of the symplectic space (ker ωa, daω). Therefore one can find apermutation (i1, . . . , ik , ik+1, . . . , in) of the indexes (1, . . . , n) in such a way that

daqi1 , . . . , daqik , dapik+1 , . . . , dapin

are linear independent on TaL, and functions qI = (qi1 , . . . , qis), pIc =(pis+1 , . . . , pin), where I = (i1, . . . , ik), Ic = (ik+1, . . . , in), can be chosen aslocal coordinates in some neighborhood of a ∈ L.

Let us rewrite the structure form

ω = du− pI dqI − pIc dqIc

in the following way:

ω = d(u− qIc pIc)− pI dqI + qIc dpIc .

It follows from the above remark that L can be represented in someneighborhood of the point a as follows:

pI = ∂S

∂qI, qIc = − ∂S

∂pIc, u = S − pIc

∂S

∂pIc

for some smooth function S = S(qI , pIc).This function S is called a generating function of the integral manifold L.Let us use the just-obtained local description of integral manifolds for a

description of contact transformations.For this purpose, we will represent graphs of contact transformations as

integral manifolds.Namely, any contact transformation

ϕ : (q, u, p) −→ (A(q, u, p), C(q, u, p), B(q, u, p)),

defines a function λ such that ϕ∗(ω) = λω.


The last equation means that

−λω + ϕ∗(ω) = dC − B dA− λ du+ λp dq = 0. (10.11)

Consider now the (2(2n+1)+1)-dimensional manifold R4n+3 with coordinates

(Q1, . . . , Qn, U, P1, . . . , Pn,λ, q1, . . . , qn, u, p1, . . . , pn)

and the following 1-form:

θ = dU − P dQ+ λp dq − λ du.

It is easy to check that θ determines a contact structure on R4n+3.A graph of a contact transformation ϕ is the following (2n+ 1)-dimensional

submanifold Lϕ :

Q = C(q, u, p), U = C(q, u, p), P = B(q, u, p), λ = λ(q, u, p).

This manifold is integral for θ , and therefore one could use generating functionsto describe contact transformations.

Example 10.4.1 Assume that functions Q, q and u are independent on Lϕ . Thenthe contact transformation is defined by the generating function S = S(Q, q, u)in the following way:

P = ∂S

∂Q, p = −∂S

∂q

/∂S

∂u, U = S.

10.4.2 A description of contact transformations in R3

We give the complete description of contact transformations in the space of1-jets J1R = R3 with the standard coordinates (q, u, p) and the Cartan formω = du− p dq.

The description can be divided into the following eight types in accordancewith the types of function which can be chosen as coordinates on the graph.

1. The generating function is S = S(Q, u, q). Then the contact transformation is

P = ∂S

∂Q, p = − ∂S

∂q

/∂S

∂u, U = S.

10.4 A local description of contact transformations 223

2. The generating function is S = S(P, u, q). Then the contact transformation is

Q = − ∂S

∂P, p = − ∂S

∂q

/∂S

∂u, U = S − P

∂S

∂P.

3. The generating function is S = S(Q, λ, q). Then the contact transformation is

P = ∂S

∂Q, u = −∂S

∂λ, p = −λ−1 ∂S

∂q, V = S − λ

∂S

∂λ.

4. The generating function is S = S(P, λ, q). Then the contact transformation is

Q = − ∂S

∂P, u = −∂S

∂λ, p = −λ−1 ∂S

∂q, V = S − λ

∂S

∂λ− P

∂S

∂P.

5. The generating function is S = S(Q, λ, R), where R = −λp. Then the contacttransformation is

P = ∂S

∂Q, u = −∂S

∂λ, q = − ∂S

∂R, p = −R/λ, V = S − λ

∂S

∂λ− R

∂S

∂R.

6. The generating function is S = S(P, u, R), where R = −λp. Then the contacttransformation is

Q = − ∂S

∂P, q = − ∂S

∂R, p = −R

/∂S

∂u, V = S − P

∂S

∂P− R

∂S

∂R.

7. The generating function is S = S(P, λ, Q). Then the contact transformation is

Q = − ∂S

∂P, u = −∂S

∂λ, p = −λ−1 ∂S

∂u, V = S − P

∂S

∂P− λ

∂S

∂λ.

8. The generating function is S = S(P, λ, R), where R = −λp. Then the contacttransformation is

Q = − ∂S

∂P, u = −∂S

∂λ, p = −λ−1 ∂S

∂R, V = S − P

∂S

∂P− λ

∂S

∂λ.

Example 10.4.2 Take the following generating function S(Q, u, q) = Qu +1/2q2. The corresponding contact transformation has the following form:

ϕ : (q, p, u)→(−q

p, u,

q2

2− qu

p

).

11

Monge–Ampère equations

11.1 Monge–Ampère operators

Let M be an n-dimensional smooth manifold and let J1M be the manifold of1-jets equipped with the standard contact structure ω0 ∈ �1(J1M).

With any differential n-form ω ∈ �n(J1M) we associate a differentialoperator (see [69])

�ω : C∞(M)→ �n(M),

which acts as

�ω(v) = ω|j1(v)(M) (11.1)

where v ∈ C∞(M) is a smooth function on the manifold M, and j1(v)(M) ⊂J1M is the graph of 1-jet j1(v) : M → J1M. We call such operators Monge–Ampère.

To explain this terminology let us consider a few examples of such operatorsand the corresponding differential forms.

Example 11.1.1 Let

ω = 3p2 dp− dx (11.2)

be the differential 1-form on J1R. The corresponding operator �ω acts as

�ω(v) = (3(v′)2v′′ − 1) dx. (11.3)

Indeed,

�ω(v) = (3p2 dp− dx)|j1(v)(M) = 3(v′)2 d(v′)− dx = (3(v′)2v′′ − 1) dx.

224

11.1 Monge–Ampère operators 225

Example 11.1.2 The differential 2-form

ω = dp1 ∧ dp2

on J1R2 corresponds to the operator

�ω(v) = d(vq1) ∧ d(vq2)

= (vq1q1 dq1 + vq1q2 dq2) ∧ (vq2q1 dq1 + vq2q2 dq2)

= (vq1q1 vq2q2 − v2q1q2

) dq1 ∧ dq2

= (det Hess v) dq1 ∧ dq2,

where

Hess v =∥∥∥∥ vq1q1 vq1q2

vq1q2 vq2q2

∥∥∥∥is the Hessian of the function v.

Thus,

�ω(v) = (det Hess v) dq1 ∧ dq2. (11.4)


ω = dq1 ∧ dp1 ∧ dp3 (11.5)

on J1R3 produces the following differential operator:

�ω(v) = dq1 ∧ d(vq1) ∧ d(vq3) = (vq1q2 vq3q3 − vq1q3 vq2q3) dq1 ∧ dq2 ∧ dq3.


ω = dp1 ∧ dq2 − dp2 ∧ dq1

on J1R2 represents the two-dimensional Laplace operator

�ω(v) = (vq1q1 + vq2q2) dq1 ∧ dq2.

In the general case, the form

ω =n∑

i=1

dq1 ∧ · · · ∧ dqi−1 ∧ dpi ∧ dqi+1 ∧ · · · ∧ dqn

226 Monge–Ampère equations

on J1Rn generates the Laplace operator

�ω(v) =(

n∑i=1

vqiqi

)dq1 ∧ · · · ∧ dqn. (11.6)

Example 11.1.5 Any differential n-form

ω = ω0 ∧ α + dω0 ∧ β

on J1M, where α ∈ �n−1(J1M), β ∈ �n−2(J1M) and ω0 is the Cartan form,gives the zero operator.

Example 11.1.6 Two differential 2-forms

ω = dq1 ∧ du and " = p2 dq1 ∧ dq2

on J1R2 generate the same operator:

�ω(v) = dq1 ∧ (vq1 dq1 + vq2 dq2) = vq2 dq1 ∧ dq2,�"(v) = vq2 dq1 ∧ dq2.

11.2 Effective differential forms

Differential forms on J1M vanishing on any integral manifold, and thereforeproducing zero differential operators, form a graded ideal of the exterior algebra�∗(J1M).

We denote this ideal by I∗ = ⊕s≥0Is, Is ⊂ �s(J1M).It follows from the Lepage theorem that the ideal I∗ is generated by forms

ω0 ∧ α + dω0 ∧ β. (11.7)

Note also that I0 = 0 and Is = �s(J1M) for s ≥ n+ 1.We call elements of the quotient module �s(J1M)/Is effective s-forms

(s ≤ n):

�sε(J

1M) ∼= �s(J1M)/Is. (11.8)

One should use these forms to define Monge–Ampère operators.The choice of the Cartan formω0 on J1M gives us a more explicit description

of effective forms.

11.2 Effective differential forms 227

First of all, let us note that the choice of ω0 gives us the decomposition ofthe tangent space:

Ta(J1M) = C(a)⊕ RX1,a (11.9)

and the cotangent space:

T∗a (J1M) = C∗(a)⊕ Rω0a (11.10)

for any point a ∈ J1M.So, if we identify the dual space C∗(a)with a subspace of T∗a (J1M) consisting

of 1-forms degenerating along X1,a:

C∗(a) ∼= {α ∈ T∗a (J1M) | α(X1,a) = 0} (11.11)

then decomposition (11.10) also generates the decomposition of s-forms:

�s(T∗a (J1M)) = �s(C∗(a))⊕ ω0a ∧�s−1(C∗(a)), (11.12)

where

�s(C∗(a)) def= {α ∈ �s(T∗a (J1M)) |X1,a�α = 0}.

Let us denote by �s(C∗) the module of differential s-forms on J1Mdegenerating along the vector field X1.

Note that in the canonical local coordinates (q, u, p) = (q1, . . . , qn, u,p1, . . . , pn) elements ω ∈ �s(C∗) have the following representation:

ω =∑I ,J

ωI ,J(q, u, p) dqI ∧ dpJ , (11.13)

where ωI ,J are smooth functions, I , J are multi-indices: I = (i1, . . . , ik), J =( j1, . . . , js−k) and

dqI def= dqi1 ∧ · · · ∧ dqik , dpJ def= dp j1 ∧ · · · ∧ dp js−k .

The corresponding decomposition of modules of differential forms

�s(J1M) = �s(C∗)⊕ ω0 ∧�s−1(C∗) (11.14)

follows from (11.12).


On the formula level this decomposition can be expressed as

α = α + ω0 ∧ α1 (11.15)

where α ∈ �s(J1M), α ∈ �s(C∗) and

α1 = X1�α ∈ �s−1(C∗).

Let us define a projection

p : �s(J1M)→ �s(C∗),

by

p(α)def= α

or

p(α) = α − ω0 ∧ (X1�α). (11.16)

Example 11.2.1 If α = du, then p(α) = p dq.

Example 11.2.2 p(ω0) = 0 and p(dω0) = dω0.

The module

�∗(C∗) def=n⊕

s=0

�s(C∗)

is obviously an algebra with respect to the operation of exterior multiplication.In algebra�∗(C∗)we consider two operators� and⊥. Namely, at each point

a ∈ J1M the restriction� of daω0 on C(a) determines the symplectic structure,and, therefore, the corresponding operators:�(ω) = �∧ω and⊥(ω) = X��ωfor all ω ∈ �∗(C∗(a)). We shall use these operators on differential forms

� : �s(C∗)→ �s+2(C∗) ⊥: �s(C∗)→ �s−2(C∗)

where

�(ω) def= dω0 ∧ ω,

⊥(ω)def= X��ω

11.2 Effective differential forms 229

and X� ∈ �2(C) is a bivector field on the Cartan distribution dual to dω0 withrespect to the symplectic structure on it.

We call a differential form ω ∈ �s(C∗), s ≤ n, effective if

⊥(ω) = 0.

In other words, effective forms in the old sense have representatives in�∗(C∗)which are effective in the above sense and which are effective in correspondencewith linear algebra notions.

Note also that a realization of effective forms as forms on the Cartan distri-bution depends heavily on decomposition (11.10) and therefore depends on thechoice of a generator of the Cartan distribution C.

It follows from the Lepage theorem that for any differential s-form ω ∈�s(C∗), s ≤ n, there exists a unique effective part ωε ∈ �s

ε(J1M), and this part

can be found by the formula

ωε =∑

s≥0

(−1)s1

s+ 1�s ⊥s

(ω).

Example 11.2.3 Any 1-form on the Cartan distribution is effective.

Example 11.2.4 Let M = R2 and ω = dq1 ∧ dp1 ∈ �2(C∗). Then,

ωε = ω − 1

2� = 1

2(dq1 ∧ dp1 − dq2 ∧ dp2)

is the effective part of ω.

Lemma 11.2.1 Any differential formω∈ Is admits the following decomposition:

α = ω0 ∧ α1 + dω0 ∧ α2, (11.17)

where the form α1 ∈ �s−1(C∗) is uniquely determined by α, and the formα2 ∈ �s−2(C∗) is also uniquely determined if s < n+ 2.

Proof Indeed, from (11.15) it follows that

α = p(α)+ ω0 ∧ (X1�α).

Since α ∈ Is, we see that p(α) vanishes on all integral manifolds. Therefore,p(α) = �α2, where α2 is uniquely determined if s < n+ 2. �


It follows from this lemma that for any differential form α ∈ �s(J1M)

we have

α = αε + ω0 ∧ α1 + dω0 ∧ α2 (11.18)

where αε is the effective part of p(α) and α1 ∈ �s−1(C∗), α2 ∈ �s−2(C∗).

Example 11.2.5 A basis of the module �2ε(J

1R2) consists of the followingdifferential 2-forms:

dq1 ∧ dq2, dq1 ∧ dp2, dq2 ∧ dp1, dp1 ∧ dp2, dq1 ∧ dp1 − dq2 ∧ dp2.

11.3 Calculus on �∗(C∗)

The ideal I∗ is differential in the sense that the de Rham operator preservesthis ideal, dIs ⊂ Is+1, and therefore induces an operator dε : �s+1

ε (J1M) →�s+1ε (J1M) on the module of effective forms.We represent this operator in terms of decomposition given by choice of the

Cartan form ω0. To simplify formulas we shall use Lf and ιf instead of LXf

and ιXf .At first we introduce the operator dp : �s(C∗)→ �s+1(C∗) by

dpdef= p ◦ d

or dpα = dα − ω0 ∧ (ι1 dα).It is easy to check that this operator satisfies the following relations.

1. dp(ω1 ∧ ω2) = dpω1 ∧ ω2 + (−1)degω1ω1 ∧ dpω2.

2. d2p + � ◦ L1 = 0.

3. dp ◦ d = 0.

4. � ◦ dp = dp ◦ �.

5. ιf (dω0) = −dpf .

6. L1 ◦ dp = dp ◦ L1.

Using these properties it is easy to find an action dp in the canonical localcoordinates q, u, p on J1M.

11.3 Calculus on �∗(C∗) 231

Indeed, for any function f ∈ C∞(J1M) we have

dpf = df − X1( f )ω0 =n∑

i=1

df

dqidqi +

n∑i=1

∂f

∂pidpi,

where

df

dqi

def= ∂f

∂qi+ pi

∂f

∂u,

and for any s-form

α =∑I ,J

αI ,J(q, u, p) dqI ∧ dJp.

we have

dpα =∑I ,J

dp(αI ,J(q, u, p)) ∧ dqI ∧ dJp.

Lemma 11.3.1 Let the differential form α ∈ Is have the following form:

α = ω0 ∧ α1 + dω0 ∧ α2.

Then,

dα = −ω0 ∧ dp(α1 + dpα2)+ dω0 ∧ (α1 + dpα2). (11.19)

Proof We have

dα = −ω0 ∧ dα1 + dω0 ∧ (α1 + dα2) (11.20)

and

dpαi = p(dαi) = dαi − ω0 ∧ ι1(dαi).

Replacing dαi by dpαi + ω0 ∧ ι1(dαi), we obtain

dα = −ω0 ∧ dpα1 + dω0 ∧ (α1 + dpα2)+ dω0 ∧ ω0 ∧ ι1(dα2). (11.21)

From the above properties of dp it follows that

d2pα2 = −dω0 ∧ L1(α2).


Since α2 ∈ �s−2(C∗), we see that X1�α2 = 0, L1(α2) = ι1 dα2 and

−dω0 ∧ ι1(dα2) = d2pα2.

Therefore (11.19) now follows from (11.21). �

Theorem 11.3.1 The complex

0 → I1 d→ I2 → · · · → Is d→ Is+1 → · · · → I2n+1 → 0

is exact for all s except the case where s = dim M + 1.

Proof Suppose that the s-form α ∈ Is is closed. Then α = ω0∧α1+dω0∧α2

and using Lemma 11.3.1, we obtain

dp(α1 + dpα2) = 0

and

�(α1 + dpα2) = dω0 ∧ (α1 + dpα2) = 0.

If s < n+ 1, then the second relation implies that α1 + dpα2 = 0.Then

α1 = −dpα2 = dα2 − ω0 ∧ ι1(dα2)

and, therefore, by Lemma 11.2.1,

α = −ω0 ∧ dα2 + α2 ∧ dω0 = d(ω0 ∧ α2).

This means that the complex is exact in the term.Note that if s = dim M+ 1, then the equality α1+ dpα2 = 0 does not follow

from the formula

dω0 ∧ (α1 + dpα2) = 0.

In dimensions s ≥ dim M+1 we have Is = �s(J1M) and therefore the complexcoincides with the de Rham complex, but Hs(J1M, R) = 0 if s ≥ dim M + 1.

�

Corollary 11.3.1 Cohomology H ·ε(J1M) of the factor-complex

C∞(J1M)dp→ �1

ε(J1M)

dε→ �2ε(J

1M)→ · · ·dε→ �n−1

ε (J1M)dε→ �n

ε(J1M)→ 0

11.4 The Euler operator 233

is isomorphic to the de Rham cohomology, Hsε(J

1M) ∼= Hs(M, R) ifs �= dim M.

11.4 The Euler operator

In this section we shall investigate the highest effective cohomology group

Hnε (J

1M) = �nε(J

1M)/dε(�n−1ε (J1M))

where n = dim M.Let α ∈ �n

ε(C∗) be an effective n-form such that the class α + In contains

a closed differential n-form, say, α + ω0 ∧ x + dω0 ∧ y, where x ∈ �n−1(C∗)and y ∈ �n−2(C∗).

Then

d(α + ω0 ∧ x + dω0 ∧ y) = 0,

and using Lemma 11.3.1, we obtain

dα − ω0 ∧ dp(x + dpy)+ dω0 ∧ (x + dpy) = 0.

Since

dα = dpα + ω0 ∧ L1(α),

we obtain

dpα + ω0 ∧ (L1(α)− dp(x + dpy))+ dω0 ∧ (x + dpy) = 0.

Note that� : �n−1(C∗)→�n+1(C∗) is the isomorphism and therefore dpα =dω0 ∧ α′, where α′ =⊥ dpα.

Therefore

ω0 ∧ (L1(α)− dp(x + dpy))+ dω0 ∧ (α′ + x + dpy) = 0,

and the class α + In contains a closed form if and only if the following systemis solvable with respect to forms x and y:

dp(x + dpy) = L1(α),

x + dpy = −α′.


The necessary condition for solvability of this system is

L1(α)+ dpα′ = 0

and the condition takes the form

L1(α)+ dp ◦ ⊥ ◦ dp(α) = 0.

Let us denote by

E def= L1 + dp ◦ ⊥ ◦ dp

the last operator. We call E Euler’s operator.We shall see later on that the Euler operator is the main operator in calculation

of conservation laws and in the calculus of variations. This will explain ourterminology.

To summarize, we obtain the following.

Theorem 11.4.1 A class α + In of an effective form α ∈ �nε(C

∗) contains aclosed form if and only if

E(α) = 0.

Corollary 11.4.1 Let α be an effective n-form such that E(α) = 0. Then,

d(α − ω0∧ ⊥ dpα) = 0.

Proof Indeed, one can take the following solution of the system: x = −α′,y = 0. �

We now list the main properties of the Euler operator.

Lemma 11.4.1 The operator E maps �sε(J

1M) to itself.

Proof First, let us prove that

E : �n(C∗)→ �n(C∗).

Suppose that α ∈ �s(C∗). Then, since ι1 α = 0, we see that

L1(α) = ι1 dα,

11.4 The Euler operator 235

and

ι1 L1(α) = 0,

i.e., L1(ω) ∈ �s(C∗).Furthermore, operators ⊥ and dp act on �∗(C∗). Therefore, E(α) ∈ �s(C∗)Secondly, we note that the operator� commutes with differential dp, the Lie

derivative L1 and [⊥,�] = −1 on �n+1(C∗). Then �(α) = 0 because α iseffective, and

� ◦ E(α)=� ◦ L1(α)+� ◦ dp ◦ ⊥ ◦ dp(α)=dp ◦ � ◦ ⊥ ◦ dp(α)

=dp ◦ ⊥ ◦ � ◦ dp(α)+d2p(α)=dp ◦ ⊥ ◦ dp ◦ �(α)−� ◦ L1(α)

=0.

Therefore, E(α) is effective. �

Corollary 11.4.2 The following sequence

0 → ker E/d(�n−1ε (J1M))→ Hn

ε (J1M)→ �n

ε(J1M)/ ker E → 0

is exact.

Theorem 11.4.2 The Euler operator E : �nε(C

∗) → �nε(C

∗) satisfies thefollowing relations:

1. E ◦ L1 = L1 ◦ E ;2. dp ◦ E = 0;3. E2 = L1 ◦ E ;4. E( f α) = f E(α)+ X1( f )α + dpf∧ ⊥ dpα + dp(ιf α), where α ∈ �n

ε(J1M),

and f ∈ C∞(J1M);5. p ◦ Lf = f E+dp ◦ (ιf − f ⊥ ◦ dp)+� ◦ ιf ◦ ⊥ ◦ dp on �n

ε(J1M).

Proof The proof of the first property is obvious.One has

dp ◦ E = dp ◦ L1 + d2p ◦ ⊥ ◦ dp = dp ◦ L1 −� ◦ L1 ◦ ⊥ ◦ dp

= dp ◦ L1 + [⊥,�] ◦ dp ◦ L1− ⊥ ◦ � ◦ dp ◦ L1

= dp ◦ L1 − dp ◦ L1− ⊥ ◦ dp ◦ L1 ◦ � = 0.


We have

dp ◦ ⊥ ◦ dp ◦ dp ◦ ⊥ dp = −L1 ◦ dp ◦ ⊥ ◦ � ◦ ⊥ ◦ dp

= −L1 ◦ dp ◦ [⊥,�] ◦ ⊥ ◦ dp

= −L1 ◦ dp ◦ ⊥ ◦ dp,

and therefore

E2 = L21+ 2L1 ◦ dp ◦ ⊥ ◦ dp − L1 ◦ dp ◦ ⊥ ◦ dp = L1 ◦ E .

One has

dp ◦ ⊥ ◦ dp( fω) = dp ◦ ⊥ (dpf ∧ ω + fdpω)

= dp(Xf �ω)+ dp( f ⊥ (dpω))

= dp(Xf �ω)+ dpf∧ ⊥ (dpω)+ fdp ⊥ (dpω).

Using the formula LX = d ◦ ιX + ιX ◦ d, we obtain

p(Lf α) = dp(ιf α)+ p(ιf dα) = dp(ιf α)+ ιf dpα + f L1(α)

= f E(α)− fdp ◦ ⊥ ◦ dp(α)+ dp(ιf α)+ ιf dpα

= f E(α)+ dp(ιf α − f ⊥ (dpα))+ ιf dpα + dpf∧ ⊥ (dpα)

for any effective n-form α.As we have seen

dpα = � ⊥ dpα.

Therefore,

ιf dpα = −dpf∧ ⊥ (dpα)+�(ιf ⊥ dpα). �

11.5 Solutions

Let us return to Monge–Ampère operators. From Example 11.1.5 and from(11.16) and (11.18) we see that forms ω, p(ω) and ωε generate the samedifferential operator:

�ω = �p(ω) = �ωε .

11.5 Solutions 237

In general, two differential n-formsω and" generate the same Monge–Ampèreoperator if and only ifω−" ∈ In. This means that the operator�ω is uniquelydetermined by the effective part of p(ω).

So, from now on we shall suppose that operators�ω are generated by effectiveforms ω ∈ �n

ε(J1M).

Any Monge–Ampère differential operator �ω defines a differential equation�ω(v) = 0 which we call the Monge–Ampère equation.

Example 11.5.1 For differential operator (11.3) the equation has the form

3(v′)2v′′ − 1 = 0. (11.22)

In the more general case, the effective differential form A(x, u, p) dx +B(x, u, p) dp generates the following ordinary differential equation:

A(x, v, v′)+ B(x, v, v′)v′′ = 0.

Example 11.5.2 For differential operators (11.4)–(11.6) the equations havethe forms:

det Hess v = 0,

vq1qq2vq3q3 − vq1q3 vq2q3 = 0

and

n∑i=1

vqiqi = 0

respectively.

Example 11.5.3 The following equation is known as the Von Karman equationon a plane [44]. It arises in aerodynamics and it describes the behavior of thevelocity potential v in the transonic approximation of gas (see [44]):

vq1 vq1q1 − vq2q2 = 0. (11.23)

The effective form

ω = p1 dq2 ∧ dp1 + dq1 ∧ dp2 (11.24)

corresponds to the equation.


Example 11.5.4 The form

ω = uα dq1 ∧ dp2 + (αuα−1p22 − p1 + F(u))dq1 ∧ dq2

corresponds to the equation

vt = (vαvx)x + F(v).

Here q1 = t, q2 = x, α is a real number and F is a smooth function. Thisequation is often called the reaction–diffusion equation [6, 87] or the equationof non-linear thermal conductivity [20] . Later on it will be considered in moredetail.

Example 11.5.5 An effective differential 4-form that corresponds to theKhokhlov–Zabolotskaya equation [98]

∂

∂q1

(∂v

∂q2− v

∂v

∂q1

)= ∂2v

∂q23

+ ∂2v

∂q24

(11.25)

is

ω = 1

2dp1 ∧ dq1 ∧ dq3 ∧ dq4 − 1

2dp2 ∧ dq2 ∧ dq3 ∧ dq4

+ udp1 ∧ dq2 ∧ dq3 ∧ dq4 + p21dq1 ∧ dq2 ∧ dq3 ∧ dq4

+ dp3 ∧ dq1 ∧ dq2 ∧ dq4 − dp4 ∧ dq1 ∧ dq2 ∧ dq3.

This equation describes the propagation of a bounded three-dimensionalsound beam [98] and later on it will be considered in more detail.

Let us write the coordinate representation of the equation �ω = 0 for theclassical case M = R2.

It is not hard to see that any effective 2-formω on J1R2 has the following form:

ω =E dq1 ∧ dq2 + B(dq1 ∧ dp1 − dq2 ∧ dp2)

+ C dq1 ∧ dp2 − A dq2 ∧ dp1 + D dp1 ∧ dp2, (11.26)

where A, B, C, D and E and are smooth functions on J1R2.The corresponding equation then takes the form of the classical Monge–

Ampère equation:

Avxx + 2Bvxy + Cvyy + D(vxxvyy − v2xy)+ E = 0,

where x = q1, and y = q2.

11.5 Solutions 239

In general case we denote the Monge–Ampère differential equation �ω = 0by Eω.

Note that for any non-vanishing smooth function h on J1M two forms ω andhω generate the same differential equations:

Eω = Ehω.

A classical (or regular) solution of the Monge–Ampère equation Eω in anopen domain N ⊂ M is a function v ∈ C∞(N) such that

�ω(v) ≡ 0.

A multivalued, or generalized solution of the Monge–Ampère equation Eω

is an integral submanifold i : L ↪→ J1M, dim L = dim M of the curtandistribution, such that ω|L = 0.

As we have seen a function v ∈ C∞(M) is a classic solution of the Monge–Ampère equation if and only if

= Lvdef= j1(v)(M)

is a multivalued solution.

Example 11.5.6 The curve

L :

x = t3,

u = 3

4t4,

p = t,

t ∈ R, is a multivalued solution of the ordinary differential equation (11.22).Indeed, the map

i : L ↪→ J1M = R3, t −→(

t3,3

4t4, t

)is an immersion, and

ω0|L = 0, ω|L = 0.

So, L is a multivalued solution of Monge–Ampère equations (11.22).

Example 11.5.7 The Lissajou figure u = cos(at), x = sin(bt), corresponds toa multivalued solution of the differential equation (see Figure 11.1)

(1− x2)y′′ + xy′ + λy = 0

with λ = a2/b2.


1

1

0.5

–0.5

–1 –0.5 0.5 1x

y

Figure 11.1. The Lissajous figure.

Example 11.5.8 The surface

L :

q1 = 1

2p2

1 + C1,

q2 = p2 + C2,

u = 1

3p3

1 +1

2p2

2,

L ⊂ J1R2, is a multivalued solution of (11.23). Here C1, C2 are constants. Theprojection of L to J0R2 is a two-valued solution.

Similarly,

L :

q1 = 1

2p2

1 + C1,

q2 = p2 + C2,

u = 1

3p3

1 +1

2p2,

is a multivalued solution of the non-homogeneous Von Karman equation (seeFigure 11.2)

vxvxx − vyy = 1.

11.6 Monge–Ampère equations of divergent type 241

q1

q2

u

Figure 11.2. The multivalued solution of the non-homogeneous Von Karmanequation.

Note that both of these solutions represented by smooth manifolds L are notsmooth in the projection on the space J0R2.

11.6 Monge–Ampère equations of divergent type

We call elements of ker E divergent forms, and the corresponding operators �ω

and equations Eω we call divergent type operators and equations respectively.The next examples show the relations between this definition and commonly

used notions.

Example 11.6.1 The classical quasi-linear differential equation of divergenttype has the form

n∑i=1

∂

∂qi

(Ai

(q, v,

∂v

∂q

))= 0. (11.27)

Consider the following effective form:

θ =n∑

i=1

(−1)i+1 Ai(q, u, p)dq1 ∧ · · · ∧ dqi ∧ · · · ∧ dqn.


Then the n-formω = dpθ defines the equation, and therefore E(ω) = 0, becausethe class of ω contains the exact form.

Example 11.6.2 The Von Karman equation (see [44])

vxvxx − vyy − vzz = 0

is of divergent type, because it can be written as

1

2

∂

∂x(v2

x)+∂

∂y(−vy)+ ∂

∂z(−vz) = 0.

For the corresponding form we have

ω = p1 dp1 ∧ dq2 ∧ dq3 − dq1 ∧ dp2 ∧ dq3 − dq1 ∧ dq2 ∧ dp3

and ω = dpθ , where

θ = 1

2p2

1 dq2 ∧ dq3 + p2 dq1 ∧ dq3 − p3 dq1 ∧ dq2.

Therefore E(ω) = 0.

Example 11.6.3 The classical Monge–Ampère equation det Hess v = 1 is ofdivergent type. Indeed, one can take

θ = p1 dp2 ∧ · · · ∧ dpn − q1 dq2 ∧ · · · ∧ dqn.

Then ω = dpθ is the effective form of divergent type that corresponds to theequation.

12

Symmetries and contact transformations ofMonge–Ampère equations

12.1 Contact transformations

Let us define an action of the group of contact diffeomorphisms Ct(J1M) onMonge–Ampère operators.

Let F be a contact transformation, F ∈ Ct(J1M), then F∗ preserves theCartan ideal I , F∗(Is) = Is, and therefore determines a map of effective forms:

F∗ : ω + Is −→ F∗(ω)+ Is.

But F, in general, does not preserve the Cartan form ω0, and therefore, does notact directly on representatives in �s(C∗) of the effective forms. Thus we shalldefine the action F on ω = ω ∈ �s

ε(C∗) by taking the effective part F∗(ω)ε.

In a similar way, we define an action of the contact diffeomorphism F ∈Ct(J1M) on the Monge–Ampère operator �ω, where ω ∈ �n

ε(J1M), by

F(�ω)def= �F∗(ω)ε .

Note that, for any F ∈ Ct(J1M) and any integral manifold L ⊂ J1M we have

F∗(ω)ε|L = ω|F(L).

Hence, if L is a multivalued solution of the equation EF∗(ω)ε , then F(L) is asolution of the equation Eω.

We say that two Monge–Ampère operators �ω1 and �ω2 , where ω1,ω2 ∈�nε(J

1M), are contact equivalent if there exists a contact transformation F,such that

ω2 = F∗(ω1)ε,

243

244 Symmetries and transformations of Monge–Ampère equations

and that two Monge–Ampère equations Eω and E" are contact equivalent ifthere exists a contact transformation F and a non-vanishing function λF ∈C∞(J1M), such that

ω2 = λFF∗(ω1)ε.

As a corollary of our interpretation of Monge–Ampère equations we obtainthe following theorem.

Theorem 12.1.1 (Sophus Lie) The class of Monge–Ampère equations is closedwith respect to contact transformations.

Example 12.1.1 The Legendre transformation

ϕ : (x, u, p) → (p, u− px, −x)

translates the form (11.2) to the form ϕ∗(ω) = 3x2 dx−dp. Then the non-linearequation (11.22) turns into the following linear equation:

v′′ = 3x2.

Example 12.1.2 (Quasilinear equations) Let us consider a quasi-linearequation of the form

A(vx , vy)vxx + 2B(vx , vy)vxy + C(vx , vy)vyy = 0.

This equation is represented by the following effective form:

ω = B(p1, p2)(dq1 ∧ dp1 − dq2 ∧ dp2)

+ C(p1, p2) dq1 ∧ dp2 − A(p1, p2) dq2 ∧ dp1.

After the Legendre transformation

ϕ : (q1, q2, u, p1, p2) → (p1, p2, u− p1q1 − p2q2, −q1, −q2) (12.1)

we obtain the following effective form:

ϕ∗(ω) = B(−q1,−q2)(dq1 ∧ dp1 − dq2 ∧ dp2)

− A(−q1,−q2)dq1 ∧ dp2 + C(−q1,−q2)dq2 ∧ dp1,

which corresponds to the linear equation

−A(−q1,−q2)vq2q2 + 2B(−q1,−q2)vq1q2 − C(−q1,−q2)vq1q1 = 0.

12.1 Contact transformations 245

Example 12.1.3 The Von Karman equation

vq1 vq1q1 − vq2q2 = 0 (12.2)

becomes the linear equation

q1vq2q2 + vq1q1 = 0 (12.3)

after a Legendre transformation (12.1). The last equation is known as theTriccomi equation.

Example 12.1.4 (The Monge–Ampère equation) This equation has the form

det Hess v = 1

and is generated by the effective differential 2-form

ω = dp1 ∧ dp2 − dq1 ∧ dq2.

After the Euler transformation

ϕ : (q1, q2, u, p1, p2) → (p1, q2, u− p1q1, −q1, p2)

it becomes

ω = dq2 ∧ dp1 − dq1 ∧ dp2

and corresponds to the Laplace equation

vq1q1 + vq2q2 = 0. (12.4)

Example 12.1.5 (TM equation [84, 21, 110]) The Titeica–Morimoto equation(the TM equation) has the following form:

det Hess v = (v− q1vq1 − q2vq2)4. (12.5)

The effective differential 2-form corresponding to (12.5) is

ω = (u− q1p1 − q2p2)4 dq1 ∧ dq2 − dp1 ∧ dp2.

This equation can be simplified by using the Legendre transformation (12.1)into the form

F∗(ω) = u4 dp1 ∧ dp2 − dq1 ∧ dq2.


The corresponding equation has a simpler form than (12.5):

v4 det Hess v = 1. (12.6)

Example 12.1.6 The following equation

v2yvxx − 2vxvyvxy + v2

xvyy + R(vx , vy) = 0

where R(vx , vy) is a homogeneous polynomial of degree 3 describes linearplanar webs and their generalizations. This equation transforms into itselfunder any diffeomorphism φ : R →R, u → φ(u).

Let us explain how contact transformations can be used in finding solutions.Let us consider, for example, the following non-linear wave equation [55]

vyy + (f (vx))x = 0. (12.7)

It corresponds to the form

ω = dq1 ∧ dp2 − f ′(p1) dq2 ∧ dp1.

The contact transformation

ϕ : (q1, q2, u, p1, p2) −→ (p1, q2, u− p1q1,−q1, p2)

takes ω to

ϕ∗(ω) = dp1 ∧ dp2 + f ′(q1) dq1 ∧ dq2. (12.8)

We obtain the equation

det Hess v+ f ′(q1) = 0. (12.9)

Consider, for example, solutions v of the last equation such that vq1q2 = 0.Then we obtain

v(q1, q 2) = c0 + c1q1 + c2q2 + q22

2c3− c3

∫f (q1) dq1, (12.10)

where c0, c1, c2, c3 are constants.


Let us write down the integral manifold L that corresponds to the solution:

u = c0 + c1q1 + c2q2 + q22

2c3− c3

∫f (q1) dq1,

p1 = c1 − c3f (q1),

p2 = c2 + q2

c3.

Then ϕ(L) gives a solution of the original equation. Thus we obtain

u = c0 + c1p1 + c2q2 + q22

2c3+ p1q1 + c3

∫f (p1) dp1,

q1 = −c3f (p1)− c1,

p2 = q2

c3+ c2.

Note that these formulae represent a multivalued solution and (p1, q2) arecoordinates on ϕ(L).

Let us take, for example, the function

f (p1) = 1

2p2

1,

which corresponds to the equation

vq1 vq1q1 + vq2q2 = 0. (12.11)

In this case the corresponding multivalued solution takes the form (seeFigure 12.1)

u = c0 + c1p1 + c2q2 + q22

2c3+ p1q1 + c3

p31

3,

q1 = −c3p2

1

2− c1,

p2 = q2

c3+ c2.

We obtain another type of solution for (12.9) from the assumption thatvq2q2 = 0.

In this case we find the following class of solutions:

v(q1, q2) = A(q1)+ q2

∫(f ′(q1))

1/2 dq1.


q1

q2

u

Figure 12.1. The multivalued solution of the Von Karman equation.

The corresponding integral manifold L is

u = A(q1)+ q2

∫(f ′(q1))

1/2 dq1,

p1 = A′(q1)+ q2 (f′(q1))

1/2,

p2 =∫(f ′(q1))

1/2 dq1

and the image ϕ(L) takes the form

u = A(p1)+ q2

∫(f ′(p1))

1/2 dp1 − p1A′(p1)− p1q2(f′(p1))

1/2,

q1 = −A′(p1)− q2(f′(p1))

1/2,

p2 =∫(f ′(p1))

1/2 dp1.

For example, for the Von Karman equation, f (p1) = 12 p2

1, taking A = 0, weobtain a multivalued solution of the type

u = 1

2ts3, q1 = −ts, q2 = t, p1 = s2, p2 = 2

3s3

(see Figure 12.2).


q1q2

u

5

Figure 12.2. The multivalued solution of the Von Karman equation.

Theorem 12.1.2 (Jörgens [69]) Any smooth solution v(q1, q2) of the classicalMonge–Ampère equation

det Hess v = 1 (12.12)

is a polynomial of second degree.

Proof First of all, let us note that the Euler transformation ϕ transforms ourequation into the Laplace equation.

Moreover, if v0 = v0(q1, q2) is a smooth solution of (12.12) and Ldef= Lv0 =

j1(v0)(R2) is the corresponding integral manifold, then the image ϕ(L) is asolution of the Laplace equation, and the integral manifold ϕ(L) is projectedwithout singularities onto the (q1, q2)-plane.

Indeed, if it not so, a non-zero vector

Xa = λ1∂

∂p1+ λ2

∂

∂p2

should be tangent to ϕ(L) at some point a ∈ ϕ(L).


Therefore, the vector

dϕ−1a (Xa) = −λ1

(∂

∂q1+ p1(a)

∂

∂u

)+ λ2

∂

∂p2

should be tangent to L at ϕ−1(a).But this is impossible, since at the point of contact ϕ−1(a) we should have∥∥∥∥∥∥∥∥∥∥

∂2v0

∂q21

∂2v0

∂q1∂q2

∂2v0

∂q1∂q2

∂2v0

∂q22

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥−λ1

0

∥∥∥∥∥ =∥∥∥∥∥ 0

λ2

∥∥∥∥∥ ,

for the solution v0 = v0(q1, q2) of (12.12) and

∂2v0

∂q21

= 0,∂2v0

∂q1∂q2= −λ2

λ1

at the point.Therefore,

det

∥∥∥∥∥∥∥∥∥∥∂2v0

∂q21

∂2v0

∂q1∂q2

∂2v0

∂q1∂q2

∂2v0

∂q22

∥∥∥∥∥∥∥∥∥∥= −

(−λ2

λ1

)2

± 1.

Moreover, observe that the second derivatives ∂2v/∂q21 and ∂2v/∂q2

2 of thefunction v are non-zero.

In fact, if at a some point

∂2v

∂q21

= ∂2v

∂q22

= 0 and∂2v

∂q1∂q2= b,

then at this point the vector(∂

∂q1+ p1

∂

∂u

)+ b

∂

∂p2

is tangent to ϕ(L), and so the vector

− ∂

∂p1+ b

∂

∂p2

is tangent to L, which is impossible.

12.2 Lie equations for contact symmetries 251

But the functions ∂2v/∂q21 and ∂2v/∂q2

2 are harmonic. Hence, by the Liouvilletheorem

∂2v

∂q21

= − ∂2v

∂q22

= constant �= 0.

Thus, ∂2v/∂q1∂q2 = constant and v(q1, q2), like v0(q1, q2), is a polynomial ofsecond degree. �

12.2 Lie equations for contact symmetries

We say that a contact transformation ϕ ∈ Ct(J1M) is a contact symmetry orsimply symmetry of a Monge–Ampère equation Eω if the form ϕ∗(ω) generatesthe same equation Eω.

It follows from the Lepage theorem that in this case

ϕ(�ω) = hϕ�ω

for some smooth not vanishing function hϕ on J1M.In terms of effective forms this means that the contact transformation ϕ is a

symmetry if the effective part of the form ϕ∗(ω) is proportional to the effectiveform ω, i.e.,

ϕ∗(ω)ε = hϕω,

for some smooth function hϕ ∈ C∞(J1M).In other words,

ϕ∗(ω) = hϕω + θ

for some differential form θ ∈ In.We consider now infinitesimal symmetries of Monge–Ampère equations.

Let X = Xf be a contact vector field. Then the Lie derivative Lfdef= LXf

preserves the Cartan ideal I∗ because Lf (ω0) = X1(f )ω0 and Lf (dω0) =dLf (ω0) = d(X1f ) ∧ ω0 + X1(f ) dω0.

We say that a contact vector field Xf on the manifold J1M is an infinitesimalsymmetry of the Monge–Ampère equation Eω if

Lf (ω) = hωmod In (12.13)

for some smooth function h on J1M.


Obviously, the set of all infinitesimal symmetries forms a Lie algebra withrespect to the commutator of vector fields.

We denote this algebra by Sym(Eω).

Example 12.2.1 The Legendre transformation (12.1) is a symmetry of the waveequation

vq1q2 = 0.

Indeed, the corresponding effective differential 2-form is

ω = dq1 ∧ dp1 − dq2 ∧ dp2

and

ϕ∗(ω) = dp1 ∧ (−dq1)− dp2 ∧ (−dq2) = ω.

Example 12.2.2 The translations

Tc : (q, u, p) −→ (q + c, u, p),

where c ∈ Rn are symmetries of equations Eω ⊂ J1Rn if their coefficients donot depend on q. In this case, T∗c (ω) = ω.

Example 12.2.3 The contact vector field

Xf = q4∂

∂q3− q3

∂

∂q4− p4

∂

∂p3+ p3

∂

∂p4,

with generating function f = p4q3 − p3q4, is an infinitesimal symmetry of theKhokhlov–Zabolotskaya equation (11.25).

Theorem 12.2.1 A contact vector field Xf is a symmetry of the Monge–Ampèreequation Eω, defined by an effective n-form ω, if and only if

dp(ιf ω)+ ιf dpω + fL1(ω) = hω (12.14)

for some smooth function h ∈ C∞(J1M).

Proof Note that

ιf (ω0 ∧ ι1 dω) = f ι1 d ω − ω0 ∧ (ιf ι1 dω).


Therefore,

p(ιf dω) = ιf dω − ω0 ∧ (ι1ιf dω) = ιf dω + ω0 ∧ (ιf ι1 ω)

= ιf dω − ιf (ω0 ∧ ι1 dω)+ f ι1 dω = ιf dpω + f ι1 dω.

Using this relation we obtain

p(Lf (ω)) = p(ιf dω + d(ιf ω)) = dp(ιf ω)+ ιf dpω + fL1(ω).

Note now that Xf is a symmetry if p(Lf (ω)) is proportional to ω. �

We call (12.14) the Lie equation for contact symmetries of Eω. We also saythat a contact vector field Xf is a symmetry of the Monge–Ampère differentialoperator if the generating function f satisfies the Lie equation with h = 0.

Example 12.2.4 (The Laplace equation) As a first example, let us calculateinfinitesimal contact symmetries of the Laplace operator on R2. As we haveseen the effective differential 2-form corresponding to the Laplace operator is

ω = dq1 ∧ dp2 − dq2 ∧ dp1.

Therefore,

ιf ω = −fp1 dp2 + fp2 dp1 + (fq1 + p1fu) dq2 − (fq2 + p2fu) dq1

and

dp(ιf ω) = (fq1q1 + fq2q2 + 2p1fuq1 + 2p2fq2u + fuu(p21 + p2

2)) dq1 ∧ dq2

+ (fp2q1 + fq2p1 + p1fp2u + p2fp1u)(dq1 ∧ dp1 − dq2 ∧ dp2)

+ (fu − fp1q1 + fq2p2 − p1fp1u + p2fup2) dq1 ∧ dp2

+ (−fp1p1 − fp2p2) dp1 ∧ dp2.

Note that dω = 0, LX1(ω) = 0, and hence

ιf dpω + fL1(ω) = 0.

The Lie equation for the generating function f takes the following form:

fq1q1 + fq2q2 + 2p1fuq1 + 2p2fq2u + fuu(p21 + p2

2) = 0,

fq1p1 − fq2p2 + p1fup1 − p2fup2 = 0,

fp2q1 + fq2p1 + p1fp2u + p2fp1u = 0,

fp1p1 + fp2p2 = 0.


Differentiating the second equation in p1 and the third one in p2 and summingobtained equalities, we obtain

fup1 = 0.

In the similar way, differentiating the second equation in p2 and the third onein p1, and subtracting obtained equalities, we obtain

fup2 = 0.

So, we see that function f has the following form:

f = a(q1, q2, u)+ b(q1, q2, p1, p2)

for some functions a and b.Rewriting the Lie equation for these functions we get the following system:

aq1q1 + aq2q2 + bq1q1 + bq2q2 + 2(p1aq1u + p2aq2u)+ (p21 + p2

2)auu = 0,

bq1p1 − bq2p2 = 0,

bq1p2 + bq2p1 = 0,

bp1p1 + bp2p2 = 0.

Differentiating the first equation in p1 and in p2, and using the last threeequations, we obtain

aq1u + p1auu = 0,

aq2u + p2auu = 0.

From these two equations it follows that the function a has the form

a = Cu+ g(q1, q2),

where C is an arbitrary constant and g is an arbitrary function in q1, q2.Let w = g+ b. Then the generating functions f has the form

f = Cu+ w,

where the function w satisfies the following system of differential equations:

wq1q1 + wq2q2 = 0,

wq1p2 + wq2p1 = 0,

wq1p1 − wq2p2 = 0,

wp1p1 + wp2p2 = 0.


So, we see that the Lie algebra of infinitesimal symmetries of the Laplaceoperator is infinite dimensional.

Example 12.2.5 (The TM equation) As a second example we consider (12.5).Recall that after the Legendre transformation this equation takes the form

v4 det Hess v = 1. (12.15)

Solving the Lie equation we find that a basis of generating functions of the Liealgebra symmetries gives the following functions (see [21] ):

f1 = up1, f2 = up2, f3 = u+ 2p1q1 − p2q2, f4 = u+ 2p2q2 − p1q1,

f5 = p2q1, f6 = p1q2, f7 = q1, f8 = q2.

The corresponding contact vector fields are:

Y1 = −u∂

∂q1+ p2

1∂

∂p1+ p1p2

∂

∂p2,

Y2 = −u∂

∂q2+ p1p2

∂

∂p1+ p2

2∂

∂p2,

Y3 = −2q1∂

∂q1+ q2

∂

∂q2+ u

∂

∂u+ 3p1

∂

∂p1,

Y4 = q1∂

∂q1− 2q2

∂

∂p2+ u

∂

∂u+ 3p2

∂

∂p2,

Y5 = −q1∂

∂q2+ p2

∂

∂p1,

Y6 = −q2∂

∂q1+ p1

∂

∂p1,

Y7 = q1∂

∂u+ ∂

∂p1,

Y8 = q2∂

∂u+ ∂

∂p2.

The commutator table is shown on Table 12.1.


Table 12.1. Commutator table

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8

Y1 0 0 3Y1 0 Y2 02Y3 + Y4

30

Y2 0 0 0 3Y2 0 Y1 Y5Y3 + 2Y4

3Y3 −3Y1 0 0 0 3Y5 −3Y6 3Y7 0

Y4 0 −3Y2 0 0 −3Y5 3Y6 0 3Y8

Y5 Y2 0 −3Y5 3Y5 0Y3 − Y4

30 Y7

Y6 0 −Y1 3Y6 −3Y6Y4 − Y3

30 Y8 0

Y7 − 2Y3 + Y4

3−Y5 −Y7 0 0 −Y8 0 0

Y8 0 −Y3 + 2Y4

30 −3Y8 −Y7 0 0 0

Note that the Lie algebra Sym(Eω) in isomorphic to the Lie algebra sl(3, R)by the mapping π : Sym(Eω)→sl(3, R), where

Y1 −→ −E31, Y5 −→ E23, Y2 −→ E21, Y6 −→ E32,

Y3 −→ E11 + E22 − 2E33,

Y7 −→ E13, Y4 −→ E11 − 2E22 + E33, Y8 −→ −E12,

here Eij are the 3×3 elementary matrices with the only non-zero element beingequal to 1 at (i, j).

12.3 Reduction

Let Xf be an infinitesimal symmetry of a Monge–Ampère equation Eω and letAt : J1M → J1M be the corresponding one-parameter group of contact trans-formations. It follows from the definition that At(L) is a multivalued solutionof Eω whenever it is the integral manifold L ⊂ J1M.

Indeed, Xf ∈ Sym(Eω) means that

A∗t (ω) = htω + In

and therefore

ω|At(L) = A∗t (ω)|L = 0

if ω|L = 0.

12.3 Reduction 257

Therefore, from a known solution L and a symmetry Xf one can construct aone-parameter family of solutions Lt = At(L).

Consider the special case when Lt = L for all t; that is, the case when theintegral manifold L is invariant with respect to the contact vector field Xf . Suchsolutions are said to be f -automodel solutions.

Note that the function f vanishes on the f -automodel solutions. Indeed, bydefinition of a multivalued solution, the restriction of the Cartan form to Lequals zero: ω0|L = 0. But ω0(Xf ) = f and the vector field Xf is tangent to L,hence f |L = 0.

So, any f -automodel solution of the Monge–Ampère equation is also asolution of the first-order partial differential equation f−1(0) ⊂ J1M.

Now let us consider a set of contact symmetries Xf1 , . . . , Xfk instead of onesymmetry Xf . Suppose that these vector fields are linear independent at eachpoint of J1M and that they span a Lie algebra

gdef= 〈Xf1 , . . . , Xfk 〉.

In other words, assume that

[Xfi , Xfj ] =k∑

l=1

clijXfl ,

where clij are the structure constants of the Lie algebra.

First of all, let us note that these vector fields are tangent to the submanifold

Egdef= { f1 = 0, . . . , fk = 0} ⊂ J1M.

Indeed, by the properties of Lagrange brackets, we obtain

[fi, fj] =k∑

l=1

clij fl = Xfi(fj)− X1(fi)fj.

This implies that the functions

Xfi(fj) =k∑

l=1

clij fl + X1(fi)fi

vanish on the submanifold Eg.We call solutions L of Eω g-invariant if they are invariant with respect to Lie

algebra symmetries g.


The same reasons as above show that L ⊂ Eg, or that any g-invariant solutionis a solution of the following system of first-order partial differential equations:f1 = 0, . . . , fk = 0.

To find g-invariant solutions we consider the following construction.Assume that the space of g-orbits on Eg form a smooth manifold Bg, and let

π : Eg → Bg be the natural projection. Contact vector fields Xf ∈ g are tangentto fibres of the projection and belong to the restriction of the Cartan distributionC on Eg. Therefore, one could define a projection Cg of the restriction of theCartan distribution on Bg as follows:

Cg([a]) = π∗(Ca ∩ TaEg)

where [a] is the orbit of a ∈ Eg.Let

ωg = ιf1 ◦ · · · ◦ ιfk (ω),

and consider the restrictions θb of ωg on Cb ∩ TbEg for b ∈ [a]. If Xf ∈ g andAt is the corresponding flow then A∗t (θb) is proportional to θA−t(b). Thereforeωg determines a one-dimensional subbundle lg in the bundle of (n− k)-formson Cg.

Note also that Cg is a contact distribution on Bg. Indeed, if L′ ⊂ Bg is ans-dimensional integral manifold of Cg then by construction π−1(L′) ⊂ Eg is anintegral manifold of the Cartan distribution, and s+ k ≤ n where n = dim M.

Therefore, the maximum dimension of the integral manifolds is n − k, anddim Bg = 2(n− k)+ 1, dim Cg = 2(n− k).

Let us now define a Monge–Ampère differential equation on a contact mani-fold (N2r+1, θ) as a one-dimensional subbundle l of the bundle effective r-formson the distribution ker θ , and call a solution of the equation any r-dimensionalintegral manifold L such that l|L = 0.

Returning to the above construction we shall call lg the reduction of theMonge–Ampère equation Eω by the contact symmetry algebra g.

Summarizing we obtain the following.

Theorem 12.3.1 Let g be an algebra Lie of contact symmetries, g ⊂Sym(Eω),such that the orbit space Bg = Eg/g is smooth. Then (Bg, Cg) is a contactmanifold and ωg determines a Monge–Ampère equation lg in such a way thatprojection of any g-invariant solution L of Eω with smooth orbit space L/g is asolution of lg, and the preimage π−1(L′) of any solution L′ of lg is a g-invariantsolution of Eω.

12.4 Examples 259

We now consider the special case of the construction. Let us assume that thereis a smooth submanifold ⊂ Eg such that each g-orbit of g on Eg intersects at only one point and that is transversal to the orbits.

Then the restriction of the Cartan distribution on gives us a contactdistribution C , and the restriction of the form ωg on gives the reducedMonge–Ampère equation for g-invariant solutions of Eω.

As we have seen, for any symmetry Xf and a multivalued solution L of adifferential equation the one-parameter family Lt = At(L) consists of solutionsof the equation. One could use this observation to reproduce new solutions fromknown ones.

12.4 Examples

12.4.1 The boundary layer equation

The differential equation which describes the flow in a boundary layer inaerodynamics (see [64]) has the form:

vxx = vvt . (12.16)

With the notation q1 = t, q2 = x the corresponding effective differential 2-formis as follows:

ω = up1 dq1 ∧ dq2 − dq1 ∧ dp2.

Since the coefficients of this form do not depend on q1 and q2, it has symmetriesof the form

f = a1p1 + a2p2.

The corresponding f -invariant solution takes the form of the traveling wave:

v = y(a2t − a1x), (12.17)

for some function y.By substituting this expression in (12.16), we obtain the following equation:

a21y′′ = a2yy′, (12.18)

which is the reduced Monge–Ampère equation for given translation symmetry.


Figure 12.3. The traveling wave solution of the boundary layer equation.

If a1 �= 0 and a2 �= 0, then the general solution of this ordinary equation is

y(τ ) = C1a21

√2

a2tan

(C1√

2(τ + C2)

),

where C1 and C2 are constants.The traveling wave solution is

v(t, x) = C1b2√

2

atan

(C1√

2(at + bx + C2)

). (12.19)

(see Figure 12.3).Besides translations the boundary layer equation also has scale symmetry

with the generation function

f = q2p2 + 2q1p1

which corresponds to the following scale transformations:

u −→ u, q1 −→ α2q1, q2 −→ αq2.

The corresponding equation for f -invariant solutions is

2q1∂v

∂q1+ q2

∂v

∂q2= 0.

12.4 Examples 261

The solutions of the last equation can be represented in the form v(q1, q2) =y(τ ), where τ = q1/q2

2, and y satisfies the following ordinary differentialequation:

4τ 2y′′ − yy′ + 6τy′ = 0.

The last equation is the reduced Monge–Ampère equation for a given scalesymmetry.

One could also use the symmetry f = q2p2 + 2q1p1 to obtain new solutionsfrom (12.19).

The flow for Xf consists of point transformations

Aτ : (q1, q2, u) → (q1 exp(−2τ), q2 exp(−τ), u).

Therefore if v = w(q1, q2) is a solution, then the functions

vτ = w(exp(−2τ), q2 exp(−τ))

are also solutions. In particular, the functions

vτ (t, x) = C1b2√

2

a

(tan

(C1√

2(a exp(−2τ)t + b exp(−τ)x + C2)

))are solutions.

12.4.2 The thermal conductivity equation

The non-linear thermal conductivity equation has the form

vt = (v−4/3vx)x + γ v, (12.20)

where α �= 0 and γ are constants [112, 99], q1 = t and q2 = x.Later on we shall investigate this equation in more detail, and now we consider

the case of symmetry with a generating function

f = q22p2 + 3q2u.

To find f -invariant solutions we obtain the system of two partial differentialequations:

vq1 = (v−4/3vq2)q2 + γ v,

q2vq2 = −3v.


Hence, f -invariant solutions have the form

v(q1, q2) = cexp(γ q1)

q32

where c is a constant.

12.4.3 The Petrovsky–Kolmogorov–Piskunov equation

This equation [48]

vt = Dvxx + F(v), (12.21)

where F is a smooth function and D �= 0 is a constant, can be defined by thefollowing effective 2-form ω:

ω = (F(u)− p1) dq1 ∧ dq2 + D dq1 ∧ dp2,

where q1 = t and q2 = x.This equation is invariant with respect to translations in x and t, and one can

consider the traveling wave solutions in the form

v = y(at + bx).

The reduced equation is as follows

y′′ = sy′ + rF(y), (12.22)

where y = y(τ ), τ = at + bx, s = a/Db2 and r = −1/Db2.Let us choose s = 1, i.e., a = Db2.The last equation has been considered in Part I. Using the results, we can

construct f -invariant solutions of (12.21).We consider here the case of (12.21):

F(v) = kv(1− v).

This corresponds to the so-called Fisher equation (see [87]) .If we take a = 6

5 and b = √5k/6D, then the reduced equation has the form

y′′ = y′ + 6

25y2 − 6

25y. (12.23)

12.4 Examples 263

Figure 12.4. The f -invariant solutions of the Fisher equation.

The last equation itself has symmetries with generation functions:

f1 = p and f2 =(

p− 2

5

)exp(−τ

5

).

Using these symmetries we find a first integral:

exp

(−6

5τ

)(25p2 + 4y2 − 20py− 4y3) = C1.

Therefore

p = 2

5y± 1

5

√(C1 exp

(6

5τ

)+ 4y3

).

If C1 = 0, then we obtain the equation

y′ − 2

5y(1±√y) = 0

and solutions

y(x) =(

C exp(−τ

5

)± 1)−2

where C > 0 is a constant (see Figure 12.4).The corresponding f -invariant solution of (12.21) is

v(t, x) =(

C exp

(− 6t

25− 1

5

√5k

6Dx

)± 1

)−2

.


Table 12.2. Generators of the infinite-dimensional Lie algebra Sym (Eω) andtheir action.

Symmetry The action At on (q, u, p)

f1 = q2p2 + q3p3 + 2u (q1, e−tq2, e−tq3, e2tu, e2tp1, e3tp2, e3tp3)

f2 = 2q1p1 + 3(p2q2 + p3q3) (e−2tq1, e−3tq2, e−3tq3, u, e2tp1, e3tp2, e3tp3)

f3 = q3p2 − q2p3(q1, q2 cos t − q3 sin t, q2 sin t + q3 cos t, u, p1,

p2 cos t − p3 sin t, p2 sin t + p3 cos t)f4 = p1 (q1 − t, q2, q3, u, p1, p2, p3)

f5 = p2 (q1, q2 − t, q3, u, p1, p2, p3)

f6 = p3 (q1, q2, q3 − t, u, p1, p2, p3)

f7 = H (q1, q2, q3, u+ tH, p1, p2 + tHq2 , p3 + tHq3)

12.4.4 The Von Karman equation

This equation describes a behavior of the velocity potential v in the tran-sonic approximation of gas motion. It also defines a spreading of rays in theneighborhood of a caustic in the homogeneous atmosphere [44]:

vxvxx − vyy − vzz = 0. (12.24)

The corresponding effective differential 3-form is

ω = p1 dp1 ∧ dq2 ∧ dq3 + dp2 ∧ dq1 ∧ dq3 + dp3 ∧ dq2 ∧ dq1,

where q1 = x, q2 = y and q3 = z.Table 12.2 contains generators of the infinite-dimensional Lie algebra

Sym(Eω) [113] and their action.In the table H = H(q2, q3) is a harmonic function:

∂2H

∂q22

+ ∂2H

∂q23

= 0.

If we restrict ourselves to contact symmetries that preserve the classof solutions vanishing at infinity then one obtains a six-dimensional Liesubalgebra

g6 = 〈f1, . . . , f6〉.

12.4 Examples 265

Table 12.3. Table of commutators for g6.

f1 f2 f3 f4 f5 f6

f1 0 0 0 0 −f5 −f6f2 0 0 0 −2f4 −3f5 −3f6f3 0 0 0 0 f6 −f5f4 0 2f4 0 0 0 0f5 f5 3f5 −f6 0 0 0f6 f6 3f6 f5 0 0 0

The structure of this algebra can be seen from the commutators given inTable 12.3.

Selecting a two-dimensional subalgebra g2 ⊂ Sym(Eω), we shall reducethe problem of finding the g-invariant solutions to integration of an ordinarydifferential equation.

Let us consider, for example, the subalgebra g2 generated by the spiralsymmetry f4 + γ f3, γ ∈ R and the scale symmetry f5.

Thus,

g2=〈f1, f4 + γ f3〉.

To find the g2-invariant solutions we should integrate the following system ofdifferential equations:

vq1 vq1q1 − vq2q2 − vq3q3 = 0

vq1 + γ (q3vq2 − q2vq3) = 0

q2vq2 + q3vq3 + 2v = 0.

The last two equations give

v = (q22 + q2

3)−1g(η),

where

η = q1 − 1

γarctan

q2

q3,

and the reduced equation takes the following form:

(γ 2g′2 − 1)g′′ − 4γ 2g = 0.


Figure 12.5. The solution of von Karman equation at t = −1, 0 and 1.

or

1

2(g′2)′ = 2γ 2(g2)′

(γ 2g′2 − 1).

Then

(γ 2g′2 − 1)′ = 4γ 4(g2)′

and

g′2 − 4γ 2g2 = c1

for some constant c.This equation has two families of solutions

g−(τ ) = − c1

2γexp(2γ (τ − c2))+ 1

8γexp(−2γ (τ − c2))

and

g+(τ ) = − c1

2γexp(−2γ (τ + c2))+ 1

8γexp(2γ (τ + c2)).

Finally, g2-invariant solutions take one of the following two forms:

v+(q1, q2, q3) = − c1

2γ (q22 + q2

3)exp(−2γ (η + c2))

+ 1

8γ (q22 + q2

3)exp(2γ (η + c2)),

v−(q1, q2, q3) = − c1

2γ (q22 + q2

3)exp(2γ (η − c2))

+ 1

8γ (q22 + q2

3)exp(−2γ (η − c2)).

12.5 Symmetries of the reduction 267

12.5 Symmetries of the reduction

As we have seen, the reduction of Monge–Ampère equations by an effective andcontact action of a Lie algebra g decreases the number of independent variablesof a Monge–Ampère equation on dim g.

In particular, if for an equation with n independent variables we know an (n−1)-dimensional subalgebra with effective contact action, then we can reduce thisequation to the ordinary one. To solve the last equation we also need symmetries.

This leads us to finding subalgebras g ⊂ Sym(Eω) such that the reduction lghas symmetries inherited from Sym(Eω).

For example, it is easy to see that the function f = p2q1 − q2 is a generat-ing function of a symmetry for the Titeica–Morimoto equation (12.5). To findf -invariant solutions one should solve the system

det Hess v = (v− q1vq1 − q2vq2)4,

q1vq2 − v = 0.

From the second equation we get

v(q1, q2) = q22

2q1+ w(q1),

and the reduced equation takes the form

w′′ − q1w4 + 4w3w′q21 − 6w2(w′)2q3

1 + 4w(w′)3q41 − (w′)4q5

1 = 0.

To solve this equation we need two symmetries.In order to find them we should know which elements of Sym(Eω) = sl3(R)

give us symmetries of the reduced equation.To this end let us consider the reduction π : Eg → Bg and let Xf ∈ Sym(Eω)

be a contact symmetry. If we expect that this symmetry produces a sym-metry for the reduction we should assume first of all that Xf preserves thebundle π . It certainly happens when [f , g] ⊂ g, that is, when f belongs to thenormalizer

Ng = {f ∈ Sym(Eω), [f , g] ⊂ g}

of the subalgebra g into the algebra Sym(Eω).


Indeed, in this case the flow At of Xf transforms g-orbits to g-orbits, and therelation

Xf (g) = [f , g] + X1(f )g = 0 on Eg

for g ∈ g shows that the vector field Xf is tangent to Eg.Thus the flow At preserves the restriction of the Cartan distribution on

Eg. Denote by At : Bg → Bg the induced transformations, and by Xf thecorresponding vector field, that is,

π∗ : Xf −→ Xf .

Note that transformations At preserve the distribution Cg and therefore Xf isa contact vector field on the contact manifold (Bg, Cg).

Moreover, the formula

Lf ◦ ιg = ιg ◦ Lf + ι[f ,g]

shows that the effective part of Lf (ωg) is proportional to ωg and therefore At

preserves the bundle lg.Summarizing we find the following.

Theorem 12.5.1 The projections Xf = π∗(Xf ) of contact vector fields Xf ,f ∈ Ng, are contact symmetries of the reduced Monge–Ampère equation lg.

Corollary 12.5.1 π∗ : Ng/g→ Sym lg is a Lie algebra monomorphism.

Let us continue with the above example for the TM equation (see [21] ). Onecan easily check that functions f7 and 2f3+ f4 belong to Nf5−f8 , and f = f5− f8is the symmetry we discussed above.

We identify Bg with a hypersurface :

= {p2 = 0, q2 = 0}.

Then the restriction of the forms ω0 and ιf ω on are

ω′0 = du− p1 dq1

and

ω′ = −q1(u− p1q1)4 dq1 + dp1.

12.5 Symmetries of the reduction 269

Figure 12.6. The f -invariant solution of the TM equation.

Denote q1 by x and p1 by p, then

ω′0 = du− p dx,

ω′ = dp− x(u− px)4 dx.

The corresponding vectors fields Xf7 and 13 X2f1+f4 are

1

3X2f1+f4 = −x

∂

∂x+ u

∂

∂u+ 2p

∂

∂p,

Xf7 = x∂

∂u+ ∂

∂p.

The distribution 〈ω′0,ω′〉 corresponds to the reduction of the TM equation. Thisdistribution is a completely integrable distribution with two shuffle symmetries,and therefore can be integrated.

The corresponding f -invariant solutions have the following form:

v(q1, q2) = q22

2q1+ (q3

1 + a3)2/3

a3− q3

1

2a4 2F1

(2

3,

1

3,

5

3,−q3

1

a3

)+ bq2

where 2F1 is the hypergeometric function and a, b are constants (seeFigure 12.6).


12.6 Monge–Ampère equations in symplectic geometry

Let us begin with the following observation. If coefficients of theMonge–Ampère equation Eω do not depend on the variable v, then coef-ficients of the effective form ω do not depend on u and therefore X1 ∈Sym(Eω). This allows us to use the symplectic manifold T∗M instead of thecontact J1M.

Namely, in this case

ι1(ω) = 0 and L1(ω) = 0

and therefore

ω = π∗(ω′)

for some formω′ ∈ �n(T∗M). Here π : J1M → T∗M is the natural projection.The form ω′ is clearly effective, i.e., ⊥ (ω′) = 0, where

⊥: �s(T∗M)→ �s−2(T∗M)

is the operator of interior multiplication by the bivector field X� dual to thestructure form � = −dρ on T∗M.

Let us denote the C∞(T∗M)-modules of effective s-forms on T∗M by�sε(T

∗M).The projection π(L) of any multivalued solution L of Eω is an embedded

Lagrange manifold such that the restriction ω′|π(L) = 0. Conversely, if L′ isthe Lagrange manifold such that ω′|L′ = 0, then there is an integral manifoldL ⊂ J1M, such that L′ = π(L) and L′ is a multivalued solution of Eω.

For this reason Lagrange manifolds L′ ⊂ T∗M such that ω′|L′ = 0; we alsocall these multivalued solutions.

Example 12.6.1 The effective form

ω′ = p1 dq2 ∧ dp1 + dq1 ∧ dp2,

on T∗M corresponds to the Von Karman equation on a plane. The followingtwo-parameter family of Lagrangian manifolds with constants c1 and c2

L′c :

q1 = 1

2p2

1 − c1,

q2 = p2 − c2

(12.25)

satisfies the relation ω′|L′c = 0.

12.6 Monge–Ampère equations in symplectic geometry 271

To find Lc we note that ω0|Lc = 0 and

du = p1 dq1 + p2 dq2 = π∗(ρ) (12.26)

on Lc. Let γ be some path on L′c, then the integral∫γρ depends on the ends of

γ because �|L′c = 0. If we choose p1 and p2 to be coordinates on L′c. Then

u(a, b) =∫γ

ρ =∫ a

c1

p21 dp1 +

∫ b

c2

p2 dp2 = 1

3a3 + 1

2b2 + c3.

Finally, we obtain a three-parameter family

u = 1

3p3

1 +1

2p2

2 + c3,

q1 = 1

2p2

1 − c1,

q2 = p2 − c2

of solutions of the Von Karman equation. The projection of the family into J0R2

gives the following family of two-valued solutions:

u(q1, q2) = ±1

3(2q1 + c1)

3/2 + 1

2(q2 + c2)

2 + c3.

Proposition 12.6.1 Let functions f0, f1 ∈ C∞(J1M) be such that f0(a) =f1(a) �= 0. Then there exists a local contact diffeomorphism ϕ such thatϕ∗(Xf0) = Xf1 and ϕ(a) = a.

Proof We use the path lifting method. Without loss of generality we canassume that Xf1,a = Xf2,a. Consider the path

ft = (1− t)f0 + tf1.

Let us show that there exists a path of contact diffeomorphisms ϕt , such thatϕ0 = id, and

ϕt∗(Xft ) = Xf0 .

Let Yt be the path of vector fields which corresponds to ϕt . Since ϕt is contactwe have Yt = XHt for some path Ht of functions.

Differentiating in t, we obtain

d

dtϕt∗(Xft ) = 0 (12.27)


or

[Xft , XHt ] + Xft= 0,

where ft = dft/dt = f1 − f0.In terms of generating functions this equation takes the form

[ ft , Ht] +·f t = 0

or

Xft (Ht)− X1(ft)Ht + f1 − f0 = 0. (12.28)

Since the vectors Xf0,a = Xf1,a �= 0, the vector field

Xft = (1− t)Xf0 + tXf1

is regular in some neighborhood of the point a, for all t ∈ [0, 1].Therefore (12.28) has a smooth solution Ht such that Ht(a) = 0 and

daHt = 0. �

Corollary 12.6.1 Let f be a contact symmetry of the equation Eω such thatf (a) �= 0. Then there exists a local contact diffeomorphism ϕ such ϕ(a) = a,and the equation Eϕ∗(ω) has the contact symmetry X1.

In other words, for such a type of equation one could use the symplecticgeometry on T∗M instead of the contact one on J1M.

13

Conservation laws

13.1 Definition and examples

We say that a differential (n− 1)-form θ on J1M is a conservation law for theMonge–Ampère equation Eω, if

dθ = gωmod In (13.1)

for some smooth function g.In this case the restriction θ |L to an arbitrary multivalued solution L of Eω is

a closed form:

dθ |L = 0. (13.2)

We denote by CL(ω) the space of all conservation laws for theMonge–Ampère equation Eω.

Example 13.1.1 Let us consider a case when n = 1. Then a conservation lawis a 0-form, or a first integral. Indeed, the second-order differential equation

v′′ = F(x, v, v′) (13.3)

can be represented by the following effective differential form:

ω = dp− F(x, u, p) dx.

Let H ∈ C∞(J1M) be a conservation law, and let L be a solution. Then thevector field

D = ∂

∂x+ p

∂

∂u+ F

∂

∂p

is tangent to L.

273

274 Conservation laws

The definition of a conservation law means that

dH = 0 mod〈ω0,ω〉

or

D(H) = Hx + pHu + FHp = 0. (13.4)

Example 13.1.2 Consider the model equation of one-dimensional gas dynam-ics [111, 70]:

∂ρ

∂t+ c(ρ)

∂ρ

∂x= 0. (13.5)

The following effective 2-form

ω = ( p1 + c(u)p2) dq1 ∧ dq2.

corresponds to this equation, q1 = t, q2 = x, u = ρ. Then the differential1-form

θ = u dq2 − C(u) dq1 (13.6)

is a conservation low of (13.5). Here C′(u) = c(u).

Example 13.1.3 A divergent-type Monge–Ampère equation can be consideredas a differential equation defined by a conservation law.

13.2 Calculus for conservation laws

Here we describe a method for computation of conservation laws.Let θ be a conservation law. Then dθ − gω ∈ In for some function g ∈

C∞(J1M). In other words, the operator �gω is of divergent type, and therefore

E(gω) = 0.

Conversely, if E(gω) = 0, then from Corollary 11.4.1 if follows that the n-form

gω − ω0∧ ⊥ (dp(gω))

is closed.

13.2 Calculus for conservation laws 275

Therefore, if there are no topological obstructions (for example, ifHn(M, R) = 0), then the form gω − ω0∧ ⊥ dp(gω) is exact and defines aconservation law θ :

dθ = gω − ω0∧ ⊥ dp(gω).

If M is a compact orientable manifold then the condition for the n-formgω − ω0∧ ⊥ dp(gω) to be exact, is equivalent to the fact that∫

M(gω)|j1(h) = 0

for some smooth function h ∈ C∞(J1M).Note that this condition is always satisfied if there exists at least one smooth

solution of the Monge–Ampère equation.Thus, we obtain the following theorem.

Theorem 13.2.1 The conservation laws for the Monge–Ampère equation Eω

are in one-to-one correspondence with solutions of the equation

E(gω) = 0 (13.7)

if either Hn(M, R) = 0 or if M is a compact orientable manifold and thereexists at least one smooth solution of the equation Eω.

To each function g ∈ C∞(J1M) such that E(gω) = 0 there is a correspondingconservation law θg that satisfies the relation

dθg = gω − ω0∧ ⊥ (dp(gω)).

The function g is called a generating function of the conservation law θg.

Corollary 13.2.1 If the operator �ω is of divergent type then the generatingfunctions of conservations laws satisfy the following equation:

dp(ιgω)+ dpg∧ ⊥ (dpω)+ X1(g)ω = 0.

Now we give an explicit formula for conservation laws.Let us fix a solution h0 ∈ C∞(M) of the Monge–Ampère equation,

�ω(h0) = 0, and let u ∈ C∞(J1M) be the function such that u([h]1a) = h(a).Denote by At : J1M → J1M the flow along the contact vector field Xu−h0 .Note that

At : (q, u, p) −→(

q, (u− h0)et + h0,

(p− ∂h0

dq

)et + ∂h0

∂q

).


Let now θg be a conservation law with a generating function g.Then from

d

dt(A∗t (θg)) = A∗t (Lu−h0(θg))

and

Lu−h0(θg) = ιu−h0 dγg + d(ιu−h0 θg)

= gιu−h0ω − (u− h0) ⊥ (dp(gω))

+ ω0 ∧ ιu−h0 (⊥ (dp(gω)))+ d(ιu−h0 θg),

we obtain

θg − A∗−∞(θg) =∫ 0

−∞d

dtA∗t (θg) dt

=∫ 0

−∞A∗t (gιu−h0ω − (u− h0) ⊥ dp(gω)) dt

+ ω0 ∧∫ 0

−∞etA∗t (ιu−h0(⊥ (dp(gω))) dt

+ d

(∫ 0

−∞A∗t (ιu−h0 ω) dt

).

Put

θg =∫ 0

−∞A∗t [gιu−h0ω − (u− h0) ⊥ dp(gω)] dt.

Then the forms θg and θg determine the same conservation law.

Theorem 13.2.2 Let h0 be a smooth solution of a Monge–Ampère equationEω, and let g be a generating function of a conservation low, E(gω) = 0. Theconservation low corresponding to g is given by the formula

θg =∫ 0

−∞A∗t (gιu−h0ω − (u− h0) ⊥ dp(gω)) dt. (13.8)

Example 13.2.1 (The SG equation) The Sine-Gordon equation

vxy = sin v

13.2 Calculus for conservation laws 277

can be defined by the effective 2-form ω = ωSG

ω = 1

2dq1 ∧ dp1 − 1

2dq2 ∧ dp2 − sin u dq1 ∧ dq2

where q1 = x, q2 = y.Let us find the generating functions g for conservation laws.In our case

dω = − cos u du ∧ dq1 ∧ dq2

and dpω = 0. The action of the Euler operator on gω can be written as

E(gω) = gL1(ω)+ dp(ιgω)+ L1(g)ω. (13.9)

Computing the last three terms in this last formula, we obtain

gL1(ω) =− g cos u dq1 ∧ dq2,

dp(ιgω) = dp −(

1

2(gq1 + p1gu)+ gp2 sin u

)dq1

+(

1

2(gq2 + p2gu)+ gp2 sin u

)dq2 − 1

2gp1 dp1 + 1

2gp2 dp2,

L1(g)ω = guω.

Then straightforward calculations give the following expression for g:

g = (aq1 + b)p1 + (− aq2 + c)p2,

where a, b and c are constants.After substituting it in the formula

dθg = gω − ω0 ∧ (ιgω)

and taking the integral (13.8), we find conservation laws:

θg =(

1

4ugq1 − gp2 cos u− 1

2p1p2gp2

)dq1

+(−1

4ugq2 + gp1 cos u+ 1

2p1p2gp1

)dq2

+ 1

4( p2gp2 − p1gp1) du+ 1

4ugp1 dp1 − 1

4ugp2 dp2.


Thus, the space of conservation laws of the Sine-Gordon equation is three-dimensional. The following conservation laws

θ1 =(−1

4uq1 − uq2 − cos u− 1

4sin u

)dq1 +

(−1

4uuq2q2 +

1

4u2

q2

)dq2,

θ2 =(

1

4uq1 + uq2 + cos u+ 1

4sin u

)dq2 +

(1

4uuq1q1 −

1

4u2

q1

)dq1,

θ3 =(

1

4uuq1 + q2 cos u+ 1

4q2uq1 uq2+

1

4uq1uq1q1+

1

4uq2 sin u−1

4q1u2

q1

)dq1

+(

1

4uuq2 + q1 cos u+ 1

4q1uq1 uq2+

1

4uq2uq2q2+

1

4uq1 sin u−1

4q2u2

q2

)dq2

give a basis in CL(ωSG) (see [69]).

Example 13.2.2 (The Von Karman equation) Generating functions for con-servation laws of the Von Karman equation in the three-dimensional space havethe following form:

g = g0(q1, q2)+ c1p1 + c2p2 + c3p3 + c4(q3p2 − q2p3)

where g0 (q1, q2) is an arbitrary harmonic function and c1, c2, c3, c4 areconstants. The corresponding to the function g conservation law θg is

θg = f1 dq2 ∧ dq3 − f2 dq1 ∧ dq3 + f3 dq1 ∧ dq2,

where

f1 =1

3c1p3

1 +1

2(c2p2 + c3p3)+ 1

2c4p2

1(q3p2 − q2p3)+ 1

2c1(p

22 + p2

3),

f2 =1

6( p3

1 + ( p22 − p2

3))(c2 + c4q3)+ p2p3(c3 − c4q2)+ c1p1p2

+ p2g0 + q1p1∂g0

∂q2,

f3 =(

1

6p3

1 + ( p22 − p2

3)

)(c3 − c4q3)− p2p3(c2 + c4q2)− c1p1p2

− p3g0 − q1p1∂g0

∂q3.

13.4 Symmetries and conservations laws 279

13.3 Symmetries and conservations laws

The famous Noether theorem establishes the relationship between symmetriesand conservation lows in the calculus of variations. In this section we show thesimilar relations for the divergent-type Monge–Ampère equations.

Theorem 13.3.1 Let ω be an effective n-form of divergent type. Then for anycontact symmetry Xf ∈ Sym(Eω) the following differential (n− 1)-form

θ = ιf ω − f ⊥ (dpω)

is a conservation law.

Proof By definition of symmetry, we have

( p ◦ Lf )ω = hf ω,

for some function hf ∈ C∞(J1M). Using the formula

( p ◦ Lf )ω = f E(ω)+ dp(ιf ω − f ⊥ (dpω))+ dω0 ∧ (ιf ⊥ (dpω))

and the condition E(ω) = 0, we obtain

dp(ιf ω − f ⊥ (dpω)) = hf ω + dω0 ∧ (ιf ⊥ (dpω)) = hf ωmod In. �

Theorem 13.3.2 Let ω be an effective n-form of divergent type such thatdpω = 0 and p(L1(ω)) = hω for some function h ∈ C∞(J1M). Then forany conservation law θg with the generating function g the contact vector fieldXg is a symmetry of the equation Eω.

Proof We have

E(gω) = gE(ω)+ X1(g)ω + dp(ιgω) = 0.

Hence,

dp(ιgω) = −X1(g)ω

and

p(LXg(ω)) = dp(ιgω)+ gLX1(ω) = (−X1(g)+ h)ω. �


13.4 Shock waves and the Hugoniot–Rankine condition

13.4.1 Shock Waves for ODEs

In this section we consider discontinuous solutions of Monge–Ampèreequations on R.

Proposition 13.4.1 Let Eω = {y′′ = F(x, y, y′)} be an ordinary differentialequation. A smooth function H ∈ C∞(J1R) is a first integral for Eω if andonly if

∫R

j∗1(h)(H) ds = 0, (13.10)

for any test function s ∈ C∞0 (R) and any smooth solution h ∈ C∞(R) of Eω.

Proof Indeed (13.10) gives

∫R

j∗1(h)(H) ds =∫

R

( j∗1(h)(H))′s dx = 0

for all s ∈ C∞0 (R), and therefore, j∗1(h)(H) is a constant for any solution h. �

This proposition shows us the way to extend a concept of solution forODEs (see Figure 13.1). Let us define a discontinuous solutions of the equa-tion Eω. To this end we fix a point a ∈ R. This point divides R into twoparts: R− = {q| q < a} and R+ = {q| q > a}. Consider a discontinuous func-tion v = v+∪v−, where v+ ∈ C∞(R+) and v− ∈ C∞(R−). Here R± = R\R∓.

�

�−

�+

a q

Figure 13.1. A discontinuous solution of an ODE.

13.4 Shock waves and the Hugoniot–Rankine condition 281

The function v is called a discontinuous solution corresponding to theintegral H if ∫

R

H|j1(v) ds = 0

for any test function s ∈ C∞0 (R).

Theorem 13.4.1 A function v is a discontinuous solution corresponding to anintegral H if and only if �ω(v) = 0 in R\{a} and at the point a the followingcondition

H([v+]1a) = H([v−]1a) (13.11)

holds.

Proof Let supp s be the support of s. If a /∈ supp s, say supp s ⊂ R−, then∫R

j∗1(v−)(H) ds =∫

R

( j∗1(v−)(H))′s dx = 0

and therefore j∗1(v−)(H) = constant.If a ∈ supp s, then∫

R

H|j1(v) ds = H([v+]1a) s(a)− H([v−]1a) s(a)

and therefore v is a solution if and only if H([v+]1a) = H([v−]1a). �

Condition (13.11) is called the Hugoniot–Rankine condition.

13.4.2 Discontinuous solutions

Let us extend the notion of discontinuous solutions for Monge–Ampèreequations in arbitrary dimension.

Suppose that the smooth manifold M is divided into two domains M+ andM− by a submanifold M0, codim M0 = 1:

M = M+ ∪M0 ∪M−.

Consider a discontinuous function v = v+ ∪ v−. We assume that v+ and v−are smooth functions in the closures M+ and M−, v+ ∈ C∞(M+) and v− ∈C∞(M−).


The function v is called a discontinuous solution corresponding to aconservative law θ if ∫

Mds ∧ θ |j1(v) = 0

for any test function s ∈ C∞0 (M).

Let us denote by θ+ def= θ |j1(v+) and θ− def= θ |j1(v−) the form θ evaluated onthe solutions v+ and v− respectively.

Theorem 13.4.2 A function v is a discontinuous solution corresponding to aconservatively low θ if and only if �ω(v) = 0 in the domain M\M0 and atpoints of M0 the Hugoniot–Rankine condition

θ+|M0 = θ−|M0 (13.12)

holds.

Proof If supp s ⊂ M+ then∫M

ds ∧ θ |j1(v) =∫

Md(sθ+)−

∫M

sθ+ = −∫

Msθ+ = 0

for all such test functions, and therefore θ+ = 0.If Ms = supp s ∩M0 �= ∅, then∫

Mds ∧ θ |j1(v) =

∫Ms

s(θ+ − θ−

)∣∣M0= 0

implies the Hugoniot–Rankine condition (see [70]). �

Remark 13.4.1 In the case of submanifold M0 = S−1(0) defined by a func-tion S ∈ C∞(M), where dS �= 0 at the points of M0, the Hugoniot–Rankinecondition is equivalent to the following one:

[θ+|M0 − θ−|M0 ] ∧ dS = 0

at points of M0.

Example 13.4.1 Suppose that the discontinuity boundary (transfer curve) M0

for the model equation (13.5) is given by

M0 = {q2 = ϕ(q1)}.

13.4 Shock waves and the Hugoniot–Rankine condition 283

Then the restriction of θ to M0 is

θ |M0 = u dϕ − C(u) dq1 = (uϕ′ − C(u))dq1

and (13.12) takes the form of the classical Hugoniot–Rankine condition:

(v+ − v−)ϕ′ = C(v+)− C(v−).

13.4.3 Shock waves

The Hugoniot–Rankine condition allows us to describe the rise of shock waves[70] as some kind of symbiosis of geometrical and analytical approaches. Wewill illustrate this on multivalued solutions of (13.5). Suppose that the multi-valued solution describes a turning-over wave. The corresponding multivaluedsolution L is shown in Figure 13.2.

In the hatched region the solution is three-valued and we must select alongthe three branches.

First, we divide the solution at time t = t2 into three parts: I, II and III(see Figure 13.3). These parts can be thought of as an incident, a reflected (orself-reflected) and again an incident (double-reflected) wave respectively.

Consider a transfer curve M0 and lift it to branches I and II; we obtain twocurves, M1 and M3. The Hugoniot–Rankine condition says that θ |M1 and θ |M3 ,viewed as 1-forms on M0, must coincide.

If we unfold surface L the region bounded by M1, M3 and the section q1 = t2becomes a curvilinear triangle C1C3P (see Figure 13.4).

Figure 13.2. The multivalued solution of ∂ρ/∂t + c(ρ)∂ρ/∂x = 0.


Figure 13.3. Formation of a turned-over wave.

Figure 13.4. Unfolding of a multivalued solution.

Next, integrating the form θ along this triangle, we obtain

∫M1

θ +∫

q1=t2θ +

∫M3

θ = 0.

The Hugoniot–Rankine condition gives

∫q1=t2

θ =∫

q1=t2u dq2 = 0

or that areas S1 and S2 (see Figure 13.5) coincide.This is the so-called Maxwell rule, i.e., the areas cut off by the line x = c

are equal: S1 = S2. This rule lets us construct discontinuous solutions (shockwaves) from multivalued ones.

An application of the Hugoniot–Rankine condition to the description ofsingularities in non-linear acoustics will be considered later on.

13.5 Calculus of variations and the Monge–Ampère equation 285

Figure 13.5. The Maxwell rule.

13.5 Calculus of variations and the Monge–Ampère equation

In practice, the most interesting differential equations of the second order ariseas Euler–Lagrange equations of variational problems. Here we give an overviewof the calculus of variations from the point of view of contact geometry.

We consider the class of variational problems associated with the Monge–Ampère equations. More exactly, we consider the functionals associated witheffective n-forms. We will justify the notion of the Euler operator given aboveand we will show that the Euler–Lagrange equations for this class of vari-ational problems are Monge–Ampère equations corresponding to the formsE(ω), whereω is an effective form which defines the initial variational problem.

We also describe the symmetries and conservation laws for such variationalproblems and give a proof of a contact analog of Noether’s theorem.

Finally, we discuss the Euler operator from the cohomology viewpoint. Weuse a filtration in the exterior form algebra of any contact manifold (a “non-holonomic” filtration introduced and studied in [75]) and show that the Euleroperator is the second differential in the spectral sequence based on thisfiltration.

13.5.1 The Euler operator

Below we give a sketch of an invariant exposition of variational calculus, usingthe machinery of effective forms developed in the previous chapters.

We associate to each differential n-form ω ∈ �nε(J

1M) the functional ω,acting as follows:

ω(h) =∫

Mj1(h)

∗(ω),

where h ∈ C∞0 (M).


Note that the Lagrangian of the functional ω, written in the canonical localcoordinates, contains a non-linearity of the same type as the Monge–Ampèreoperators.

The next theorem shows why the operator E was called the Euler operator.

Theorem 13.5.1 Extremals of the functional ω are solutions of the Monge–Ampère equation associated with the n-form E(ω).

Proof Let h0 ∈ C∞0 (M) be an extremal of ω. Fix a test function s ∈ C∞0 (M)

and let

ϕ(t) =∫

Mj1(h0 + ts)∗(ω).

Since h0 is an extremal, ϕ′(0) = 0.On the other hand,

ϕ′(0) =∫

Mj1(h0)

∗Lsω.

It follows from Theorem 11.4.2 that

ϕ′(0) =∫

Mj1(h0)

∗(s E(ω)) =∫

Ms j1(h0)

∗(E(ω)) = 0

for all test functions s, and therefore

j1(h0)∗(E(ω)) = �E(ω)(h0) = 0.

�

13.5.2 Symmetries and conservation laws in variational problems

We call a contact vector field Xf ∈ ct(J1M) a symmetry of the functional ω ifLf (ω) ∈ In.

To give a motivation for this definition, let us suppose that the manifold Mis compact. Let At : J1M → J1M be a one-parameter group of shifts alongthe contact vector field Xf . Then for an arbitrary function h ∈ C∞(M) andsufficiently small t, the integral manifolds Lt = At(L), L = j1(h)(M), canbe projected diffeomorphically on M and, therefore, Lt = j1(ht)(M) for somefamily of smooth functions ht ∈ C∞(M).

Moreover, from the relation Lf (ω) ∈ In it follows that A∗t (ω)− ω ∈ In and

j1(h)(ω) = j1(h)(A∗t ω) = j1(ht)(ω),

ω(ht) = ω(h).

13.5 Calculus of variations and the Monge–Ampère equation 287

Using Theorem 11.4.2 we obtain the following proposition.

Proposition 13.5.1 A contact vector field Xf is a symmetry of the variationalproblem if and only if the generating function f satisfies the following equation:

dpιf ω + ιf dpω + f L1ω = 0.

Theorem 13.5.2 (Noether) Let a contact vector field Xf be a symmetry of thefunctional ω. Then the differential (n − 1)-form ιf ω − f⊥ dpω represents theconservation law for the Euler equation EE(ω).

Proof Using 11.4.2 we get

0 = p(Lf ω) = f E(ω)+ dp(ιf ω − f⊥ dpω)+�(ιf⊥dpω), (13.13)

i.e.,

f E(ω)+ d(ιf ω − f⊥dpω) ∈ In. (13.14)

�

Theorem 13.5.3 Suppose that L1(E(ω)) = 0. Then every conservation lawfor the Euler equation EE(ω) is determined by some contact symmetry of theequation.

Proof Note that the Monge–Ampère operator �E(ω) is of divergent type.Indeed,

E2(ω) = L1E(ω) = 0.

The statement of the theorem follows now from Theorem 13.3.2 and the relationdp ◦ E = 0. �

13.5.3 Classical variational problems

Let ω0 ∈ �n(M) be a volume form. Then any function L ∈ C∞(J1M) definesan effective form ω = L π∗1 (ω0) and the corresponding variational problem.This is the classical case, where L is the Lagrangian.

In this case

E(ω) = dpιL(π∗1ω0)+ X1(L)π

∗1 (ω0).


The correspondence between contact symmetries and conservation laws hasthe form

f −→ θ = kιf (π∗1ω0)− f ιL(π

∗1ω0).

The contact symmetries of the variational problem can be found from thefollowing Lie equation:

dp(Lιf π∗1ω0)+ Xf (L)π

∗1 (ω0) = 0.

The equation for generating functions of conservation laws of the Eulerdifferential equation takes the form

dpιgE(ω)+ L1(gE(ω)) = 0.

13.6 Effective cohomology and the Euler operator

Let M be a contact manifold of dimension 2n+1 and C be the contact distribu-tion. Denote by J the ideal in the algebra of differential forms�∗(M) generatedby the module C0 = Ann(C) ⊂ �1(M), that is,

J = C0 ∧�∗(M).

We consider the following filtration in the algebra of differential forms �∗(M)

associated with the ideal:

Fp,q = C−q0 ∧�p+2q(M), q ≤ 0, p ≥ 0.

This is a decreasing filtration of �∗(M),

Fp,q ⊂ �p+q(M), �k(M) =⋃

p+q=k

Fp,q

and

Fp, q ⊃ Fp+1, q−1,

and this filtration is compatible with the de Rham differential d:

dFp, q ⊂ Fp, q+1.

Consider now a spectral sequence {Ep, qr , dp, q

r : Ep, qr → Ep+r, q−r+1

r } associ-ated with this filtration. This sequence is converged to the de Rham cohomologyof the contact manifold.


The term s of {Ep, qr , dp, q

r } are located in the fourth quadrant:

0 ≤ p+ 2q ≤ 2n+ 1, p ≥ 0, q ≤ 0.

The multiplicative structure in the spectral sequence is inherited from thestructure in the algebra of differential forms.

The E0-term is

Ep, q0 = Fp, q/Fp+1, q−1.

Note that Ep, q0 are non-trivial only for q = 0 and q = −1, because C0 has

dimension 1.In this case

Ep, 00 = Fp, 0

Fp+1,−1= �p(M)

C0 ∧�p−1(M)' �p(C∗)

is the module of p-forms on distribution C, and

Ep,−10 = Fp,−1

Fp+1,−2= C0 ∧�p−2(M) ≈ �p−2(C∗)

where we have used the identification

C∗ = �1(M)/C0.

The complexes dp, q0 : Ep, q

0 → Ep, q+10 are two-term sequences

0 → �p−2(C∗) ≈ C0 ∧�p−2(M)dp,−1

0→ �p(C∗)→ 0

where the differential acts as follows:

dp,−10 : ϑ ∧ ω −→ class(d(ϑ ∧ ω)) = dϑ |C ∧ ω|C

for any ϑ ∈ C0.Note that any local basis 1-form ϑ in C0 defines a symplectic structure

dϑ |C ∈ �2(C∗) on the contact distribution. If ϑ ′ = h ϑ is another basis in C0

then dϑ ′∣∣C = h dϑ |C and therefore any two of these symplectic structures are

proportional.Thus

Ep,−11 = ker dp,−1

0 = ker((� : �p−2(C∗)→ �p(C∗)))


and

Ep,01 = coker dp,−1

0 = coker(� : �p−2(C∗)→ �p(C∗))

where � is the operator defined by one of the symplectic structures.Using the properties of operator � and the Hodge–Lepage decomposition

we get the following description:

Ep,−11 =

{0, if p ≤ n+ 1,

�n−p+2(�p−2ε (C∗)) if p ≥ n+ 2,

and

Ep,01 =

{0, if p ≥ n+ 1,

�pε(C∗) if p ≤ n.

Elements of

coker(� : �p−2(C∗)→ �p(C∗)) = ker(⊥: �p(C∗)→ �p−2(C∗))

are effective p-forms, and the elements of

ker((� : �p(C∗)→ �p+2(C∗)))

we shall call co-effective p-forms.As we have seen co-effective (n + k)-forms are exactly the forms in

�k(�n−kε (C∗)). We denote the module of co-effective p-forms we denote by

�pκ(C∗).Note that we have �p

ε(C∗) = 0 if p ≥ n+ 1 and �pκ(C∗) = 0 if p ≤ n− 1,

in the middle dimension �nκ(C

∗) = �nε(C

∗).The differentials dp, q

1 : Ep, q1 → Ep+1, q

1 give us two complexes for calculationof Ep, q

2 ; the first one for effective forms when q = 0, is

0 → �0ε(C

∗) dε→ �1ε(C

∗) dε→ �0ε(C

∗) dε→ · · · dε→ �nε(C

∗)→ 0

and the second one for co-effective forms when q = −1, is

0 → �nε(C

∗) =�nκ(C

∗) dκ→ �n+1κ (C∗) dκ→ �n+2

κ (C∗) dκ→ · · ·dκ→ �2n

κ (C∗)→ 0.


We have denoted by dε and dκ the corresponding differentials dp,01 and dp,−1

1 .The cohomology of the complexes we call the effective and the co-effectivecohomology of the contact manifold and denote these by Hp

ε (M), 0 ≤ p ≤ n,and Hp

κ(M), n ≤ p ≤ 2n respectively.

Example 13.6.1 If M = J1N is the contact manifold, then Hpκ(M) = 0 when

p ≥ n+ 1.

Let us look at the term Ep, q2 . First of all the above description shows that

Ep,−12 =

{0, if p ≤ n+ 1,

Hp−2κ (M), if p ≥ n+ 2,

and

Ep, 02 =

{0, if p ≥ n+ 1,

Hpε (M), if p ≤ n.

The differential dp, q2 : Ep, q

2 → Ep+2, q−12 is non-trivial only for p = n, q = 0,

and acts as follows:

dn, 02 : Ep, 0

2 = �nε(C

∗)→ �nε(C

∗) = En+2,−12 .

Therefore we obtain

Ep,−13 =

0, if p ≤ n+ 1,

coker dn, 02 , if p = n+ 2,

Hp−2x (M), if p ≥ n+ 3,

and

Ep, 03 =

0, if p ≥ n+ 1,

ker dn, 02 , if p = n,

Hpε (M), if p ≤ n− 1.

The next differentials dp, q3 : Ep, q

3 → Ep+3, q−23 are obviously trivial. Therefore,

Ep, q∞ = Ep, q3 .


Proposition 13.6.1 Let M = J1N, n = dimN, and let �nε(C

∗) → Hnε (M) =

�nε(C

∗)\Im dε be the natural projection and Hnκ(M) = ker dκ → �n

ε(C∗) =

�nκ(C

∗) be the embedding. Then the following diagram commutes.

Λn (C*) �

dn,0

Λn (C*)

Hn (M)Hn (M)2

�

�

�

�

Proof It follows from the original construction of the Euler operator. �

Owing to this proposition we call differential dn,02 the Euler operator on the

contact manifold M and denote it by E .Summing up we arrive at the following result.

Theorem 13.6.1 The spectral sequence (Ep,qr , dp,q

r ) on a contact manifold Mcollapses in the term E3 and has the following description.

1. The term Ep, q0

Ep, q0 =

�p(C∗), if q = 0,

�p−2(C∗), if q = −1,

0, otherwise.

2. The term Ep, q1

Ep, q1 =

�

pε(C∗), if 0 ≤ p ≤ n, q = 0,

�n−p+2(�p−2ε (C∗)), if n+ 2 ≤ p, q = −1,

0, otherwise.

3. The term Ep, q2

Ep, q2 =

Hpε , if 0 ≤ p ≤ n, q = 0,

Hp−2κ (M), if n+ 2 ≤ p, q = −1,

0, otherwise.


4. The term Ep, q3 = Ep, q∞

Ep, q3 =

Hpε (M), if p ≤ n− 1; ker E , if p = n, q = 0,

coker E , if p = n+ 2; Hp−2κ (M), if p ≥ n+ 3, q = −1,

0, otherwise.

5. The de Rham cohomology Hk(M, R) of the contact manifold coincides withthe effective cohomology Hk

ε (M) if k ≤ n − 1, or with the co-effectivecohomology Hk−1

κ (M) if k ≥ n + 3, dim M = 2n + 1. In the middledimensions Hn(M, R) = ker E and Hn+1(M, R) = coker E .

14

Monge–Ampère equations on two-dimensionalmanifolds and geometric structures

In this chapter we consider Monge–Ampère equations on a two-dimensionalmanifold M. In this case Monge–Ampère equations endow the space J1M withstructures that are very important in differential geometry: the non-holonomicalmost product structure, the non-holonomic almost complex structure (a partialcase of the Cauchy–Riemann (CR) structure) and the non-holonomic almosttangent structure. We use this link in order to find the geometrical nature forthe differential equations.

We also discuss some methods for integration of the Cauchy problem forMonge–Ampère equations.

294

14.1 Non-holonomic geometric structures 295

14.1 Non-holonomic geometric structures associated withMonge–Ampère equations

14.1.1 Non-holonomic structures on contact manifolds

Let N be a 2n-dimensional smooth manifold and let A be a smooth field ofendomorphisms on N :

Ax : TxN → TxN

for x ∈ N .Recall that A is called a (classial) almost product structure, an almost complex

structure or an almost tangent structure if A2 = 1,−1 or 0, respectively.Let � be a (2n+ 1)-dimensional contact manifold with a contact structure

P : � � a −→ P(a) ⊂ Ta�.

Suppose that on each subspace P(a) (a ∈ �) of the tangent space Ta� anendomorphism Aa : P(a) → P(a) is given and the map a −→ Aa is smooth.Then we call the field A : � � a −→ Aa ∈ End(P(a)) a non-holonomic fieldof endomorphisms on the contact manifold �.

If for a non-holonomic field of endomorphisms A one has A2 = 1, −1 or0, then the field A is called a non-holonomic almost product structure , anon-holonomic almost complex structure or a non-holonomic almost tangentstructure, respectively.

A non-holonomic almost complex structure is a partial case of so-calledCauchy–Riemann structure or simply CR-structure.

Let A be a non-holonomic almost complex structure. A two-dimensionalintegral submanifold L ↪→ J1M is called a complex curve of the non-holonomicalmost complex structure A if Ax : TxL → TxL for all x ∈ L.

14.1.2 Non-holonomic fields of endomorphisms generated byMonge–Ampère equations

As we have seen any effective differential form ω ∈ �2ε(J

1M) generates thenon-holonomic field of endomorphisms

Aω : J1M � a −→ Aωa ∈ End(Ca), (14.1)

where

X �ω = AωX� �.

for any vector field X ∈ D(C).

296 Monge–Ampère equations on 2D manifolds and geometric structures

Here

�adef= dω0|C(a)

is the standard symplectic structure on C(a) and Aω(X)a = Aω,a(Xa) for alla ∈ J1M.

A function Pf(ω) ∈ C∞(J1M) defined pointwise by the formula

Pf(ω)(a)def= Pf(ωa)

is called a Pfaffian of the form ω.The following properties of the operator field Aω follow from the properties

of the linear operator Aω ,a.

Proposition 14.1.1

1. Aω is symmetric with respect to �, i.e.,

�(AωX , Y) = �(X, AωY)

for any vector field X , Y ∈ D(C).2. Vector fields X , AωX ∈ D(C) are skew-orthogonal, i.e.,

�(AωX , X) = 0.

3. For any multivalued solution L of the equation Eω one has

Aωa : TaL → TaL

for all a ∈ L.4. The square of Aω is scalar:

A2ω + Pf(ω) = 0. (14.2)

5. Let q1, q2, u, p1, p2 be the canonical local coordinates in J1M. The vectorfields

d

dq1

def= ∂

∂q1+ p1

∂

∂u,

d

dq2

def= ∂

∂q2+ p2

∂

∂u,

∂

∂p1,

∂

∂p2(14.3)


form a local basis in the module D(C). Then

Aω =(

Bd

dq1+ C

d

dq2− E

∂

∂p2

)⊗ dq1 +

(B

∂

∂p1+ D

d

dq2−A

∂

∂p2

)⊗ dp1(

−Ad

dq1+ E

∂

∂p1− B

d

dq2

)⊗ dq2

+(−D

d

dq1+ C

∂

∂p1− B

d

dq2

)⊗ dp2

or

Aω =

∥∥∥∥∥∥∥∥B −A 0 −DC −B D 00 E B C

−E 0 −A −B

∥∥∥∥∥∥∥∥ (14.4)

if

ω = A dp1 ∧ dq2 + B(dq1 ∧ dp1 + dp2 ∧ dq2)+ C dq1 ∧ dp2

+ D dp1 ∧ dp2 + E dq1 ∧ dq2

and

Pf(ω) = DE − AC + B2.

We say that a Monge–Ampère equation Eω is non-degenerate at a pointa ∈ J1M if Pf(ω)(a) �= 0.

If Pf(ω)(a) = 0, then an equation Eω is said to be degenerate at the point a.Using classification of effective forms we say that a Monge–Ampère equation

Eω is hyperbolic, elliptic or parabolic at the point a ∈ J1M if Pf(ω)(a) isnegative, positive or zero at the point.

An equation Eω is called a mixed type (or variable type) at a point a if Pf(ω)changes sign at the point.

Example 14.1.1 The Von Karman equation

vxvxx − vyy = 0

has the Pfaffian −vx, and therefore, it is hyperbolic in the domain vx > 0,elliptic in the domain vx < 0 and of mixed type at points of the hypersurfacevx = 0.


Note that a Monge–Ampère equation is determined by an effective form ω

up to multiplication on a non-vanishing smooth function h, i.e., Eω = Ehω.Since the multiplication of ω by h does not change the sign of the Pfaffian:

Pf(hω) = h2 Pf(ω),

we see that the above definitions do not depend on the representative ω.If Pf(ω) �= 0, we can normalize the form ω:

ω → 1√|Pf(ω)|ω.

If |Pf(ω)| = 1, then the form ω is called normed. For a normed form ω itsPfaffian Pf(ω) is equal to −1 for a hyperbolic equation and +1 for an ellipticequation. The operator Aω corresponding to the normed form ω is also callednormed and is denoted by A.

It is clear that A2 = 1 for hyperbolic equations and A2 = −1 for elliptic equa-tions. Therefore, we obtain a non-holonomic almost product structure on themanifold J1M for hyperbolic equations and a non-holonomic almost complexstructure for elliptic equations.

For parabolic equations we have A2ω = 0. This is a non-holonomic almost

tangent structure on the manifold J1M.

14.1.3 Non-degenerate equations

In this section we consider hyperbolic and elliptic equations together. Let Eω

be a non-degenerate Monge–Ampère equation, where ω is a normed effectiveform. Then Pf(ω) = ε and ε = −1 for a hyperbolic equation and ε = 1 forelliptic ones. Moreover A2 = −ε.

Denote by CCa the complexification of the Cartan subspace Ca and by AC

a thecomplexification of the operator Aa. Then CC

a splits into the direct sum

CCa = C+(a)⊕ C−(a),

where

C±(a)def= {X ∈ CC

a | ACa X = ∓ι√εX}

are complex eigensubspaces of the operator ACa . We will call these distributions

characteristic distributions.The subspaces C+(a) and C−(a) are skew-orthogonal and since the com-

plexification �Ca of the form �a is non-degenerate, each of them is a complex


Figure 14.1. A decomposition of TCa (J

1M) into a direct summ.

symplectic plane. So, we obtain two two-dimensional (of complex dimension)complex distributions on J1M:

C± : J1M � a → C±(a) ⊂ TCa (J1M).

Let us consider the derived distributions C(1)+ and C(1)

− . Let X and Y be basicvector fields in the distribution C− or in the distribution C+. From the non-degeneracy of the form �|C± it follows that

ωC0 ([X, Y ]) = X(ωC

0 (Y))− Y(ωC0 (X))− dωC

0 (X, Y) = −dωC0 (X, Y)

= −�C(X , Y) �= 0.

Therefore, the derived distributions C(1)+ and C(1)

− are three-dimensional and ateach point a ∈ J1M the intersection

l(a)def= C(1)

+ (a) ∩ C(1)− (a)

is a complex line. We obtain a one-dimensional complex distribution

l : J1M � a → l(a) ⊂ TCa (J1M).

The complexified tangent space to J1M at any point a splits into the direct sum:

TCa (J1M) = C+(a)⊕ l(a)⊕ C−(a)

(see Figure 14.1). It is clear that in the hyperbolic case the line l(a) is a realline. But in the elliptic case it is also real. Indeed, since operator A is real, the


subspaces C+(a) and C−(a) are complex conjugate: C+(a) = C−(a). Then thesubspaces C(1)

+ (a) and C(1)− (a) are also complex conjugate and its intersection

is complex conjugate to itself: l(a) = l(a). Therefore this complex line isgenerated by a real vector Z: l(a) = CZ , Z ∈ Ta(J1M).

We see that non-degenerate Monge–Ampère operators generate AP-structures on J1M, where n = 5, P1 = C+, P2 = l, P3 = C−.

Note that the non-degenerate Monge–Ampère equation, in contrast to theMonge–Ampère operator, generates an AP-structure (C+, l, C−) up to thechange C+ and C−. Indeed, effective 2-forms ω1 = ω and ω2 = −ω generatethe same equation, but C1− = C2+ and C1+ = C2−. Here Ci± are eigensubspacesof the operator AC

ωi(i = 1, 2).

So, for a non-degenerate Monge–Ampère equation on J1M there exists aninvariant defined distribution l of real lines. The lines of this distribution aretransversal to the Cartan space at each point. We will use this fact for contactclassification of Monge–Ampère equations.

Example 14.1.2 Let us consider the equation

vxx + εvyy = f (x, y, v, vx , vy),

where ε = −1 or ε = 1. In the case where ε = −1 we have the nonlinear waveequation and in the case where ε = 1 we have the nonlinear Laplace equation.

The effective form is

ω = f dq1 ∧ dq2 − ε dq1 ∧ dp2 + dq2 ∧ dp1.

The Pfaffian Pf(ω) = ε. In basis (14.3) of the module D(C) the operator A hasthe form

Aω =

∥∥∥∥∥∥∥∥0 1 0 0−ε 0 0 00 f 0 −ε−f 0 1 0

∥∥∥∥∥∥∥∥ .

Then

C+ =⟨ε

d

dq2+ f

∂

∂p2− ι√ε

d

dq1,

d

dq1+ f

∂

∂p1+ ι√ε

d

dq2

⟩,

C− =⟨ε

d

dq2+ f

∂

∂p2+ ι√ε

d

dq1,

d

dq1+ f

∂

∂p1− ι√ε

d

dq2

⟩and the distribution l is generated by the vector field

Z = 2∂

∂u+ fp1

∂

∂p1+ εfp2

∂

∂p2.


So, a hyperbolic Monge–Ampère equation defines a pair of distributions ofreal planes on J1M. On the other hand, any unordered pair of arbitrary realdistributions C1,0 and C0,1 on J1M such that

1. dim C1,0 = dim C0,1 = 2;2. at each point a ∈ J1M C(a) = C1,0(a)⊕ C0,1(a);3. at each point a ∈ J1M the subspaces C1,0(a) and C0,1(a) are skew-

orthogonal with respect to the symplectic structure �;

determines the operator A up to the sign. Therefore a hyperbolic Monge–Ampère equation can be regarded as such an unordered pair {C1,0, C0,1}.

In a similar way, an elliptic Monge–Ampère equation can be viewed as anunordered pair {C1,0, C0,1} of complex conjugate distributions on J1M suchthat:

1. dimC C1,0 = dimC C0,1 = 2;2. at each point a ∈ J1M CC

a = C1,0(a)⊕ C0,1(a);3. at each point a ∈ J1M the subspaces C1,0(a) and C0,1(a) are skew-

orthogonal with respect to the complexification �Ca of the symplectic

structure �a.

Let L be a multivalued solution of a non-degenerate Monge–Ampère equationEω and let a ∈ L. Due to Proposition 14.1.1, the tangent space TaL is invariantunder operator Aω and the restriction of the symplectic form � on TaL is zero.Therefore the complexification of TaL is intersected with the complex planesC+(a) and C−(a). By dimension reason, the intersection (TaL)C with each ofC±(a) is a complex line and (TaL)C splits into the direct sum

(TaL)C = h+(a)⊕ h−(a),

where

h±(a)def= (TaL)C ∩ C±(a).

It is clear that for hyperbolic equations h+(a) and h−(a) are real lines. Forelliptic equations these lines are complex. These are called characteristic dir-ections of L at the point a and the corresponding distributions h+ and h−are called characteristic distributions of the multivalued solution L (see breakFigure 14.2).

Remark 14.1.1 For linear hyperbolic Monge–Ampère equations of the form

Avxx + 2Bvxy + Cvyy + E = 0,


Figure 14.2. Characteristic directions.

where A, B, C, E ∈ C∞(M), the constructed characteristic distributions can beprojected to the manifold M and we get the classical characteristics.

Theorem 14.1.1 A two-dimensional manifold L ⊂ J1M is a multivalued solu-tion of a hyperbolic Monge–Ampère equation Eω if and only if at any pointa ∈ L the tangent plane TaL has one-dimensional intersections with subspacesC+(a) and C−(a).

For a multivalued solution of the elliptic Monge–Ampère equations we havethe following result.

Theorem 14.1.2 A two-dimensional manifold L is a multivalued solution of anelliptic Monge–Ampère equation Eω if and only if L is a complex curve of thenon-holonomic complex structure A.

14.1.4 Parabolic equations

Now we consider a parabolic Monge–Ampère equation Eω. We assume thatωa �= 0 for all a ∈ J1M.

In this case the operator Aω is nilpotent (A2ω = 0) and ω ∧ ω = 0.

This means that the 2-form ωa on the Cartan subspace Ca is decomposable,i.e.,ωa = αa∧βa for some 1-formsα andβ on C. The intersection of the kernelsof the 1-forms αa and βa generates a two-dimensional subspace P(a) ⊂ C(a)(see Figure 14.3):

Pω(a)def= Ann(αa) ∩ Ann(βa).

Moreover, the 2-formωa is effective if and only if the plane Pω(a) is Lagrangian.


aP(a)

Figure 14.3. An almost tangent structure for a parabolic equation.

Indeed, let Xa,Ya ∈ P(a) be two vectors, then Xa�ωa = Ya�ωa = 0, and

(Xa ∧ Ya)�(ωa ∧�a) = �a(Xa, Ya) ωa = 0.

Therefore, �a(Xa, Ya) = 0.In other words, a parabolic effective differential form ω generates a two-

dimensional distribution

Pω : J1M � a → Pω(a) ⊂ C(a)

of Lagrangian planes (see Figure 14.3), and any such distribution defines aparabolic Monge–Ampère equation.

Let L be a multivalued solution of a parabolic Monge–Ampère equation Eω.Then ω|L = 0 and therefore 1-forms α|L and β|L are linear dependent, and

Pω(a) ∩ TaL �= 0

for all a ∈ L.

Theorem 14.1.3 An integral two-dimensional manifold L is a multivalued solu-tion of a parabolic Monge–Ampère equation Eω if and only if the tangent bundleof L has a non-trivial intersection with the distribution Pω.

If L is a multivalued solution of a parabolic Monge–Ampère equation Eω,then the line

h(a)def= L ∩ Pω(a)

is called the characteristic direction of L at the point a and the correspond-ing distribution h is called the characteristic distribution of the multivaluedsolution L.


14.2 Intermediate integrals

14.2.1 Classical and non-holonomic intermediate integrals

Let us consider a second-order partial differential equation

F(x, y, v, vx , vy, vxx , vxy, vyy) = 0. (14.5)

A function H ∈ C∞(J1M) is called a (classical) intermediate integral of thisequation if all functions v = v(x, y) satisfying H(x, y, v, vx , vy) = constant aresolutions of (14.5).

Let us formulate the definition of an intermediate integral for Monge–Ampèreequations in terms of effective differential 2-forms on J1M.

Let ω be an effective differential 2-form on J1M. A function H ∈ C∞(J1M)

is called an intermediate integral of the Monge–Ampère equation Eω if

ω = (dH ∧ θ)ε

for some differential 1-form θ or, equivalently,

ω = dH ∧ θ + λ0 dω0 + λ1 ∧ ω0 (14.6)

for some λk ∈ �k(J1M).

Example 14.2.1 The function H = eq1(p2 + u) is an intermediate integral ofthe equation

vxy + vx + vy + v = 0. (14.7)

Here q1 = x, q2 = y, p1 = vx , p2 = vy, u = v. Indeed, if v is a solution of thefamily of equations ex(vy + v) = constant, then

ex(vxy + vx + vy + v) = ∂

∂x(ex(vy + v)) = 0

and therefore the function v is a solution of (14.7).The effective differential 2-form corresponding to (14.7)

ω = dq1 ∧ dp1 − dq2 ∧ dp2 + 2(p1 + p2 + u) dq1 ∧ dq2

can be represented in the form ω = (dH ∧ θ)ε with

θ = ((p1 + p2 + u)γ1 − p1γ2) dq1 + (2e−q1 + p2(γ1 − γ2)) dq2

+ γ1 dp2 + γ2 du,

where γ1 and γ2 are smooth functions on J1M.

14.2 Intermediate integrals 305

Let us introduce a notion of a non-holonomic intermediate integral which isa natural generalization of the classical one mentioned above (see [76]).

A three-dimensional distribution I on J1M is called a non-holonomic inter-mediate integral of the Monge–Ampère equation Eω if the following conditionshold:

1. I(a) is a subspace of the Cartan subspace at any point a ∈ J1M, i.e., I(a) ⊂C(a);

2. any two-dimensional integral submanifold L of I is a multivalued solutionof Eω;

3. I has a characteristic symmetry.

Any classical intermediate integral H generates a non-holonomic intermedi-ate integral I according to the rule

I = F〈ω0, dH〉. (14.8)

The contact vector field XH is a characteristic symmetry of the distribution I .In the general case a non-holonomic intermediate integral is locally generated

by the Cartan form ω0 and some 1-form ϑ :

I = F〈ω0,ϑ〉.

Since the Cartan distribution on five-dimensional manifolds has no three-dimensional integral submanifolds, we see that the distribution I is notcompletely integrable and therefore dim I(1) ≥ 4.

It is not difficult to verify that the distribution I has a characteristic symmetryif and only if dim I(1) = 4. In this case for ω0 = du− p dq and ϑ = ϑ1 dq1 +ϑ2 dq2 + ϑ3 du+ ϑ4 dp1 + ϑ5 dp2 the vector field

V = −ϑ4∂

∂q1− ϑ5

∂

∂q2− (p1ϑ4 + p2ϑ5)

∂

∂u+ (ϑ1 + p1ϑ3)

∂

∂p1

+ (ϑ2 + p2ϑ3)∂

∂p2

is a characteristic symmetry of I .Moreover, the distribution I is generated by a certain function H according

to the rule (14.8) if and only if dim I(1) = 4 and I(2) = I(1), i.e., the distributionI(1) is completely integrable.

Example 14.2.2 Let us consider a hyperbolic Monge–Ampère equation Eω.It generates the real AP-structure (C+, l, C−). Suppose that the distribution


C(1)+ is not completely integrable and dim C(2)

+ = 4. Then the intersection

C(2)+ (a) and C−(a) is a line for any point a ∈ J1M. We get a one-dimensional

distribution

l1 : a → l1(a)def= C(2)

+ (a) ∩ C−(a).

Construct a three-dimensional distribution Idef= C+ ⊕ l1. If dim I(1) = 4, then

I is a non-holonomic intermediate integral of Eω. Indeed, I(a) is a subspace ofthe Cartan space C(a) at any point a ∈ J1M. Moreover, since the distributionC+ is not completely integrable, any two-dimensional integral manifold L of Iintersects the planes of the distributions C+ and C−. Therefore ω|L = 0 andwe see that L is a multivalued solution of Eω.

Example 14.2.3 The Martin potential of one-dimensional isentropic flows ofa politropic gas satisfies the following Monge–Ampère equation:

Hess v+ q−α2 = 0 (14.9)

with α = 1+ 1/γ , where γ is the adiabatic exponent (see [93]). We considerthe case where γ = 3. The corresponding effective differential 2-form is

ω = q−4/32 dq1 ∧ dq2 + dp1 ∧ dp2.

The Pfaffian of ω is Pf(ω) = −q−4/32 < 0, therefore the equation is hyperbolic.

The corresponding normed operator in the basis (14.3) is

A = 1

q2/32

∥∥∥∥∥∥∥∥0 0 0 −10 0 1 00 1 0 0−1 0 0 0

∥∥∥∥∥∥∥∥ .

The distributions C+ and C− are generated by the vector fields

X+ = ∂

∂p2− q2/3

2d

dq1, Y+ = ∂

∂p1+ q2/3

2d

dq2

and

X− = ∂

∂p2+ q2/3

2d

dq1, Y− = ∂

∂p1− q2/3

2d

dq2

14.2 Intermediate integrals 307

respectively. The distribution l is generated by the vector field

Z = ∂

∂p2+ 3q2

∂

∂u

and l1 = F〈X−〉. Therefore I = F〈ω0, θ〉, where θ = dp1 − q−2/32 dq2. We see

that θ = dH, where H = p1−3q1/32 . The corresponding first-order differential

equation is vq1 = 3q1/32 + c1, where c1 is an arbitrary constant. Any solution

of this equation has the form

v(q1, q2) = 2q1q1/32 + c1q1 + h(q2)

and is a solution of (14.9). Here h is an arbitrary smooth function. Theconsidered equation has also non-holonomic intermediate integrals Iϕ =F〈ω0, θϕ〉, where

θϕ = q−2/32 (dq1 − kϕ dq2)+ kϕ dp1 + dp2,

kϕ = 13 q−1/3

2 (ϕ(H) − p2) and ϕ is an arbitrary smooth function on R. Thecharacteristic symmetry of Iϕ is

Vϕ = −kϕ∂

∂q1− ∂

∂q2− (p1kϕ + p2)

∂

∂u+ q−2/3

2

(∂

∂p1− kϕ

∂

∂p2

).

More details on the method of finding classical and non-holonomic interme-diate integrals can be found in [76, 79, 93].

14.2.2 Cauchy problem and non-holonomic intermediate integrals

Non-holonomic intermediate integrals can be used for solving the Cauchyproblem for Monge–Ampère equations.

Let Eω be a Monge–Ampère equation and let I be its non-holonomicintermediate integral with a characteristic symmetry V . A smooth curveK ⊂ J1M is called corresponding to I Cauchy data for Eω if the followingtwo conditions hold:

1. K is an integral curve of the distribution I;2. the tangent line to K is transversal to the vector field V at each point a ∈ K,

i.e., Va �⊂ TaK.

A multivalued solution L of Eω is called a solution of the Cauchy problem ifL ⊃ K.


Let K be corresponding to I Cauchy data and At be a translations group alongthe vector field V . Then the surface L = ∪tAt(K) is a multivalued solution ofthe Cauchy problem.

14.3 Symplectic Monge–Ampère equations

Let ω be an effective differential 2-form on the smooth manifold J1M which isinvariant with respect to the contact vector field X1. Thenω is a lift of differentialform ω′ on T∗M with respect to the projection π : J1M → T∗M,ω = π∗(ω′).If L ⊂ J1M is a solution of Eω, then the projection L′ = π(L) is a submersedLagrangian manifold, and ω′|L′ = 0.

This observation allows us to consider a special class of Monge–Ampèreequations defined by X1-invariant effective forms.

We call such equations symplectic, because they can be studied in terms ofthe symplectic geometry of T∗M.

14.3.1 A field of endomorphisms Aω on T∗MThe cotangent bundle T∗M is a four-dimensional symplectic manifold with astructure differential 2-form � = dρ. A differential 2-form ω ∈ �2(T∗M) iseffective if and only if

ω ∧� = 0. (14.10)

We define a Pfaffian of the differential 2-form ω on T∗M as a function Pf(ω) ∈C∞(T∗M) such that

Pf(ω)(a)def= Pf(ωa).

The operator field associated with ω generates the field of endomorphismsAω on T∗M:

Aω : T∗M � a −→ Aωa ∈ End(Ta(T∗M))

by the rule

X�ω = AωX��.

The field of endomorphisms Aω on T∗M inherits all properties of the non-holonomic linear operator Aω on J1M (see Proposition 14.1.1).

14.3 Symplectic Monge–Ampère equations 309

Proposition 14.3.1

1. Vector fields X , AωX are skew-orthogonal, i.e.,

�(AωX , X) = 0.

2. Any multivalued solution L of the equation Eω, i.e., a Lagrangian submani-fold such that ω|L = 0, is invariant with respect to Aω:

Aωa : TaL → TaL

for any point a ∈ L.3. The square of Aω is scalar and

A2ω + Pf(ω) = 0.

4. Let

ω = E dq1 ∧ dq2 + B(dq1 ∧ dp1 − dq2 ∧ dp2)

+ C dq1 ∧ dp2 − A dq2 ∧ dp1 + D dp1 ∧ dp2,

in canonical coordinates, where A, B, C, D, E are smooth functions on T∗M.Then in the basis

∂

∂q1,∂

∂p1,∂

∂q2,∂

∂p2(14.11)

of the module D(T∗M) we obtain the following representation of Aω:

Aω =(

B∂

∂q1+ C

∂

∂q2− E

∂

∂p2

)⊗ dq1+

(B

∂

∂p1+ D

∂

∂q2− A

∂

∂p2

)⊗dp1

+(−A

∂

∂q1+ E

∂

∂p1− B

∂

∂q2

)⊗ dq2

+(−D

∂

∂q1+ C

∂

∂p1− B

∂

∂q2

)⊗ dp2.

Equation Eω is called non-degenerate at a point a if Pf(ω)(a) �= 0 anddegenerate if Pf(ω)(a) = 0.

We say that the symplectic Monge–Ampère equation Eω withω ∈ �2ε(T

∗M)

is hyperbolic, elliptic or parabolic at a point a ∈ T∗M if Pf(ω)(a) is negative,positive or zero, and ωa �= 0.


The equation Eω is called mixed-type (or variable type) at a point a, ifωa �= 0and Pf(ω) changes sign at this point.

Example 14.3.1 The Triccomi equation

xvxx + vyy = 0

has Pf(ω) = x, and therefore is hyperbolic in the domain x < 0, elliptic in thedomain x > 0 and of mixed type on the hypersurface x = 0.

As above, in the case where Pf(ω) �= 0, we can normalize the form ω in sucha way that |Pf(ω)| = 1. We call such forms ω normed. For a normed form ω

its Pfaffian Pf(ω) is−1 for a hyperbolic equation and is+1 for an elliptic one.The operator Aω corresponding to the normed form ω is also called normed

and is denoted by A.Then A2 = 1 for hyperbolic equations and A2 = −1 for elliptic equations.

For parabolic equations we have A2ω = 0.

14.3.2 Non-degenerate symplectic equations

Let A be the normed operator that corresponds to a non-degenerate Monge–Ampère equation Eω.

It follows from A2 = ε, where ε = Pf(ω) = ±1 that the operator A is analmost product structure for the hyperbolic equation and an almost complexstructure for elliptic equations.

The complexification of the tangent space Ta(T∗M) at a ∈ T∗M splits intothe direct sum of two skew-orthogonal complex symplectic planes:

TCa (T

∗M) = V+(a)⊕ V−(a), (14.12)

where

V+(a)def= {X ∈ Ta(T

∗M)C | ACa X = −ι√εX},

V−(a)def= {X ∈ Ta(T

∗M)C | ACa X = ι

√εX}.

The corresponding distributions V+ and V− we will call characteristicdistributions (as in the contact situation).

Example 14.3.2 Let us consider the equation

vxx + εvyy = f (x, y, vx , vy),

14.3 Symplectic Monge–Ampère equations 311

–+

Figure 14.4. A decomposition of Ta(T∗M) into a direct sum.

where ε = −1 or ε = 1. The effective form is

ω = f dq1 ∧ dq2 − ε dq1 ∧ dp2 + dq2 ∧ dp1.

The Pfaffian Pf(ω) = ε. In basis (14.11) of the module D(T∗M) the operatorA has the form

Aω =

∥∥∥∥∥∥∥∥0 1 0 0−ε 0 0 0

0 f 0 −ε−f 0 1 0

∥∥∥∥∥∥∥∥ .

Then,

V+ =⟨ε∂

∂q2+ f

∂

∂p2− ι√ε∂

∂q1,

∂

∂q1+ f

∂

∂p1+ ι√ε∂

∂q2

⟩,

V− =⟨ε∂

∂q2+ f

∂

∂p2+ ι√ε∂

∂q1,∂

∂q1+ f

∂

∂p1− ι√ε∂

∂q2

⟩.

For hyperbolic equations V+(a) and V−(a) can be considered as real planes.On the other hand, any unordered pair of arbitrary real distributions V1,0 andV0,1 on T∗M such that:

1. dim V1,0 = dim V0,1 = 2;2. at each point a ∈ T∗M Ta(T∗M) = V1,0(a)⊕ V0,1(a);3. at each point a ∈ T∗M the subspaces V1,0(a) and V0,1(a) are skew-

orthogonal with respect to the symplectic structure �a,

determines a hyperbolic operator A up to sign.In the elliptic case the operator A is defined (up to sign) by any unordered

pair {V1,0, V0,1} of complex conjugate distributions on T∗M such that:

1. dimC V1,0 = dimC V0,1 = 2;2. at each point a ∈ T∗M Ta(T∗M)C = V1,0(a)⊕ V0,1(a);


3. at each point a ∈ T∗M the subspaces V1,0(a) and V0,1(a) are skew-orthogonal with respect to the complexification �C

a of the symplecticstructure �a.

Therefore a non-degenerate symplectic Monge–Ampère equation can beviewed as such unordered pairs of distributions {V1,0, V0,1}.

We see that hyperbolic and elliptic symplectic Monge–Ampère equationsgenerate AP-structures on T∗M with n = 4, P1 = V+, P2 = V−.

The following two theorems explain the geometrical sense of multivaluedsolutions for non-degenerate symplectic Monge–Ampère equations.

Theorem 14.3.1 A Lagrangian two-dimensional manifold L ⊂ T∗M is a solu-tion of a symplectic hyperbolic equation Eω,ω ∈ �2

ε(T∗M), if and only if the

tangent spaces TaL for all a ∈ L have one-dimensional intersections h+(a) andh−(a) with real symplectic planes V+(a) and V−(a):

TaL = h+(a)⊕ h−(a).

The lines h±(a)def= TaL ∩ V±(a) are called the characteristic directions of

the multivalued solution L at the point a and the corresponding distributions h±are called characteristic distributions of the multivalued solution L.

Theorem 14.3.2 A Lagrangian surface L ⊂ T∗M is a multivalued solution ofa symplectic elliptic Monge–Ampère equation Eω if and only if L is a complexcurve of the almost complex structure A.

14.3.3 Symplectic parabolic equations

Now we consider a symplectic parabolic Monge–Ampère equation Eω, whereω ∈ �2

ε(T∗M).

Suppose that ωa �= 0 at all points a ∈ T∗M. In this case ω ∧ ω = 0 and theform ω is decomposable, i.e., ω = α ∧ β for some differential 1-forms α andβ on T∗M.

Therefore, the equation Eω determines the two-dimensional Lagrangiandistribution Pω = ker α ∩ ker β on T∗M.

Theorem 14.3.3 A Lagrangian two-dimensional manifold L ⊂ T∗M is a solu-tion of a symplectic parabolic equation Eω, if and only if tangent spaces TaLhave non-trivial intersections with the Lagrangian distribution Pω for all a ∈ L.

The intersection h(a)def= L ∩ Pω,a mentioned in this theorem is called the

characteristic direction of L at the point a and the corresponding distribution his called the characteristic distribution of the multivalued solution L.

14.4 Cauchy problem for hyperbolic Monge–Ampère equations 313

14.3.4 Intermediate integrals

Let us discuss the definition of intermediate integral for symplectic Monge–Ampère equations in terms of effective differential 2-forms on T∗M.

Letω be an effective differential 2-form on T∗M. A function H ∈ C∞(T∗M)

is called an intermediate integral of the Monge–Ampère equation Eω if

ω = (dH ∧ θ)ε (14.13)

for some differential 1-form θ on T∗M or, equivalently, ω = dH ∧ θ + λ� forsome function λ and some 1-form θ .

Example 14.3.3 For the wave equation

vxy = 0

the function H = q1 + p1 is an intermediate integral. Indeed, if v is a solutionof the equation x + vx = 0, then v is also a solution of the wave equation.On the other hand, the effective 2-form corresponding to the wave equationω = dq1 ∧ dp1 − dq2 ∧ dp2 can be represented in the form (14.13) withθ = (2− γ )dq1 + γ dp1. Here γ is an arbitrary smooth function on T∗M.

If a Monge–Ampère equation Eω admits two intermediate integrals H1 andH2 then ω = h(dH1 ∧ dH2)ε, i.e.,

ω = λ1 dH1 ∧ dH2 + λ2�

for some smooth functions h, λ1 and λ2.

14.4 Cauchy problem for hyperbolic Monge–Ampèreequations

We call the form ω (+)-integrable or (−)-integrable if the distribution C(1)+ or

C(1)− is completely integrable respectively (see [72]).If the form ω is (+)-integrable and (−)-integrable simultaneously, then the

form ω is called (±)-integrable.Let Eω be a hyperbolic Monge–Ampère equation. A smooth curve K ⊂ J1M

is called a Cauchy data for Eω if the following two conditions hold:

1. K is an integral curve of the Cartan distribution, i.e., ω0|K = 0;2. the tangent line to K at each point a ∈ K is not characteristic, i.e., TaK �⊂

C−(a) and TaK �⊂ C+(a).


A multivalued solution L of Eω is called a solution of the Cauchy problem ifL ⊃ K.

14.4.1 Constructive methods for integration ofCauchy problem

A-methodSuppose that ω is (+)-integrable. Let H ∈ C∞(J1M) be an integral of thedistribution C(1)

+ and let Mc = H−1(c). Since, l ⊂ C(1)+ , we see that l(a) ⊂

Ta(Mc) for any point a ∈ Mc.Hence, the restriction of the Cartan distribution on Mc is a three-dimensional

distribution. Denote this restriction by Cc.The restriction of the distribution C− on Mc is a one-dimensional distribution,

denote it by l−.Therefore,

Cc(a) = C+(a)⊕ l−(a).

Moreover, the skew-orthogonality of distributions C+ and C− implies that thedistribution l− is the characteristic distribution for Cc.

Now we can solve the Cauchy problem in the following way.

1. Consider two independent integrals H1, H2 ∈ C∞(J1M) of the distributionC(1)+ . Functions H1|K, H2|K are functionally dependent. Therefore, we can

find a smooth function F(x, y) ∈ C∞(R2) such that F(H1|K, H2|K) = 0.2. Let H = F(H1, H2) and let M0 = H−1(0) ⊂ J1M be a smooth hypersur-

face. Denote by L(a) � a the integral curves of the distribution l−. ThenK ⊂ M0, and

L = ∪a∈K

L(a)

is a smooth surface in some neighborhood of K.This surface is integral because l− is a characteristic distribution for Cc, andby construction it has one-dimensional intersections with the characteristicdistributions C±. Therefore L is a solution of the Cauchy problem.

B-methodLet ω be a (±)-integrable effective form, and let H and G be integrals of thedistributions C(1)

+ and C(1)− respectively.


Consider a submanifold

Nc = {H = c1, G = c2} ⊂ J1M,

where c = (c1, c2) ∈ R2.Let Cc be the restriction of the Cartan distribution on Nc. This is a two-

dimensional distribution and

Cc(a) = l+(a)⊕ l−(a)

where l± are the restrictions of the characteristic distributions C± on Nc.Moreover,

TaNc = l(a)⊕ l−(a)⊕ l+(a).

If X ∈ D(l+), Y ∈ D(l−), then

ω0([X, Y ]) = X(ω0(Y))− Y(ω0(X))− dω0(X, Y) = 0.

Therefore, Cc is a completely integrable distribution and its two-dimensionalintegral manifolds are solutions of the Monge–Ampère equation.

This observation gives the following method of solution to the Cauchyproblem.

1. As above we find integrals H (for the distribution C+) and G (for thedistribution C−) such that H|K = 0, G|K = 0.

2. Find an integral F ∈ C∞(N0) of the distribution C0 on N0 = {F = G = 0}.3. On the curve K we have F|K = c. Then the two-dimensional manifold

L = {H = 0, G = 0, F = c}

is the solution of the Cauchy problem.

ExampleConsider the non-linear wave equation

vxy = f (x, y, v, vx , vy).

This equation is represented by the following effective form:

ω = −2f dq1 ∧ dq2 + dq1 ∧ dp1 − dq2 ∧ dp2.


Then Pfaffian Pf(ω) = −1 and operator Aω has the form

Aω =

∥∥∥∥∥∥∥∥∥1 0 0 0

0 −1 0 0

0 −2f 1 0

2f 0 0 −1

∥∥∥∥∥∥∥∥∥in basis (14.3).

The characteristic distributions are

C+ =⟨P1 = d

dq1+ f

∂

∂p2, P2 = ∂

∂p1

⟩,

C− =⟨Q1 = d

dq2+ f

∂

∂p1, Q2 = ∂

∂p2

⟩.

The distribution l is generated by the vector field

Z = ∂

∂u+ fp2

∂

∂p1+ fp1

∂

∂p2.

Moreover,

[Q1, Z] = (−fu + p2 fp2u − 2fp1 fp2 + ffp1 p2 + fq2p2)∂

∂p1

+ (p2 fp1u + ffp1p1 + fq2 p1)∂

∂p2− fp1

∂

∂u,

[Q2, Z] = fp2 p2

∂

∂p1+ fp1p2

∂

∂p2.

Therefore the distribution C(1)− is completely integrable if and only if the

following conditions hold:

fp2p2 = 0,

−fu + p2fp2u − fp1 fp2 + ffp1p2 + fq2 p2 = 0.

From the first equation it follows that

f = s(q, u, p1)p2 + r(q, u, p1),


and from second one that

rq2 − su + srp1 − rsp1 = 0.

For example, differential equations of the form

vxy = αvx + βvy − αβu+ γ ,

where α,β, γ ∈ R are (±)-integrable and hence can be solved by the B-method.

15

Systems of first-order partial differentialequations on two-dimensional manifolds

In this chapter we consider Jacobi differential equations on two-dimensionalmanifolds. They are systems of two non-linear first-order partial differentialequations of the following form:

A1 + B1∂v1

∂x− C1

∂v1

∂y− D1

∂v2

∂y+ E1

∂v2

∂x+ F1 det Jv = 0,

A2 + B2∂v1

∂x− C2

∂v1

∂y− D2

∂v2

∂y+ E2

∂v2

∂x+ F2 det Jv = 0,

where

det Jv =

∣∣∣∣∣∣∣∣∣∂v1

∂x

∂v1

∂y

∂v2

∂y

∂v2

∂x

∣∣∣∣∣∣∣∣∣is the Jacobian and Ai, Bi, Ci, Di, Ei and Fi (i = 1, 2) are functions in x, y andv. For the first time the geometry of such systems of first-order differentialequations on two-dimensional manifolds was discussed in [71]. Classificationproblems for Jacobi equation was studied in [25].

Although the coordinate representation of Monge–Ampère equations andJacobi equations looks different, their geometies are very closed.

We associate the same geometrical structures with Jacobi equations: almostproduct structures for hyperbolic equations, almost complex structures forelliptic equations and almost tangent structures for parabolic ones.

318

15.1 Non-linear differential operators of first order 319

15.1 Non-linear differential operators of first order ontwo-dimensional manifolds

We start with a simple observation. Let M = N × Rm and N be a two-dimensional manifold. Any differential 2-form ω ∈ �2(M) generates adifferential operator

�ω : C∞(N , Rm)→ �2(N)

in the following way:

�ω(v)def= ω|Lv ,

where Lv ⊂ N × Rm is a graph of the vector-function v.Suppose that (x, u) = (x1, x2, u1, . . . , um) are local coordinates on M such that

x1, x2 are local coordinates on the manifold N and u1, . . . , um are coordinatesin Rm. Let

ω =m∑

i,j=1(i<j)

Fij(x, u)dui ∧ duj +m∑

i=1

Bi(x, u)dui ∧ dx1

+m∑

i=1

Ci(x, u)dui ∧ dx2 + A(x, u)dx1 ∧ dx2

be a coordinate representation of the 2-form ω and let v = (v1, . . . , vm) ∈C∞(N , Rm) be a vector-function. Then

Lv = {u1 = v1(x1, x2), . . . , um = vm(x1, x2)} ⊂ M

and, therefore,

�ω(v) = m∑

i,j=1(i<j)

Fij(x, v) det Ji,j +m∑

i=1

(Ci(x, v)

∂vi

∂x1− Bi(x, v)

∂vi

∂x2

)

+ A(x, v)

dx1 ∧ dx2,

where

det Ji,j = det

∥∥∥∥∥∥∥∥∂vi

∂x1

∂vj

∂x1∂vi

∂x2

∂vj

∂x2

∥∥∥∥∥∥∥∥ .

320 Systems of first-order partial differential equations

If we consider k differential 2-forms ω1, . . . ,ωk on M, we obtain a systemof k partial differential equations of first order:

�ω1(v) = 0,

......

......

...

�ωk (v) = 0.

(15.1)

It is clear, that if aij are smooth functions on the manifold M and matrix‖aij‖i,j=1,...,k is non-degenerate at each point of M, then the system

a11�ω1(v)+ · · · + a1k�ωk (v) = 0,

......

......

a1k�ω1(v)+ . . .+ akk�ωk (v) = 0

is equivalent to (15.1). Therefore, we can identify the system (15.1) with asmooth field of k-dimensional subspaces

E : M � a −→ E(a) ⊂ �2(T∗a M),

A two-dimensional submanifold L ⊂ M is a multivalued solution of thesystem of first-order partial differential equations E if L is integral for eachωi ∈ �(E)(i = 1, . . . , k).

Example 15.1.1 (Variational calculus) Any differential 2-form θ ∈ �2(M)

also defines a functional � on orientable two-dimensional submanifolds:

� : L →∫

L�.

Let X be a vector field on M with the one-parameter group ϕt : M → M. Thenfor variation of � we find

�(ϕt(L))− �(L) =∫ϕt(L)

�−∫

L� =

∫L(ϕ∗t (�)−�)

= t∫

LX(�)+ o(t) = t∫

X�d�+ o(t).

Hence, a submanifold L ⊂ M is an extremal of � if the restriction ω|L = 0

for all 2-forms ωdef= X�d� and for all vector field X on M. The system of

15.2 Jacobi equations 321

differential equations E�, where

E�(a) = {Xa�da� ∈ �2(T∗a M)|∀Xa ∈ TaM}

is called the Euler equation for the functional �.

15.2 Jacobi equations

If we suppose that m = k = 2 we obtain a system of two partial differen-tial equations. In order to find its coordinate representation we consider twodifferential 2-forms

ωi = Ai dx1 ∧ dx2 + Bi du1 ∧ dx1 + Ci du1 ∧ dx2 (15.2)

+ Di du2 ∧ dx1 + Ei du2 ∧ dx2 + Fi du1 ∧ du2

(i = 1, 2) on the manifold M. The corresponding system of differentialequations is

A1 − B1∂v1

∂x2+ C1

∂v1

∂x1− D1

∂v2

∂x2+ E1

∂v2

∂x1+ F1 det Jv = 0,

A2 − B2∂v1

∂x2+ C2

∂v1

∂x1− D2

∂v2

∂x2+ E2

∂v2

∂x1+ F2 det Jv = 0,

(15.3)

where det Jv is the determinant of the Jacobi matrix

Jv =

∥∥∥∥∥∥∥∥∥∂v1

∂x1

∂v1

∂x2

∂v2

∂x1

∂v2

∂x2

∥∥∥∥∥∥∥∥∥ .

This gives rise to the following definitions.Any smooth field of planes

E : M � a −→ E(a) ∈ �2(T∗a M)

is called a Jacobi system of partial differential equations on manifold M dimM =Y or a Jacobi equation.

Planes E(a) are called Jacobi planes.


A two-dimensional submanifold L ⊂ M is called a multivalued solution ofthe Jacobi equation E = F〈ω1,ω2〉 if L is an integral manifold for each ω, i.e.,ω1|L = ω2|L = 0.

As we have seen in 6.3 each non-degenerate equation E generates a field ofendomorphisms AE on the smooth manifold M by the following rule:

AE : M � a −→ AE(a) ∈ End(TaM),

where the endomorphism AE(a) is defined by the formula

X�ω2,a = AE(a)X�ω1,a,

X ∈ TaM for an orthogonal basis ω1,ω2. The endomorphism AE(a) does notdepend on the choice of (oriented) orthogonal basis of the plane E(a).

Example 15.2.1 (Cauchy–Riemann systems) The Cauchy–Riemann system

∂v1

∂x2− ∂v2

∂x1= 0,

∂v1

∂x1+ ∂v2

∂x2= 0

(15.4)

is a Jacobi equation which is generated by the 2-forms

ω1 = du1 ∧ dx1 + du2 ∧ dx2,

ω2 = du1 ∧ dx2 − du2 ∧ dx1.

The corresponding field of endomorphisms in the basis

∂

∂x1,∂

∂u1,∂

∂x2,∂

∂u2(15.5)

of the module D(M) is

AE =

∥∥∥∥∥∥∥∥0 0 1 00 0 0 −1−1 0 0 00 1 0 0

∥∥∥∥∥∥∥∥ .

Example 15.2.2 (Wave equation) The wave equations

∂v1

∂x2− ∂v2

∂x1= 0,

∂v1

∂x1− ∂v2

∂x2= 0

(15.6)


are generated by the forms

ω1 = du1 ∧ dx1 + du2 ∧ dx2,

ω2 = du1 ∧ dx2 + du2 ∧ dx1.

The field of endomorphisms in the basis (15.5) is

AE =

∥∥∥∥∥∥∥∥0 0 1 00 0 0 11 0 0 00 1 0 0

∥∥∥∥∥∥∥∥ .

Note that in these examples ω1 is a standard symplectic structure on M.

Example 15.2.3 (Equation of the Chaplygin gas) The following systemdescribes the non-stationary flow of the Chaplygin gas [14, 109]:

∂ρ

∂t+ ∂

∂x(ρv) = 0,

∂v

∂t+ v

∂v

∂x+ c2

2

∂

∂x(ρ−2) = 0.

(15.7)

Here v and ρ are the velocity and the normalized density of a gas, t and x are thetime and a space coordinate, while c is a constant. This equation also describeprocesses in unstable plasmas. The corresponding 2-forms ω1, ω2 and the fieldof endomorphisms are the following

ω1 = −u2 du1 ∧ dx1 + du1 ∧ dx2 − u1 du2 ∧ dx1,

ω2 = c2u−31 du1 ∧ dx1 − u2 du2 ∧ dx1 + du2 ∧ dx2

and

AE =

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

u2

u10 − 1

u10

0 0 0 1

c2 + u21u2

2

u31

0 −u2

u10

0 − c2

u41

0 0

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥.

Here x1 = t, x2 = x, u1 = ρ and u2 = v.


Example 15.2.4 (The Gibbons–Tsarev system) The system (see [27])

v2∂v1

∂x1− ∂v1

∂x2+ 1

v1 − v2= 0,

v1∂v2

∂x1− ∂v2

∂x2+ 1

v2 − v1= 0.

(15.8)

is generated by the differential 2-forms

ω1 = u2 du1 ∧ dx2 + du1 ∧ dx1 + 1

u1 − u2dx1 ∧ dx2,

ω2 = u1 du2 ∧ dx2 + du2 ∧ dx1 + 1

u2 − u1dx1 ∧ dx2.

The following example establishes a connection between Jacobi equationsand Monge–Ampère equations.

Example 15.2.5 (Equation of one-dimensional gas flow) A one-dimensionalnon-stationary flow of barotropic non-viscous gas is described by the followingsystem:

∂v

∂t+ ∂u

∂x= 0,

∂u

∂t+ ∂v

∂x(p(v)) = 0,

(15.9)

where u is the velocity of the gas, v = 1/ρ, ρ and p are the density and thepressure of the gas respectively. The function p = p(v) is the equation of stateof a gas [93]. Then,

ω1 = du1 ∧ dx1 + du2 ∧ dx2,

ω2 = du1 ∧ dx2 − p′(u2) du2 ∧ dx1

and

AE =

∥∥∥∥∥∥∥∥∥∥

0 0 −1 0

0 0 0 p′(u2)

p′(u2) 0 0 0

0 −1 0 0

∥∥∥∥∥∥∥∥∥∥.


Example 15.2.6 (Monge–Ampère equations) Any Monge–Ampère equation

Avxx + 2Bvxy + Cvyy + D(vxxvyy − v2xy)+ E = 0

on T∗M can be considered as a Jacobi equation as follows

∂v2

∂x1− ∂v1

∂x2= 0,

A∂v1

∂x1+ 2B

∂v1

∂x2+ C

∂v2

∂x2+ D

(∂v1

∂x1

∂v2

∂x2− ∂v1

∂x2

∂v2

∂x1

)+ E = 0.

(15.10)

Here x1 = x, x2 = y, v1 = vx and v2 = vy.

Let E1 = F〈ω11,ω1

2〉 and E2 = F〈ω21,ω2

2〉 be two Jacobi equations. We willsay that equations E1 and E2 are equivalent if there exists a diffeomorphism Fof the smooth manifold M such that F∗(E1) = E2, i.e.,

F∗(ω11) = a11ω

21 + a12ω

22,

F∗(ω12) = a21ω

21 + a22ω

22.

Note that since F is a diffeomorphism we have det ‖aij‖ �= 0.One can write the last formulae without the functions aij:

F∗(ω1i ) ∧ ω2

1 ∧ ω22 = 0 (i = 1, 2).

Theorem 15.2.1 The class of Jacobi equations is invariant with respect todiffeomorphisms of the smooth manifold M.

This theorem is also true for general systems (15.1).The equation E is called elliptic, hyperbolic or parabolic at a point a ∈ M if

the Jacobi plane E(a) is so. The equation E is called non-degenerate at a pointa ∈ M if the Jacobi plane E(a) is elliptic or hyperbolic.

Example 15.2.7 Cauchy–Riemann equations (15.4) and the wave equation(15.6) are elliptic and hyperbolic respectively.

Example 15.2.8 Equation (15.7) is elliptic. Indeed, if we choose the volumeform

µ = dx1 ∧ dx2 ∧ du1 ∧ du2, (15.11)


then the matrix Q = ‖q(ωi,ωj)‖ has the form

Q =

∥∥∥∥∥∥∥−2u1 0

0 −2c2

u31

∥∥∥∥∥∥∥and

ε(E) = sign det Q = 1.

Example 15.2.9 Equation (15.8) is hyperbolic. Indeed, for the volume form(15.11) the matrix Q is

Q =∥∥∥∥ 0 u2 − u1

u2 − u1 0

∥∥∥∥ .

Hence,

ε(E) = sign(−(u2 − u1)2) = −1.

Note that ω1 ∧ ω1 = ω2 ∧ ω2 = 0 and ω1 ∧ ω2 = hµ �= 0. Let us introducethe orthogonal basis ω′1, ω′2 of E:

ω′1 = ω1 + ω2,

ω′2 = ω1 − ω2.

Then we have ω′1 ∧ ω′2 = 0 and ω′1 ∧ ω′1 = −ω′2 ∧ ω′2 �= 0. The matrixQ′ = ‖q(ω′i,ω′j)‖ is diagonal:

Q′ =∥∥∥∥2(u2 − u1) 0

0 −2(u2 − u1)

∥∥∥∥ .

We assume u2 �= u1, then

AE =

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

u1 + u2

u1 − u20

2u1u2

u1 − u20

− 2

(u1 − u2)21 − 2u1

(u1 − u2)20

− 2

u1 − u20 −u1 + u2

u1 − u20

2

(u1 − u2)20

2u2

(u1 − u2)2−1

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥. (15.12)


We see that A2� = 1 and det A� = 1. Note that the form ω′1 is closed, i.e.,

dω′1 = 0, hence ω′1 is a symplectic structure on M. To represent this sym-plectic structure in canonical coordinates let us construct a diffeomorphismF : R2 ×D → M = R4,:

F :

x1 = X1,

x2 = X2,

u1 = 1

2(U1 −

√U2

1 − 4U2),

u2 = 1

2(U1 +

√U2

1 − 4U2).

where X1, X2 are coordinates on R2, D = {(U1, U2) ∈ R2|U21 − 4U2 > 0}.

The inverse transformation is

F−1 :

X1 = x1,

X2 = x2,

U1 = u1 + u2,

U2 = u1u2.

Then,

F∗(ω′1) = dU1 ∧ dX1 + dU2 ∧ dX2,

F∗(ω′2) =1√

U21 − 4U2

(−U1 dU1 ∧ dX1 − 2U2 dU1 ∧ dX2 + 2 dU2 ∧ dX1

+ U1 dU2 ∧ dX2 − 2 dX1 ∧ dX2).

The 2-formsω′′1def= F∗(ω′1) andω′′2

def= F∗(ω′2) form a basis of E. Note thatω′′1is a canonical symplectic structure on R2 × D. Therefore, (15.8) is equivalentto the following system:

∂v2

∂x1− ∂v1

∂x2= 0,

2v2∂v1

∂x1− v1

∂v1

∂x2− v1

∂v2

∂x1+ 2

∂v2

∂x2+ 2 = 0,

(15.13)

if v21 − 4v2 > 0.


For this equation we have

AE = 1√(u2

1 − 4u2)

∥∥∥∥∥∥∥∥∥u1 0 2u2 0

0 u1 −2 −2

−2 0 −u1 0

2 2u2 0 −u1

∥∥∥∥∥∥∥∥∥ .

(We denoted ui = Ui, i = 1, 2).Note that A2

� = 1 and det A� = 1.The corresponding Monge–Ampère equation is (see (15.10)):

vyvxx − 1

2vxvxy + vyy + 1 = 0.

15.3 Symmetries of Jacobi equations

A diffeomorphism F of the manifold M is called a symmetry of the Jacobiequation E if it takes E to itself.

If E = F〈ω1,ω2〉, this means that

F∗(ωi) ∧ ω1 ∧ ω2 = 0

or, equivalently,

F∗(ω1) = a11ω1 + a12ω2,

F∗(ω2) = a21ω1 + a22ω2,

for some aij ∈ C∞(M) such that det ‖aij‖ �= 0, (i, j = 1, 2).If F is a symmetry of E, and L is a multivalued solution of E, then F(L) is

also a multivalued solution of E.An infinitesimal symmetry of the Jacobi equation E = F〈ω1,ω2〉 is a vector

field X on M such that

LX(ωi) ∧ ω1 ∧ ω2 = 0 (i = 1, 2)

or, equivalently,

LX(ω1) = a11ω1 + a12ω2,

LX(ω2) = a21ω1 + a22ω2,

for some aij ∈ C∞(M) such that det ‖aij‖ �= 0.

15.3 Symmetries of Jacobi equations 329

Note that the functions aij in the last formula are not arbitrary. Indeed, if the2-forms ω1 and ω2 are orthogonal, then

LX(ω1 ∧ ω2) = LX(ω1) ∧ ω2 + ω1 ∧ LX(ω2) = 0,

LX(ω1 ∧ ω1) = ε(�)LX(ω2 ∧ ω2).

Hence,

a12 + ε(�)a21 = 0,

a11 = a22.

Example 15.3.1 The translation

F : (x1, x2, u1, u2)→ (x1 + c1, x2 + c2, u1 + c3, u2 + c4),

where c1, . . . , c4 ∈ R, is a symmetry of the Cauchy–Riemann system (15.4) andof the wave equation (15.6). The translation

F : (x1, x2, u1, u2)→ (x1 + c1, x2 + c2, u1, u2)

is a symmetry of the Jacobi system (15.3), where the coefficients Ai, Bi, Ci, Di, Ei

and Fi (i = 1, 2) do not depend on x1 and x2.

Example 15.3.2 Note that

ω1 = du1 ∧ dx1 + du2 ∧ dx2

in Examples 15.2.1 and 15.2.2 is a symplectic structure on R4. Therefore aHamiltonian vector field XH, which is defined by the equality

XH�ω1 = −dH ,

is a symmetry of the Cauchy–Riemann system and the wave equation if

LXH (ω2) = aω1 + bω2

for some smooth functions a and b on R4.

Theorem 15.3.1 Each element At of the translation group {At}t∈I along tra-jectories of an infinitesimal symmetry X of E is a symmetry of the Jacobiequation E.


Proof From the formula

d

dtA∗t (ω)

∣∣∣∣t=t0

= A∗t0(LX(ω))

it follows that

dωi

dt= A∗t (LX(ωi)) = Ai1(t)ω1(t)+ Ai2(t)ω2(t), (15.14)

where

ωj(t)det= A∗t (ωj), Aij(t)

det= A∗t (aij) (i, j = 1, 2).

Note that

ωi(0) = ωi (i = 1, 2). (15.15)

Therefore, we can solve the Cauchy problem (15.15) for the ordinary differentialequation (15.14):

ωi(t) = Bi1(t)ω1 + Bi2(t)ω2

(i = 1, 2). Here ‖Bij(t)‖i,j=1,2 is a fundamental matrix of the linear system(15.14). So, At is a symmetry of Jacobi equation E for each t. �

15.4 Geometric structures associated withJacobi’s equations

Let E = F〈ω1,ω2〉 be a non-degenerate Jacobi equation. We can choose a basisof E such that ω1 ∧ ω2 = 0 and ω1 ∧ ω1 = εω2 ∧ ω2, where ε = −1 for theelliptic equation and ε = 1 for the hyperbolic one. Then A2

E = ε. We get analmost complex structure in the elliptic case and almost product structure in thehyperbolic case.

The complexification of the tangent space TaM at a ∈ M splits in to the directsum of two skew-orthogonal complex symplectic planes:

(TaM)C = V+(a)⊕ V−(a), (15.16)

15.4 Geometric structures associated withJacobi’s equations 331

where

V+(a)def= {X ∈ (TaM)C | AC

E,aX = −ι√εX},V−(a)

def= {X ∈ (TaM)C | ACE,aX = ι

√εX}.

We call the corresponding distributions V+ and V− characteristic distributions.For hyperbolic equations these distributions can be viewed as being real.

Therefore the non-degenerate Jacobi equation generate a AP-structure on Mwith n = 4, P1 = V+, P2 = V−.

Theorem 15.4.1 A two-dimensional manifold L ⊂ M is a multivalued solutionof a hyperbolic Jacobi equation E, if and only if the tangent spaces TaL for alla ∈ L have one-dimensional intersections h+(a) and h−(a) with real planesV+(a) and V−(a): TaL = h+(a)⊕ h−(a).

The lines h±(a)def= TaL ∩ V±(a)(i, j = 0, 1; i �= j) are called the char-

acteristic directions of the multivalued solution L at the point a and thecorresponding distributions h± are called characteristic distributions of themultivalued solution L.

Theorem 15.4.2 A surface L ⊂ M is a solution of an elliptic Jacobi equationE if and only if L is a complex curve of the almost-complex structure AE .

Let E be a non-degenerate Jacobi equation. Due to (15.16) (see Section 3.5),the modules �s(M)C fall into the direct sum

�s(M)C =⊕

p+q=s

�p,q(M)C (15.17)

and the de Rham differential d : �s(M)C → �s+1(M)C splits in the direct sum

d = d1,0 ⊕ d0,1 ⊕ d2,−1 ⊕ d−1,2,

where d2,−1 and d−1,2 are tensor invariants of the Jacobi equation.


15.5 Conservation laws of Jacobi equations

A differential 1-form θ is called a conservation law of the Jacobi equationE = F〈ω1,ω2〉 if its differential dθ vanishes on each solution of E:

dθ |L = 0.

This means that

dθ = aω1 + bω2

for smooth functions a, b on M.Let L be a solution of E and let θ be a conservation law. Assume that D ⊂ M

is a domain in L, and ∂D is a boundary of D. From the Stokes theorem itfollows that ∫

∂Dθ =

∫D

dθ = 0.

Example 15.5.1 (Cauchy–Riemann systems) The differential 1-forms

θ1 = u1 dx1 + u2 dx2

and

θ2 = u1 dx2 − u2 dx1

are the conservation law of (15.4). Indeed, dθ1 = −ω1 and dθ2 = ω2.

Theorem 15.5.1 If a Jacobi equation E has a conservation law θ such that theform dθ is non-degenerate, then locally E can be written as a Monge–Ampèreequation on T∗M.

Proof The form dθ determines a symplectic structure on M. Due to the Dar-boux theorem, locally there exist canonical coordinates x1, x2, u1, u2 on M suchthat dθ has the form

dθ = du1 ∧ dx1 + du2 ∧ dx2.

Letω1 be a 2-form, such that E = F〈ω1, dθ〉. If we write E in local coordinates,we get Monge–Ampère equations (15.10). �

Let us describe a method of calculation of conservation laws for hyperbolicJacobi equations.

15.5 Conservation laws of Jacobi equations 333

The next theorem follows from the decompositions (15.17).

Theorem 15.5.2 A differential 1-form θ ∈ �1(M) is a conservation law of thehyperbolic Jacobi equation E if and only if

dθ ∈ �2(V+)⊕�2(V−), (15.18)

or using decomposition θ = θ+ + θ−, where θi,j ∈ �1(V±), if and only if

d0,1θ+ + d1,0θ− = 0.

Here P1,0 = V+ and P0,1 = V−.

Example 15.5.2 Let us find conservation laws for (15.13). The distributionsV+ and V− are defined as

V+ =⟨(√

u21 − 4u2 + u1

)∂

∂x1− 2

∂

∂x2+ 2

∂

∂u2,

(√u2

1 − 4u2 + u1

)∂

∂u1+ 2u2

∂

∂u2

⟩,

V− =⟨(√

u21 − 4u2 − u1

)∂

∂x1+ 2

∂

∂x2− 2

∂

∂u2,

(√u2

1 − 4u2 − u1

)∂

∂u1− 2u2

∂

∂u2

⟩.

Moreover, the modules �1(V+) and �1(V−) are generated by the forms

2 dx1 +(

u1 −√

u21 − 4u2

)du1,

2u2 dx2 +(√

u21 − 4u2 − u1

)du1 +

(√u2

1 − 4u2 − u1

)du2,

and

2 dx1 +(

u1 +√

u21 − 4u2

)du1,

− 2u2 dx2 +(√

u21 − 4u2 + u1

)du1 +

(√u2

1 − 4u2 + u1

)du2

respectively.


Therefore, the module �1,1(M) is generated by:

dx1 ∧ du1;

dx2 ∧ du1 + dx2 ∧ du2;

du2 ∧ dx1 − u2 du1 ∧ dx2;

2u2 dx1 ∧ dx2 − 2u2 du1 ∧ du2 + u1 du2 ∧ dx1 + u1u2 du1 ∧ dx2.

Let

θ = a dx1 + b dx2 + c du1 + h du2

be a conservation law of (15.13). Then formula (15.18) implies that the projec-tion of the differential dθ to the free module �1,1(M) is zero. It is easy to seethat

∂a

∂u1= ∂c

∂x1,

∂b

∂u1= ∂c

∂x2,

∂a

∂x2= ∂b

∂x1,

∂a

∂u2= ∂h

∂x1.

15.6 Cauchy problem for hyperbolic Jacobi equations

Let E be a hyperbolic Jacobi equation.A smooth curve K ⊂ M is called a Cauchy data if the tangent to K at each

point a ∈ K is contained neither in V+(a) nor in V−(a). A multivalued solutionL of the hyperbolic Jacobi equation E is called the solution of the Cauchyproblem if L ⊃ K.

A hyperbolic Jacobi equation E is called a (+)-type equation (or (−)-typeequation) if the distribution V+ (or V−) is completely integrable.

Example 15.6.1 Suppose that in (15.9) we have p′(u2) < 0. Then this equation

is hyperbolic. Let f (u2)def= √−p′(u2). The corresponding distributions V± for

15.6 Cauchy problem for hyperbolic Jacobi equations 335

(15.9) are generated by the vector fields

X± = ∂

∂x1∓ f (u2)

∂

∂x2, Y± = f (u2)

∂

∂u1∓ ∂

∂u2.

Here X±, Y± ∈ D(V±). If f ′ = 0 (i.e., p′′ = 0), then Z±def= [X±, Y±] = 0 and

distributions V± are completely integrable.

Let E be a (+)-type equation and let H1, H2 ∈ C∞(M) be functionally inde-pendent 1st integrals of the distribution V+. Then for any multivalued solutionL of E we can find a function H = F(H1, H2) such that the restriction H toL is a constant: H|L = c = constant. Let us consider a three-dimensionalsubmanifold

Mcdef= {H = c} ⊂ M.

Then,

1. the distribution V− is tangent to the submanifold Mc;2. the restriction l− of the distribution V− on Mc is a one-dimensional

distribution:

l− : Mc � a −→ l−(a)def= V−(a) ∩ TaL.

Hence, any invariant with respect to the l− two-dimensional submanifoldL′ ⊂ Mc is a multivalued solution of E.

From the above observation we obtain the following method for the Cauchyproblem.

Let K be a Cauchy data. Consider two independent integrals H1, H2 ∈C∞(M) of the distribution V+. For the functions H1|K, H2|K one can find asmooth function F(x, y) ∈ C∞(R2) such that F(H1|K, H2|K) = 0. By setting

Hdef= F(H1, H2) and M0

def= {H = 0} ⊂ M one obtains on the manifold M0 acharacteristic distribution l−. But K ⊂ M0, hence the manifold

Ldef=⋃a∈K

L(a)

is a solution of the Cauchy problem. Here we denote by L(a) the integral curveof the distribution l− passing through the point a.

PART IV

Applications

16

Non-linear acoustics

In this chapter an application of symmetries and conservation laws to non-linearacoustics are given. This is based on the Khokhlov–Zabolotskaya (KZ) equa-tion, which is the main equation of three-dimensional non-linear acoustics.We investigate the structure of the Lie algebra of contact symmetries of

339

340 Non-linear acoustics

this equation, conservation laws and invariant solutions. This leads to somenon-linear effects: the rise of the gradient catastrophy and self-diffraction.

Here we follow results obtained in [41, 42, 69, 70].

16.1 Symmetries and conservation laws of the KZ equation

16.1.1 KZ equation and its contact symmetries

The propagation of a bounded three-dimensional sound beam is described,taking into account the non-linear properties of medium, by the Khokhlov–Zabolotskaya equation [98]:

∂

∂q1

(∂ρ

∂q2− ρ

∂ρ

∂q1

)= ∂2ρ

∂q23

+ ∂2ρ

∂q24

. (16.1)

Here ρ is a deviation of the density from an equilibrium state,

q1 = c

ε

(t − x

c

), q2 = x, q3 =

√2

εy, q4 =

√2

εz,

and x is the coordinate in the direction of the propagation, y, z are transversecoordinates, c is the velocity of sound in the medium, ε = 1

2 (γ + 2) and γ isthe adiabatic exponent.

The effective differential 4-form, which represents the KZ equation is

ω =1

2dp1 ∧ dq1 ∧ dq3 ∧ dq4 − 1

2dp2 ∧ dq2 ∧ dq3 ∧ dq4

+ u dp1 ∧ dq2 ∧ dq3 ∧ dq4 + p21 dq1 ∧ dq2 ∧ dq3 ∧ dq4

+ dp3 ∧ dq1 ∧ dq2 ∧ dq4 − dp4 ∧ dq1 ∧ dq2 ∧ dq3.

Straightforward calculations show that the Lie algebra gKZ of symmetries forthe KZ equation Eω is generated by the contact vector fields with the followinggenerating functions:

A =(

q1q2 + 3(q23 + q2

4)

4

)p1 + 5

2q2

2p2 + 3q2(q3p3 + q4p4)+ 4q2u+ q1,

S1 = q2p2 + q3p3 + q4p4

2+ u,

S2 = q1p1 + q3p3 + q4p4

2− u,

16.1 Symmetries and conservation laws of the KZ equation 341

R = q4p3 − q3p4,

T = p2,

T1(k) = kp1 + k′,

T3(l) = 2lp3 + l′q3p1 + q3l′′,

T4(m) = 2mp4 + m′q4p1 + q4m′′,

where k = k(q2), l = l(q2) and m = m(q2) are arbitrary smooth functions.Therefore, the Lie algebra of contact symmetries of the KZ equation is infinite

dimensional and consists of point symmetries only.Note that the symmetries S1 and S2 correspond to scale transformations and

R and T correspond to rotation in the plane (q3, q4) and translation along q2

respectively.Below we listed the transformations on the solutions which correspond to

the above symmetries:A

u(q) → u(q)

(1+ (5/2)tq2)8/5− tq1

(1+ (5/2)tq2)2

where

q1 =q1(1+ (5/2)tq2)− 3t/4(q2

3 + q24)

(1+ (5/2)tq2)7/5,

q2 =q2

1+ (5/2)tq2,

q3 =q3

(1+ (5/2)tq2)6/5,

q4 =q4

(1+ (5/2)tq2)6/5.

S1

u(q) → etu(q1, etq2, et/2q3, et/2q4).

S2

u(q) → e−tu(etq1, q2, et/2q3, et/2q4).


R

u(q) → u(q1, q2, q3 cos t − q4 sin t, q4 cos t + q3 sin t).

T

u(q) → u(q1, q2 + t, q3, q4)

T1(k)

u(q) → u(q1 + tk(q2), q2, q3, q4)+ tk′(q2).

T3(k)

u(q) → u(q1 + tk′(q3 + kt), q2, q3 + 2kt, q4)+ tk′′(q3 + kt).

T4(k)

u(q) → u(q1 + tk′(q4 + kt), q2, q3, q4 + 2kt)+ tk′′(q4 + kt).

16.1.2 The structure of the symmetry algebra

As a linear space, the symmetry algebra gKZ is the direct sum

gKZ = g5 ⊕ g∞

of the five-dimensional subspace

g5def= 〈A, S1, S2, R, T〉

and the infinite-dimensional subspace

g∞def= 〈T1(k), T3(l), T4(m)〉.

In this sense, the KZ equation is similar to linear equations, which also have∞-dimensional symmetry algebras. Roughly speaking, the KZ equation canbe thought of as an intermediary between linear equations and truly non-linearequations, which have a few symmetries if any.

Structure of g5

First of all we remark that g5 is closed with respect to the Lagrange bracketand therefore forms a five-dimensional Lie algebra. Moreover all commutators


with S2 and R are vanishing and therefore symmetries S2 and R belong to thecentre of the Lie algebra.

The commutation relations between generators A, S1 and T are given by thetable

A S1 T

A 0 A 5S1 + S2

S1 −A 0 TT −5S1 − S2 −T 0

.

Let

b+ = −2T

5, b− = A, h = 2S1 + 2S2

5,

be the new generators instead of A, S1 and T , then commutation relationsbetween them take the form

[h, b+] = 2b+, [h, b−] = −2b−, [b+, b−] = h.

In other words, we obtain the following description of the Lie algebra.

Proposition 16.1.1 The Lie algebra g5 is a direct sum of the Lie algebra sl2and the abelian two-dimensional Lie algebra:

g5 = sl2 ⊕ R2,

where

sl2 = 〈b+, b−, h〉 and R2 = 〈S2, R〉.

Structure of g∞First we remark that g∞ is closed with respect to the Lagrange bracket andtherefore forms an∞-dimensional Lie algebra. Moreover [g5, g∞] ⊂ g∞, andtherefore g∞ is an ideal in the symmetry Lie algebra.

We can represent g∞ as a g5-invariant direct sum of three copies of C∞(R)under realizations T1, T3 and T4.

Then realization T1 : k(x) −→ k(q2)p1 + k′(q2) gives us a commutativealgebra:

[T1(k), T1(l)] = 0

for all k, l ∈ C∞(R).


Elements of algebra g5 under realization T1 are represented by the followinglinear differential operators:

b+ ⇒ 2

5

d

dx, b− ⇒ −5x2

2

d

dx+ x, h ⇒ −2x

d

dx+ 2

5,

S2 ⇒ 1, R ⇒ 0.

Realization T3 : k(x) −→ 2k(q2)p3+k′(q2)q3p1+q3k′′(q2)gives the followingrepresentation for elements of g5:

b+ ⇒ 2

5

d

dx, b− ⇒ −5x2

2

d

dx+ 3x, h ⇒ −2x

d

dx+ 6

5,

S2 ⇒ 1

2, R ⇒ T3 →−T4

and for T4 : k(x) −→ 2k(q2)p4 + k′(q2)q4p1 + q3k′′(q2) we obtain

b+ ⇒ 2

5

d

dx, b− ⇒ −5x2

2

d

dx+ 3x, h ⇒ −2x

d

dx+ 6

5,

S2 ⇒ 1

2, R ⇒ T4 → T3.

We unify the last two into a single realization of C∞(R, C) : B : k(x)+ιl(x) −→T3(k)+ T4(l).

Then elements b±, h and S2 will have the same representations as above butwith R = −ı.

Denote by �α(R) = { f (x)(dx)α} the module of α-forms on the line. Thenthe Lie derivative gives a representation of the Lie algebra of vector fields onR by differential operators of the form

�α : k(x)d

dx→ k(x)

d

dx+ αk′(x).

This observation shows that T1 corresponds to �−1/5 but T3, and T4 to �−3/5

[B3(k), B4(l)] = 0

and

[B3(k), B3(l)] = 2B1(kl′ − k′l), [B3(k), B3(l)] = 2B3(kl′ − k′l).


This shows that

1. g5 and g∞ are subalgebras of gKZ .2. g∞ is an ideal of gKZ , that is [g5, g∞] ⊂ g∞.3. g∞ = g0 ⊕ g1, where

g0 def= 〈T1(m)〉 and g1 def= 〈T3(k), T4(l)〉.

This decomposition is subject to the commutation relations

[g1, g1] ⊂ g0, [g∞, g0] = 0.

4. [g5, g0] ⊂ g0, [g5, g1] ⊂ g1.5. g5 = g2 ⊕ g3, where

g2def= 〈S2, R〉

is the center of g5, i.e., [g2, g5] = 0, and g3 is the commutator subalgebra, i.e.,

g3def= [g5, g5].

We have seen that g3 is isomorphic to the Lie algebra sl2.

16.1.3 Classification of one-dimensional subalgebras of sl(2, R)

Here we classify elements of g = sl(2, R), with respect to the adjoint action.We recall that each element of sl(2, R) has the from

X = x0h+ x+b+ + x−b−,

where x0, x+, x− ∈ R.Using the commutation relations, we find

adX(h) = [h, X] = 2x+b+ − 2x−b−,

adX(b+) = [b+, X] = −2x0b+ + x−h,

adX(b−) = [b−, X] = 2x0b− − x+h.


Hence, the matrix representation of the operator adX in the basis (h, b+, b−)has the form

adX =∥∥∥∥∥∥

0 x− −x+2x+ −2x0 0−2x− 0 2x0

∥∥∥∥∥∥ .

We remind that the bilinear form

k(Y1, Y2)def= tr(adY1 ◦ adY2),

is called the Killing form.The Killing form has the following important property:

k(adX Y1, Y2)+ k(Y1, adX Y2) = 0,

which says that adX are skew symmetric with respect to k.This property is equivalent to the condition

d

dtk(exp(t adX)Y1, exp(t adX)Y2) = 0,

which means that the Killing form is invariant under the adjoint representation.As a result we see that if Y1 and Y2 are on the same orbit, then

k(Y1, Y1) = k(Y2, Y2).

Moreover, the orbits of the adjoint representation are level surfaces of thequadratic form k(Y , Y).

A straightforward computation gives

k(Y , Y) = 8(y20 + y−y+),

if Y = y0h+ y+b+ + y−b−.This quadratic form is of signature (+,+,−). Therefore the adjoint rep-

resentation can be viewed as the group of Lorenz transformations on thethree-dimensional Minkowski space sl(2, R).

The light cone k(Y , Y) = 0 divides this space into disconnected parts. Finally,we obtain the following list of inequivalent one-dimensional subspaces of thealgebra sl(2, R):

1. 〈b+〉, k = 0,2. 〈b+ − b−〉, k < 0,3. 〈h〉, k > 0.


16.1.4 Classification of symmetries

Now we apply this result to the symmetry algebra gKZ of the KZ equation. Werecall that

gKZ = g5 ⊕ g∞ and g5 = g2 ⊕ g3,

where g∞ is an ideal of the algebra gKZ and g2 is a center of g5 ∼= gKZ/g∞, andg3 is the commutator subalgebra of g5, i.e., g3 = [g5, g5].

Since g3 ∼= sl(2, R), our previous result can be viewed as the list ofinequivalent one-dimensional subalgebras of g3:

1. 〈T〉,2. 〈2T + 5A〉,3. 〈5S1 + S2〉of the Lie algebra g3 under the adjoint representation.

Let us classify one-dimensional subalgebras of g5. Since g5 has a non-trivialcenter, the corresponding Lie group of transformations g5 is not isomorphicto the adjoint representation Ad g5. The Lie algebra of Ad g5 is isomorphicto the quotient algebra g5/g2 ∼= g3 ∼= sl(2, R). The group Ad g5 coincideswith the adjoint action of the group g3 on g5 and leaves invariant each elementof the center g2. Therefore the list of inequivalent one-dimensional subalgebrasof g5 is given by the following list:

1. 〈T + c1S2 + c2R〉,2. 〈2T + 5A+ c1S2 + c2R〉,3. 〈5S1 + S2 + c1S2 + c2R〉,4. 〈c1S2 + c2R〉,where c1, c2 ∈ R.

Let us discuss the classification problem for the full algebra gKZ . Since g∞is an ideal if gKZ and gKZ/g∞ ∼= g5, the obtained results can be viewed asthe list of inequivalent one-dimensional subalgebras of the quotient algebragKZ/g∞:

1. 〈T + c1S2 + c2R+ g∞〉,2. 〈2T + 5A+ c1S2 + c2R + g∞〉,3. 〈5S1 + S2 + c1S2 + c2R + g∞〉,4. 〈c1S2 + c2R+ g∞〉.


This gives the following preliminary classification of one-dimensionalsubalgebras of the full algebra gKZ :

1. 〈T + c1S2 + c2R+ Y〉,2. 〈2T + 5A+ c1S2 + c2R + Y〉,3. 〈5S1 + S2 + c1S2 + c2R+ Y〉,4. 〈c1S2 + c2R+ Y〉,5. 〈Y〉.Here c1, c2 ∈ R and Y ∈ g∞.

In other words, there are five equivalence types, and two subalgebras fromtwo different classes cannot be transformed to each other under the adjointrepresentation of the group generated by the Lie algebra gKZ . On the otherhand, two subalgebras from the same class can be equivalent to each other.

16.1.5 Conservation laws

To find conservation laws for the KZ equation one should solve the equa-tion E(gω) = 0 with respect to the generating function g. Straightforwardcomputations show that

g = a(q2, q3, q3)+ q1b(q2, q3, q4)

where a(q2, q3, q4) is biharmonic with respect to q3, q4 and

b(q2, q3, q4) = w(q3, q4)+q2∫0

(∂2a

∂q23

+ ∂2a

∂q24

)dq2,

where w(q3, q4) is a harmonic function.The corresponding conservation laws are

θg =((up1 − p2)g− 1

2

∂g

∂q1u2)

dq2 ∧ dq3 ∧ dq4 + u∂g

∂q1dq1 ∧ dq3 ∧ dq4

+(

p3g− ∂g

∂q3u

)dq1 ∧ dq2 ∧ dq4 +

(u∂g

∂q4− p4g

)dq1 ∧ dq2 ∧ dq3.

Recall that the solution of the KZ equation represents the deviation from theequilibrium density which, for bounded sound beams, is proportional to thevelocity of the sound beam along the q2-axis. Thus, θg may be viewed as

16.2 Singularities of solutions of the KZ equation 349

the “energy” of a medium whose density is proportional to ∂g/∂q1. Con-sequently, the conservation law θg for g = q1 represents the conservation ofenergy, while for g = 1 it represents the conservation of momentum.

16.2 Singularities of solutions of the KZ equation

16.2.1 Caustics

Here we construct solutions invariant under the abelian subalgebra

g =⟨q3

∂

∂q1+ 2q2

∂

∂q3, q4

∂

∂q1+ 2q2

∂

∂q4

⟩.

The corresponding generating functions of the vector fields are

f = p1q3 + 2p3q2 and g = p1q4 + 2p4q2

respectively.Each g-invariant solution ρ(q1, . . . , q4) has the form

ρ = v(x, y),

where

x = q2, y = 4q1q2 − (q23 + q2

4)

are invariants for g.For such type of solutions the KZ equation has the form

2∂v

∂y− 4x2

(v∂2v

∂y2+(∂v

∂y

)2)+ x

∂2v

∂x∂y+ y

∂2v

∂y2= 0.

The order of this equation can be reduced by integrating with respect to y:

x∂v

∂x+ (y − 4x2v)

∂v

∂y+ v = h(x),

where h = h(x) is an arbitrary function.We shall consider solutions v(x, y) such that v(x, y)→ 0 when (x, y)→∞.

For these solutions h(x) ≡ 0.


This leads us to the following first-order equation

x∂v

∂x+ (y− 4x2v)

∂v

∂y+ v = 0.

After substitution

v = 1

xw(η, θ), η = y

x, θ = ln x

we obtain the classical Euler equation

∂w

∂θ= 4w

∂w

∂η.

Solutions of this equation can be described as follows: for a given initial density

w|θ=0 = w0(η)

one can find w = w(η, θ) from the equation

w = w0(η + 4wθ)

or from the system

η + 4θw = ς ,

w = w0(ς).(16.2)

Geometrically it can represented as follows. Let us fix η and θ and let us ploton the plane (ς , w) the graph of the line l = l(η, θ) given by the first equationin (16.2) and the graph K of the function w = w0(ς).

If the line l and the curve K have only one common point, then the system(16.2) has only one solution (ζ0, w0). This means that w (and, therefore, v also)is a one-valued function. We can see this situation if the initial curve w0(η) isconvex or concave.

If the line l intersects the curve K at several points then w is a multivaluedfunction. In this case caustics arise. Physically the appearance of a causticmeans additional acoustic waves in a domain that is bounded by this caustic(see Figure 16.1).

Let us describe the process causing the appearance of caustics in geometricalterms.


w

l

0

K

��0

Figure 16.1. How multivalued solutions arise.

Let us now fix θ only. It is equivalent to the corresponding cross-section ofthe sound beam by the plane q2 = constant and η shall increase monotonic-ally in time. From the geometrical point of view this means that we fix theslope of the line l and then translate it parallel to itself by increasing the ordin-ate w. Then, up to a certain moment, K and l will intersect at only one point:as η increases further, the solution becomes multivalued. The correspondingtimes when “gradient” catastrophes occur can be computed using the givensystem (16.2).

16.2.2 Contact shock waves

Here we use the conservation laws to describe the propagation of perturbationsin non-linear acoustics.

Let M0 ⊂ R4 be a three-dimensional submanifold that divide R4 into twodomains M+ and M−:

R4 = M+ ∪M0 ∪M−.

Let us consider so-called contact shock waves.They are continuous solutions of the KZ equation. That is, we consider solu-

tions of the form v = v+ ∪ v−, where v+ is defined on M+ and v− on M−respectively.


Assume that the boundary of the beam is given by the function S(q). In thiscase the Hugoniot–Rankine condition

[θ+|M0 − θ−|M0 ] ∧ dS = 0

yields the Hamilton–Jacobi equation

v+(q)(∂S

∂q1

)2

− ∂S

∂q1

∂S

∂q2+(∂S

∂q3

)2

+(∂S

∂q4

)2

= 0 (16.3)

independent of the choice of conservation law.Given an arbitrary background solution v+(q), the solution S(q) of this

equation can only be found by numerical methods.We shall examine in more detail the case of a zero background v+ = 0,

assuming that the initial perturbation at q2 = 0 is localized inside a disk whoseradius changes according to a given law R(q1), that is,

S|q2=0 = S0(q1, q3, q4) = q23 + q2

4 − R2(q1).

Solving the Hamilton–Jacobi equation with this initial value, we find that atthe cross-section q2 = c = constant the perturbation is localized in a disk ofradius

R1(q1, q2) =∣∣∣∣R(z)+ 2q2

R′(z)

∣∣∣∣ (16.4)

where z, 0 � z � q1 is a solution of the equation:

q1 − z

q2=(

1

R′(z)

)2

. (16.5)

Consider now several particular cases of R(q1).

1. R(q1) = R0 + εq1, the radius of the perturbation domain grows with linearspeed ε for the cross-section q2 = 0. In this case, at the cross-section q2 =constant > 0 the radius of the perturbation disc also grows with speed ε intime and ε−1 in q2:

R1(q1, q2) = R0 + εq1 + 1

εq2.


3π2

– 3π2

π2

– πO

Figure 16.2. How self-diffraction arises.

Figure 16.3. Self-diffraction.

It follows that, in order to get a stable beam, it is necessary that for theboundary of the initial perturbation to perform self-sustained oscillations.

2. R(q1) = R0 + ε sin(ωq1). In this case (16.5) for z takes the form

ε2ω2 q1 − z

q2= 1

cos2(ωz).

Depending on

γdef= ε2ω2 q1

q2

the equation admits a distinct number of solutions on the interval 0 � z � q1.Fixing q2 we see that as q1 grows, the sound beam necessarily separatesinto layers – the phenomenon of “self-diffraction” of sound beams (seeFigure 16.2).The time at which separation occurs and the radius of the rings that are thusformed can be computed using relation (16.4) (see Figure 16.3).


3. S0(q1, q3, q4) = (Aq, q)− η(q1), where q = (q3, q4), A is a symmetric pos-itive matrix, (·, ·) is the scalar product and η(q1) is the given law governingthe evolution of the boundary. Then

S0(q1, q2, q) = S0(q∗1, q∗),

where q∗1, q∗ and q1, q2, q are connected by

q1 − q∗1 =4q2

(η′(q∗1))2(Aq∗, q∗),

q =(

1+ 4q2

η′(q∗1)A

)q∗.

Let us consider in more detail the case

η(q1) = η0 + εq1,

where η0 ∈ R. Then we have

S(q1, q2, q) = (ABq, q)+ η0 − εq1

with

B =(

1+ 4q2

εA

)−1

.

From this relation it follows that at the cross-section q2 = constant theperturbation is localized in the domain obtained from the section q2 = 0 byapplying the affine transformation with the matrix

(1+ 4q2

εA

)−1/2

.

This means that if we look, say, at the section

q2 = −4λ

ε,

where λ is an eigenvalue of A, then the shock wave shall be focused in thecorresponding eigendirection (see Figure 16.4).


–

Figure 16.4. The shock wave collapses.

For example, for a circular perturbation A = a1 the shock wave collapses atthe section

q2 = −4a

ε.

17

Non-linear thermal conductivity

Here we consider a problem of group classification for the non-linear thermalconductivity equation. This is the model equation for many processes: heatand mass transfer, theory of combustion, biology and ecology. It describesspecies propagation in natural habitat, membrane ion transport, nerve impulsepropagation, the spread of chemical concentration waves, self-organization inbiochemical systems, the formation of apex zones in plants, non-linear effectsin plasma and so on (see [6, 20, 87, 99]).

17.1 Symmetries of the TC equation

17.1.1 TC equation

The equation

vt = (vαvx)x + F(v) (17.1)

356

17.1 Symmetries of the TC equation 357

is often called the reaction–diffusion equation or the equation of non-linearthermal conductivity.

We will call it the thermal conductivity equation (TC equation). This isthe Kolmogorov–Petrovsky–Piskunov equation [48] with non-linear diffusioncoefficient.

Here α is an arbitrary real number, vα is a diffusion coefficient and F(v) is theso-called source function. We consider only the case of non-linear diffusion, i.e.,

α �= 0.

For the differential TC equation we have the following effective differentialform:

ω = uα dq1 ∧ dp2 + (αuα−1p22 − p1 + F(u))dq1 ∧ dq2.

Here q1 = t and q2 = x.

17.1.2 Group classification of TC equation

Here we follow [63] and [49]. Note that (17.1) has infinitesimal symmetrieswith generation functions f1 = p1 and f2 = p2 for all α and F(v).

Below we list equations with non-trivial algebras of symmetries. There areten classes of the equtations, all others have the symmetries p1 and p2 only.

Case 1

F(v) = 0, α �= 0, α �= −4

3.

The TC equation has the form

vt = (vαvx)x .

The basis of gTC is

f1 = p1, f2 = p2, f3 = u+ αq1p1, f4 = 2u− αq2p2,

with the following Lie algebra structure:

X1 X2 X3 X4

X1 0 0 −αX1 0X2 0 0 0 αX2

X3 αX1 0 0 0X4 0 −αX2 0 0

358 Non-linear thermal conductivity

Here Xidef= Xfi are contact vector fields with the generating functions fi,

i = 1, 2, 3, 4.

Case 2

F(v) = 0, α = −4

3.


vt = (v−4/3vx)x .

The basis of gTC is

f1 = p1, f2 = p2, f3 = 3u− 4q1p1, f4 = 3u+ 2q2p2, f5 = q22p2 + 3q2u,


X1 X2 X3 X4 X5

X1 0 0 4X1 0 0X2 0 0 0 −2X2 −X4

X3 −4X1 0 0 0 0X4 0 2X2 0 0 −2X5

X5 0 X4 0 2X5 0

Case 3

F(v) = γ v, α �= 0, α �= −4

3, γ �= 0.


vt = (vαvx)x + γ v.

The basis of gTC is

f1 = p1, f2 = p2, f3 = αq2p2 − 2u, f4 = (p1 − γ u) exp(−αγ q1),


X1 X2 X3 X4

X1 0 0 0 αγX4

X2 0 0 −αX2 0X3 0 αX2 0 0X4 −αγX4 0 0 0


Case 4

F(v) = γ v, α = −4

3, γ �= 0.


vt = (v−4/3vx)x + γ v.

The basis of gTC is

f1 = p1, f2 = p2, f3 = 3u+ 2q2p2, f4 = q22p2 + 3q2u,


X1 X2 X3 X4

X1 0 0 0 0X2 0 0 −2X2 −X3

X3 0 2X2 0 −2X4

X4 0 X3 2X4 0

Case 5

F(v) = µvβ , α �= 0, β �= α + 1, β �= −1

3, β �= 1.


vt = (vαvx)x + µvβ . (17.2)

The basis of gTC is

f1 = p1, f2 = p2, f3 = 2β − 1

β − α − 1q1p1 + q2p2 + 2

β − α − 1u,


X1 X2 X3

X1 0 0 −2β − 1

β − α − 1X1

X2 0 0 −X2

X3 2β − 1

β − α − 1X1 X2 0

(17.3)


Case 6

F(v) = µv−1/3, α = −4

3, µ > 0.


vt = (v−4/3vx)x + µv−1/3.

The basis of gTC is

f1 = p1, f2 = p2, f3 = 3u− 4q1p1,

f4 =(

u+ 1√3µ

p2

)exp

(2

√µ

3q2

),

f5 =(

u− 1√3µ

p2

)exp

(−2

√µ

3q2

),


X1 X2 X3 X4 X5

X1 0 0 4X1 0 0

X2 0 0 0 −2

3

√3µX4

2

3

√3µX5

X3 −4X1 0 0 0 0

X4 02

3

√3µX4 0 0 − 4

3√

3µX2

X5 0 −2

3

√3µX5 0

4

3√

3µX2 0

Case 7

F(v) = µv−1/3, α = −4

3, µ < 0.


vt = (v−4/3vx)x + µv−1/3.


The basis of gTC is

f1 = p1, f2 = p2, f3 = 3u− 4q1p1,

f4 = u cos

(2

√−µ

3q2

)+ p2

1√−3µsin

(2

√−µ

3q2

),

f5 = u sin

(2

√−µ

3q2

)− p2

1√−3µcos

(2

√−µ

3q2

),


X1 X2 X3 X4 X5

X1 0 0 4X1 0 0

X2 0 0 02

3

√−3µX5 −2

3

√−3µX4

X3 −4X1 0 0 0 0

X4 0 −2

3

√−3µX5 0 0 − 2

3√−3µ

X2

X5 02

3

√−3µX4 02

3√−3µ

X2 0

Case 8

F(v) = γ v + µv1+α , α �= 0, α �= −4

3.


vt = (vαvx)x + γ v + µv1+α .

The basis of gTC is

f1 = p1, f2 = p2, f3 = (p1 − γ u) exp (−αγ q1) ,


X1 X2 X3

X1 0 0 αγX3

X2 0 0 0X3 −αγX3 0 0


Case 9

F(v) = γ v + µv−1/3, α = −4

3, γ �= 0, µ > 0.


vt = (v−4/3vx)x + γ v + µv−1/3.

The basis of gTC is

f1 = p1, f2 = p2, f3 = (p1 − γ u) exp

(4

3γ q1

),

f4 =(

u+ 1√3µ

p2

)exp

(2

√µ

3q2

),

f5 =(

u− 1√3µ

p2

)exp

(−2

√µ

3q2

),


X1 X2 X3 X4 X5

X1 0 0 −4

3γX3 0 0

X2 0 0 0 −2

3

√3µX4

2

3

√3µX5

X34

3γX3 0 0 0 0

X4 02

3

√3µX4 0 0 − 4

3√

3µX2

X5 0 −2

3

√3µX5 0

4

3√

3µX2 0

Case 10

F(v) = γ v + µv−1/3, α = −4

3, γ �= 0, µ < 0.


vt = (v−4/3vx)x + γ v + µv−1/3.


The basis of gTC is

f1 = p1, f2 = p2, f3 = (p1 − γ u) exp

(4

3γ q1

),

f4 = u cos

(2

√−µ

3q2

)+ p2

1√−3µsin

(2

√−µ

3q2

),

f5 = u sin

(2

√−µ

3q2

)− p2

1√−3µcos

(2

√−µ

3q2

),


X1 X2 X3 X4 X5

X1 0 0 −4

3γX3 0 0

X2 0 0 02

3

√−3µX5 −2

3

√−3µX4

X34

3γX3 0 0 0 0

X4 0 −2

3

√−3µX5 0 0 − 2

3√−3µ

X2

X5 02

3

√−3µX4 02

3√−3µ

X2 0

17.2 Invariant solutions

Here we find exact solutions using symmetry algebras gTC .We begin with some elementary remarks. Let us consider, for example, the

equation

vt = (vvx)x + µ, (17.4)

where µ is a constant. This equation is a special case of the classification whenα = 1 and β = 0.

First of all solutions which are invariant with respect to the symmetry f =bp1 − ap2, a, b ∈ R, are exactly traveling wave solutions, i.e., the solutions ofthe form

v(t, x) = �(at + bx).


The structure of the Lie algebra gTC of (17.4) is as follows:

X1 X2 X3

X1 0 0 −X1

X2 0 0 −X2

X3 X1 X2 0

Note that

Xf = bX1 − aX2 = −b∂

∂q1+ a

∂

∂q2.

Since

[Xf , X1] = [Xf , X2] = 0

and

[Xf , X3] = b[X1, X3] − a[X2, X3] = −bX1 + aX2 = −Xf ,

we see that the normalizer of the subalgebra g = 〈Xf 〉 coincides with gTC :Ng = gTC . Let

g1 = f2 = p2, g2 = f3 = q1p1 + q2p2 − u

and

= { f = bp1 − ap2 = 0, q1 = 0} .

Then is invariant under the vector fields

Xg1 = −q1∂

∂q1− q2

∂

∂q2− u

∂

∂u

and

Xg2 = −∂

∂q2.

The variables

ydef= u, x

def= q2, pdef= p2


can be choosen as coordinates on . Note that

Xf �ω = −bu dp2 − b(p22 − p1 + µ) dq2 − a(p2

2 − p1 + µ) dq1

and

ω2 = −1

b(Xf �ω)| = y dp+

(p2 − a

bp+ µ

)dx.

Therefore the reduced equation on has the form

(yy′)′ − a

by′ + µ

b= 0. (17.5)

This equation, by construction, has a two-dimensional Lie algebra of contactsymmetries. Namely, the Lie algebra generated by vector fields

Y0def= Xg1 | = −

∂

∂x, Y1

def= Xg2 | = −x∂

∂x− y

∂

∂y.

The algebra can be used for integrating equation (17.5) in quadratures. Usingthe method described in Part I, we pass to the forms

ω1 = dx − x

ydy− b

xp− y

p2b− ap+ µdp,

ω2 = −1

ydy− bp

p2b− ap+ µdp.

Note that dω2 = 0 and, therefore, there exists a function H1 such that ω2 = dH1.If we put µ = a2/4b, then

H1(x, y, p) = 2ab(1− p)

(2bp− a)(2b− a)− ln(y(2bp− a)).

To find the restriction of the form ω1 to the surface H1(x, y, p) = C1, wenote that

y = 1

C1(2bp− a)exp

(2ab(1− p)

(2bp− a)(2b− a)

)and

ω1|H1=C1 = dx + 4b2 exp

(2ab(1− p)

(2bp− a)(2b− a)

)[C1(2bp− a)3]dp.


The last form is closed too, and its integral is

H0(x, p) = x + 4b(bp− a)

(2bp− a)a2C1exp

(2ab(1− p)

(2bp− a)(2b− a)

).

From the equation H0(x, p) = C2 we get

p(x) = a(2+ LambertW(a2C1/2b exp(2(a− b)/(2b− a))(x − C2)))

2b(LambertW(a2C1/2b exp(2(a− b)/(2b− a))(x − C2))+ 1).

(17.6)

Here LambertW is the special function (see Figure 17.1) defined by

LambertW(x) exp(LambertW(x)) = x.

From (17.6) we obtain the general solution of the differential equation (17.5):

y(x) = a

2b(x + c2)

(1+ 1

LambertW(c1(x + c2))

), (17.7)

where

c1 = 1

2ba2C1 exp

(−2

b− a

2b− a

), c2 = −C2.

So, the curve L has the form (17.7). We get the corresponding solution of(17.4) as the solution of the equation f = 0, i.e.,

but − aux = 0,

– – – –

Figure 17.1. The function y(x) = x(1+ (1/(LambertW(x)))).


with the Cauchy data:

u(0, x) = a

2b(x + c2)

(1+ 1

LambertW(c1(x + c2))

).

This solution is

u(t, x) = a

2b2(at + bx + c2)

(1+ 1

LambertW(c1(at + bx + c2))

).

Below we list some other exact solutions of the non-linear diffusion equation.Here C, C1 and C2 are arbitrary constants.

1. The equation:

vt = (vαvx)x . (17.8)

The symmetry:

f = 2u− αq2p2.

The f -invariant solution:

v(t, x) = x2/α(

C − 2(α + 2)

αt

)−1/α

.

2. The equation:

vt = (v−4/3vx)x + γ v.

The symmetry:

f = q22p2 + 3q2u.


v(t, x) = Cexp(γ t)

x3.

3. The equation:

vt = (vαvx)x + γ v.


Figure 17.2. The equation vt = (vvx)x + v, with the solution v(t, x) =− 1

6 x2et+2/(et+2 − 1).

The symmetry:

f = 2u− αq2p2.


v(t, x) = x2/α exp((t + C)γ )

(−1

2

α2γ

(2+ α)(exp((t + C)αγ )− 1)

)1/α

.

Figure 17.2 presents the graph of the f -automodel solution.

4. The equation:

vt = (vvx)x + γ v+ µv2.

The generating function of symmetry:

f = ( p1 − γ u) exp(−γ q1).

If µ > 0, then the f -invariant solution:

v(t, x) = exp(γ t) 4

√2C1

µ(tan2(C2 −√

2√µx)+ 1)

.


Figure 17.3. The equation vt = (vvx)x + v + 12 v2, with the solution v(t, x) =√

2et(tan2 x + 1)−1/4.

If µ < 0, then

v(t, x) = exp(γ t)

×√

1√−2µexp(−√−2µx − C2)(exp(2

√−2µx + 2C2)+ C1).

Figure 17.3 shows the solution.Note that if we put γ = 1 and µ = −1, we obtain the Fisher equation witha non-linear diffusion coefficient:

vt = (vvx)x + v(1− v).

The corresponding solution is

v(t, x) = exp(t)

√1√2

exp(−√2x − C2)(exp(2√

2x + 2C2)+ C1).

5. The equation:

vt = (v2vx)x + γ v+ µv3.

The symmetry:

f = (p1 − γ u) exp(−2γ q1).

If µ > 0, then the f -invariant solution:

v(t, x) = exp(γ t)(C1 cos(√

3µx)− C2 sin(√

3µx))1/3.


If µ < 0, then

v(t, x) = exp

(γ t −

√−3µ

3x

)(C1 exp(2

√−3µx)+ C2)1/3.

6. The reduced equation for traveling wave solutions v(t, x) = y(at + bx) of(17.2) is

( yαy′)′ − a

b2y′ + µ

b2yβ = 0.

Here y = y(z), z = at + bx. This ODE inherits the symmetries (17.2)

X1 = ∂

∂z,

X2 = −z∂

∂z− 1

αy∂

∂y+(

1− 1

α

)p∂

∂p.

For β = 1− α and a = 0, we get a first integral

H = ( y′)2y2α + µ

b2y2

of the reduced equation

( yαy′)′ + µ

b2y1−α = 0.

For more solutions of (17.1) see [49].

18

Meteorology applications

The results in this section are based on the recent contributions [93, 94] tothe ongoing research into balanced models in geophysical fluid dynamics, inwhich the objective is to understand the properties of the equations that assistsolution strategies. The relationships between the ellipticity of the operators,the Hamiltonian structure of the evolution equations and the stability of flowsdescribed by such balanced models are subtle and our understanding of theseissues is far from complete. This chapter only illustartes how geometry can offerinsights into some of these problems. For example, the quaternionic structureassociated with a Monge–Ampère equation facilitates an understanding of whycomplex coordinates arise in a natural way in these models – something thatwould be difficult to deduce from the physics alone.

371

372 Meteorology applications

18.1 Shallow water theory and balanced dynamics

The motion of a typical particle in shallow water theory can be described byexpressing the current Cartesian horizontal coordinates

x = x(a, b, t), y = y(a, b, t) (18.1)

as functions, on the right, of the Lagrangian particle labels a, b and the time t.The incompressibility hypothesis requires the current depth, or height, h to bea function h(a, b, t) with the property

h(a, b, 0)

h(a, b, t)= ∂(x, y)

∂(a, b), (18.2)

where the Jacobian on the right is that of the mapping (18.1). The time derivativeof (18.2) following a particle gives the differential continuity equation.

The equations of horizontal momentum balance for flows over a bedwhich is rotating with position-dependent Coriolis parameter f (y) (where they-coordinate denotes north) are

x + g∂h

∂x− yf = 0,

y+ g∂h

∂y+ xf = 0.

(18.3)

Here g is a given constant, representing the combined effect of the accelera-tion due to gravity and a centrifugal component due to the Earth’s rotation andthe overdot denotes the time derivative following a particle.

Another important kinematic concept is the so-called potential vorticitydefined by

ξ = 1

h

(∂ y

∂x− ∂ x

∂y+ f

), (18.4)

which is conserved on particles. Common choices of the Coriolis parameter aref = constant or f = βy, (β ∈ R) as approximations to 2� sin φ, depending onthe purpose. Here β and � are constants1 (related to the spin of the Earth) andφ is latitude.

The semi-geostrophic approximation to (18.3), in the case when f is a con-stant, is the replacement of the true acceleration by the time derivative of another

1 Note that to avoid confusion we also use � to denote a symplectic structure.

18.1 Shallow water theory and balanced dynamics 373

vector

ug = −g

f

∂h

∂y, vg = g

f

∂h

∂x, (18.5)

following the particle. This vector is a notional velocity, called the geostrophicvelocity. The semi-geostrophic approximation seeks to find motions satisfying

ug + g∂h

∂x− yf = 0,

vg + g∂h

∂y+ xf = 0,

(18.6)

together with the continuity equation obtained from (18.2). Associated withthese equations, the potential vorticity

% = 1

h

[f + ∂vg

∂x− ∂ug

∂y+ 1

f

∂(ug, vg)

∂(x, y)

](18.7)

is conserved.We introduce a transformation of coordinates

X = x + g

f 2

∂h

∂x, Y = y+ g

f 2

∂h

∂y. (18.8)

For a thorough discussion of this transformation, including its Legendre andcontact properties, see Roulstone and Sewell [97]. The coordinates (X, Y) arecalled geostrophic coordinates because, when f is a constant, X = ug, Y = vg.Motivated by the issue of the dependence of the Coriolis parameter, f , on latit-ude, Salmon [101] studied certain generalized semi-geostrophic equations withpseudo-Hamiltonian form in (X , Y)-space, namely

X = − 1

f (X , Y)

∂�

∂Y, Y = 1

f (X , Y)

∂�

∂X, (18.9)

where

�(X , Y , t) = 1

2(u2

g + v2g)+ gh. (18.10)

The results of Salmon [102] are also of interest when f is independentof position (in which case (18.9) can easily be written in canonical form)


and henceforth, in this chapter, we consider f to be a given constant. In thecoordinates (18.8) the potential vorticity takes the Jacobian form (noting (18.5))

% = f

h

∂(X , Y)

∂(x, y)= 1

h

[f + ∂vg

∂x− ∂ug

∂y+ 1

f

∂(ug, vg)

∂(x, y)

]. (18.11)

The definition (18.11) can be rewritten as

% = gf

(H − 12 f (x2 + y2))

det Hess(H), (18.12)

where

H = 1

2f (x2 + y2)+ gh

and

Hess(H) =

∥∥∥∥∥∥∥∥∥∂2H

∂x2

∂2H

∂x∂y

∂2H

∂y∂x

∂2H

∂y2

∥∥∥∥∥∥∥∥∥ .

This is the starting point for a Monge–Ampère type equation in which H(x, y, t)is the unknown, to be found in conjunction with suitable boundary conditions.

18.2 A geometric approach to semi-geostrophic theory

Consider fluid flow in a region D ⊂ R2. The cotangent bundle T∗R2 hascoordinates q1

def= x, q2def= y, p1 = p, p2 = q which span the phase space of a

fluid obeying Newton’s second law (the shallow water equations as above).The transfromation

F : (q1, q2, p1, p2) → (Q1 = q1 + p1, Q2 = q2 + p2, P1 = p1, P2 = p2)

is a symplectic, and the function

fdef= 1

2(p2

1 + p22)

is a generating function for F.

18.2 A geometric approach to semi-geostrophic theory 375

Taking a suitably scaled geopotential function φ in most applications, weobtain a natural lift Fc of the transformation F to the contact space of 1-jets:

Fc : F : (q1, q2, u, p1, p2)

→(

Q1 = q1 + p1, Q2 = q2 + p2, V = u+ 1

2(p2

1 + p22),

P1 = p1, P2 = p2

).

The map Fc is a contact transformation.The graph of the geopotential φ is a Legendrian submanifold

L ={

u = φ, p1 = ∂φ

∂q1, p2 = ∂φ

∂q2

}in J1D.

Let ω be an effective 2-form in J1D such that being evaluated on the graphof the geopotential φ yields a Monge–Ampère equation

�ω(φ) = 0.

We note that the contact transformation Fc, when restricted to the graphof φ, is the geostrophic momentum transformation, and in meteorologicalterminology, the vector field

S = −p2∂

∂q1+ p1

∂

∂q2= ug

∂

∂q1+ vg

∂

∂q2

is known as the geostrophic wind.The semi-geostrophic equations can be written in the “pseudo-Hamiltonian”

form

Q1 = − ∂�

∂Q2, Q2 = ∂�

∂Q1, (18.13)

where

d

dt= ∂

∂t+ q1

∂

∂q1+ q2

∂

∂q2= ∂

∂t+ Q1

∂

∂Q1+ Q2

∂

∂Q2. (18.14)

The mapping {q1, q2} → {Q1, Q2} is a Legendre transformation

Q1 = ψq1 , Q2 = ψq2 ,


where

ψ = φ + 1

2(q2

1 + q22)

and

q1 = �Q1 , q2 = �Q2

with � = 12 (Q

21 + Q2

2)−�.The singularities of this map can be interpreted as atmospheric fronts [15, 16].The conservation law for the potential vorticity can be expressed (using the

Hamiltonian representation of the semi-geostrophic equations and a continuityequation), as

d

dt

(∂(q1, q2)

∂(a, b)

)= d

dt(1+ φq1q1 + φq2q2 + det Hessφ) = 0.

The Monge–Ampère operator corresponding to the conservation law canbe described as a pull-back of 2-form dQ1 ∧ dQ2 ∈ �2(J1(D)) under thetransformation F:

F∗(dQ1 ∧ dQ2) = d(q1 + p1) ∧ d(q2 + p2)

= dq1 ∧ dq2 + dq1 ∧ dp2 + dp1 ∧ dq2 + dp1 ∧ dp2

and the corresponding Monge–Ampère equation is

1+ φq1q1 + φq2q2 + det Hessφ = 0.

Now we show how this equation relates to a classical geometric structure onthe phase space T∗R2 = R4. Namely, it is a part of hyper-Kähler triplet of theMonge–Ampère equations introduced in [94, 95] which we will explain in thenext section.

18.3 Hyper-Kähler structure and Monge–Ampère operators

Letω ∈ �2ε(R

4) be an effective 2-form representing an elliptic Monge–Ampèreoperator �ω : C∞ → �2(R2) with constant coefficients.

Positivity of the Pfaffian Pf(ω) > 0 (in an open domain of a point x ∈R4) provides the ellipticity condition for the corresponding Monge–Ampèreoperator�ω and as was shown the existence of almost complex structure in R4.

18.3 Hyper-Kähler structure and Monge–Ampère operators 377

We will identify the points (x, p, y, q) ∈ R4 with quaternions l ∈ H:

l = x + ip+ jy + kq

and the points (x, p), (y, q) ∈ R2 with z, w ∈ C × C = C2. Then using thestandard multiplication of quaternions we have

l = z + jw, z = x + ip, w = y + iq.

Let

l = x − ip− jy − kq = z − jw

be the quaternion conjugated to l. Then there are three Kähler formsωI ,ωJ ,ωK ∈ �2(R4) such that

−1

2dl ∧ dl = iωI + jωJ + kωK .

These forms are expressed in the coordinates in R4 as

ωI = dx ∧ dp+dy ∧ dq, ωJ = dx ∧ dy+dq ∧ dp, ωK = dx ∧ dq+dp ∧ dy.

The 2-forms have a many remarkable features, the most important of which istheir self-duality with respect to the Hodge star-operator∗ : �2(R4) → �2(R4)

via the Euclidian standard metric on R4.Let us consider the corresponding Monge–Ampère operators. It is easy to

see that

ωI = dx ∧ dp+ dy ∧ dq = �

and hence �ωI ≡ 0.The other two operators are elliptic:

�ωK = �, �ωJ = 1− detHess,

where � is the usual Laplace operator in R2 and detHess is a determinant ofthe second derivatives matrix for a “potential” φ. It is easy to check that thetransposed quaternion wedge product dl ∧ dl produces an anti-self-dual triple


of 2-forms:

−1

2dl ∧ dl = iωI + jωJ + kωK .

ωI = dx ∧ dp− dy ∧ dq,

ωJ = dx ∧ dy − dq ∧ dp,

ωK = dx ∧ dq − dp ∧ dy.

The latter triple encodes the hyperbolic Monge–Ampère operator with con-stant coefficients in R2. In complex coordinates z, w self-dual forms have thefollowing expressions:

ωI = ι

2(dz ∧ dz + dw ∧ dw),

ωJ = 1

2(dz ∧ dw+ dz ∧ dw),

ωK = − ι

2(dz ∧ dw− dz ∧ dw),

and we have that

ωJ + ιωK = dz ∧ dw ∈ �2,0(C2)

is a holomorphic symplectic form on C2 with respect to the complex structure i.We consider C2 as a hyper-Kähler manifold endowed with three complex

structures I , J , K subjected to quaternionic algebra relations. The twistor spaceof C2 is the product P : CP1. It admits the structure of a complex contactthree-dimensional manifold.

As we mentioned above, the set of all elliptic Monge–Ampère operators arein one-to-one correspondence with the set of all (almost) complex structuresin C2 and this correspondence is realized as a homeomorphism between theGrassmanian LGω associated with an elliptic Monge–Ampère operator �ω andthe projective space CP1. The precise correspondence goes as follows. Let usparametrize the points of CP1 = S2 by a non-zero complex number ξ such thatthe Cartesian coordinates (a, b, c) ∈ R3, a2 + b2 + c2 = 1 are related to ξ by

(a, b, c) =((1− ξ ξ )

1+ ξ ξ,ξ + ξ

1+ ξ ξ,ι(ξ − ξ )

1+ ξ ξ

).

18.3 Hyper-Kähler structure and Monge–Ampère operators 379

Denote the corresponding complex structure endomorphism by Iξ :

Iξ :

(zw

)→

ι(1− ξ ξ )

1+ ξ ξz + 2ξ

1+ ξ ξw

−2ξ

1+ ξ ξz − ι(1− ξ ξ )

1+ ξ ξw

.

It is easy to check that the self-dual triple ωI ,ωJ ,ωK is invariant with respectto the transformation induced by Iξ :

I∗ξ (ωI) = ωI , I∗ξ (ωJ) = ωJ , I∗ξ (ωK ) = ωK .

The “general” elliptic Monge–Ampère operator �ωξ corresponding to thepoint ξ ∈ CP1 is defined by the effective 2-form

ωξ = ξ + ξ

1+ ξ ξωJ + ι(ξ − ξ )

1+ ξ ξωK .

Its Pfaffian is

Pf(ωξ ) = 4(α2 + β2)

(1+ α2 + β2)2,

where ξ = α + ιβ.Introduce a new “complex coordinates” associated with Iξ :

zξ = z + ξ w, wξ = w− ξ z.

We have

dzξ ∧ dwξ = d(z + ξ w) ∧ d(w− ξ z)

= dz ∧ dw− ξ(dz ∧ dz + dw ∧ dw)+ ξ2 dz ∧ dw)

= ωJ + ιωK + 2ιξωI + ξ2(ωJ − ιωK )

= (1+ ξ2)ωJ + (1− ξ2)ιωK + 2ιξωI .

The corresponding effective 2-form is given by a quadratic pencil ofMonge–Ampère operators

ωξ = (1+ ξ2)ωJ + (1− ξ2)ιωK

within the initial hyper-Kähler triplet of forms.


Now we consider the canonical transformation for “near-local balancedmodels” (see the next subsection):

X = x + ιcq + ap, Y = y − ιcp+ aq, (18.15)

where a, c ∈ R, ι = √−1 and (p, q) = (g ∂h/∂x, g ∂h/∂y) if we define z =x+ ιcq, w = y− ιcp on a two-dimensional complex manifold, then there existsa hyper-Kähler triple of 2-forms

ωI = *(dz ∧ dw) = dx ∧ dy− c2 dp ∧ dq, (18.16)

ωJ = ι

2(dz ∧ d ˆz + dw ∧ d ˆw) = c(dx ∧ dq + dp ∧ dy), (18.17)

ωK = ,(dz ∧ w) = −c(dx ∧ dp+ dy ∧ dq) = −c�, (18.18)

where � is the canonical 2-form. The first two 2-forms, (18.16) and (18.17),represent Legendre-conjugate elliptic Monge–Ampère operators. The trans-formation (z, w) → (X , Y), implicit in (18.15), has the following action on thehyper-Kähler triple:

ωI → dx ∧ dy+ a(dx ∧ dq + dp ∧ dy)+ (a2 − c2) dp ∧ dq, (18.19)

ωJ → c(dx ∧ dq + dp ∧ dy)+ 2ac dp ∧ dq, (18.20)

ωK → ωK (= −c�). (18.21)

18.4 Monge–Ampère operators with constant coefficientsand plane balanced models

The main meteorological output of this geometric description is that the generalfamily of the (elliptic) Monge–Ampère operators with constant coefficientscontains all known plane balance models:

1+ φq1q1 + aφq2q2 + (a2 − c2) det Hessφ = ζC/f , (18.22)

where ζC = h% is constrained vorticity (see the discussion of this notionin [82]).

It also gives a geometric explanation to the rather miraculous appearance ofa complex structure in one such model.

18.4 Monge–Ampère operators with constant coefficients 381

The ellipticity condition for this general Monge–Ampère operator is satisfiedfor the models with

(c2 − 1)ζC/f < c2.

The most popular among them correspond to the following choice ofparameters:

(a) the semi-geostrophic model (a = 1, c = 0 for a positive ζC/f );(b) the L1-dynamics of Salmon [100] with a = c = 1;(c) the

√3-dynamics of McIntyre and Roulstone [81] which corresponds to

a = 1, c = √3 if ζC/f < 32 .

Our classification theorem in d = 2 gives in the same time a geometricdescription of all plane “near-balanced models (0 < c <

√3) with uniform

potential vorticity if ζC/f < 32 .

They:

• correspond geometrically to the points of the Monge–Ampère elliptic Lag-rangian Grassmanian (CP1). In other words, all such models correspond toa choice of complex structure on R4 which is parametrized by CP1;

• all of them are linearizable by the choice of a proper canonical coordinatetransformation.

PART V

Classification of Monge–Ampère equations

19

Classification of symplectic MAOs ontwo-dimensional manifolds

The problem of equivalence and classification of Monge–Ampère equationsgoes back to Sophus Lie’s papers from the 1870s and 1880s (see [65, 66, 67]).Sophus Lie have raised the following problem. Find equivalence classes of non-linear second-order differential equations with respect to the group of contacttransformations.

Sophus Lie himself had found conditions to transform a Monge–Ampèreequation (MAE) to a quasi-linear one and to a linear equation with constantcoefficients. The important steps in a solution of this problem were made by

385

386 Classification of symplectic MAOs on two-dimensional manifolds

Darboux and Goursat [29], who had basically treated the hyperbolic Monge–Ampère equations.

As far as we know, a complete proof of Lie’s theorems had never beenpublished. The first results in this direction were obtained in [73, 74, 77].

In this part we consider the problem of local equivalence for Monge–Ampèreequations and Monge–Ampère operators.

In [53, 54, 55, 59, 60, 61] it was shown that for Monge–Ampère equationsof general type this problem can be reduced to the equivalence problem fore-structures. This leads to a solution of the equivalence problems and to aclassification of Monge–Ampère equations and Monge–Ampère operators.

In this chapter we consider the problem of classification of symplecticMonge–Ampère operators of hyperbolic, elliptic and mixed types.

Let ω1 and ω2 be two effective differential 2-forms on the cotangent bundleT∗M.

We say that two symplectic Monge–Ampère operators �ω and �ω (or twosymplectic effective differential 2-forms ω and ω) are equivalent if there existsa symplectic transformation ϕ of T∗M, such that ϕ∗(ω) = ω. We say thattwo Monge–Ampère operators �ω and �ω (or two effective differential 2-forms ω and ω) are locally symplectic equivalent at a point a ∈ T∗M if thereexists a symplectic transformation ϕ of some neighborhood Oa of a such thatϕ∗(ω) = ω and ϕ(a) = a.

19.1 e-Structures

Recall that an e-structure (or an absolute parallelism) on a smoothn-dimensional manifold � is a collection of n vector fields X1, . . . , Xn that arelinearly independent at each point. This means that X1, . . . , Xn form a globalbasis of the module D(�).

Let X1, . . . , Xn be an e-structure on the n-dimensional smooth manifold �.Since the vector fields X1, . . . , Xn are linearly independent at each point thencommutators [Xi, Xj] are linear combinations of the fields and we obtainstructure functions of the e-structures. They are functions f k

ij such that

[Xi, Xj] =n∑

k=1

f kij Xk .

Here we restrict ourselves to the case when n = 4 and � is a symplecticmanifold with a structure form �.

19.2 e-Structures 387

Let us define differential 1-forms on � by

θ1 = −X2��, θ2 = X1��, θ3 = −X4��, θ4 = X3��.

Assume that the vector fields X1, . . . , X4 form the four-dimensional Liealgebras over R:

g =4⊕

i=1

RXi.

Then the structure functions f kij are constant and we denote f k

ij by ckij.

The Maurer–Cartan equations take the form

dθk = −∑

1�i<j�4

ckijθi ∧ θj.

Indeed, since the bases X1, . . . , X4 and θ1, . . . , θ4 are dual, we get

dθk(Xi, Xj) = Xi(θk(Xj))− Y(θk(Xi))− θk([Xi, Xj]) = −θk([Xi, Xj]) = −ckij.

Proposition 19.1.1 Let X11 , . . . , X1

4 and X21 , . . . , X2

4 be two e-structures on afour-dimensional smooth symplectic manifold �. Suppose that they form four-dimensional Lie algebras g1 and g2 over R. Suppose also that the correspondingstructure constant of g1 and g2 coincide. Then for any point a ∈ � thereexists its neighborhood Oa and a preserving the point a local diffeomorphismϕ : Oa → ϕ(Oa) such that ϕ∗(θ2

i ) = θ1i and ϕ∗(X1

i ) = X2i (i = 1, . . . , 4).

Proof Instead of the diffeomorphismϕ we will construct its graph . Considerthe natural projection π : �2 → � and denote θ l

i = π∗(θ li ). Consider on �2

a four-dimensional distribution F〈θ11 − θ2

1 , . . . , θ14 − θ2

4 〉. This distribution iscompletely integrable. Indeed

d(θ1k − θ2

k ) = −∑

1�i<j�4

ckij(θ

1i ∧ θ1

j − θ2i ∧ θ2

j )

= −∑

1�i<j�4

ckij(θ

1i ∧ (θ1

j − θ2i )+ (θ1

i − θ2i ) ∧ θ2

j ).

The point (a, a) belongs to one of the maximal integral manifolds of F , say .Let γ : → � be the projection on �. Then γ ∗(θ2

i ) = θ2i | = θ1

i | . We have

(γ−1)∗π∗(θ2i ) = θ1

i

and the map π1 ◦ γ−1 is required. �


19.2 Classification of non-degenerate Monge–Ampèreoperators

19.2.1 Differential invariants of non-degenerate operators

Let ω be an effective differential 2-form such that Pf(ω) = ε = ±1 and let Abe the operator that corresponds to ω. Then A2 = ε and the complexificationof the tangent space Ta(T∗M) at a point a ∈ T∗M splits in to the direct sum oftwo skew-orthogonal complex symplectic planes:

Ta(T∗M)C = V+(a)⊕ V−(a), (19.1)

where

V±(a) = {X ∈ Ta(T∗M)C | AC

a X = ∓ι√εX}.

Let us denote by V1 = V+ and V2 = V− the corresponding distributions, andV (i)

j their derivations. Let us assume that V (i)j are also distributions . Then for

each j = 1, 2 one has the following choices:

(0) V (1)j = Vj, that is Vj is a completely integrable distribution;

(1) Vj �= V (1)j but V (1)

j = V (2)j , that is Vj is not completely integrable but V (1)

jis a three-dimensional completely integrable distribution; and

(2) Vj �= V (1)j �= V (2)

j , distributions Vj, V (1)j and V (2)

j are not completelyintegrable.

We say that the symplectic Monge–Ampère operator (or equation) belongsto class Hk,l at a point a ∈ T∗M(k, l = 0, 1, 2; k ≤ l) if case (k) holds for oneof Vj and case (l) holds for the other.

Classes Hk,l are obviously invariant under symplectic transformations.As we have seen in Part I, there are the following decompositions of the de

Rham complex:

�s(T∗M)C =⊕

p+q=s

�p,q(T∗M),

d = d1,0 ⊕ d0,1 ⊕ d2,−1 ⊕ d−1,2,

where �p,q(T∗M)=�p(V+) ⊗ �q(V−) and di,j :�p,q(T∗M)→�p+i,q+k

(T∗M).Remark that d−1,2 and d2,−1 are tensor invariants of the Monge–Ampère

equation Eω.

19.2 Classification of non-degenerate Monge–Ampère operators 389

Proposition 19.2.1

1. The distribution V+ is completely integrable if and only if d2,−1 = 0.2. The distribution V− is completely integrable if and only if d−1,2 = 0.

Proof The distribution V+ is generated by a pair of 1-forms α−,β− ∈�0,1(T∗M), with α− ∧ β− �= 0. This distribution is completely integrableif and only if dα− = α− ∧ γ1 + β− ∧ δ1 and dβ− = α− ∧ γ2 + β− ∧ δ2

for some 1-forms γi, δi ∈ �1(T∗M)C (i = 1, 2). The latter means thatdα−, dβ− /∈ �2,0(T∗M). �

The formula

Wω��2 = 2dω

uniquely determines the real vector field Wω. Let µω be the the dual (real)differential 1-form

µωdef= Wω��.

Using decomposition (19.1), we obtain Ww = W+ + W−, where W+ ∈D(V+) and W− ∈ D(V−). Since the distributions V+ and V− are skew-orthogonal, 1-forms W+��C and W−��C belong to�1,0(T∗M) and�0,1(T∗M)

correspondingly. Denote them by

µ+def= W+��, µ−

def= W−��.

Two three-dimensional distributions kerµ+ and kerµ− define the two-dimensional distribution ker µ+ ∩ ker µ−. Note that

µ+(W+) = µ+(W−) = µ−(W+) = µ−(W−) = 0,

therefore the distribution kerµ+ ∩ ker µ− is generated by the vector fields W+and W−: ker µ+ ∩ ker µ− = F〈W+, W−〉.

Let Q be their commutator

Qdef= [W+, W−].

Then using decomposition (19.1) again, we get two vector fields Q+ ∈ D(V+)Cand Q− ∈ D(V−)C such that Q = Q+ + Q−.

Suppose now that the three-dimensional distributions ker µ+ and ker µ− areboth integrable, i.e., in a some neighborhood Oa of a point a the equationbelongs to the class H1,1.


Since this distribution F〈W+, W−〉 is also completely integrable, one candefine two functional invariants g+ and g− of the form ω by the followingformula:

[W+, W−] = g+W+ + g−W−.

Distributions kerµ+ and kerµ− are completely integrable, therefore

µ+ ∧ dµ+ = µ− ∧ dµ− = 0.

Then

µ+ ∧ (W+�dµ+) = W+�(µ+ ∧ dµ+) = 0,

i.e. the 1-forms µ+ and W+�dµ+ are linear dependent. Therefore

W+�dµ+ = g0µ+

for some function g0.This function is an invariant of ω. Note also that W−�dµ− = −g0µ−.Since µ+ ∈ �1,0, we have

dµ+ = d1,0µ+ + d0,1µ+ + d−1,2µ+.

and µ+ ∧ d1,0µ+ = 0.Then,

µ+ ∧ dµ+ = µ+ ∧ d0,1µ+ + µ+ ∧ d−1,2µ+.

Since µ+ ∧ dµ+ = 0, µ+ ∧ d0,1µ+ ∈ �2,1 and µ+ ∧ d−1,2µ+ ∈ �1,2, wesee that

µ+ ∧ d0,1µ+ = 0 (19.2)

and µ+ ∧ d−1,2µ+ = 0.Since d−1,2µ+ ∈ �0,2 and µ+ ∈ �1,0, the last equality is realized if and

only if d−1,2µ+ = 0.Then,

dµ+ = d1,0µ+ + d0,1µ+. (19.3)


In the similar way we obtain

dµ− = d1,0µ− + d0,1µ−. (19.4)

From (19.2) it follows that

d0,1µ+ = µ+ ∧ γ−,

for some uniquely determined differential 1-form γ− ∈ �0,1. In a similar waywe obtain a uniquely determined differential 1-form γ+ ∈ �1,0 such that

d1,0µ− = µ− ∧ γ+.

We denote by X+ and X− the dual vector fields:

X±�� = γ±. (19.5)

Lemma 19.2.1 γ−(W−) = g+ and γ+(W+) = −g−.

Proof Since (19.3) and the fact that

W−�d1,0µ+ = W+�d0,1µ− = 0,

we obtain

W−�dµ+ = W−�d0,1µ+ = W−�(µ+ ∧ γ−) = −γ−(W−)µ+

and

W+�dµ− = W+�d1,0µ− = W+�(µ− ∧ γ+) = −γ+(W+)µ−.

Moreover

[W+, W−]�� = [LW+ , ιW−](�)= LW+(W−��)−W−�LW+(�)

= W+�dµ− −W−�dµ+= −γ+(W+)µ− + γ−(W−)µ+.

On the other hand,

[W+, W−]�� = g+µ+ + g−µ−.

Therefore, g+ = γ−(W−) and g− = −γ+(W+). �


19.2.2 Hyperbolic operators

In the hyperbolic case we can assume that above functions, vector fields,differential forms and tensor fields are real.

Consider two cases.

Case 1Vectors W+,a, W−,a, Q+,a, and Q−,a are linear independent at a point a ∈ T∗M.

In this case the vector fields W+, W−, Q+, and Q− are linear independent insome neighborhood Oa of a.

We define two functional invariants of the form ω:

r+def= �(W+, Q) and r−

def= �(W−, Q).

From non-degeneracy of the form � on the distributions V+ and V− it fol-lows that the functions r± do not vanish in some neighborhood of the point a.Therefore one can normalize the vector fields:

X1def= W+, X2

def= 1

r+Q+

X3def= W−, X4

def= 1

r−Q−.

We have

�(X1, X2) = �(X3, X4) = 1

and since the distributions V+ and V− are skew-orthogonal,

�(X1, X3) = �(X1, X4) = �(X2, X3) = �(X2, X4) = 0.

Define differential 1-forms

θ1def= −X2��, θ3

def= −X4��,

θ2def= X1��, θ4

def= X3��

(i = 1, . . . , 4).The local bases X1, X2, X3, X4 and θ1, θ2, θ3, θ4 of D(T∗M) and �1(T∗M)

are dual and we have the following representation of the symplectic form:

� = θ1 ∧ θ2 + θ3 ∧ θ4.


Let us now find values ofω on the vector fields X1, X2, X3, X4. Since X1, X2 ∈D(V+) and X3, X4 ∈ D(V−), we see that AX1 = X1, AX2 = X2, AX3 = −X3

and AX4 = −X4, where A = Aω.Therefore

ω(X1, X2) = �(AX1, X2) = �(X1, X2) = 1.

In the similar way we find values of the form ω on the rest of the pairs of thevector fields:

ω(↑,←) X1 X2 X3 X4

X1 0 1 0 0X2 −1 0 0 0X3 0 0 0 −1X4 0 0 −1 0

This gives us the following representation of the form ω:

ω = θ1 ∧ θ2 − θ3 ∧ θ4.

The following theorem provides the canonical representation of hyperboliceffective 2-forms.

Theorem 19.2.1 Let ω be a hyperbolic normed effective differential 2-form onT∗M and a ∈ T∗M. Suppose that the vectors W+,a, W−,a, Q+,a and Q−,a arelinear independent at the point a. Then in a some neighborhood of a there existsan e-structure X1, . . . , X4 such that we have the following representation of theforms � and ω:

� = θ1 ∧ θ2 + θ3 ∧ θ4, (19.6)

ω = θ1 ∧ θ2 − θ3 ∧ θ4.

Case 2In a neighborhood Oa � a the Monge–Ampère operator belongs to the classH1,1. In this case the three-dimensional distributions kerµ+ and kerµ− areintegrable in Oa.


As a corollary of Lemma 19.2.1 one get a table of values for the 1-formsµ+,µ−, γ+, γ− on the vector fields W+, W−, X+, X−:

W+ X+ W− X−µ+ 0 g− 0 0γ+ −g− 0 0 0µ− 0 0 0 −g+γ− 0 0 g+ 0

Lemma 19.2.2 µ+ ∧ γ+ ∧ µ− ∧ γ− = g+g−�2.

Proof We have:W+��2 = 2µ+ ∧�;

W−�(W+��2) = 2W−�(µ+ ∧�) = 2µ− ∧ µ+;

X+�(W−�(W+��2)) = 2X+�(µ− ∧ µ+) = 2g−µ−;

�2(W+, W−, X+, X−) = 2g−X−�µ− = −2g+g−. �

Corollary 19.2.1 The vector fields W+, W−, X+, X− are linear independentin a neighborhood of the point a ∈ T∗M if and only if g+g− �= 0 at thepoint.

Suppose now that (g+g−)(a) �= 0. Due to Lemma 19.2.2, the vector fieldsW+, W−, X+, X− form a basis of the module D(T∗M) in some neighborhood ofthe point a.

Let us consider the normed basis of D(T∗M):

X1def= W+, X2

def= 1

g−X+,

X3def= W−, X4

def= − 1

g+X−

and a basis of the module �1(T∗M) of differentia 1-forms on T∗M:

θ1def= −X2��, θ2

def= X1��,

θ3def= −X4��, θ4

def= X3��.


These bases are dual and the symplectic structure � and the effective formω have the following canonical forms:

� = θ1 ∧ θ2 + θ3 ∧ θ4,

ω = θ1 ∧ θ2 − θ3 ∧ θ4.

Local coordinatesLet us find a coordinate representation of hyperbolic Monge–Ampère operatorsof H1,1 class.

Note that for an effective differential 2-form

ω =E dq1 ∧ dq2 + B(dq1 ∧ dp1 − dq2 ∧ dp2)

+ C dq1 ∧ dp2 − Adq2 ∧ dp1

we have

Wω = a∂

∂q1+ b

∂

∂q2+ c

∂

∂p1+ e

∂

∂p2,

where

a = Ap2 − Bp1 − Dq2 , b = Bp2 − Cp1 + Dq1 ,

c = Bq1 + Cq2 − Ep2 , e = Ep1 − Aq1 − Bq2 .

Theorem 19.2.2 Any normed hyperbolic 2-form ω of H1,1 class is (locally)symplectic equivalent to the differential 2-form

−2f dq1 ∧ dq2 + dq1 ∧ dp1 − dq2 ∧ dp2,

where f = f (q, p) is a smooth function.

Proof Let H+ and H− be integrals of the distributions F〈µ+〉 and F〈µ−〉respectively. Without loss generality, we can suppose that H+ = q1, H− = q2.Then µ+ = h+ dq1 and µ− = h− dq2 for some functions h±, and the vectorfields W+ and W− have the forms W+ = −h+∂/∂p1 and W− = −h−∂/∂p2.


Note that ω(W+, W−) = 0, and therefore the form ω does not contain theterm dp1 ∧ dp2. Then the distributions V+ and V− have the following forms:

V+ = F⟨X+ = (1+ B)

∂

∂q1+ C

∂

∂q2− E

∂

∂p2, Y+ = (1+ B)

∂

∂p1− A

∂

∂p2

⟩,

V− = F⟨X− = (1− B)

∂

∂q1− C

∂

∂q2+ E

∂

∂p2,

Y− = A∂

∂q1+ (1+ B)

∂

∂q2− E

∂

∂p1

⟩.

Since the distributions V+ and V− are skew-orthogonal, we have kerµ+ ⊃V− and kerµ− ⊃ V+. Therefore, µ+(X−) = h+(1 − B) = 0 ⇒ B = 1 andµ−(X+) = C = 0. Then,

W+ = Ap2

∂

∂q1− Ep2

∂

∂p1+ 1

2(Ep2 A− EAp2)

∂

∂p2= −h+

∂

∂p1.

Therefore Ep2 = h+, A = 0. We see that

W− = Ep1

∂

∂p2= −h−

∂

∂p2,

so

Ep1 = −h−,

and

ω = −2 fdq1 ∧ dq2 + dq1 ∧ dp1 − dq2 ∧ dp2, (19.7)

where 2f = −E. �

For this form we obtain the following coordinate representation of theoperator Aw and the projection operators Pr+ and Pr− corresponding to the


distributions V+ and V− :

Aw =

∥∥∥∥∥∥∥∥1 0 0 00 −1 0 00 −2f 1 02f 0 0 −1

∥∥∥∥∥∥∥∥ ,

Pr+ = 1

2

∥∥∥∥∥∥∥∥1 0 0 00 0 0 00 −f 1 0f 0 0 0

∥∥∥∥∥∥∥∥ ,

Pr− = 1

2

∥∥∥∥∥∥∥∥0 0 0 00 1 0 00 f 0 0−f 0 0 1

∥∥∥∥∥∥∥∥ .

in the basis ∂/∂q1, ∂/∂q2, ∂/∂p1, ∂/∂p2

The differential forms µ+ and µ− are

µ+ = −2fp2 dq1,

µ− = 2fp1 dq2

and

V+ = F⟨∂

∂q1+ f

∂

∂p2,∂

∂p1

⟩,

V− = F⟨∂

∂q2+ f

∂

∂p1,∂

∂p2

⟩.

The vector field W has the form

W = 2fp2

∂

∂p1− 2fp1

∂

∂p2

and its projections to the distributions V+ and V− are

W+ = 2fp2

∂

∂p1, W− = −2fp1

∂

∂p2

respectively.


Then

[W+, W−] = 4fp1 fp2p2

∂

∂p1− 4fp2 fp1p1

∂

∂p2

and we have the following functional invariants:

g+ = 2fp1 fp2p2

fp2

, g− = 2fp2 fp1p1

fp1

, g0 = 2fp1p2 .

Bases for modules �p,q are the following:

�1,0 =〈dq1, dp1 − f dq2〉,

�0,1 =〈dq2, dp2 − f dq1〉,

�1,1 =〈dq1 ∧ dq2, dq1 ∧ dp2, dq2 ∧ dp1, f 2 dq1 ∧ dq2 − f dq1 ∧ dp1

+ f dq2 ∧ dp2 − dp1 ∧ dp2〉,

�2,0 =〈dq1 ∧ dp1 − f dq1 ∧ dq2〉,

�0,2 =〈dq2 ∧ dp2 + f dq1 ∧ dq2〉,

�1,2 =〈dq1 ∧ dq2 ∧ dp2, dq2 ∧ dp1 ∧ dp2 − f dq1 ∧ dq2 ∧ dp1〉,

�2,1 =〈dq1 ∧ dq2 ∧ dp1, dq1 ∧ dp1 ∧ dp2 − f dq1 ∧ dq2 ∧ dp2〉,

�2,2 =〈dq1 ∧ dq2 ∧ dp1 ∧ dp2〉,

and

d0,1µ+ = 2( f fp1p2 + fq2p2) dq1 ∧ dq2 + 2fp2p2 dq1 ∧ dp2,

d1,0µ− = 2( f fp1p2 + fq1p1)dq1 ∧ dq2 − 2fp1p1 dq2 ∧ dp1.

Therefore we have

γ+ = − fp1p1

fp1

(dp1 − f dq2)− f fp1p2 + fq1p1

fp1

dq1,

γ− = − fp2p2

fp2

(dp2 − f dq1)− f fp1p2 + f fq2p2

fp2

dq2


and the corresponding vector fields:

X+ = − fp1p1

fp1

(∂

∂q1+ f

∂

∂p2

)+ f fp1p2 + fq1p1

fp1

∂

∂p1,

X− = − fp2p2

fp2

(∂

∂q2+ f

∂

∂p1

)+ f fp1p2 + fq2p2

fp1

∂

∂p2,

and

X1 = 2fp2

∂

∂p1,

X2 = 1

2fp2

∂

∂q1− f fp1p2 + fq1p1

2fp2 fp1p1

∂

∂p1+ f

2fp2

∂

∂p2,

X3 = −2fp1

∂

∂p2,

X4 = 1

2fp1

∂

∂q2+ f

2fp1

∂

∂p1− f fp1p2 + fq2p2

2fp1 fp2p2

∂

∂p2.

The coordinate representations of the tensors d2,−1 and d−1,2 are:

d2,−1 = fp1(dq1 ∧ dp1 − f dq1 ∧ dq2)⊗ ∂

∂p2,

d−1,2 = fp2(dq2 ∧ dp2 + f dq1 ∧ dq2)⊗ ∂

∂p1.

Theorem 19.2.3 Let ω be an effective symplectic differential 2-form of classH1,1 and let g0 = g+ = g− = 0 in a neighborhood Oa � a. Then the form ω

is locally equivalent to the form

ω = −2(A(q)p1 + B(q)p2 + C(q)) dq1 ∧ dq2 + dq1 ∧ dp1 − dq2 ∧ dp2

(19.8)

for some functions A, B and C.

Proof From the coordinate representations of the functions g± and g0 itfollows that fp1p1 = fp2p2 = fp1p2 = 0. Therefore,

f = A(q)p1 + B(q)p2 + C(q). �


Now consider the case when the vector fields X1, X2, X3, X4 form a four-dimensional real Lie algebra g, i.e.,

[Xi, Xj] =4∑

k=1

ckijXk ,

where i, j = 1, . . . , 4 and ckij ∈ R are structure constants of the Lie algebra g.

Note that ckij = θk([Xi, Xj]) and from the formula

[X1, X3] = [W+, W−] = g+W+ + g−W− = g+X1 + g−X3

it follows that g+ = c113 and g− = c3

13. Using the definition of the functionalinvariant g0, we obtain X1�dθ2 = g0θ2.

Then θ2([X1, X2]) = −dθ2(X1, X2) = g0 and we see that g0 = c212.

Thus, if the vector fields X1, X2, X3, X4 form a four-dimensional real Liealgebra, then the functional invariants g0, g+, g− are constants.

One can check that under the condition g+g− �= 0 there exists only the Liealgebra and with the structure presented in the following table.

X1 X2 X3 X4

X1 01

2X1 − 4κX2 + 1

2X3 4κ(X1 + X3)

1

2X1 + 1

2X3

−4κX4

X2 − 1

2X1 + 4κX2 − 1

2X3 0

1

2X1 − 4κX2 + 1

2X3 0

X3 −4κ(X1 + X3) − 1

2X1 + 4κX2 − 1

2X3 0 − 1

2X1− 1

2X3

+4κX4

X4 − 1

2X1 − 1

2X3 + 4κX4 0

1

2X1 + 1

2X3 − 4κX4 0

The function f which generates this Lie algebra is:

f (q, p) = (p1 + κp2)2.

Here κ is a non-zero real number, and g0 = g− = g+ = 2κ .

Theorem 19.2.4 Suppose that ω is an effective symplectic differential 2-formof class H1,1 in a neighborhood Oa of a point a, with g+g−(a) �= 0, and thevector fields X1, X2, X3, X4 form a Lie algebra.


Then the form ω is a locally symplectic equivalent to the differential 2-form

ω = −2(p1 + κp2)2dq1 ∧ dq2 + dq1 ∧ dp1 − dq2 ∧ dp2,

where κ is a non-zero real number.The corresponding Monge–Ampère operator has the form

�ω(v) = (vq1q2 − (vq1 + κvq2)2)dq1 ∧ dq2.

19.2.3 Elliptic operators

In this case the vector fields W+, W−, Q+ and Q− are complex. Moreover, W+and W− (Q+ and Q−) are complex conjugate, i.e., W+ = W− and Q+ = Q−.

Note that W and Vdef= AW are linear independent at each point of real vector

fields.Define a real vector field X: if Re Q+ �= 0 we put X

def= Re Q+ and Xdef=

Im Q+ otherwise. Moreover, set

Zdef= AX, η

def= V��, ξdef= X��, τ

def= Z��.

The table below indicates values of the 1-forms ξ , τ , µ, η on the vector fieldsX, Z , W , V :

X Z W V

ξ 0 0 w υ

τ 0 0 v −w

µ −w −υ 0 0η −v w 0 0

(19.9)

where vdef= �(X, V), w

def= �(X , W).Note that

�2(W , V , X , Z) = 2(v2 + w2)

and therefore the vector fields W , V , X , Z (and the differential 1-forms ξ , τ ,µ, η) are linear independent if and only if v2 + w2 �= 0.

As above we will consider two cases.


Case 1v2 + w2 �= 0.

In this case the vector fields W , V , X , Z form a basis of the module D(Oa) ina some neighborhood Oa of a.

Let

X1def= − 1

v2 + w2(υV + wW),

X2def= X ,

X3def= 1

v2 + w2(υW − wV), (19.10)

X4def= −Z .

Then X3 = AX1, X4 = −AX2, and

�(X1, X2) = �(X3, X4) = 1,

�(X1, X3) = �(X1, X4) = �(X2, X3) = �(X2, X4) = 0.

Let (θ1, . . . , θ4) be the dual basis for (X1, . . . , X4). Then

� = θ1 ∧ θ2 + θ3 ∧ θ4.

Calculating values of ω on the vector fields X1, X2, X3, X4:

ω(↑,←) X1 X2 X3 X4

X1 0 0 0 1X2 0 0 1 0X3 0 −1 0 0X4 −1 0 0 0

we get the following representation of the form ω:

ω = θ1 ∧ θ4 − θ2 ∧ θ3

and the following result.

Theorem 19.2.5 Let ω be an elliptic normed effective differential 2-form onT∗M and a ∈ T∗M. Suppose that v2(a)+w2(a) �= 0. Then in a some neighbor-hood of a there exists an e-structure X1, . . . , X4 such that we have the following


representation of the forms � and ω:

� = θ1 ∧ θ2 + θ3 ∧ θ4,

ω = θ1 ∧ θ4 − θ2 ∧ θ3. (19.11)

Remark 19.2.1 Similar e-structures for hyperbolic and elliptic equationswere obtained by Kruglikov in [53, 54, 55].

Case 2The Monge–Ampère operator belongs to the classH1,1.

This means that the three-dimensional complex distributions ker µ+ andker µ− are both integrable in Oa.

Let Xdef= Re X− and Z

def= AX . These are real vector fields. As above we set

ηdef= V��, ξ

def= X��, τdef= Z��.

Then we have the same table (19.9) of values of the 1-forms on the vector fields,

where as above vdef= �(P, V), w

def= �(P, W).We define the new basis of D(Oa) by (19.10) and get the same canonical

representation (19.11) of the form � and ω.

Theorem 19.2.6 Let ω be an elliptic normed effective differential 2-form onT∗M and a ∈ T∗M. Suppose that at each point of some neighborhood of a theMonge–Ampère operator belongs to the class H1,1. Then in a some neighbor-hood of a there exists an e-structure X1, . . . , X4 such that we have the followingrepresentation of the forms � and ω:

� = θ1 ∧ θ2 + θ3 ∧ θ4,

ω = θ1 ∧ θ4 + θ2 ∧ θ3.

Example 19.2.1 As an example we consider the following non-linear Laplaceoperator:

�ω(v) = (vq1q1 + vq2q2 − f (q, p))dq1 ∧ dq2,

which corresponds to the non-linear Laplace equation

vxx + vyy = f (x, y, vx , vy). (19.12)


For this operator the corresponding effective differential 2-form is

ω = −f (q, p) dq1 ∧ dq2 + dq1 ∧ dp2 + dq2 ∧ dp1.

In the basis ∂/∂q1, ∂/∂q2, ∂/∂p1, ∂/∂p2 the operator A has the following matrixrepresentation:

A =

∥∥∥∥∥∥∥∥0 1 0 0−1 0 0 00 f 0 −1−f 0 1 0

∥∥∥∥∥∥∥∥ .

The complex distributions V+ and V− are

V+ =⟨∂

∂q2+ f

∂

∂p2+ ι

∂

∂q1,

∂

∂q1+ f

∂

∂p1− ι

∂

∂q2

⟩,

V− =⟨∂

∂q2+ f

∂

∂p2− ι

∂

∂q1,

∂

∂q1+ f

∂

∂p1+ ι

∂

∂q2

⟩.

The vector field

W = −fp2

∂

∂p1+ fp1

∂

∂p2

falls into two components

W± = 1

2

(−fp2

∂

∂p1+ fp1

∂

∂p2∓ ι

(fp1

∂

∂p1+ fp2

∂

∂p2

)).

Therefore,

µ± = 12 (fp2 dq1 − fp1 dq2 ± ι(fp1 dq1 + fp2 dq2)).

We see that µ+ ∧ dµ+ = 0 and µ− ∧ dµ− = 0, therefore the distributionsker µ+ and ker µ− are completely integrable and (19.12) belongs to the classH2,2. The vector field

Q = ι

((2fp1 fp1p2 + fp2(fp2p2 − fp1p1))

∂

∂p1

+(−2fp2 fp1p2 + fp1(fp2p2 − fp1p1))∂

∂p2

)


is a linear combination of the vector fields W+ and W− with coefficients

g+ = (fp2 − ιfp1)(2fp1p2 + ι(fp1p1 − fp2p2))

2(fp1 − ιfp2),

g− = (fp1 − ιfp2)(fp2p2 − fp1p1 − ι2fp1p2)

2(fp2 − ιfp1)

respectively.

Example 19.2.2 Let us construct an e-structure for the following case of thenon-linear Laplace operator:

�ω(v) = (vq1q1 + vq2q2 − v2q1)dq1 ∧ dq2.

For this operator

W = 2p1∂

∂p2,

V = −2p1∂

∂p1,

W± = p1∂

∂p2∓ ιp1

∂

∂p1,

γ± = 1

2p1(−dp1 ± ι(p2

1 dq2 − dp2)),

v = 1,

w = 0.

Therefore we have the following e-structure:

X1 = 2p1∂

∂p1,

X2 = − 1

2p1

∂

∂q1,

X3 = 2p1∂

∂p2,

X4 = − 1

2p1

∂

∂q2− p1

2

∂

∂p2.


The dual basis is

θ1 = 1

2p1dp1,

θ2 = −p1 dq1,

θ3 = 1

2p1dp2 − p1

2dq2,

θ4 = −2p1 dq2

and

θ1 ∧ θ2 + θ3 ∧ θ4 = dq1 ∧ dp1 + dq2 ∧ dp2 = �,

θ1 ∧ θ4 + θ2 ∧ θ3 = −p21 dq1 ∧ dq2 + dq1 ∧ dp2 + dq2 ∧ dp1 = ω.

19.3 Classification of degenerate Monge–Ampère operators

19.3.1 Non-linear mixed-type operators

Let ω be an effective differential 2-form such that at a point a ∈ T∗M thePfaffian Pf(ω) vanishes. Assume also that

Pf(ω) = f n,

where n is a natural number and f is a smooth function such that its differentialdoes not vanish at the point a: dfa �= 0.

Then in a neighborhood of the point a the surface {f = 0} is a smooth

manifold. Let Xdef= Xf be a Hamiltonian vector field with a Hamiltonian f .

Define the vector field

Zdef= AωX ,

τdef= Z��,

and two smooth functions

υdef= V(f ), w

def= W(f ).

19.3 Classification of degenerate Monge–Ampère operators 407

Then we have

X�ω = τω, Z�ω = f n df , V�ω = −f nµ.

The table below shows the values of the 1-forms df , τ , µ, η on the vectorfields X, Z , W , V .

X Z W V

df 0 0 w υ

τ 0 0 −υ Fnw

µ w υ 0 0η υ −Fnw 0 0

Note that dω = µ ∧�.

Indeed,

2dω = W��2 = 2(W��) ∧� = 2µ ∧�.

Lemma 19.3.1 For the differential 1-forms df , τ , µ, η the following propertieshold:

1. τ ∧� = df ∧ ω;2. η ∧� = −µ ∧ ω;3. f nµ ∧� = η ∧ ω;4. τ ∧ µ ∧� = η ∧ df ∧�;5. τ ∧ η ∧� = −f nµ ∧ df ∧�;6. τ ∧ µ ∧ df = −wτ ∧�− υ df ∧�.

Proof Since the form ω is effective, ω ∧� = 0. Then we have

0 = X� (ω ∧�) = τ ∧�− df ∧ ω

and the first property is proved.In the similar way one obtains properties 2 and 3.Properties 4 and 5 follow from properties 1 and 2 by

µ ∧ τ ∧� = µ ∧ df ∧ ω = −df ∧ µ ∧ ω = df ∧ η ∧�,

−η ∧ τ ∧� = −η ∧ df ∧ ω = df ∧ η ∧ ω = FndF ∧ µ ∧�,


and property 6 follows from property 4:

X� (τ ∧ µ ∧�) = X� (η ∧ df ∧�),

then

υ df ∧� = −wτ ∧�− τ ∧ µ ∧ df . �

Lemma 19.3.2 One has

2 df ∧ τ ∧ η ∧ µ = g�2,

where

gdef= Fnw2 + υ2.

Proof From property 6, we get

(f nw2 + υ2)� = υ df ∧ η + wτ ∧ η − υτ ∧ µ− f nµ ∧ df

and multiplying both parts by � and using properties 7 and 4 of the previouslemma, we find

(f nw2 + υ2)�2 = 2(wτ ∧ η ∧�− υτ ∧ µ ∧�) = 2df ∧ τ ∧ η ∧ µ.�

Suppose now that at the point a the function υ does not vanish:

υ(a) �= 0.

Then the vector fields X , Z , W , V are linear independent in a neighborhood ofthe point a and these vector fields form a local basis.

Let

X1def= 1

g(FnwW + υV),

X2def= X ,

X3def= 1

g(υW − wV),

X4def= Z


and

θ1def= df ,

θ2def= 1

g(f nwµ+ υη),

θ3def= −τ ,

θ4def= 1

g(υµ− wη).

Then bases X1, . . . , X4 and θ1, . . . , θ4 are dual and

X1�� = θ2, X2�� = −θ1, X3�� = θ4, X4�� = −θ3.

We see that AωX3 = X1 and AωX2 = X4.In the basis θ1, . . . , θ4 the symplectic form � and the effective form ω have

the following canonical representations:

� = θ1 ∧ θ2 + θ3 ∧ θ4, (19.13)

ω = θ3 ∧ θ2 + f nθ4 ∧ θ1.

This representation allows us to construct normal forms of mixed-typeeffective differential 2-forms.

Letω be a mixed-type effective differential 2-form and (19.13) be a canonicalrepresentation, and let vector fields X1, . . . , X4 form a four-dimensional Liealgebra g.

Since d� = 0, we obtain the following extra conditions for the structureconstants:

c223 + c4

12 = 0, c224 − c3

12 = 0, c323 + c4

24 = 0,

c313 + c2

12 = 0, c334 + c2

24 = −w, c234 − c3

13 − c414 = 0,

c314 − f n

ωc424 = υω, c4

23 = c214 = c4

34 = 0, c1ij = 0 (i, j = 1, . . . , 4)

(here w = wω). Therefore in this case υω = constant. From υω(a) �= 0 itfollows that c3

14 �= 0.Let

a1 = c212, a2 = c2

13, a3 = c223,

a4 = c312, a5 = c2

34, a6 = c323,

a7 = c324, a8 = c4

13, a9 = c314.


Finally we get eight Lie algebras.

(1) Lie algebra g1 (a1 �= 0, wω �= 0):

[↑,←] X1 X2 X3 X4

X1 0 a1X2 −2a21w−1X2 − a1X3 a9X3

X2 −a1X2 0 0 0

X3 2a21w−1X2 + a1X3 0 0 −a1X2 − wX3

X4 −a9X3 0 a1X2 + wX3 0

(2) Lie algebra g2:

[↑,←] X1 X2 X3 X4

X1 0 0 0 a9X3

X2 0 0 0 a7X3

X3 0 0 0 −wX3

X4 −a9X3 a7X3 wX3 0

(3) Lie algebra g3 (a4 �= 0, β = 2λ2a2 �= 2a4):

[↑,←] X1 X2 X3 X4

X1 0 −λa2X2 + a4X3 a2X2 + λa2X3 a9X3

X2 λa2X2 − a4X3 0 0 a4X2 + λa4X3

X3 −a2X2 − λa2X3 0 0 λa2X2 + (β + a4)X3

X4 −a9X3 −a4X2 − λa4X3 −λa2X2 − (β + a4)X3 0

(4) Lie algebra g4 (a7 �= 0, σ �= 0):

[↑,←] X1 X2 X3 X4

X1 0 σ 2a7X2 + σa7X3 −σ 3a7X2 − σ 2a7X3 a9X3

X2 −σ 2a7X2 − σa7X3 0 0 σa7X2 + a7X3

X3 σ 3a7X2 + σ 2a7X3 0 0 −σ 2a7X2 − σa7X3

X4 −a9X3 −σa7X2 − a7X3 σ 2a7X2 + σa7X3 0


(5) Lie algebra g5 (a1 �= 0):

[↑,←] X1 X2 X3 X4

X1 0 a1X2 a2X2 − a1X3 + a8X4 a9X3 + 2a1X4

X2 −a1X2 0 0 0

X3 −a2X2 + a1X3 − a8X4 0 0 a1X2

X4 −a9X3 − 2a1X4 0 −a1X2 0

(6) Lie algebra g6:

[↑,←] X1 X2 X3 X4

X1 0 a1X2 a2X2 − a1X3 + a8X4 a9X3 + a1X4

X2 −a1X2 0 0 0

X3 −a2X2 + a1X3 − a8X4 0 0 0

X4 −a9X3 − a1X4 0 0 0

(7) Lie algebra g7 (a4 �= 0, a9 �= 0, γ = a24a−1

9 ):

[↑,←] X1 X2 X3 X4

X1 0 a4X3 0 a9X3

X2 −a4X3 0 γX3 a4X2 + a7X3 − γX4

X3 0 −γX3 0 −a4X3

X4 −a9X3 −a4X2 − a7X3 + γX4 a4X3 0

(8) Lie algebra g8 (a6 �= 0):

[↑,←] X1 X2 X3 X4

X1 0 0 0 0

X2 0 0 a6X3 a7X3 − a6X4

X3 0 −a6X3 0 0

X4 0 −a7X3 + a6X4 0 0


These Lie algebras admit the following representations by vector fields.

1. Lie algebra g1:

X1 = ∂

∂q1− (a1p1 + 2a2

1w−1p2)∂

∂p1+ (a1p2 + a9q2)

∂

∂p2,

X2 = ∂

∂p1,

X3 = − ∂

∂p2,

X4 = ∂

∂q2+ (a1p2 + a9q2)

∂

∂p1− (wp2 + a9a−1

1 wq2)∂

∂p2.

2. Lie algebra g2:

X1 = ∂

∂q1,

X2 = ∂

∂p1,

X3 = − ∂

∂p2,

X4 = ∂

∂q2− (a9q1 + a7p1 + wp2)

∂

∂p2.

3. Lie algebra g3:

X1 = ∂

∂q1+ (λa2p1 + a2p2 − a2a9a−1

4 q1)∂

∂p1+ (a4p1 − λa2p2)

∂

∂p2,

X2 = ∂

∂p1,

X3 = − ∂

∂p2,

X4 = ∂

∂q2+ (a4p1 − λa2p2)

∂

∂p1+ (−λa4p1

+(2λ2a2 + a4)p2 − a9q1)∂

∂p2.


4. Lie algebra g4:

X1 = ∂

∂q1+ σ 2(a9q1 − a7p1 − σa7p2)

∂

∂p1+ σa7(p1 + σp2)

∂

∂p2,

X2 = ∂

∂p1,

X3 = − ∂

∂p2,

X4 = ∂

∂q2+ σa7(p1 + σp2)

∂

∂p1− (a7p1 + σa7p2 + a9q1)

∂

∂p2.

5. Lie algebra g5:

X1 = ∂

∂q1+ (a8p2 − 2a1q2)

∂

∂q2+ (a2p2 − a1p1)

∂

∂p1

+(a9q2 + a1p2)∂

∂p2,

X2 = ∂

∂p1,

X3 = (a8p2 − 2a1q2)∂

∂p1− ∂

∂p2,

X4 = ∂

∂q2+ (a9q2 + a1p2)

∂

∂p1.

6. Lie algebra g6:

X1 = ∂

∂q1+ (a8p2 − a1q2)

∂

∂q2

+(−a2p2 − a1p1 + 1

2a1a9q2

2 −1

2a1a8p2

2 + a21p2q2

)∂

∂p1

+(a9q2 + a1p2)∂

∂p2,

X2 = ∂

∂p1,

X3 = (a1q2 − a8p2)∂

∂p1+ ∂

∂p2,

X4 = − ∂

∂q2− (a9q2 + a1p2)

∂

∂p1.


7. Lie algebra g7:

X1 = −a9∂

∂q2− a4a9p2

∂

∂p1,

X2 = a4a−19

∂

∂q1− a2

4(a−19 q2 + a7p2)

∂

∂q2+ a−1

9 (a24p2 − 1)

∂

∂p1,

X3 = a4∂

∂q2+ (a2

4p2 − 1)∂

∂p1,

X4 = ∂

∂q1− (a4q2 + a4a7a9p2)

∂

∂q2+ (a7a9p2 + q2)

∂

∂p1+ a4p2

∂

∂p2

8. Lie algebra g8:

X1 = ∂

∂q1,

X2 = ∂

∂p1+ (a7q2 − a6p2)

∂

∂p2+ a6q2

∂

∂q2,

X3 = − ∂

∂p2− a6q2

∂

∂q1,

X4 = (a6p2 − a7q2)∂

∂q1+ ∂

∂q2.

These representations lead us to the following normal forms of effectivedifferential 2-forms.

1. Lie algebra g1:

ω =(−qn1 + a2

1p22 − a2

2q22 + wa1p1p2 + a9wq2p1)dq1 ∧ dq2 + (a1p2

+ a9q2)(dq1 ∧ dp1 − dq2 ∧ dp2)− (wp2 + a9a−11 wq2)dq2 ∧ dp1

+ (a1p1 + 2a21wp2)dq1 ∧ dp2+dp1∧dp2.

2. Lie algebra g2:

ω = dp1 ∧ dp2 − (a9q1 + a7p1 + wp2)dq2 ∧ dp1 − qn1dq1 ∧ dq2


3. Lie algebra g3:

ω =(−qn1 + a2a2

9a−14 q2

1 − αa4p21 + αa2p2

2 + 2αλa2p1p2

− 2αa2a9a−14 q1p2)dq1 ∧ dq2 + (a4p1 − λa2p2)(dq1 ∧ dp1

− dq2 ∧ dp2)+ (a2a9a−14 q1 − λa2p1 − a2p2)dq1 ∧ dp2

+ ((2λ2a2 + a4)p2 − λa4p1 − a9q1)dq2 ∧ dp1 + dp1 ∧ dp2.

Here αdef= a4 + λ2a2.

4. Lie algebra g4:

ω =− (qn1 + σ 2a9q1)dq1 ∧ dq2 + σa7(p1 + σp2)(dq1 ∧ dp1 − dq2 ∧ dp2)

+ σ 2(a7p1 + σa7p2 − a9q1)dq1 ∧ dp2

− (a7p1 + σa7p2 + a9q1)dq2 ∧ dp1 + dp1 ∧ dp2.

5. Lie algebra g5:

ω =− (qn1 + (a9q2 + a1p2)

2)dq1 ∧ dq2 + (a9q2 + a1p2)(dq1 ∧ dp1

− dq2 ∧ dp2)+ (a1p1 − a2p2 + (a8a9 − 2a21)q2p2 − 2a1a9q2

2

+ a1a8p22)dq1 ∧ dp2 + dp1 ∧ dp2.

6. Lie algebra g6:

ω =(qn1 + (a9q2 + a1p2)

2)dq1 ∧ dq2 − (a9q2 + a1p2)(dq1 ∧ dp1

− dq2 ∧ dp2)+ ((2a21 − a8a9)q2p2 + 3

2 a1a9q22 − 3

2 a1a8p22 − a1p1

− a2p2)dq1 ∧ dp2 − dp1 ∧ dp2.

7. Lie algebra g7:

ω =(−a24a9p2

2 − a−19 (1− a2

4p2)2f n)dq1 ∧ dq2

+ (a4a9p2 − a4a−19 (1− a2

4p2)fn)(dq1 ∧ dp1 − dq2 ∧ dp2)

× (q2 + a7a9p2)(1− a24p2)(a

24a−1

9 f n − a9)dq1 ∧ dp2

+ (a9 + a24a−1

9 f n)dp1 ∧ dp2.

Here fdef= −a−1

9 q2 + a24a−1

9 p2q2 − a4a−19 p1 + 1

2 a24a7p2

2.


8. Lie algebra g7:

ω =f n(−dq1 ∧ dq2 + a6q2(dq1 ∧ dp1 − dq2 ∧ dp2)

+ a6q2(a7q2 − a6p2)dq2 ∧ dp1)+ (Fna26q2

2 + 1)dp1 ∧ dp2.

Here fdef= q1 + 1

2 a7q22 − a6p2q2.

19.3.2 Linear mixed-type operators

Let us start from the following remark. If an effective differential 2-form ω

is non-degenerate, then one can normalize it by multiplication by the function1/√|Pf(ω)|. If a form ω is of mixed type, then this is not possible. But if the

Pfaffian of ω can be written in the form Pf(ω) = f n, where n ∈ N and f is asmooth function such that f (a) = 0 and its differential dfa �= 0 (a ∈ T∗M),then one can find a symplectic transformation such that Pf(ω) transforms to afunction of the form qn

1. This fact is a corollary of the following lemma.

Lemma 19.3.3 Let F and G be smooth functions on T∗M and let F(a) =G(a) = 0 but daF �= 0, daG �= 0. Then there exists a symplectic diffeomorphismϕ of a neighborhood of the point a such that ϕ(a) = a and ϕ∗(F) = G.

Proof Without loss of generality we can assume that daF = daG. Considera path

Ftdef= G+ t(F − G),

with F0 = G and F1 = F, and find a path ϕt in the pseudo-group of localsymplectic diffeomorphisms such that ϕt(a) = a and ϕ∗t (Ft) = G. This isequivalent to

d

dtϕ∗t (Ft) = 0 (19.14)

or

XHt (Ft)+ F − G = 0, (19.15)

where Ht is the Hamiltonian of the vector field Xt corresponding to the path φt .The last equation can be written as

XFt (Ht) = F − G


and has a smooth solution Ht with daHt = 0, because XFt �= 0 at thepoint a. �

Suppose two differential forms ω1 and ω2 have the Pfaffians fω1 = Pf(ω1)

and fω2 = Pf(ω2) respectively, and the point a is the point where fω1(a) =fω2(a) = 0, but dfω1,a �= 0, dfω2,a �= 0. Then from Lemma 19.3.3 there existsa local symplectic diffeomorphism that takes fω1 and fω2 , and moreover we canassume that

fω1 = fω2 = q1

in canonical coordinates.There are two classical equations of mixed-type:

vxx + xvyy + αvx + βvy + γ (x, y) = 0 (19.16)

and

xvxx + vyy + αvx + βvy + γ (x, y) = 0. (19.17)

The first is known as the Triccomi equation [108] and the second as theKeldysh one [46]. The effective differential forms corresponding to (19.16)and (19.17) are

ωT = q1 dq1 ∧ dp2 − dq2 ∧ dp1 + (αp1 + βp2 + γ (q))dq1 ∧ dq2

and

ωK = dq1 ∧ dp2 − q1 dq2 ∧ dp1 + (αp1 + βp2 + γ (q))dq1 ∧ dq2

respectively.

The corresponding differential operators �Tdef= �ωT and �K

def= �ωK wecall the Triccomi operator and the Keldysh operator respectively.

In the both cases the Pfaffians are the same:

Pf(ωT ) = Pf(ωK ) = q1.

Lemma 19.3.4 If ω �= 0, then

Z(f ) = 0.


Proof Using Lemma 20.4.1, we obtain

df ∧ ω = (Z��) ∧�.

Hence,

Z� (df ∧ ω) = Z(f )ω − df ∧ (Z�ω).But

df ∧ (Z�ω) = df ∧ (AωZ��) = df ∧ (A2ωX��) = f df ∧ dfF = 0.

Therefore, Z(f )ω = 0. Since ω �= 0, we find that Z(f ) = 0. �

Theorem 19.3.1 The mixed-type effective differential 2-form ω is locallysymplectic equivalent at the point a ∈ {Pf(ω) = 0} to the form

ωT = q1 dq1 ∧ dp2 − dq2 ∧ dp1 + h(q, p) dq1 ∧ dq2 (19.18)

if and only if the vector field Z (see (20.13)) is Hamiltonian and the point a isits regular point, i.e., Za �= 0. If, in addition,

W + αZ + βX = 0, (19.19)

then this form is equivalent to the Triccomi form ωT .

Proof For the form ωT we have XT = XT = ∂/∂p1 and ZT = ZT = −∂/∂p2.We see that ZT is a Hamiltonian vector field with Hamiltonian q2. The path-lifting method allows us to construct a symplectic transformation ϕ thattransforms the Hamiltonian g of the vector field Z to −q2 and preserves thepoint a and the Pfaffian f = q1. Taking into account coordinate representationsof the vector field Z and the Pfaffian f , we obtain A = 1, B = D = 0, C = q1.Hence, ω has the form

ω = q1 dq1 ∧ dp2 − dq2 ∧ dp1 + E dq1 ∧ dq2 (19.20)

and the corresponding equation is (19.18), where E = h. The first part of thetheorem is proved.

Let us prove the second part. For the form ω and for the Triccomi form ωT

we have

W = −Ep2

∂

∂p1+ Ep1

∂

∂p2,

WT = −β ∂

∂p1+ α

∂

∂p2


respectively. Since (19.19), we obtain Ep1 = α and Ep2 = β. Then E =αp1 + βp2 + k(q). �

Remark 19.3.1 The conditions (19.19) can be written in terms of the form ω:

dω = df ∧ (β�− αω).

If α = β = 0, we obtain

dω = 0.

For the Keldysh form we have

XK = ∂

∂p1, ZK = −q1

∂

∂p2,

WK = −β ∂

∂p1+ (α − 1)

∂

∂p2, VK = (α − 1)

∂

∂p1+ βq1

∂

∂p2.

We see that the vector field ZK vanishes on the surface {q1 = 0}. Therefore, forthe form ω we should define other vector fields:

Ydef= 1

FZ .

For the Keldysh form YK = −∂/∂p2 is a Hamiltonian regular vector fields.Now we can formulate an analog of Theorem 19.3.1.

Theorem 19.3.2 The mixed-type effective differential 2-form ω is locallysymplectic equivalent at the point a ∈ {Pf(ω) = 0} to the form

ωK = dq1 ∧ dp2 − q1 dq2 ∧ dp1 + f (q, p) dq1 ∧ dq2

if and only if the vector field Y is a Hamiltonian and the point a is its regularpoint, i.e., Ya �= 0. If, in addition,

W + (α − 1)Y + βX = 0, (19.21)

then this form is equivalent to the Keldysh form ωK .

The proof is similar to that for the previous theorem.

Remark 19.3.2 Received results can be generalized to the case when Pf(ω) =Fn, where F(a) = 0 and dFa �= 0 (see [58]).


The method, which was used to prove this theorem, gives us an algorithmfor constructing symplectic transformations, which send Eω to the Triccomiequation.

Example 19.3.1 (Flow of a multicomponent gas mixture) Consider the equa-tion of gas dynamics, which describes two-dimensional transonic stationaryflow on a multicomponent gas mixture:

(vx + b)vxx − vyy = 0, (19.22)

where v is the velocity potential. If b = 0, we get the Von Karman equation[44]. The corresponding effective form for (19.22) is as follows:

ω = (p1 + b)dq2 ∧ dp1 + dq1 ∧ dp2.

The Pfaffian of ω is Pf(ω) = −(p1 + b) and dω = 0. Simple calculations give

X = ∂

∂q1, Z = ∂

∂q2, W = 0.

We see that the field Z is Hamilton with Hamiltonian g = −p2. So, all theconditions of Theorem 19.3.1 are realized. Hence, (19.22) can be reduced tothe Triccomi equation (19.16) withα = β = 0. To construct this transformationwe put Q1 = −p1 − b and Q2 = −g = p2. We get the rest of the variables byP1 = q1 and P2 = −q2. In new coordinates ω has the form

ωT = Q1 dQ1 ∧ dP2 − dQ2 ∧ dP1. (19.23)

Example 19.3.2 (The Maxwell–Einstein equation) The following equationarises in the theory of gravitation and describes an axially symmetric stationarygravitation field (see [23, 8])

((x2 − 1)vx)x + ((1− y2)vy)y = 0. (19.24)

The corresponding effective differential 2-form

ω = 2(p1q1 − p2q2)dq1 ∧ dq2 + (1− q22)dq1 ∧ dp2 + (1− q2

1)dq2 ∧ dp1

satisfies Theorem 19.3.2 for α = 1, β = 0.Indeed,

f = −(1− q21)(1− q2

2)


and

X = −2q1(1− q22)

∂

∂p1− 2q2(1− q2

1)∂

∂p2,

Y = 2q2∂

∂p1− 2q1

∂

∂p2,

W = 0.

We see that at points (±1,±1) and (±1,∓1) on the plane q1, q2 the zero levelof the Pfaffian is not smooth. Hence, at points a which do not coincide with oneof them (19.24) transforms to the equation

xvxx + vyy − vx = 0.

20

Classification of symplectic MAEs ontwo-dimensional manifolds

We say that two symplectic Monge–Ampère equations Eω and Eω are equival-ent if there exists a symplectic transformation ϕ of T∗M and a non-vanishingfunction hϕ ∈ C∞(T∗M), such that ϕ∗(ω) = hϕω.

We say that two symplectic Monge–Ampère equations Eω and Eω are locallyequivalent at a point a ∈ T∗M if there exists a symplectic transformation ϕ ofa neighborhood Oa of a and a function hϕ ∈ C∞(Oa), hϕ(a) �= 0, such thatϕ∗(ω) = hϕω in a neighborhood.

20.1 Monge–Ampère equations with constant coefficients

The class of Monge–Ampère equations

Avxx + 2Bvxy + Cvyy + D(vxxvyy − v2xy)+ E = 0, (20.1)

that can be reduced to Monge–Ampère equations with constant coefficientsA, B, C, D, E by symplectic transformations is described by the followingtheorem.

Theorem 20.1.1 A non-degenerate Monge–Ampère equation Eω is symplect-ically equivalent to a Monge–Ampère equation with constant coefficients if andonly if

dω = 1

2d(ln |Pf(ω)|) ∧ ω. (20.2)

In this case a hyperbolic equation is locally equivalent to the wave equation

vxy = 0

422

20.1 Monge–Ampère equations with constant coefficients 423

and an elliptic one is equivalent to the Laplace equation

vxx + vyy = 0.

Remark 20.1.1 The conditions of the theorem can be formulated in terms ofthe de Rham complex: a non-degenerate Monge–Ampère equation Eω is sym-plectically equivalent to a Monge–Ampère equation with constant coefficientsif and only if d−1,2 = d2,−1 = 0.

Proof First of all note that if a non-degenerate Monge–Ampère equation Eω1

with the normed form ω1 can be reduced to another Monge–Ampère equation(20.1) with constant coefficients A, B, C, D, E by a symplectic diffeomorphismthen dω1 = 0.

Secondly, any non-degenerate effective differential 2-form ω can be writtenas ω = hω1, where h = √| Pf(ω1)|, and ω1 is an effective normed differential2-form. If the equation Eω is equivalent to a Monge–Ampère equation withconstant coefficients, then dω1 = 0. But

dω1 = d

(1

hω

)= − 1

h2dh ∧ ω + 1

hdω = 0,

and we obtain

dω − 1

2d(ln | Pf(ω)|) ∧ ω = dω − 1

hdh ∧ ω = 0. �

Proofs of sufficiency will be given for hyperbolic and elliptic equationsseparately.

20.1.1 Hyperbolic equations

Let ω be a hyperbolic form such that the condition (20.2) is realized and ω1 bea normed hyperbolic form: ω1

def= 1/√− Pf(ω)ω. Then Pf(ω1) = −1 and the

form ω1 is closed: dω1 = 0. From the definition of the Pfaffian it follows thatω1 ∧ ω1 +� ∧� = 0 and, since ω1 is effective and take into account the factthat ω1 is an effective form, i.e., ω1 ∧� = 0, the last equality can be rewrittenas (ω1 ±�) ∧ (ω1 ±�) = 0. Therefore, the 2-forms ω1 +� and ω1 −� aredecomposable and, therefore,

ω1 ±� = γ±1 ∧ γ±2 (20.3)

for some differential 1-forms γ±1 and γ±2 .

424 Classification of symplectic MAEs on 2D manifolds

Denote by V∓ distributions generated by γ±1 and γ±2 . We have

X� (ω1 ±�) = X�ω1 ± X�� = (AX ± X)��

for any vector field X . Therefore,

X ∈ ker(ω1 ±�)⇔ AX = ∓X

and V∓ = ker(A∓ 1).

Lemma 20.1.1 The distributions V− and V+ are completely integrable if andonly if dω1 = 0.

Proof Since

dω1 = dγ±1 ∧ γ±2 − γ±1 ∧ dγ±2 .

Condition dω1 = 0 implies

dγ±i ∧ γ±1 ∧ γ±2 = 0, (20.4)

for i = 1, 2.On the other hand, if the distributions V− and V+ are both completely

integrable then dγ±i = f±γ±1 ∧ γ±2 and

dω1 = dγ±1 ∧ γ±2 − γ±1 ∧ dγ±2 = 0.

Therefore, if dω1 = 0, then distributions V− and V+ are completely integ-rable. Take functionally independent integrals f1, f2 and g1, g2 for V− and V+respectively. Then

ω1 +� = df1 ∧ df2 and ω1 −� = dg1 ∧ dg2.

Therefore,

� = 1

2(df1 ∧ df2 + dg1 ∧ dg2),

ω1 = 1

2(df1 ∧ df2 − dg1 ∧ dg2).

If we put

q1def= 1

2f1, q2

def= 1

2g1, p1

def= f2, p2def= g2,


we get the following representations of ω1 and �:

ω1 = dq1 ∧ dp1 − dq2 ∧ dp2,

� = dq1 ∧ dp1 + dq2 ∧ dp2. �

Note that the form ω1 corresponds to the wave equation

vxy = 0, (20.5)

where x = q1, y = q2.

20.1.2 Elliptic equations

First of all we recall the definition of the Nijenhuis tensor (or the so-calledNijenhuis bracket [26]).

With each field of endomorphisms A on a smooth manifold � one can asso-ciate a so-called Nijenhuis tensor NA ∈ T� ⊗ �2(�) with the followingformula:

NA(X, Y)def= [AX , AY ] − A[AX , Y ] − A[X , AY ] + A2[X, Y ].

It is easy to check that NA is a tensor.Let A be an almost complex structure, A2 = −1, then the Newlander–

Nirenberg theorem [88] states that the almost complex structure A is integrableif and only if the Nijenhuis tensor NA is zero. The integrability of A means thereis a complex structure such that

A :

∂

∂xk−→ ∂

∂yk,

∂

∂yk−→ − ∂

∂xk

in local complex coordinates z1 = x1 + ιy1, . . . , zn = xn + ιyn.Let ω be an elliptic form such that condition (20.2) is realized and ω1 be a

normed elliptic form: ω1def= 1/

√Pf(ω)ω. Then Pf(ω1) = 1 and the form ω1

is closed. The corresponding operator Adef= Aω1 generates an almost complex

structure on T∗M: A2 = −1.Let us prove that this complex structure is integrable. Let X, Y be arbitrary

vector fields. The formulas ι[X,Y ] = [LX , ιY ] and LX = ιXd + dιX imply that

ι[X,Y ] = ιX dιY + dιX ιY − ιY ιX d − ιY dιX .


Let Z = NA(X, Y). To prove that the vector field Z is zero for any vector fieldsX and Y it is enough to show that ιZ(�) = 0. We have:

ι[X, Y ](�) = ιX dιY (�)− d�(X , Y)− ιY dιX(�),

ιA[X, AY ](�) = ι[X, AY ](ω1) = ιX dιAY (ω1)− ιAY dιX(ω1)− dω1(X, AY)

= −ιX dιY (�)+ d�(X , Y)− ιAY dιX(ω1),

ιA[AX, Y ](�) = ιAX dιY (ω1)+ ιY dιX(�)+ d�(X, Y),

ι[AX, AY ](�) = ιAX dιAY (�)+ d�(AY , AX)− ιAY dιAX(�)

= ιAX dιY (ω1)+ d�(X , Y)− ιAY dιX(ω1).

Adding these equalities, we find ιZ(�) = 0 and Z = 0, and, therefore, NA = 0.Due to the theorem of Newlander–Nirenberg, we see that the almost complex

structure A is integrable.Let

θdef= �C − ιωC

1 ,

where �C and ωC1 are complexifications of � and ω1 respectively.

Then θ(AX, Y) = θ(X , AY) = ιθ(X , Y) and dθ = 0. Therefore θ definesa complex symplectic structure on the complex manifold T∗M. Due to thecomplex version of the Darboux theorem one can find local complex coordinatesz1, z2 such that θ = dz1 ∧ dz2. Then � = dx1 ∧ dx2 − dy1 ∧ dy2 and ω1 =−dx1 ∧ dy2 − dy1 ∧ dx2, or

� = dq1 ∧ dp1 + dq2 ∧ dp2,

ω1 = dq1 ∧ dp2 + dp1 ∧ dq2

in coordinates q1 = x1, p1 = x2, q2 = y1, p2 = −y2.Note that the form ω1 corresponds to the Laplace equation vxx + vyy = 0

where x = q1, y = q2. Hence, the elliptic equation Eω can be transformed intothe Laplace equation by a symplectic transformation.

20.1.3 Parabolic equations

Now we consider the parabolic Monge–Ampère equation Eω with the corres-ponding field of endomorphisms Aω. Suppose that ω �= 0, but

Pf(ω) = 0


and

A2ω = 0.

Thus, we have ω ∧ ω = 0, i.e., the form ω is decomposable:

ω = α ∧ β

for some differential 1-forms α and β. Therefore, the equation Eω defines atwo-dimensional distribution

P = F〈α,β〉 (20.6)

on T∗M. This distribution is Lagrangian because ω is effective.In terms of the operator Aω the plane P(a) is a kernel of the linear

operator Aω,a.Indeed, if X ∈ P(a), then

Aω,aX��a = X�ωa = α(X)β − β(X)α = 0.

On the other hand, if Aω,aX = 0, then α(X) = β(X) = 0 and, therefore,X ∈ P(a).

So, we have P = ker Aω. Since A2ω = 0, we see that Pa = ker Aω,a ⊃ Im Aω,a

at any point a ∈ T∗M. Therefore, by reason of a dimensional, at any pointa ∈ T∗M we have

Im Aω,a = ker Aω,a = P(a). (20.7)

Theorem 20.1.2 The distribution P is completely integrable if and only if theNijenhuis tensor NAω = 0. If this condition is realized, then there exists a (local)coordinate system q, p on T∗M such that

� = dq1 ∧ dp1 + dq2 ∧ dp2,

ω = dq1 ∧ dp2.

In this case the equation Eω is locally symplectic equivalent to the equation

vxx = 0.

Proof Assume that

NAω = 0. (20.8)


From (20.7), we can choose non-zero vector fields X ∈ D(P) and Z /∈ D(P)such that AωZ �= X at any point of T∗M. Then the vector fields X and Y = AωZform a basis of the module D(P). Note that AωX = AωY = 0. Therefore we have

[X, Y ] = NAω(X , Y)− [AωX , AωY ] + Aω[AωX, Y ] + Aω[X, AωY ] = 0.

Now suppose P is completely integrable and f1 and g2 are its independentintegrals, i.e., ω = df1 ∧ dg2. Let f1 and g1 be some functions on T∗M thatare functionally independent on f1 and g2. Then we can choose the following(local) coordinates:

q1def= f1, q2

def= f2, p1def= g1, p2

def= g2.

In these coordinates ω = dq1 ∧ dp2 and � = dq1 ∧ dp1 + dq2 ∧ dp2. �

20.2 Non-degenerate quasilinear equations

The Monge–Ampère equation (20.1) is called quasilinear if D = 0, i.e., it hasthe following form:

Avxx + 2Bvxy + Cvyy + E = 0. (20.9)

The following problem was posed by Sophus Lie: when is the Monge–Ampèreequation (20.1) equivalent to a quasilinear one?

The following lemma gives necessary and sufficient conditions for such anequivalence.

Lemma 20.2.1 A Monge–Ampère equation Eω is locally equivalent to a quasi-linear one if and only if there exists a two-dimensional Lagrangian foliation ofT∗M by solutions of Eω.

Proof Suppose the equation Eω is quasilinear. It means that the form ω has noterm D dp1∧dp2. Hence, ω vanishes on the following two-parameter family ofLagrangian planes

{p1 = C1, p2 = C2}. (20.10)

Suppose now that there exist a two-dimensional Lagrangian foliation π suchthat each of its fibres is integral for the form ω. As we have seen all Lagrangianfoliations are locally equivalent. Hence, there exists a symplectic diffeomorph-ism ϕ which transforms π to the foliation ϕ(π) with fibres (20.10). It means

20.3 Intermediate integrals and classification 429

that the form (ϕ−1)∗(ω) vanishes on the foliation ϕ(π) and, because of this, ithas no term D dp1 ∧ dp. �

The following theorem is a simple corollary of the above lemma.

Theorem 20.2.1 [73]Let M be a real analytic two-dimensional manifold. Anynon-degenerate Monge–Ampère equation (20.1) with real analytic coefficientsA, B, C, D, E is locally equivalent to a quasilinear equation.

Proof Since the equation is non-degenerate at the point a, the Pfaffian does notvanish at a and therefore, in a neighborhood of a the Monge–Ampère satisfiesthe Cauchy–Kovalevsky theorem. Therefore, there exists a two-parameter ana-lytic family of solutions, which defines a Lagrangian foliation. Lemma 20.2.1completes the proof. �

Note that Lemma 20.2.1 and Theorem 20.2.1 hold not only on T∗M, but alsoon J1M. The proof is the same, but one has to consider Legendre foliationsinstead of Lagrangian ones.

Remark 20.2.1 Monge–Ampère equations on manifolds of dimension higherthan two are, in general, not quasilinearizable.

20.3 Intermediate integrals and classification

Let ω be an effective differential 2-form on the cotangent bundle T∗M. Assumethat the Monge–Ampère equation Eω is of hyperbolic or parabolic type andadmits two independent intermediate integrals f , g ∈ C∞(T∗M). This meansthat Eω can be represented by the effective part of the form df ∧ dg.

Theorem 20.3.1 Suppose that a Monge–Ampère equation Eω admits interme-diate integrals f , g such that the subspace

I def= {h ∈ C∞(T∗R2) | h = C1 + C2 f + C3g, (C1,C2, C3 ∈ R)}

of the vector space C∞(T∗R2) generated by a triplet (1, f , g) over R is aPoisson subalgebra with respect to the canonical Poisson algebra structure onC∞(T∗R2). Then the equation Eω is symplectically equivalent to one of thefollowing normal forms:

1. the wave equation vxy = 0 for hyperbolic equations;2. the differential equation vxx = 0 for parabolic equations.


Proof Since I is closed with respect to the Poisson brackets, we have

{ f , g} = c1 + c2f + c3g

for some c1, c2, c3 ∈ R. Then, up to an isomorphism, there are three possiblecases:

1. { f , g} = 1;2. { f , g} = 0;3. { f , g} = f .

The functions f and g can be transformed by a symplectic diffeomorphisminto p1 and q1 in the first case, and into p1 and q2 in the second case. The thirdcase is reduced to the first one by an evident re-scale: g is replaced by g0 = g/f .Then we have { f , g0} = 1. ω takes the form ω = dp1 ∧ dq1 + λ2� in the firstcase, and the form ω = dp1∧ dq2+λ2� in the second one. The correspondingequations are vxy = 0 and vxx = 0 respectively. �

20.4 Classification of generic Monge–Ampère equations

20.4.1 Monge–Ampère equations and e-structures

Here we consider a classification of generic symplectic Monge–Ampèreequations.

In order to solve this problem we construct e-structures for Monge–Ampèreequations as was done for Monge–Ampère operators above. Our constructiondoes not depend on the type of equation (see [61]).

Thus, let us consider a Monge–Ampère equation Eω, where ω is an effective2-form on T∗M. Suppose h is a smooth non-vanishing function on the cotangentbundle T∗M. Then the forms ω and hω generate the same Monge–Ampèreequation. Because of this, a tensor Tω is an invariant of the Monge–Ampèreequation if it does not change when we multiply the formω by h, i.e., Thω = Tω,and Tϕ∗ω = ϕ∗(Tω) for all symplectic diffeomorphisms ϕ.

Remark that the Pfaffian is not an invariant of the Monge–Ampère equations,but a relative one. Indeed,

Pf(hω) = h2 Pf(ω). (20.11)

In general, a tensor Tω is called a relative invariant of the Monge–Ampèreequation Eω if Thω = hkTω for some real number k and for any non-vanishing

20.4 Classification of generic Monge–Ampère equations 431

function h, and Tϕ∗(ω) = ϕ∗(Tω). If k = 0, then the tensor Tω is called anabsolute invariant or invariant.

Let us denote the Pfaffian of the form ω by Fω: Fω = Pf(ω). Let Xωdef= XFω

be the Hamiltonian vector field: Xω�� = −dFω. Then Fω and Xω are invariantsofω with respect to symplectic diffeomorphisms. The same holds for the vectorfields Zω, Wω, Vω.

Let us check what happens with them when we multiply the form ω by anon-vanishing function h.

Lemma 20.4.1 For any smooth function h on T∗M we have:

dh ∧ ω = (AωXh��) ∧�,

where Xh is the Hamiltonian vector field: Xh�� = −dh.

Proof The form ω is effective, i.e., ω ∧� = 0. Then

0 = Xh� (ω ∧�) = (AωXh��) ∧�− dh ∧ ω. �

Since the vector field Xω is Hamiltonian and Ahω = hAω, we see that

Xhω = Xh2Fω= h2Xω + 2hFωXh (20.12)

and

Zhω = AhωXhω = h3Zω + 2h2FωAωXh. (20.13)

Using Lemma 20.4.1, we have

Whω��2 = 2d(hω) = 2(dh ∧ ω + hdω) = 2((AωXh + hWω)��) ∧�.

Therefore

Whω = hWω + AωXh. (20.14)

For the vector field Vhω we have

Vhω = h2Vω − hFωXh. (20.15)

Let us now construct relative invariant vector fields Y1,ω, . . . , Y4,ω.We define

Y1,ωdef= Xω + 2Vω,

Y2,ωdef= Zω − 2FωWω.


Note that

Y2,ω = AωY1,ω.

Define two 1-forms σ1 and σ2:

σi,ωdef= Yi,ω�� (i = 1, 2)

and construct vector fields Y3,ω and Y4,ω and differential 1-forms σ3 and σ4

using the following formulas:

Y3,ω��2 = 2σ1,ω ∧ dσ1,ω,

Y4,ωdef= AωY3,ω,

σi,ωdef= Yi,ω�� (i = 3, 4).

From (20.12)–(20.15) it follows that

Y1,hω = Xhω + 2Vhω = h2(Xω + 2Vω) = h2Y1,ω,

Y2,hω = Zhω − 2h2FωWhω = h3(Zω − 2FωWω) = h3Y2,ω.

Therefore,

σ1,hω ∧ dσ1,hω = h2σ1,ω ∧ d(h2σ1,ω) = h4σ1,ω ∧ dσ1,ω.

Hence, the vector fields Yi,ω(i = 1, . . . , 4) are relative invariants and

Yi,hω = hi+1Yi,ω.

From now on we write the vector fieldss Yi,ω and 1-forms σi,ω without ω:Yi = Yi,ω and σi = σi,ω.

Let us find values of the forms σj on the vector fields Yi. By definition, put

rωdef= �(Y1, Y3), sω

def= �(Y2, Y4).


Then:

σi(Yi) = 0 (i = 1, . . . , 4),

σi(Yi+1) = �(Yi, AωYi) = 0 (i = 1, 3),

σi+1(Yi) = �(AωYi, Yi) = 0 (i = 1, 3),

σ2(Y3) = �(AωY1, Y3) = sω,

σ2(Y4) = �(AωY1, AωY3) = −Fωrω.

We collect this in the following table:

Y1 Y2 Y3 Y4

σ1 0 0 rω sωσ2 0 0 sω −Fωrωσ3 −rω −sω 0 0σ4 −sω Fωrω 0 0

Lemma 20.4.2 The following relations:

1. Yi�ω = σi+1 (i = 1, 3),2. Yi+1�ω = −Fωσi (i = 1, 3),3. σ3 ∧� = σ1 ∧ dσ1,4. σi ∧ ω = −σi+1 ∧� (i = 1, 3),5. σi+1 ∧ ω = Fωσi ∧� (i = 1, 3),6. σ2 ∧ σ3 ∧� = σ1 ∧ σ4 ∧�,7. σ4 ∧ σ2 ∧� = Fωσ1 ∧ σ3 ∧�,8. σ1 ∧ σ2 ∧ σ3 = sωσ1 ∧�− rωσ2 ∧�

hold.

Proof The proofs of relations 1 and 2 are similar. For example,

Yi+1�ω = −AωYi+1�� = A2ωYi�� = −FωYi�� = −Fωσi

proves relation 2, and

2σ1 ∧ dσ1 = Y3��2 = 2(Y3��) ∧� = 2σ3 ∧�.

proves relation 3.The proofs of relations 4 and 5 are similar.


Let us prove relation 4. Since ω ∧� = 0, we have

0 = Yi� (ω ∧�) = σi+1 ∧�+ σi ∧ ω.

To prove relation 6. we note that from relations 4 and 5 it follows that

σ3 ∧ σ2 ∧� = −σ3 ∧ σ1 ∧ ω = σ1 ∧ σ3 ∧ ω = −σ1 ∧ σ4 ∧�.

In the similar way relation 7 can be proved. Multiplying both sides of relation 6,by Y1,ω, we get relation 8. �

Lemma 20.4.3 One has

2σ1 ∧ σ2 ∧ σ3 ∧ σ4 = gω�2,

where

gωdef= Fωr2

ω + s2ω.

Proof It follows from relation 8 that(Fωr2

ω + s2ω

)� = sωσ2 ∧ σ3 + rωFωσ1 ∧ σ3 + rωσ4 ∧ σ2 + sωσ1 ∧ σ4.

Multiplying both sides by � and using relations 7 and 8, we obtain(Fωr2

ω + s2ω

)�2 = 2(sωσ2 ∧ σ3 + rωσ4 ∧ σ2) ∧�.

Multiplying relation 8 byσ4 and using relation 6 we obtain from the last formula:

σ1 ∧ σ2 ∧ σ3 ∧ σ4 = sω σ2 ∧ σ3 ∧�+ rωσ4 ∧ σ2 ∧� = 1

2gω�

2. �

It follows from this lemma that the vectors Y1,a, . . . , Y4,a are linearlyindependent if and only if

gω(a) �= 0. (20.16)

In what follows we assume that condition (20.16) holds.Note that

rhω = h6rω, shω = h7sω, ghω = h14gω.


This enables us to construct the following.

1. Functional invariants of the equation Eω:

Fdef= Fωs−2/7

ω ,

I = rωs−6/7ω .

2. Invariant vector fields

X1def= s2/7

ω

gω(FωrωY3 + sωY4),

X2def= − 1

s2/7ω

Y1,

X3def= s3/7

ω

gω(sωY3 − rωY4),

X4def= − 1

s3/7ω

Y2.

3. Invariant differential 1-forms

θ1def= 1

s2/7ω

σ1,

θ2def= s2/7

ω

gω(Fωrωσ3 + sωσ4),

θ3def= 1

s3/7ω

σ2,

θ4def= s3/7

ω

gω(sωσ3 − rωσ4).

Note that for a general Monge–Ampère equation these vector fields (anddifferential 1-forms) are linearly independent and form bases of the modulesD(Oa) and �1(Oa) respectively in a neighborhood Oa of a point a. In otherwords they produce an e-structure in Oa related to the Monge–Ampère equation.

Let us show how to normalize the form ω. We set

ωdef= s−1/7

ω ω.


Note that the function F and the field of endomorphisms

Adef= s−1/7

ω Aω

are invariants.It is easy to see that

θ1 = −X2��, θ3 = −X4��,

θ2 = X1��, θ4 = X3�� (20.17)

and

AX2 = X4, AX3 = X1.

In addition, the bases X1, . . . , X4 and θ1, . . . , θ4 are dual.Summarizing we get the following result.

Theorem 20.4.1 Suppose that ωa �= 0, sω(a) �= 0 and gω(a) �= 0. Then in aneighborhood of the point a we have the following representation of the forms� and ω:

� = θ1 ∧ θ2 + θ3 ∧ θ4,

ω = θ3 ∧ θ2 + Fθ4 ∧ θ1. (20.18)

This theorem shows that two Monge–Ampère equations Eω1 and Eω2 arelocally equivalent if there exists a local diffeomorphism ϕ and such that

ϕ∗(F2) = F1,ϕ∗(θ2

j

)= θ1

j , j = 1, 2, 3, 4, and ϕ(a) = a.

Theorem 20.4.2 Symmetries of a Monge–Ampère equation are symmetries ofthe e-structure preserving the function F.

20.4.2 Normal forms of mixed-type equations

We use canonical representation (20.18) to construct normal forms of mixed-type Monge–Ampère equations.

We shall suppose that for the form ω the following conditions hold:

1. F(a) = 0 and the differential of the function F does not vanish at the pointa: dFa �= 0;

2. the first covariant derivatives of the function F are constants, i.e., Xj(F) =constant(j = 1, . . . , 4);


3. the vector fields X1, . . . , X4 form the four-dimensional Lie algebra g =⊕4i=1 RXi.

Denote by ckij the structure constants of the Lie algebra g:

[Xi, Xj] =4∑

k=1

ckijXk .

The condition d� = 0 gives the following extra conditions for the structureconstants:

c113 + c2

23 + c412 = 0, c1

14 + c224 − c3

12 = 0,

c234 − c3

13 − c414 = 0, c1

34 + c323 − c4

24 = 0. (20.19)

For the 2-form ω one has

σ1,ω = θ1, σ2,ω = θ3, σ3,ω = θ4 + rωθ2, σ4,ω = θ2 − Frωθ4,

and from the third property of Lemma 20.4.2 it follows that

c123 = c1

34 = 0, c124 = −1

and rω = 0.Note that

Xω = −f 2X1 + f 1X2 − f 4X3 + f 3X4,

Vω = α3X1 − α4FX2 − α1FX3 + α2X4,

where f i def= Xi(F), and

α1 = −c334 − c2

24,

α2 = c214 + (c4

34 − c113)F − f 3,

α3 = c314 + (c1

12 − c424)F + f 2,

α4 = −c212 − c3

13 + c423F.

Then,

Y1 = (2α3 − f 2)X1 + (f 1 − 2α4F)X2 − (f 4 + 2α1F)X3 + (f 3 + 2α2)X4.


Remark that sω = 1 and, therefore, Y1 = −X2, and

2α3 − f 2 = f 4 + 2α1F = f 3 + 2α2 = 0, f 1 − 2α4F = −1.

Finally,

f 1 = −1− 2(c212 + c3

13)F + 2c423F2,

f 2 = −2c314 + 2(c4

24 − c112)F,

f 3 = 2c214 + 2(c4

34 − c113)F,

f 4 = 2(c224 + c3

34)F.

Since all derivatives are constants and dF �= 0, we obtain

c423 = c2

12 + c313 = c4

34 − c113 = c4

24 − c112 = c2

24 + c334 = 0

f 1 = −1, f 2 = −2c314, f 3 = 2c2

14, f 4 = 0. (20.20)

From the Jacobi identity for the Lie algebra g we find, that

c113 − c2

23 + c434 = 0.

Note that

f ji − f ij = [Xi, Xj](F) =

4∑k=1

ckijf

k ,

where f ji def= Xi(f j). But from condition 2 we see that f ij = 0 and∑4

k=1ck

ij f k = 0.Summarizing, we obtain the following relations for the structure

constants ckij:

c114 = 0, 2(c2

24c314 − c3

24c214) = 1,

c214c3

23 − c223c3

14 = 0, c234c3

14 − c334c2

14 = 0,

c112 + 2(c2

12c314 − c3

12c214) = 0, c1

13 + 2(c213c3

14 − c313c2

14) = 0. (20.21)


There exists a unique Lie algebra g satisfying conditions (20.19)–(20.21) andits structure is given by the following multiplication table.

[↑,→] X1 X2 X3 X4

X1 0 0 0 bX2 + aX3

X2 0 0 0 −X1 − 1

2bX3

X3 0 0 0 0

X4 −bX2 − aX3 X1 + 1

2bX3 0 0

Here a �= 0 and b �= 0 are arbitrary constants. This algebra has the followingrepresentation by vector fields:

X1 = ∂

∂q1− bq1

∂

∂q2,

X2 = −p2∂

∂q2+ ∂

∂p1,

X3 = ∂

∂q2,

X4 = −p1∂

∂q1+(

aq1 − 1

2bp1

)∂

∂q2+ bq1

∂

∂p1+ ∂

∂p2.

Therefore,

θ1 = dq1 + p1 dp2,

θ2 = dp1 − bq1 dp2,

θ3 = dq2 + bq1 dq1 + p1dp1 +(

1

2bp1 − aq1

)dp2,

θ4 = dp2.

Assume that a = (0, 0, 0, 0). Then

F = −q1 + b2q21 + 2bq2 − 2ap1 + bp2

1

and we obtain the following normal effective 2-form:

ω =− (F + b2q21) dq1 ∧ dp2 + dq2 ∧ dp1

+ bq1(dq1 ∧ dp1 − dq2 ∧ dp2)+(

aq1 − 1

2bp1 − bp1q1

)dp1 ∧ dp2.


The corresponding Monge–Ampère equation E" is

vxx − 2bxvxy + (2by− x − 2avx − bv2x)vyy

+((

1

2b+ bx

)vx − ax

)Hess v = 0, (20.22)

where x = q1and y = q2.

Theorem 20.4.3 Assume that for equation Eω derivatives of the function Falong X1, . . . , X4 are constant and these vector fields form a four-dimensionalLie algebra. Then this equation at the point a ∈ {Fω = 0} is locally equivalentto (20.22).

20.5 Applications

In this section we apply the obtained results to some popular differentialequations.

20.5.1 The Born–Infeld equation

The effective differential 2-form

ω = (1− p21) dq1 ∧ dp2 + p1p2(dq1 ∧ dp1 − dq2 ∧ dp2)+ (1+ p2

2) dq2 ∧ dp1

corresponds to the Born–Infeld equation [11, 111]

(1− ϕ2t )ϕxx + 2ϕtϕxϕtx − (1+ ϕ2

x )ϕtt = 0, (20.23)

where q1 = t, q2 = x. Its Pfaffian is Pf(ω) = p21 − p2

2 − 1. We consider adomain where this equation is hyperbolic, i.e. p2

1 − p22 − 1 < 0. The normed

form is

ω = ω√1− p2

1 + p22

.

For this form

µ+ =−p1 + p2

√1− p2

1 + p22

−1+ p21 − p2

2

dp1 +p2 − p1

√1− p2

1 + p22

−1+ p21 − p2

2

dp2,

µ− =p1 + p2

√1− p2

1 + p22

−1+ p21 − p2

2

dp1 −p2 + p1

√1− p2

1 + p22

−1+ p21 − p2

2

dp2

20.5 Applications 441

and the distributions ker µ+ and ker µ− are completely integrable and

g0 = g+ = g− = 0.

Due to Theorem 19.2.3, the form ω is equivalent to the form (19.8). Integralsof the distributions ker µ+ and ker µ− are

H+ = p1 + p2

1+√

1− p21 + p2

2

and

H− = −p1 + p2

1+√

1− p21 + p2

2

respectively. Follow the proof of Theorem 19.2.2, we can introduce newcoordinates

Q1 = H+,

Q2 = H−,

P1 = −1

2

[q1

(1+ p1p2 − p2

1 +√

1− p21 + p2

2

)+ q2(1− p1p2 + p2

2 +√

1− p21 + p2

2

)],

P2 = 1

2

[q1

(1− p2

1 − p1p2 +√

1− p21 + p2

2

)− q2

(1+ p1p2 + p2

2 +√

1− p21 + p2

2

)].

The inverse symplectic transformation is

q1 = (Q1Q2 − 1)((1+ Q21)P1 − (1+ Q2

2)P2)

2(1+ Q1Q2),

q2 = (Q1Q2 − 1)((1− Q21)P1 + (1− Q2

2)P2)

2(1+ Q1Q2),

p1 = −Q1 + Q2

−1+ Q1Q2,

p2 = Q1 + Q2

1− Q1Q2


and we get the following 2-form:

ω = 4(Q1P1 + Q2P2)

(Q1Q2)2 − 1dQ1 ∧ dQ2 − dP1 ∧ dQ1 + dP2 ∧ dQ2.

This means that in the domain of hyperbolicity Born–Infeld’s equation (20.23)is equivalent to the following linear wave equation:

vQ1Q2 =2(Q1vQ1 + Q2vQ2)

(Q1Q2)2 − 1. (20.24)

20.5.2 Gas-dynamic equations

The following equation describes one-dimensional non-stationary gas flow:

ϕtt + 2ϕxϕtx +((k − 1)ϕt + k + 1

2ϕ2

x

)ϕxx = 0. (20.25)

Here ϕ is a velocity potential and k is a constant (an adiabatic index), k > 1.This equation is generated by the following differential 2-form:

ω = dq1 ∧ dp1 + (dq1 ∧ dp1 − dq2 ∧ dp2)p1

−(

k + 1

2p2

1 + (k − 1)p2

)dq2 ∧ dp1.

Here q1 = x and q2 = t.Suppose that the Pfaffian of ω is negative:

Pf(ω) = 1

2(k − 1)(p2

1 + 2p2) < 0.

Then the corresponding normed differential 2-form is

ω =√

2

(1− k)(p21 + 2p2)

ω.

For this 2-form the conditions of Theorem 19.2.3 are satisfied and

µ+ = k − 3

4(k − 1)(p21 + 2p2)

((2p1 +

√2√(1− k)(p2

1 + 2p2)

)dp1 + 2 dp2

),

µ− = k − 3

4(k − 1)(p21 + 2p2)

((2p1 −

√2√(1− k)(p2

1 + 2p2)

)dp1 + 2 dp2

).


Integrals of the distributions ker µ+ and ker µ− are

H+ = p1√2+√(1− k)(p2

1 + 2p2)

1− k,

H− = − p1√2+√(1− k)(p2

1 + 2p2)

1− k

respectively.The following symplectic transformation

q1 −→ P2((k − 3)Q2 + (1+ k)Q1 − P1((k − 3)Q1 + (1+ k)Q2))√2(k − 1)(Q1 + Q2)

,

q2 −→ 2(P1 + P1)

(k − 1)(Q1 + Q2),

p1 −→ 1√2(Q1 − Q2),

p2 −→ −1

8(2(k − 3)Q1Q2 + (k + 1)(Q2

1 + Q22))

transforms integrals H+ and H− into Q1 and Q2 and the differential 1-formsµ+ and µ− into

3− k

(k − 1)(Q1 + Q2)dQ1 and

k − 3

(k − 1)(Q1 + Q2)dQ2

respectively.In these coordinates the differential 2-form ω has the form

ω = (k − 3)(P1 + P2)

(1− k)(Q1 + Q2)dQ1 ∧ dQ2 + dQ1 ∧ dP1 − dQ2 ∧ dP2.

So, in a hyperbolicity domain equation (20.25) is equivalent to the followinglinear wave equation:

vQ1Q2 =(k − 3)(vQ1 + vQ2)

2(1− k)(Q1 + Q2). (20.26)

For (20.26) one can construct a class of partial solutions v(Q1, Q2) = �(Q1+Q2). For the function � we have an ordinary differential equation of second


order:

�′′ = k − 3

(1− k)z�′,

where z = Q1 + Q2.Its general solution is

�(z) = C1z2/(k−1) + C2

(C1 and C2 are arbitrary constants) and we obtain the following two-parameterfamily of solutions of (20.26):

v(Q1, Q2) = C1(Q1 + Q2)2/(k−1) + C2. (20.27)

To construct multivalued solutions of the non-linear equation (20.25)corresponding to (20.27), we write the multivalued solution of (20.26):

P1 = P2 = 2C1

k − 1(Q1 + Q2)

(3−k)/(k−1).

But

Q1 = p1√2+√(1− k)(p2

1 + 2p2)

1− k,

Q2 = − p1√2+√(1− k)(p2

1 + 2p2)

1− k,

P1 = − 1

2√

2

(2q1 + q2

(− 2p1 +

√2√(1− k)(p2

1 + 2p2)))

,

P2 = − 1

2√

2

(− 2q1 + q2

(2p1 +

√2√(1− k)(p2

1 + 2p2)))

,

and we obtain the following family of multivalued solutions of (20.25):

q1 − q2p1 = 0, 2p2 + p21 = z(q2, C)

where a function z(q2) is a solution of the following functional equation:

C(−z)1/(k−1) + (k − 1)k/(k−1)q2z = 0.


Figure 20.1. The solution ϕ(x, t) = x2/2t.

For example, we can put C = 0 and then for any k > 1, k �= 3 we have themultivalued solution

L = {q1 − q2p1 = 0, 2p2 + p21 = 0}.

The corresponding classical solution is (see Figure 20.1)

ϕ(x, t) = x2

2t.

In the case when k = 3 we have dω = 0 and (20.25) is equivalent to thewave equation with constant coefficients

vQ1Q2 = 0.

20.5.3 Two-dimensional stationary irrotational isentropicflow of a gas

The following equation

(ϕ2x − c2)ϕxx + 2ϕxϕyϕxy + (ϕ2

y − c2)ϕyy = 0 (20.28)

describes two-dimensional stationary irrotational isentropic flow of a gas [7].Here ϕ is the velocity potential and c2 is the square of the velocity of sound. It is


easy to see that the corresponding formω is a wave type and g0 = g+ = g− = 0.Due to Theorem 19.2.3, this equation is equivalent to the equation

vq1q2 = A(q)vq1 + B(q)vq2

for some functions A(q) and B(q). Indeed, it is easy to see that the godographtransformation

(q1, q2, p1, p2) → (p1, p2,−q1,−q2)

sends (20.28) into a linear equation.

21

Contact classification of MAEs ontwo-dimensional manifolds

Letω, ω ∈ �2ε(J

1M). We say that two Monge–Ampère equations Eω and Eω areequivalent if there exists a contact transformationϕ of J1M and a non-vanishingfunction hϕ ∈ C∞(J1M), such that ϕ∗(ω)ε = hω.

We say that two Monge–Ampère equations Eω and Eω are locally equivalentat a point a∈ J1M if there exists a contact transformation ϕ of a neighborhoodOa of a and a function hϕ ∈C∞(Oa), such that ϕ∗(ω)ε = hϕω, ϕ(a) = a,where hϕ(a) �= 0.

21.1 Classes Hk,l

In this chapter we consider Monge–Ampère equations of constant types (hyper-bolic, elliptic or parabolic). In order to unify all types, we consider a complextangent bundle TC(J1M). Let Eω be a Monge–Ampère equation and let Cj

(j = 1, 2) be a subbundle of TC(J1M) whose fibre Cj(a) at each point a ∈ J1Mis a complex eigenspace of the operator Aω, i.e.,

Cj(a) = {X ∈ TCa (J1M)| AωX = λjX},

where λj is an eigenvalue of Aω. Thus we have the complex distributions Cj onJ1M:

Cj : a ∈ J1M → Cj(a) ⊂ TCa (J1M).

Let C(k)j be the kth derivative of the distribution Cj: C(0)

jdef= Cj.

We assume that all C(k)j are distributions. Then for these distributions we

expect one of the following five cases:

(0) C(1)j = Cj;

(1) Cj �= C(1)j = C(2)

j and dimC C(1)j = 3;

447

448 Contact classification of MAEs on two-dimensional manifolds

(2) Cj �= C(1)j �= C(2)

j = C(3)j and dimC C(1)

j = 3, dimC C(2)j = 4;

(3) Cj �=C(1)j �= C(2)

j �= C(3)j = TC(J1M) and dimC C(1)

j = 3, dimC C(2)j = 4;

(4) Cj �= C(1)j �= C(2)

j = TC(J1M) and dimC C(1)j = 3.

We say that the Monge–Ampère equation belongs to the class Hk,l at a pointa ∈ J1M (k, l = 0, . . . , 4; k ≤ l) if case (k) holds for one of Cj and case (l)holds for the other. Classes Hk,l are invariant under contact transformations.

Note that the class H0,0 contains only parabolic equations. The classes Hk,k

(1 ≤ k ≤ 4) contain only parabolic or elliptic equations. For hyperbolicequations eight classes Hk,l (k, l = 1, . . . , 4; k ≤ l) are possible.

Lemma 21.1.1 Let functions F1, F2 ∈ C∞(J1R2) be such that (dF1 ∧ dF2 ∧ω0)a �= 0 and dF1 ∧ dF2 ∧ ω0 ∧ dω0 = 0. Then there exists a contactdiffeomorphism ϕ such that ϕ∗(F1) = q1, ϕ∗(F2) = q2.

Proof It follows from the conditions of the lemma that the two-dimensionaldistribution P = F〈dF1, dF2,ω0〉 is completely integrable. Let F0 bean integral of P such that dF0 ∧ dF1 ∧ dF2 �= 0. Then ω0 =∑2

i=0 ai dFi for some smooth functions ai (i = 0, 1, 2). Since (dF1 ∧ dF2 ∧ω0)a �= 0, we see that a0 �= 0. Therefore ω0 = a0(dF0 − F3 dF1 −F4 dF2), where F3

def= a1/a0, F4def= −a2/a0. Moreover, the condition

ω0 ∧ dω0 ∧ dω0 �= 0 implies that the functions F0, . . . , F4 are functionalyindependent. Then ψ1 : (q1, q2, u, p1, p2)→ (F1, F2, F0, F3, F4) is the contactdiffeomorphism. �

Let us introduce the following modules of differential 1-forms:

�1(Cj)def= {α ∈ �1(J1M)C|X1�α = 0 and X�α = 0 ∀X ∈ D(C3−j)}.

Since the form�a = dω0|C(a) is non-degenerate on C(a), we see that the map

D(Cj) � X → X�� ∈ �1(Cj)

is an isomorphism of the modules.Moreover, since C1 and C2 are skew-orthogonal

α1 ∧ α2 ∧ dω0 = 0 (21.1)

for α1 ∈ �1(C1) and α2 ∈ �1(C2).

Theorem 21.1.1 (D. Tunitskii) Assume that a Monge–Ampère equationbelongs to one of the classes Hk,l (0 ≤ k ≤ l ≤ 2). Then this equation is

21.1 Classes Hk,l 449

locally contact equivalent to a Monge–Ampère equation of type

vxx + εvyy = f (x, y, v, vx , vy), (21.2)

where ε = sgn(Pf(ω)) and f is a smooth function.

Proof Hyperbolic case. In this case the distributions C1 = C+ and C2 = C−are real and

C±(a) = {X ∈ C(a)|Aω,aX = ±λX}

where λ = √Pf(ω).Denote by F+ and F− first integrals of the distributions C(2)

+ and C(2)− respect-

ively. Then dpF+ ∈ �1(C−) and dpF− ∈ �1(C+). By (21.1), we see thatdpF+ ∧ dpF− ∧ dω0 = 0. Therefore dF+ ∧ dF− ∧ dω0 ∧ ω0 = 0. Moreover,dF+ ∧ dF− ∧ ω0 �= 0. Due to Lemma 21.1.1, suppose that F+ = q1 andF− = q2 in local canonical coordinates. We will still denote the correspondingform ϕ∗(ω)ε by ω.

Then

ω ∧ ω − Pf(ω)� ∧� = (ω + λ�) ∧ (ω − λ�) = 0.

Therefore, the 2-forms ω+ λ� and ω− λ� are decomposable and ω± λ� =α± ∧ β± for some differential 1-forms α+,β+,α−,β−. Note that α+,β+ ∈�1(C+) and α−,β− ∈ �1(C−). Indeed for a vector field X ∈ D(C−) we haveX1�(ω + λ�) = 0 and

X�(ω + λ�) = X�ω + λX�� = (AωX + λX)�� = (−λX + λX)�� = 0.

We can choose α+ = dq1 and α− = dq2.Then

ω + λ� = dq1 ∧ β+,

ω − λ� = dq2 ∧ β−

and

� = 1

2λ(dq1 ∧ β+ − dq2 ∧ β−).

On the other hand, � = dq1 ∧ dp1 + dq2 ∧ dp2, therefore

β+ = a1 dq1 + a2 dq2 + 2 λ dp1


and

β− = b1 dq1 + b2 dq2 − 2 λ dp2,

where ai, bi (i = 1, 2) are smooth functions, such that a2 + b1 = 0.Then

ω = 12 (dq1 ∧ β+ + dq2 ∧ β−) = s dq1 ∧ dq2 + dq1 ∧ dp1 − dq2 ∧ dp2,

with s = a2 − b1.Finally the following contact transformation

χ :

q1 → 1

2 (q1 − q2 + q01 + q0

2), q2 → 12 (q1 + q2 − q0

1 + q02),

u → u− 12 ((q1 − q0

1)(p01 − p0

2)+ (q2 − q02)(p

01 + p0

2)),

p1 → p1 − p2 + p02, p2 → p1 + p2 − p0

1

takes ω to the form

ω = dq1 ∧ dp2 + dq2 ∧ dp1 − f dq1 ∧ dq2

for some smooth function f .

Elliptic caseIn this case the distributions C1 = C+ and C2 = C− are complexdistributions, and

C±(a) = {X ∈ C(a)|Aω,aX = ±ıλX},

where λ = √− Pf(ω). Note that dimC C(2)± ≤ 4, C1 = C2, and the distributions

C(2)j +C

(2)j (j = 1, 2) are involutive. Due to the Frobenius–Nirenberg theorem,

the complex distributions C(2)1 and C(2)

2 are completely integrable. Let Gj be a

first integral of the distribution C(2)j . Note that G1 = G2, dpG1 ∈ �1(C2) and

dpG2 ∈ �1(C1). By (21.1), we see that dpG1 ∧ dpG2 ∧ dωC0 = 0. Therefore

dG1 ∧ dG2 ∧ dωC0 ∧ ωC

0 = 0. We set F1 = Re G1, F2 = Im G1. From theinequalities dG1 ∧ dG2 ∧ ωC

0 �= 0 and dG1 ∧ dG2 ∧ dωC0 ∧ ωC

0 = 0 it followsthat dF1 ∧ dF2 ∧ ω0 �= 0 and dF1 ∧ dF2 ∧ dω0 ∧ ω0 = 0.

Due to Lemma 21.1.1, we suppose that F1 = q1 and F2 = q2. Similar to thehyperbolic case, we have: (ωC ± ιλ�C) ∧ (ωC ± ιλ�C) = 0. Therefore, the2-forms ωC + ιλ�C and ωC − ιλ�C are decomposable: ωC + ιλ�C = α ∧ β


and ωC− ιλ�C = α∧β, where α,β ∈ �1(C1) and α,β ∈ �1(C2) = �1(C1).Let us put α = dG1and β = γ1 + ιγ2.

Then

ωC + ιλ�C = (dq1 + ι dq2) ∧ (γ1 + ιγ2),

ωC − ιλ�C = (dq1 − ι dq2) ∧ (γ1 − ιγ2)

and

�C = 1

λ(dq1 ∧ γ2 + dq2 ∧ γ1).

On the other hand�C = dq1∧dp1+dq2∧dp2, and we obtain γ1 = a dq2+λ dp2

and γ2 = b dq1 + λ dp1. Then

ω = s dq1 ∧ dq2 + dq1 ∧ dp2 − dq2 ∧ dp1,

for a smooth real function s.

Parabolic caseIn this case the real distributions C1 and C2 coincide: P

def= C1 = C2. Theequation Eω can be in one of the classes H0,0, H1,1 or H2,2 only.

Suppose that Eω belongs to one of the classes H0,0 or H1,1. In this casedim P(2) ≤ 3. Let F1 and F2 be functional independent first integrals of thedistribution P(2). The conditions of Lemma 21.1.1 hold, therefore we can putF1 = q1 and F2 = q2. Suppose now that Eω belongs to one of the classesH2,2. In this case dim P(2) = 4 and we have only one first integral F1 ofthe distribution P(2). The missing function F2 can be found from the relationdF1∧dF2∧ω0∧dω0 = 0. This relation is a first-order scalar linear differentialequation with respect to function F2, and therefore has a solution. By choosinginitial values in a suitable way the solution can be chosen in such a way thatdF1 ∧ dF2 ∧ ω0 �= 0.

For the parabolic equation Eω the form ω is decomposable: ω = α ∧ β,and the 1-forms ω0,α,β generate the distribution P: P = F(ω0,α,β). Dueto Lemma 21.1.1, we have ω = dq2 ∧ γ , where γ is some 1-form. Since ω

is effective, ω ∧ � = dq2 ∧ γ ∧ (dq1 ∧ dp1 + dq2 ∧ dp2) = 0, we obtainγ = a dq1 + b dq2 + c dp, where a, b and c are functions. Then

ω = s dq1 ∧ dq2 + t dq2 ∧ dp1. �


Corollary 21.1.1 Let a parabolic Monge–Ampère equation belong to the classeH2,2. Then it is locally contact equivalent to an equation of type

vy = g(x, y, v, vx , vxx),

where g is a smooth function.

Let us consider the following complex distributions:

ldef= C(1)

1 ∩ C(1)2

and

l1 = C1 ∩ C(2)2 ,

l2 = C2 ∩ C(2)1 .

Assume that in some domain of J1M the Monge–Ampère equation Eω

belongs to the classes H2,2 and Pf(ω) �= 0. Then l ⊕ lj (j = 1, 2) aretwo-dimensional complex distributions.

The following lemma gives a conditions of linearization with respect to thefirst derivatives of non-degenerate equations of class H2,2.

Theorem 21.1.2 (D. Tunitsky) Assume that a non-degenerate Monge–Ampèreequation Eω belongs to the class H2,2 and Pf(ω) �= 0. Then this equation islocally contact equivalent to a Monge–Ampère equation of the type

vxx + εvyy = r(x, y, v)vx + s(x, y, v)vy + g(x, y, v)

if and only if the distributions l ⊕ l1 and l ⊕ l2 are completely integrable.

Proof Due to Theorem 21.1.1, Eω is equivalent to (21.2). For this equationthe corresponding effective differential 2-form is

ω = −f (q, u, p) dq1 ∧ dq2 + dq1 ∧ dp2 + ε dq2 ∧ dp1

and the distribution l is generated by the vector field

2∂

∂u+ εfp1

∂

∂p1+ fp2

∂

∂p2.

The distributions l1,0 and l0,1 are generated by the vector fields ∂/∂p1 and∂/∂p2 respectively. Since the distributions l ⊕ l1 and l ⊕ l2 are completely


integrable, we see that fp1p1 = fp2p2 = 0. Therefore the function f has thefollowing form:

f (q, u, p) = r(q, u)p1 + s(q, u)p2 + w(q, u)(p21 + εp2

2)+ g(q, u)

for some functions r, s, w and g. To prove that (21.2) with this function f islocally contact equivalent to the same equation with the right part:

r(q, u)p1 + s(q, u)p2 + g(q, u)

for some r, s, g we apply the following contact transformation

ϕ : (q1, q2, p1, p2, u) → (q1, q2, Zq1 + p1Zu, Zq2 + p2Zu, Z)

that preserves the point a and where Z = Z(q, u) is a smooth function. Thenthe effective part of ϕ∗(ω) is

ϕ∗(ω)ε = Zu(dq2 ∧ dp1 − ε dq1 ∧ dp2)

+ (rp1Zu + sp2Zu + g− (Zuu − wZ2u )(p

21 + εp2

2)

+ sZq2 + 2εwp2ZuZq2 + εwZ2q2− 2εp2Zq2u

− εZq2q2 + rZq1 + 2wp1ZuZq1 + wZ2q1

− 2p1Zq1u − Zq1q1) dq1 ∧ dq2,

where r, s, g, w are functions in q, u. Let (q0, u0, p0) be local coordinatesof the point a. Take Z as the solution of the ordinary differential equationZuu − wZ2

u = 0 with the Cauchy data Z(q, u0) = u0, Zu(q, u0) = 1, and weobtain the required function Z . �

The following theorem is a contact analog of Theorem 20.1.1 for symplecticMonge–Ampère equations.

Theorem 21.1.3 A Monge–Ampère equation is locally contact equivalent to aMonge–Ampère equation of type

vxx + εvyy = 0

if and only if it belongs either to the class H1,1 (in the case of a non-zeroPfaffian), or to the class H0,0 (in the case where the Pfaffian is zero).


21.2 Invariants of non-degenerate Monge–Ampère equations

21.2.1 Tensor invariants

As we have seen, any non-degenerate Monge–Ampère equation determinesa triplet of the real (for hyperbolic case) and complex (for elliptic case)distributions P = ({C1,0, C0,1}, l, ) on J1M.

Moreover, Monge–Ampère equations E and E are contact equivalent if andonly if there exists a diffeomorphism ϕ such that ϕ∗(P) = P . Here P is thetriplet corresponding to the equation E.

In Part I we have constructed the tensors invariants d−1,1,1, d1,1,−1, d1,−1,1,d0,−1,2, d2,−1,0, d2,1,−2 and d−2,1,2 of the corresponding AP-structure. In thisspecial case, we have d1,−1,1 = 0 because the distributions C1,0 and C0,1 areskew-orthogonal. Moreover, tensors d2,1,−2 ,d2,0,−1,d−1,0,2 and d−2,1,2 are alsoequal to zero.

Let us prove, for example, that tensors d2,0,−1 and d−1,0,2 are trivial.Since

d2,0,−1 : �p,r,q(J1M)→ �p+2,r,q−1(J1M),

we see that d2,0,−1 = α1 ∧ α2 ⊗ Q, where αi ∈ �1,0,0(J1M) and Q ∈ D(C0,1).Let β ∈�0,0,1(J1M) be an arbitrary 1-form and P1, P2 be arbitrary vector fieldsfrom the distribution C1,0. Then since β ∈ �0,0,1(J1M) and [P1, P2] ∈ D(C(1)

1,0),we have

dβ(P1, P2) = −β([P1, P2]) = 0.

But dβ(P1, P2) = d2,0,−1β(P1, P2). Therefore, d2,0,−1β(P1, P2) = 0 for anyβ ∈ �0,0,1(J1M) and P1, P2 ∈ D(C1,0).

Thus, we have four 2-covariant and 1-contravariant tensors d−1,1,1, d1,1,−1,d0,−1,2 and d2,−1,0:

d1,1,−1 = ωC0 ∧ α1 ⊗ Q1 + ωC

0 ∧ α2 ⊗ Q2,

d−1,1,1 = ωC0 ∧ β1 ⊗ P1 + ωC

0 ∧ β2 ⊗ P2,

d2,−1,0 = α3 ∧ α4 ⊗ Z ,

d0,−1,2 = β3 ∧ β4 ⊗ Z .

(21.3)

Here αi ∈ �1(C1,0), βi ∈ �1(C0,1), Pj ∈ D(C1,0), Qj ∈ D(C0,1) and Z ∈ D(l)(i = 1, . . . 3; j = 1, 2).

21.2 Invariants of non-degenerate Monge–Ampère equations 455

These tensors can be considered as maps:

d1,1,−1 : C(1)1,0 × C(1)

1,0 → C0,1,

d−1,1,1 : C(1)0,1 × C(1)

0,1 → C1,0,

d2,−1,0 : C1,0 × C1,0 → l,

d0,−1,2 : C0,1 × C0,1 → l.

Recall that a tensor θ ⊗ X , where θ is a 2-form and X is a vector field, actson a differential form α as

(θ ⊗ X)(α) = θ ∧ (X�α),

and acts on a pair of vector fields Y1, Y2 as

(θ ⊗ X)(Y1, Y2) = θ(Y1, Y2)X.

Proposition 21.2.1 We have C(2)1,0 = C(1)

1,0 ⊕ Im d1,1,−1 and C(2)0,1 = C(1)

0,1 ⊕Im d−1,1,1.

Proof We prove the first statement only. The proof of the second one is similar.Let P and Z be vector fields from the distributions C1,0 and l respectively, and

VP,Z = d1,1,−1(Z , P) ∈ C0,1.

Using the representation (21.3), we obtain

VP,Z = ω0(Z)(α1(P)Q1 + α2(P)Q2).

Let β1, β2 be a basis in �1,0,0(J1M). Then

βi(VP,Z) = ω0(Z)(βi(Q1)α1(P)+ βi(Q2)α2(P)).

On the other hand, the commutator [Z , P] is uniquely represented in the form[Z , P] = X + hZ + Q, where X ∈ D(C1,0), Q ∈ D(C0,1), h ∈ C∞(J1M). Wesee that

βi([Z , P]) = −dβi(Z , P) = −d1,1,−1βi(Z , P)

= ω0(Z)(βi(Q1)α1(P)+ βi(Q2)α2(P)) = βi(VP,Z)


for i = 1, 2. Therefore VP,Z = Q and

[Z , P] = X + hZ + d1,1,−1(Z , P).

Since the vector fields P and Z are arbitrary and C(2)1,0 = 〈P1, P2, Z , [Z , P1],

[Z , P2]〉, we have C(2)1,0 = C(1)

1,0 ⊕ Im d1,1,−1. �

Corollary 21.2.1

1. One has:

dim C(2)1,0 = dim C(1)

1,0 + dim Im d1,1,−1

and

dim C(2)0,1 = dim C(1)

0,1 + dim Im d−1,1,1;

2. the distribution C(1)1,0 is completely integrable if and only if d1,1,−1 = 0

and the distribution C(1)0,1 is completely integrable if and only if d−1,1,1 = 0;

3. a non-degenerate Monge–Ampère equation E is locally contact equivalentto the equation vxx + εvyy = 0 if and only if d1,1,−1 = d−1,1,1 = 0.

Remark 21.2.1 Tensors d1,1,−1 and d−1,1,1 are curvatures of the distributionsC(1)

1,0 and C(1)0,1 respectively.

21.2.2 Absolute and relative invariants

Assume that the distributions C(1)1,0 and C(1)

0,1 are not completely integrable, anda non-degenerate Monge–Ampère equation Eω belongs to one of the classesHk,l(0 ≤ k ≤ l ≤ 2).

Note that the tensors d1,1,−1 and d−1,1,1 are decomposable. Indeed Eω isequivalent to (21.2) and for this case the tensors are decomposable.

Let

d1,1,−1 = ωC0 ∧ α ⊗ Q,

d−1,1,1 = ωC0 ∧ β ⊗ P,

and define 2-forms ξ1, ξ2 ∈ �1,0,1(J1M) as follows:

ξ1def= P�d2,−1,0(ω

C0 ∧ β),

ξ2def= Q�d0,−1,2(ω

C0 ∧ α).

21.2 Invariants of non-degenerate Monge–Ampère equations 457

Since d0,−1,2 and d2,−1,0 are C∞(J1M)-homomorphisms, we see that theforms ξ1 and ξ2 are invariants.

Using representations (21.3) for d2,−1,0 and d0,−1,2, we get the followingrepresentation of ξ1 and ξ2:

ξ1 = P�(α3 ∧ α4 ∧ β) = (α3(P)α4 − α4(P)α3) ∧ β,

ξ2 = Q�(β3 ∧ β4 ∧ α) = (β3(Q)β4 − β4(Q)β3) ∧ α.

Note that for elliptic equations these forms are complex conjugate: ξ1 = ξ2.Differential 1-forms

ν1def= Q�ξ1,

ν2def= P�ξ2,

are relative invariants, because vector fields Q and P are defined up to themultiplication by a function.

Example 21.2.1 (Non-linear wave equation) Consider the followingnon-linear wave equation:

vxy = f (x, y, v, vx , vy).

Vector fields

P1 = d

dq1+ f

∂

∂p2,

P2 = ∂

∂p1

constitute a basis for the distribution C1,0 and

Q1 = d

dq2+ f

∂

∂p1,

Q2 = ∂

∂p2

for the distribution C0,1.The distribution l is generated by the following vector field:

Z = ∂

∂u+ fp2

∂

∂p1+ fp1

∂

∂p2.


Thus the vector fields P1, P2, Z, Q1, Q2 form a basis in vector fields on J1M.The dual basis is

α1 = dq1,

α2 = dp1 + p1 fp2 dq1 + (p2 fp2 − f ) dq2 − fp2 du,

ω0 = du− p1 dq1 − p2 dq2,

β1 = dq2,

β2 = dp2 + (p1 fp1 − f ) dq1 + p2 fp1 dq2 − fp1 du.

For this case we have the following representation of the four constructedtensor invariants:

d−1,1,1 = (f fp2p2 dq1 ∧ du− fp2p2 dp2 ∧ du− p1fp2p2 dq1 ∧ dp2 − p2 fp2p2 dq2

∧ dp2 + (fu − p2 fp2u + fp1 fp2 − f fp1p2 − fq2p2) dq2 ∧ du

+ (p1fu − p1p2fp2u − p2f fp2p2 + p1fp1 fp2 − p1f fp1p2 − p1 fq2p2) dq1

∧ dq2)⊗ ∂

∂p1,

d1,1,−1 = (f fp1p1 dq2 ∧ du− fp1p1 dp1 ∧ du− p1 fp1p1 dq1 ∧ dp1

− p2 fp1p1 dq2 ∧ dp1 + (fu + fp1 fp2 − p1fp1u − f fp1p2 − fq1p1) dq1

∧ du+ (−p2 fu − p2 fp1 fp2 + p1p2 fp1u + p2f fp1p2 + p1f fp1p1

+ p2fq1p1) dq1 ∧ dq2)⊗ ∂

∂p2,

d2,−1,0 = (dq1 ∧ dp1 − fp2 dq1 ∧ du+ (p2 fp2 − f ) dq1 ∧ dq2)

⊗(∂

∂u+ fp2

∂

∂p1+ fp1

∂

∂p2

),

d0,−1,2 = (dq2 ∧ dp2 − fp1 dq2 ∧ du− (p1 fp1 − f ) dq1 ∧ dq2)

⊗(∂

∂u+ fp2

∂

∂p1+ fp1

∂

∂p2

).

The differential 2-forms ξ1 and ξ2 are:

ξ1 = fp2p2(fp1 dq1 ∧ du− dq1 ∧ dp2)

+ (fu − p2 fp2u + fp1 fp2 − p2fp1 fp2p2 − f fp1p2 − fq2p2) dq1 ∧ dq2,

ξ2 = fp1p1(fp2 dq2 ∧ du− dq2 ∧ dp1)

+ (−fu + p1 fp1u − fp1 fp2 + p1 fp2 fp1p1 + f fp1p2 + fq1p1) dq1 ∧ dq2

21.3 The problem of contact linearization 459

and

ν1 = fp2p2 dq1,

ν2 = fp1p1 dq2.

Example 21.2.2 For (21.2) we have:

ν1 = h1((2√εf − ιp1fp2 −

√εfp1) dq1 − p2(ιfp2 +

√εfp1) dq2

+ (ιfp2 +√εfp1) du− 2

√ε dp1 − 2ιε dp2),

ν2 = h2(εp1(√εfp1 − ιfp2) dq1 + ε(2ιf +√εp2fp1 − ιp2fp2) dq2

+ ε(ιfp2 −√εfp1) du+ 2

√ε dp1 − 2 ι dp2),

where h1 = εfp1p1 + 2ιfp1p2 − fp2p2 and h2 = εfp1p1 − 2ιfp1p2 − fp2p2 .

Using forms ν1 and ν2 we get the following generalization of Theorem 21.1.2.

Theorem 21.2.1 Assume that a non-degenerate Monge–Ampère equation Eω

belongs to one of the classes Hk,l , where 1 ≤ k ≤ l ≤ 2. Then Eω is locallyequivalent to a Monge–Ampère equation of the type

vxx + εvyy = r(x, y, v)vx + s(x, y, v)vy + g(x, y, v) (21.4)

if and only if ν1 = ν2 = 0.

Proof Due to Theorem 21.1.1 the equation Eω is locally contact equivalentto (21.2). Due to Example 21.2.2 for this equation ν1 = ν2 = 0 if and only ifh1 = h2 = 0. This means that

fp1 p2 = 0,

εfp1p1 = fp2p2 .

Therefore f (q, u, p) = r(q, u)p1+ s(q, u)p2+g(q, u) for some functions r, s, g.�

21.3 The problem of contact linearization

In this section we use the constructed tensors to solve the problem of contactlinearization of non-degenerate Monge–Ampère equations.


That is, find a class of Monge–Ampère equations that are locally contactequivalent to non-homogeneous linear equations

vxx + εvyy = r(x, y)vx + s(x, y)vy + c(x, y)v+ d(x, y). (21.5)

The following result gives a solution of the problem.

Theorem 21.3.1 (Linearizaton of non-degenerate MAEs) A non-degenerateMonge–Ampère equation Eω of the class H2,2 is locally contact equivalent tothe linear equation (21.5) if and only if the following conditions hold:

1. ν1 = ν2 = 0;2. dξ1 = dξ2 = 0.

Proof The proof of the necessity it trivial. Indeed, is easy to check that (21.5)belongs to one of the classes H2,2 and for this equation we have ν1 = ν2 = 0and dξ1 = dξ2 = 0. Let us prove the sufficiency.

Hyperbolic caseDue to Theorem 21.2.1, the equation Eω is contact equivalent to (21.4)

vxx − vyy = r(x, y, v)vx + s(x, y, v)vy + g(x, y, v).

The corresponding effective differential 2-form is

ω = dq1 ∧ dp2 + dq2 ∧ dp1 + (rp1 + sp2 + g) dq1 ∧ dq2.

The contact transformation

ϕ : q1 → q1 − q2√2

, q2 → q1 + q2√2

, u → u,

p1 → p1 − p2√2

, p2 → p1 + p2√2

takes it to the form

ω1def= ϕ∗(ω) = dq1 ∧ dp1 − dq2 ∧ dp2 + (Rp1 + Sp2 + G) dq1 ∧ dq2,

(21.6)


where

R(q1,q2, u) = 1√2

(s

(q1 − q2√

2,

q1 + q2√2

, u

)+ r

(q1 − q2√

2,

q1 + q2√2

, u

)),

S(q1,q2, u) = 1√2

(s

(q1 − q2√

2,

q1 + q2√2

, u

)− r

(q1 − q2√

2,

q1 + q2√2

, u

)),

G(q1,q2, u) = g

(q1 − q2√

2,

q1 + q2√2

, u

).

We have the following coordinate representation of the tensors:

d1,1,−1 = (Rs+ Gu + p2Su − Rq1)(dq1 ∧ du− p2 dq1 ∧ dq2)⊗ ∂

∂p2,

d−1,1,1 = (RS + Gu + p1Ru − Sq2)(dq2 ∧ du+ p1 dq1 ∧ dq2)⊗ ∂

∂p1,

d2,−1,0 = (dq1 ∧ dp1 − S dq1 ∧ du− (G+ p1R) dq1 ∧ dq2)

⊗(∂

∂u+ S

∂

∂p1+ R

∂

∂p2

),

d0,−1,2 = (dq2 ∧ dp2 − R dq2 ∧ du+ (G+ p2S) dq1 ∧ dq2)

⊗(∂

∂u+ S

∂

∂p1+ R

∂

∂p2

).

and the invariants:

ξ1 = (RS + Gu + p1Ru − Sq2) dq1 ∧ dq2,

ξ2 = −(RS + Gu + p2Su − Rq1) dq1 ∧ dq2.

Then,

dξ1 = Ru dq1 ∧ dq2 ∧ dp1 + (SRu + RSu + Guu

+ p1Ruu − Suq2) dq1 ∧ dq2 ∧ du,

dξ2 = −Su dq1 ∧ dq2 ∧ dp2 − (SRu + RSu + Guu

+ p2Suu − Ruq1) dq1 ∧ dq2 ∧ du.

We see that Ru = Su = Guu = 0 if and only if dξ1 = dξ2 = 0. In this case

G(q1, q2, u) = c(q1, q2)u+ z(q1, q2)


and we obtain

ω1= 2(R(q)p1+ S(q)p2+ c(q)u+ z(q)) dq1 ∧ dq2+ dq1 ∧ dp1− dq2 ∧ dp2.

Elliptic caseDue to Theorem 21.2.1, the equation Eω is equivalent to the equation

vxx + vyy = r(x, y, v)vx + s(x, y, v)vy + g(x, y, v).

The invariant 2-form ξ1 is

ξ1 =(

1

4

(ds

dq1− dr

dq2

)+ ι

8(r2 + s2 + 4gu

+2(p1ru − rq1 + p2su − sq2))

)dq1 ∧ dq2

and ξ2 = ξ1. Then

dξ1 = 1

4(−ru + ιsu) dq1 ∧ dq2 ∧ dp2 + 1

4(ιru + su) dq1 ∧ dq2 ∧ dp1

+ 1

4((p1suu − p2ruu − ruq2 + suq1)

+ ι(rru + ssu + 2guu + p1ruu + p2suu − sq2u − rq1u)) dq1 ∧ dq2 ∧ du.

We see that ru = su = guu = 0 if and only if dξ1 = 0. In this case

g(q1, q2, u) = c(q1, q2)u+ z(q1, q2)

and we obtain the following effective 2-form:

ω = −(r(q)p1 + s(q)p2 + c(q)u+ z(q)) dq1 ∧ dq2 + dq1 ∧ dp2 − dq2 ∧ dp1.

�

Example 21.3.1 (The generalized Hunter–Saxton equation) Let us considerthe generalized Hunter–Saxton equation

vtx = vvxx + κu2x .

This equation has applications in the theory of liquid crystals [39] and in thegeometry of Einstein–Weil spaces. This equation is hyperbolic,

ω = 2u dq2 ∧ dp1 + dq1∧dp1 − dq2 ∧ dp2 − 2 κp21 dq1∧dq2


and

Aω =

∥∥∥∥∥∥∥∥∥∥∥

1 2u 0 0

0 −1 0 0

0 −2κp21 1 0

2κp21 0 2u −1

∥∥∥∥∥∥∥∥∥∥∥in the basis

d

dq1,

d

dq2,∂

∂p1,∂

∂p2.

Let us consider the following basis:

P1 = ∂

∂q1+ p1

∂

∂u+ κp2

1∂

∂p2,

P2 = ∂

∂p1+ u

∂

∂p2,

Q1 = ∂

∂q2+ κp2

1∂

∂p1− u

∂

∂q1+ (p2 − up1)

∂

∂u

Q2 = ∂

∂p2,

Z = ∂

∂u+ (2κ − 1)p1

∂

∂p2,

and the dual one

α1 = dq1 + u dq2,

α2 = dp1 − κp21 dq2,

β1 = dq2,

β2 = dp2+(1− 2κ)p1 du+ (κ−1)p21 dq1+ (2κ − 1)p1p2 dq2 − u dp1 ,

ω0 = du− p1 dq1 − p2 dq2.

Then P1, P2 ∈ D(C1,0), Q1, Q2 ∈ D(C0,1), Z ∈ D(l), α1,α2 ∈ �1,0,0(J1M),β1,β2 ∈ �0,0,1(J1M) and we have the following representation of the


tensors:

d−1,1,1 = −(p1 dq1 ∧ dq2 + dq2 ∧ du)⊗(

∂

∂q1+ p1

∂

∂u+ κp2

1∂

∂p2

),

d1,1,−1 = 2( κ − 1)(κp31 dq1 ∧ dq2 + κp2

1 dq2 ∧ du− dp1 ∧ du− p1 dq1

∧ dp1 − p2 dq2 ∧ dp1)⊗ ∂

∂p2,

d2,−1,0 = (dq1 ∧ dp1 − κp21 dq1 ∧ dq2 + u dq2 ∧ dp1)

⊗(∂

∂u+ (2κ − 1)p1

∂

∂p2

),

d0,−1,2= (dq2 ∧ dp2+(1−2κ)p1 dq2 ∧ du+(1−κ)p21 dq1 ∧ dq2 − u dq2 ∧ dp1)

⊗(∂

∂u+(2κ − 1)p1

∂

∂p2

)and

ξ1 = −dq2 ∧ dp1,

ξ2 = 2(1− κ) dq2 ∧ dp1.

Due to theorem 21.3.1, the generalized Hunter–Saxton equation is linearized.The corresponding linear equation is the Euler–Poisson equation

vtx = 1

κ(t + x)vt + 2(1− κ)

κ(t + x)vx − 2(1− κ)

(κ(t + x))2u

(cf. [86]).

21.4 The problem of equivalence for non-degenerateequations

21.4.1 e-Structure for non-degenerate equations

In this section we construct an e-structure for generic non-degenerateMonge–Ampère equations.

Let Edef= Eω be a non-degenerate Monge–Ampère equations with triplet

P =({C1,0, C0,1}, l). Let Z be a real vector field which generates the complexdistribution l and that is normed by the condition ω0(Z) = 1.

Let us introduce submodules�s(J1M) ⊂ �s(J1M) of all differential s-formsvanishing on Z:

�s(J1M)def= {α ∈ �s(J1M)| Z�α = 0}.

21.4 The problem of equivalence for non-degenerate equations 465

Elements of the submodule �s(J1M) we will call l-horizontal forms. The setof all l-horizontal forms is an algebra �∗(J1M) with respect to the operationof exterior multiplication.

Let us define a projection � : �s(J1M)→ �s(J1M) by

�(α)def= α − ω0 ∧ (Z�α)

and an operator

∂ : �s(J1M)→ �s+1(J1M) by ∂def= � ◦ d.

The kernel of the operator � is

ker� = {ω0 ∧ α|α ∈ �∗(J1M)}.

Let Edef= Eω be another non-degenerate Monge–Ampère equation with

triplet P=({C1,0, C0,1}, l). If ϕ : J1M → J1M, ϕ∗(ω0) = λω0 is a contacttransformation such that ϕ∗(ω)ε = hω, then ϕ∗(P) = P and

ϕ−1∗ (Z) = 1

λZ .

Lemma 21.4.1 Operators � and ∂ are invariant with respect to contactdiffeomorphisms, i.e.,

ϕ∗ ◦� = � ◦ ϕ∗ (21.7)

andϕ∗ ◦ ∂ = ∂ ◦ ϕ∗. (21.8)

Proof For an s-form α ∈ �s(J1M) we have

ϕ∗(�(α)) = ϕ∗(α)− ϕ∗(ω0) ∧ ϕ∗(Z�α)= ϕ∗(α)− ω0 ∧ (ϕ−1∗ (Z)�ϕ∗(α))

= ϕ∗(α)− λω0 ∧(

1

λZ�ϕ∗(α)

)= �(ϕ∗(α)),

and for an arbitrary differential l-horizontal form α we have

ϕ∗(∂α) = ϕ∗ ◦� ◦ d(α) = � ◦ ϕ∗ ◦ d(α)

= � ◦ d ◦ ϕ∗(α) = ∂(ϕ∗(α)). �


The restriction of ∂ to the algebra �∗(J1M) we denote by δ and will call anoperator of non-holonomic differentiation

δ : �s(J1M)→ �s+1(J1M).

Note that the sequence

0δ→ �0(J1M)

δ→ �1(J1M)δ→ �2(J1M)

δ→ �3(J1M)δ→ �4(J1M)

δ→ 0(21.9)

is a complex.This complex we call a non-holonomic de Rham complex. Note that

δ(α ∧ β) = δα ∧ β + (−1)degαα ∧ δβ

for any differential forms α,β ∈ �s(J1M).A differential l-horizontal 2-form ω is called l-effective if ω ∧ ∂ω0 = 0.Note also that

∂(gω0) = g∂ω0

for any function g ∈ C∞(J1M).Let ω be an l-effective differential 2-form and let ϕ : J1M → J1M

be a contact diffeomorphism such that ϕ∗(ω) = ω. Then the form ω isl-effective also.

For any differential l-horizontal 2-form α we can construct its l-effective part

αldef= α − sα∂ω0 ∧ ∂ω0,

where the function sα is defined by

α ∧ ∂ω0 = sα∂ω0 ∧ ∂ω0.

Obviously, the effective 2-form ω and its l-effective part ωl generates thesame Monge–Ampère equation.

Thus we can reformulate the condition of contact equivalence ofMonge–Ampère equations Eω and Eω in terms of l-effective forms:

ϕ∗(ωl) = hωl.

From now on we assume that ω and ω are l-effective and l-effectivedifferential 2-forms respectively and ϕ∗(ω0) = λω0, ϕ∗(ω) = hω.


If we introduce an l-Pfaffian F of ω by

F∂ω0 ∧ ∂ω0 = ω ∧ ω,

then

ϕ∗(F) = h2

λ2F.

Here F is the l-Pfaffian of ω.Define an operator A : D(C)→ D(C) by the formula:

AX�∂ω0 = X�ω, (21.10)

where the vector field X ∈ D(C).As above the square of the operator A is scalar and A2 + F = 0.

Lemma 21.4.2 We have

A = λ

hϕ−1∗ ◦ A ◦ ϕ∗.

Proof From (21.10), for any vector field X ∈ D(C) we have:λϕ−1∗ (AX)�∂ω0 = hϕ−1∗ (X)�ω. Moreover, ϕ−1∗ (X)�ω = Aϕ−1∗ (X)�∂ω0.Since ∂ω0 is non-degenerate, we obtain λϕ−1∗ ◦ A = hA ◦ ϕ−1∗ . ThereforeA = (λ/h)ϕ−1∗ ◦ A ◦ ϕ∗. �

The differential 2-form ∂ω0 is non-degenerate on the module of vector fieldsfrom the Cartan distribution. This means that if X�∂ω0 = 0 for X ∈ D(C), thenX = 0.

For a function H ∈ C∞(J1M) the formula

XH�∂ω0 = −∂H (21.11)

uniquely defines a vector field XH ∈ D(C), and

ϕ−1∗ (XH) = 1

λXϕ∗(H).

Similarly with Lemma 20.4.1 one can prove that

∂h ∧ ω = 1

2AXh�(∂ω0 ∧ ∂ω0).

for any function h ∈ C∞(J1M).


The relations

W� (∂ω0 ∧ ∂ω0) = 2∂ω,

ω0(W) = 0.

uniquely determine a vector field W in the Cartan distribution.Let 〈Z , d1,1,−1〉 and 〈Z , d−1,1,1〉 be contractions of the vector field Z and the

tensors d1,1,−1 and d−1,1,1 respectively. Define a function k as a contraction ofthe (1, 1)-tensors 〈Z , d1,1,−1〉 and 〈Z , d−1,1,1〉:

kdef= 〈〈Z , d1,1,−1〉, 〈Z , d−1,1,1〉〉.

Then

ϕ∗(k) = 1

λ2k

and therefore k is a relative invariant.Suppose that k �= 0. Let us introduce the function

F0def= F

k

and the vector field

Vdef= 1

kAW .

Since Lemma 21.4.2, and the facts that

ϕ−1∗ (W) = 1

λ2(hW + AXh),

ϕ∗(F0) = h2F0,

we get

ϕ−1∗ (V) = 1

ϕ∗(k)(ϕ−1∗ ◦ A ◦ ϕ∗) ◦ ϕ−1∗ (W)

= h2

λV − h

λF0Xh.

For the vector fields XF0 we have

ϕ−1∗ (XF0) =h2

λXF0

+ 2h

λF0Xh.


Then the vector field

Y def=√|k|F0

(XF0 + 2V)

is an invariant (up to the sign):

ϕ−1∗ (Y) = Y .

This vector field falls into the sum

Y = Y1,0 + Y0,1,

where Y1,0 ∈ D(C1,0) and Y0,1 ∈ D(C0,1) are also invariants.

Define the vector field Z ∈ D(l) by

ω0(Z) = 1√|k| .

This vector field is an invariant (up to the sign).Applying tensors d−1,1,1 and d1,1,−1 to these invariants we obtain two

invariant vector fields from the distributions C1,0 and C0,1:

X1,0def= d−1,1,1(Z , Y0,1),

X0,1def= d1,1,−1(Z , Y1,0).

For the case of the general Monge–Ampère equation the vector fieldsZ , X1,0, Y1,0, X0,1, Y0,1 form an e-structure on J1M. Denote the constructede-structure by

eE = (Z , {X1,0, X0,1}, {Y1,0, Y0,1}).

If the equation E is hyperbolic, then this e-structure is real. If E is elliptic, thenthis e-structure is complex. In the last case we can construct a real e-structureusing the conjugate operation.

Theorem 21.4.1 Two non-degenerate Monge–Ampère equations E and E arecontact equivalent if and only if the e-structures eE and eE are equivalent.

Example 21.4.1 Let us consider the following non-linear wave equation:

vxy = (vx + vy)3.


In order to simplify formulae we assume that p1 + p2 > 0. We have k =36(p1 + p2)

2. The l-effective part of the form ω and the form ∂ω0 are

ω = dq1 ∧ dp1 − dq2 ∧ dp2 + 3(p1 + p2)2(dq2 ∧ du− dq1 ∧ du+)

+ (p1 + p2)3dq1 ∧ dq2,

∂ω0 = dq1 ∧ dp1 + dq2 ∧ dp2 − 3(p1 + p2)2(dq1 ∧ du+ dq2 ∧ du)

+ (3(p1 + p2)2(p2 − p1)) dq1 ∧ dq2

respectively. In the free basis X1,0, Y1,0, X0,1, Y1,0 the operator A has adiagonal form:

A =

∥∥∥∥∥∥∥∥∥∥∥

1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −1

∥∥∥∥∥∥∥∥∥∥∥.

The following vector fields define the e-structure:

Z = 1

6(p1 + p2)

(∂

∂u+ 3(p1 + p2)

2(

∂

∂p1+ ∂

∂p2

)),

Y1,0 = 12

((p1 + p2)

3(

∂

∂p2− 4

∂

∂p1

)+ d

dq1

),

X1,0 = −54(p1 + p2)3 ∂

∂p1,

Y0,1 = 12

((p1 + p2)

3(

∂

∂p1− 4

∂

∂p2

)+ d

dq2

),

X0,1 = −54(p1 + p2)3 ∂

∂p2.

21.4.2 Functional invariants

Let us construct two functional invariants for non-degenerate Monge–Ampèreequations.

First of all we define the contraction operations

� : �i(N)⊗ D(N)×�j(N)⊗ D(N)→ �i+j−1(N)⊗ D(N),


(see [49]) and

〈 , 〉 : �i(N)⊗ D(N)×�j(N)⊗ D(N)→ �i+j−2(N)

on a smooth manifold N .For decomposable tensors α ⊗ X and β ⊗ Y we put

(α ⊗ X)�(β ⊗ Y)def= α ∧ (X�β)⊗ Y

and

〈α ⊗ X ,β ⊗ Y〉 def= (Y�α) ∧ (X�β).

Suppose that �〈d−1,1,1�d2,−1,0, d1,1,−1�d0,−1,2〉 �= 0 and by definition, let

I1,0def= �〈d−1,1,1�d2,−1,0, d−1,1,1�d2,−1,0〉

�〈d−1,1,1�d2,−1,0, d1,1,−1�d0,−1,2〉and

I0,1def= �〈d1,1,−1�d0,−1,2, d1,1,−1�d0,−1,2〉

�〈d−1,1,1�d2,−1,0, d1,1,−1�d0,−1,2〉 .

Since the numerators and denominators in these fractions are differentialforms from �4(J1M), these definitions are correct. Therefore I1,0 and I0,1 arefunctional invariants of Monge–Ampère equations.

For elliptic equations its real and imaginary parts are real functionalinvariants.

Note that for (21.2) I1,0 and I0,1 are zero.

22

Symplectic classification of MAEs onthree-dimensional manifolds

In this chapter we consider local classification of general symplecticMonge–Ampère equations. We will discuss the conditions which we need toimpose on the effective forms ωi, i = 1, 2 such that there exist a local sym-plectic diffeomorphism F : T∗M → T∗M transforming one form into another:F∗(ω1) = ω2.

The problem is purely local so we can suppose without loss of generalitythat M = Rn. We will also restrict ourselves to the case when one of theseforms has constant coefficients. Then we will reduce the equivalence problemto integration of a system of non-linear differential equations in the space of

472

22.1 Jets and differential equations on submanifolds 473

1-jets of Lagrangian sabmanifolds in R4n. The local equivalence problem hasa solution in a neighborhood of 0 ∈ R2n if and only if this system has a localsolution.

The main sources of the results in this chapter are the original papers [74]and [77] but extended by results of Banos [5] who had adapted and simpli-fied our original proofs. In particular, he had used the original idea of [77]to replace the Cartan–Kähler integrability theorem by the Frobenius theorem,which is much easies but can be applied only under some conditions on thehigher prolongations of the Sp(n)-orbit stabilizers.

We refer interested readers to the original paper [77], where we treat moregeneral cases but in much less detail.

22.1 Jets of submanifolds and differential equations onsubmanifolds

In this section we briefly recall main structures on the manifolds of jets (a moredetailed exposition can be found in [1]).

Let M be a manifold. We consider submanifolds N ⊂ M of fixed codimensionm. Let N be such a submanifold, then we denote by [N]ka the k-jet of N at thepoint a ∈ M.

Let Jka (M, m) be the set of all k-jets of submanifolds at the point

a ∈ M, and

Jk(M, m) =⋃a∈M

Jka (M, m)

which is the set of all k-jets.Then Jk(M, m) are smooth manifolds and the natural projections πk,l :

Jk(M, m)→ Jl(M, m), k > l, are smooth bundles. Here πk,l : [N]ka −→ [N]la.Moreover, the bundles πk,k−1 are affine bundles when k ≥ 2. Namely,

fibres F(ak−1) = π−1k,k−1(ak−1), where ak−1 = [N]k−1

a , are affine spaces.

The vector spaces associated with the fibres are Sk(T∗a N) ⊗ νa(N) whereνa(N) = TaM/TaN is the normal space to the submanifold N at the pointa ∈ N . The affine structure is invariant with respect to diffeomorphisms of M.

The bundle π1,0 can be described as follows. First of all J0(M, m) = M, andJ1

a (M, m) can be identified with the Grassmanian Grm(TaM)of vector subspacesin TaM having codimension m.

Therefore, fibres π−11,0 (a) = Grm(TaM) are Grassmanians and the bundle has

Grassmanian structure.

474 Symplectic classification of MAEs on 3D manifolds

Any submanifold N ⊂ M of the codimension m defines submanifoldsjk(N) ⊂ Jk(M, m) by

jk(N) = {[N]ka | a ∈ N}.

The submanifold jk(N) can be viewed as a “derivative” of jk−1(N). Geometric-ally, a point ak = [N]ka can be identified with the tangent space to submanifoldjk−1(N) at the point ak−1 = [N]k−1

a . The tangent space obviously depends onak only, and we will denote it by L(ak).

The linear span of all subspaces L(ak) at the point ak−1 forms a Cartan sub-space C(ak−1) of the tangent space Tak−1(J

k−1(M, m)). This subspace consistsof all vectors v ∈ Tak−1(J

k−1(M, m)) such that (πk−1,k−2)∗v ∈ L(ak−1), or

C(ak−1) = (πk−1,k−2)−1∗ (L(ak−1)).

The distribution C : ak−1 −→ C(ak−1) of Cartan subspaces on Jk−1(M, m)is called the Cartan distribution.

From the construction of the Cartan distribution we see that the submanifoldsjk−1(N) are integral for C. Moreover, one can easily prove that any integralmanifold L ⊂ Jk−1(M, m) of the Cartan distribution has the form jk−1(N) ifand only if the map πk−1,0 : L → M is an embedding.

By a system of differential equations of order k we mean a submanifoldE ⊂ Jk(M, m). A solution of this system is an integral manifold L of the Cartandistribution on Jk(M, m) such that L ⊂ E and dim L = dim M −m. If the mapπk,0 : L → M is an embedding, then L = jk(N) for a submanifold N ⊂ M.

We call the last submanifold a smooth solution of E.With a system E ⊂ Jk(M, m) of order k we associate a derived or prolongated

system E(1) ⊂ Jk+1(M, m) of order k+1 as follows: a point ak+1 ∈ Jk+1(M, m)belongs to E(1) if and only if L(ak+1) ⊂ Tak (E).

The set E(1) we call first prolongation of E. If E(1) is smooth, then onecan repeat the procedure and get the second prolongation E(2) = (E(1))(1) ⊂Jk+2(M, m), and so on.

A system E ⊂ Jk(M, m) is called formally integrable if all prolongationsE(i) ⊂ Jk+i(M, m) are smooth submanifolds and all projections πk+i,k+i−1 :E(i) → E(i−1) are smooth bundles.

For any system E ⊂ Jk(M, m) fibres of the map πk+1,k : E(1) → E, if theyare not empty, are affine subspaces of the affine bundle πk+1,k . Therefore, inorder to check the smoothness of E(1) one should estimate dimensions of thefibres. This can be done in the following way.

22.1 Jets and differential equations on submanifolds 475

A symbol of E at point ak ∈ E is the subspace gk(ak) of the vertical tangentvectors, that is

gk(ak) = Tak (E) ∩ Tak (F(ak−1))

where ak−1 = πk,k−1(ak).Because Tak (F(ak−1)) = Sk(T∗a N)⊗ νa(N) the symbol is a vector subspace

of the tensor space

gk(ak) ⊂ Sk(T∗a N)⊗ νa(N).

Let g(l)k (ak+l) be the symbol of the lth prolongation at ak+l ∈ E(l), then

g(l)k (ak+l) ⊂ Sk+l(T∗a N)⊗ νa(N)

and all of these spaces can be defined by the symbol gk(ak) only.Namely, the first prolongation symbol g(1)k (ak+1) consists of tensors θ ∈

Sk+1(T∗a N) ⊗ νa(N) such that v(θ) ∈ gk(ak) for all vectors v ∈ TaN , and

g(l+1)k (ak+l+1) = (g(l)k (ak+l))

(1). Here we denote the derivative of θ along vby v(θ).

We say, that a system E has a finite type if g(l)k (ak) = 0 for all ak ∈ E andall l ≥ l0.

Let E be such a system. Assume that E(l0) is a smooth manifold andπk+l0,k+l0−1 : E(l0) → E(l0−1) is a diffeomorphism.

Note that πk+l0,k+l0−1 : E(l0) → E(l0−1) is a local diffeomorphism

if g(l0)k = 0.Denote by a′k+l0−1 ∈E(l0) the preimage of ak+l0−1 ∈E(l0−1), πk+l0,k+l0−1

(a′k+l0−1) = ak+l0−1.Then the correspondence

L : ak+l0−1 ∈ E(l0−1) −→ L(a′k+l0−1) ⊂ Tak+l0−1(E(l0−1))

defines a distribution on E(l0−1).Note that this distribution is completely integrable if and only if E(l0+1) is

not empty and πk+l0+1,k+l0 : E(l0+1) → E(l0) is a diffeomorphism.Indeed, if the distribution L is completely integrable then its integral manifold

has the form jk+l0−1(N) and [N]k+l0+1a are the only points over [N]k+l0

a ∈ E(l0).On the other hand, let πk+l0+1,k+l0 : E(l0+1) → E(l0) be a diffeomorphism.

To find value of the curvature form of L at a point b = ak+l0−1 ∈ E(l0−1)

we should find a value of the commutator [X , Y ]b for any pair of vector fields


belonging to L. Note that this value depends on 1-jets of vector fields X and Yat the point b.

Let a′′k+l0−1 ∈ E(l0+1), a′k+l0−1 ∈ E(l0) be the preimages of the point b,

and let a′′k+l0−1 = [N]k+l0+1a . Then for given 1-jets [X]1b and [Y ]1b the vector

fields X and Y can be chosen to be tangent to jk+l0−1(N). Therefore, [X, Y ]b ∈Tb( jk+l0−1(N)) = L(b), and L is completely integrable.

Summrizing we obtain the following result.

Theorem 22.1.1 Let E ⊂ Jk(M, m) be a finite type system such that

1. g(l0)k = 0,2. πk+l0+1,k+l0 : E(l0+1) → E(l0) and πk+l0,k+l0−1 : E(l0) → E(l0−1) are

diffeomorphisms.

Then

• the distribution L on E(l0−1) defined by E(l0) is completly integrable, and• for any point ak+l0−1 ∈ E(l0−1) there is a solution N ⊂ M, such that

[N]k+l0−1a = ak+l0−1.

22.2 Prolongations of contact and symplectic manifolds andoverdetermined Monge–Ampère equations

In this section we shall apply the previous constructions to Lagrangianand Legendrean submanifolds of symplectic and contact manifolds, and tooverdetermined Monge–Ampère equations.

22.2.1 Prolongations of symplectic manifolds

Let (M,�) be a symplectic manifold. We consider Lagrangian submanifoldsN ⊂ M. If N is such a submanifold, then, as above, we denote by [N]ka the k-jetof N at the point a ∈ M.

Let Jka (M,�) be the set of all k-jets of Lagrangian submanifolds at the point

a ∈ M, and

Jk(M,�) =⋃a∈M

Jka (M,�)

the set of all k-jets.Then Jk(M,�) are smooth manifolds and the natural projections πk,l :

Jk(M,�)→ Jl(M,�), k > l, are smooth bundles.

22.2 Prolongations of contact and symplectic manifolds 477

Moreover, the bundles πk,k−1 are affine bundles, when k ≥ 2.Namely, fibres F(ak−1) = π−1

k,k−1(ak−1), where ak−1 = [N]k−1a , are affine

spaces, and the vector spaces associated with the fibres are Sk+1(T∗a N). Theaffine structure is invariant with respect to symplectic diffeomorphisms of M.

The bundle π1,0 can be described as follows. First of all J0(M,�) = M, andJ1

a (M,�) can be identified with the Lagrangian Grassmanian LGr(TaM,�) ofLagrangian subspaces in TaM.

Therefore, fibres π−11,0 (a) = LGr(TaM,�) are Lagrangian Grassmanians and

the bundle has the Grassmanian structure.Any Lagrangian submanifold N ⊂ M defines submanifolds jk(N) ⊂

Jk(M,�) by

jk(N) = {[N]ka | a ∈ N}.

The submanifold jk(N) can be viewed as a “derivative” of jk−1(N).Geometrically, a point ak = [N]ka can be identified with the tangent space

to the submanifold jk−1(N) at the point ak−1 = [N]k−1a . The tangent space

obviously depends on ak only, and we will denote it by L(ak).The linear span of all subspaces L(ak) at the point ak−1 forms a Cartan sub-

space C(ak−1) of the tangent space Tak−1(Jk−1(M,�)). This subspace consists

of all vectors v ∈ Tak−1(Jk−1(M,�)) such that (πk−1,k−2)∗v ∈ L(ak−1), or

C(ak−1) = (πk−1,k−2)−1∗ (L(ak−1)).

The distribution C : ak−1 −→ C(ak−1) of Cartan subspaces on Jk−1(M,�)is called the Cartan distribution.

From the construction of the Cartan distribution we see that the submanifoldsjk−1(N) are integral for C. Moreover, one can easily prove that any integralmanifold L ⊂ Jk−1(M,�) of the Cartan distribution has the form jk−1(N) ifand only if the map πk−1,0 : L → M is an embedding, and L is Lagrangian.

By a system of differential equations of order k on Lagrangian submanifoldswe mean a submanifold E ⊂ Jk(M,�). A solution of this system is an integralmanifold L of the Cartan distribution on Jk(M,�) such that L ⊂ E and dim L =12 dim M.

If the map πk,0 : L → M is an embedding, then L = jk(N) for someLagrangian submanifold N ⊂ M. We call the last submanifold a smoothsolution of E.

With a system E ⊂ Jk(M,�) of order k we associate a derived, or pro-longated system E(1) ⊂ Jk+1(M,�) of order k + 1 as follows: a pointak+1 ∈ Jk+1(M,�) belongs to E(1) if and only if L(ak+1) ⊂ Tak (E).


The set E(1) we call first prolongation of E.If E(1) is smooth, then one can repeat the procedure and get the second

prolongation E(2) = (E(1))(1) ⊂ Jk+2(M,�), and so on.A system E ⊂ Jk(M,�) is called formally integrable if all prolongations

E(i) ⊂ Jk+i(M,�) are smooth submanifolds and all projections πk+i,k+i−1 :E(i) → E(i−1) are smooth bundles.

For any system E ⊂ Jk(M,�) fibres of the map πk+1,k : E(1) → E, if theyare not empty, are affine subspaces of the affine bundle πk+1,k . Therefore, inorder to check the smoothness of E(1) one should estimate dimensions of thefibres. This can be done in the same way as for general case.

A symbol of E at point ak ∈ E is the following subspace gk(ak) of the verticaltangent space, that is,

gk(ak) = Tak (E) ∩ Tak (F(ak−1)),

where ak−1 = πk,k−1(ak).Because Tak (F(ak−1)) = Sk+1(T∗a N) the symbol is a vector subspace of the

tensor space

gk(ak) ⊂ Sk+1(T∗a N).


g(l)k (ak+l) ⊂ Sk+l+1(T∗a N)

and all of these spaces can be defined by the symbol gk(ak) only.Namely, the first prolongation symbol g(1)k (ak+1) consist of tensors

θ ∈ Sk+2(T∗a N) such that v(θ)∈ gk(ak) for all vectors v∈TaN , and

g(l+1)k (ak+l+1)= (g(l)k (ak+l))

(1).

We say, that a system E has a finite type if g(l)k (ak)= 0 for all ak ∈Eand l≥ l0.


Denote by a′k+l0−1 ∈E(l0) the preimage of ak+l0−1 ∈E(l0−1), πk+l0,k+l0−1



defines a distribution on E(l0−1).


As above one can prove that this distribution is completely integrable ifand only if E(l0+1) is not empty and πk+l0+1,k+l0 : E(l0+1) → E(l0) is adiffeomorphism.

Summrizing we obtain the following result.

Theorem 22.2.1 Let E ⊂ Jk(M,�) be a finite type system of differentialequations on Lagrangian submanifols such that:


diffeomorphisms.

Then,

• the distribution L on E(l0−1) defined by E(l0) is completly integrable, and• for any point ak+l0−1 ∈ E(l0−1) there is a solution, that is, a Lagrangian

submanifold N ⊂ M, such that jk+l0−1(N) ⊂ E(l0−1), and [N]k+l0−1a =

ak+l0−1.

Note that any effective r-form ω ∈ �rε(M) defines a first-order system of

differential equations on Lagrangian submanifolds Eω ⊂ J1(M,�), where

Eω = {[N]1a ∈ LGr(TaM,�) | ω|TaN = 0}.

We call this system the overedermined Monge–Ampère equation.If r < 1

2 dim M then Eω has the finite type for ωs “in the general position”and one can use the above theorem in order to get smooth solvability of thesystem.

Later on we will apply this remark to overedermined Monge–Ampèreequations arising in the classification problems.

22.2.2 Prolongations of contact manifolds

Let (M, P) be a contact manifold. We consider Legendrean submani-folds N ⊂M.

Let Jka (M, P) be the set of all k-jets of Legendrean submanifolds at the point

a ∈ M, and

Jk(M, P) =⋃a∈M

Jka (M, P)

the set of all k-jets.


Then Jk(M, P) are smooth manifolds and the natural projections πk,l :Jk(M, P)→ Jl(M, P), k > l, are smooth bundles.

Moreover, the bundles πk,k−1 are affine bundles when k ≥ 2. Namely, fibresF(ak−1) = π−1

k,k−1(ak−1), where ak−1 = [N]k−1a , are affine spaces. The vector

spaces associated with the fibres are Sk+1(T∗a N) ⊗ νa, where νa = TaM/Pa.The affine structure is invariant with respect to contact diffeomorphisms of M.

The bundle π1,0 can be described as follows. First of all J0(M, P) = M,and J1

a (M, P) can be identified with the Lagrangian Grassmanian LGr(Pa) ofLagrangian subspaces in Pa.

Therefore fibres π−11,0 (a) = LGr(Pa) are Lagrangian Grassmanians and the

bundle has Grassmanian structure.Any Legendrean submanifold N ⊂ M defines submanifolds jk(N) ⊂

Jk(M, P) by

jk(N) = {[N]ka | a ∈ N}.Geometrically, a point ak = [N]ka can be identified with the tangent space

to the submanifold jk−1(N) at the point ak−1 = [N]k−1a . The tangent space

obviously depends on ak only, and we will denote it by L(ak).The linear span of all subspaces L(ak) at the point ak−1 forms a Cartan

subspace C(ak−1) of the tangent space Tak−1(Jk−1(M, P)).

This subspace consist of all vectors v ∈ Tak−1(Jk−1(M, P)) such that

(πk−1,k−2)∗v ∈ L(ak−1), or

C(ak−1) = (πk−1,k−2)−1∗ (L(ak−1)).

The distribution C : ak−1 −→ C(ak−1) of Cartan subspaces on Jk−1(M, P) iscalled the Cartan distribution.

From the construction of the Cartan distribution we see that the submanifoldsjk−1(N) are integral for C. Moreover, one can easily prove that any integralmanifold L ⊂ Jk−1(M, P) of the Cartan distribution has the form jk−1(N) ifand only if the map πk−1,0 : L → M is an embedding, and L is Legendrean.

By a system of differential equations of order k on Legendrean submanifoldswe mean a submanifold E ⊂ Jk(M, P). A solution of this system is an integralmanifold L of the Cartan distribution on Jk(M, P) such that L ⊂ E and dim L =12 (dim M − 1). If the map πk,0 : L → M is an embedding, then L = jk(N) forsome Legendrean submanifold N ⊂ M. We call the last submanifold a smoothsolution of E.

With a system E ⊂ Jk(M, P) of order k we associate a derived, or pro-longated system E(1) ⊂ Jk+1(M, P) of order k + 1 as follows: a pointak+1 ∈ Jk+1(M, P) belongs to E(1) if and only if L(ak+1) ⊂ Tak (E).


The set E(1) we call first prolongation of E. If E(1) is smooth, then wecan repeat the procedure and get the second prolongation E(2) = (E(1))(1) ⊂Jk+2(M,�), and so on.

A system E ⊂ Jk(M, P) is called formally integrable if all prolongationsE(i) ⊂ Jk+i(M, P) are smooth submanifolds and all projections πk+i,k+i−1 :E(i) → E(i−1) are smooth bundles.

For any system E ⊂ Jk(M, P) fibres of the map πk+1,k : E(1) → E, if theyare not empty, are affine subspaces of the affine bundle πk+1,k . Therefore, inorder to check smoothness of E(1) one should estimate dimensions of the fibres.This can be done in the same way as for general and symplectic cases.

A symbol of E at point ak ∈ E is the following subspace gk(ak) of the verticaltangent space, that is,

gk(ak) = Tak (E) ∩ Tak (F(ak−1))

where ak−1 = πk,k−1(ak).Because Tak (F(ak−1)) = Sk+1(T∗a N) the symbol is a vector subspace of the

tensor space

gk(ak) ⊂ Sk+1(T∗a N)⊗ νa.


g(l)k (ak+l) ⊂ Sk+l+1(T∗a N)⊗ νa

and all of these spaces can be defined by the symbol gk(ak) only.Namely, the first prolongation symbol g(1)k (ak+1) consists of tensors θ ∈

Sk+2(T∗a N) ⊗ νa such that v(θ) ∈ gk(ak) for all vectors v ∈ TaN , and

g(l+1)k (ak+l+1) = (g(l)k (ak+l))

(1).

We say that a system E has a finite type if g(l)k (ak) = 0 for all ak ∈ E andl ≥ l0.


Denote by a′k+l0−1 ∈E(l0) the preimage of ak+l0−1 ∈E(l0−1), πk+l0,k+l0−1



defines a distribution on E(l0−1).


As above one can prove that this distribution is completely integrable ifand only if E(l0+1) is not empty and πk+l0+1,k+l0 : E(l0+1) → E(l0) is adiffeomorphism.


Theorem 22.2.2 Let E ⊂ Jk(M,�) be a finite type system of differentialequations on Legendrean submanifolds such that


diffeomorphisms.

Then,

• the distribution L on E(l0−1) defined by E(l0) is completly integrable, and• for any point ak+l0−1 ∈ E(l0−1) there is a solution, that is, a Legendrean

submanifold N ⊂ M, such that jk+l0−1(N) ⊂ E(l0−1), and [N]k+l0−1a =

ak+l0−1.

Note that any effective r-form ω ∈ �rε(M) defines a first-order system of

differential equations on Legendrean submanifolds Eω ⊂ J1(M,�) where

Eω = {[N]1a ∈ LGr(TaM) | ω|TaN = 0}.

We call this system the (overdetermined) Monge–Ampère equation.If r < 1

2 (dim M − 1) then Eω has finite type for ωs “in the general position”and one can use the above theorem in order to obtain smooth solvability of thesystem.

22.3 Differential equations for symplectic equivalence

Let ω1 and ω2 be effective n-forms on T∗M, n = dim M. We reformulate theproblem of finding a symplectic diffeomorphism φ such that φ∗(ω2) = ω1 asfollows. Denote the symplectic manifold (T∗M,�) by (�,�), and let (�′,�′)be a copy. The graph Lφ ⊂ �×�′ of a symplectic diffeomorphismφ : �→ �′is a Lagrangian submanifold of the symplectic manifold (�×�′,π∗�−π ′∗�′),where π : � × �′ → �, and π ′ : � × �′ → �′ are the natural projections.Moreover, φ∗(ω2) = ω1 if and only if Lφ is a solution of the Monge–Ampèreequations corresponding to the effective form ω = π∗ω1 − π ′∗ω2.

In other words, local symplectic equivalence of effective forms ω1 and ω2

follows from the local solvability of the Monge–Ampère equation

Eω ⊂ J1(�×�′,π∗�− π ′∗�′).


Note that [φ]1(a,b) ∈ Eω if and only if the differential φ∗,a : Ta�→ Tb� sends

ω2,b to ω1,a. Therefore, π1,0 : Eω → � × �′ is onto if and only if effectiven-forms ω1,a and ω2,b are symplectically equivalent for any points a, b ∈ �. Inthis case, symbols of Eω at different points are isomorphic.

Assume, for example, that ω1and ω2 are equal at a point a0. Then the symbolat [id]1(a0,a0)

∈ Eω, which we denote by g ⊂ S2T∗a0, is isomorphic to the stabilizer

of ω1,a0 in the Lie algebra of Hamiltonian operators. It is easy to check thatsymbols of the prolongations of Eω are exactly the prolongations of g.

Assume now that the first prolongation g(1) is trivial. This means that the fibresπ2,1 : E(1)

ω → Eω are discrete or empty. Let [φ]1(a,b) ∈ Eω or φ∗a (ω2,b) = ω1,a.

Then a 1-jet of the n-form θ = φ∗(ω2)−ω1 is equal to zero at point a. Considerthe 1-jet of θ

ε = [θ ]1a ∈ �nef,a ⊗ T∗a ,

and deformations θt = (At ◦ φ)∗(ω2) − ω1 by symplectic diffeomorphismswhich preserve the point a. Then the fibre of π2,1 : E(1)

ω → Eω at [φ]1(a,b) is not

empty if one can find a deformation such that [θt]1a = 0 for some value of t.Remark that A∗t = 1+ tLH + o(t) for a Hamiltonian H with dbH = 0, and

θt = θ + tφ∗LH(ω2)+ o(t).

Therefore,

[θt]1a = [θ ]1a + t[φ∗LHω2)]1a.

Let

Ka = {[φ∗LHω2)]1a, daH = 0} ⊂ �nef ,a

be the subspace of the space of effective forms.Then [φ]1

(a,b) ∈ Eω has a prolongation if and only if [θ ]1a ∈ Ka. Consider thefollowing function:

σ1([φ]1(a,b)) = [θ ]1a mod Ka.

Then [φ]1(a,b) ∈ Eω admits a prolongation if and only if

σ1([φ]1(a,b)) = 0.

Assume now that σ1 vanishes on the equation Eω, then bundles of the projectionπ2,1 : E(1)

ω → Eω isomorphic to g(1) = 0, and, as above, we obtain a distribution


L on Eω. The curvature σ2 of the distribution defines a g-valued function onEω, and if σ2 = 0 on Eω then the classification problem has a (local) solution.

Summarizing, we obtain the following theorem.

Theorem 22.3.1 Let ω1 and ω2 be effective n-forms on T∗M, n = dim M,such that

1. exterior forms ω1,x and ω2,y are simplectically equivalent for all points x, yfrom a neighborhood of a point a ∈ T∗M,

2. the first prolongation of the stabilizer of ω1,a in sp(Ta,�a) is trivial, and3. the obstractions σ1 and σ2 vanish.

Then the n-form ω1 is symplectically equivalent to ω2 in a neighborhood ofthe point a ∈ T∗M.

Let apply this result for M = R3. Denote by ωconst an effective form withconstant coefficients on T∗R3. We say that an effective 3-form ω is regular ifthe Sp-orbit of ωa is regular for all points a in the domain of consideration. Inour case this means that it belongs to one of the five first orbits listed in theclassification table. Then the first prolongation of the stabilizer is trivial and wecan apply the previous result.

Theorem 22.3.2 Consider a Monge–Ampère equation in R3, corresponding toa regular effective 3-form ω, such that Sp-orbits of ωq do not depend on q. Letωconst be an effective form with constant coefficients with the same Sp-orbit.Assume that the obstractions σ1 and σ2, constructed for ω and ωconst, vanish.Then this differential equation is locally equivalent to one of the followingequations:

λ+ det Hess v = 0, λ �= 0;

∂2v

∂q21

− ∂2v

∂q22

+ ∂2v

∂q23

+ λ2 det Hess v = 0, λ �= 0;

∂2v

∂q21

+ ∂2v

∂q22

+ ∂2v

∂q23

− η2 det Hess v = 0, η �= 0;

∂2v

∂q21

− ∂2v

∂q22

+ ∂2v

∂q23

= 0;

∂2v

∂q21

+ ∂2v

∂q22

+ ∂2v

∂q23

= 0.

References

[1] D. V. Alekseevskij, V. V. Lychagin and A. M. Vinogradov, Basic Ideas and Con-cepts of Differential Geometry, Geometry vol. I. (Berlin: Springer–Verlag, 1991),pp. 1–264.

[2] J. S. Allen, J. A. Barth and P. A. Newberger, On intermediate models for barotropiccontinental shelf and slope flowfields. Part I: formulation and comparison of exactsolutions. Phys. Oceanogr. 20 (1990), 1017–1042.

[3] E. Artin, Geometric Algebra. (New York: John Wiley & Sons, Inc., 1988).[4] B. Banos, Nondegenerate Monge–Ampère structures in dimension 3. Lett. Math.

Phys. 62:1 (2002), 1–15.[5] B. Banos, On symplectic classification of effective 3-forms and Monge–Ampère

equations. Differ. Geom. Appl. 19:2 (2003), 147–166.[6] N. Belotelov and A. Lobanov, Migration and demographic processes. In Non-

linear Diffusion, Proceeding of Mathematics, Computer, Education, InternationalConference. (Moscow: Dubna, 1999), pp. 434–443.

[7] L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics.(New York: John Wiley & Sons, Inc., 1958).

[8] A. V. Bitsadze, Some Classes of Partial Differential Equations. (Moscow: Nauka,1981).

[9] E. Bonan, Decomposition of the exterior algebra of hyper-Kähler manifolds.C. R. Acad. Sci. Paris, Math. 320:4. (1995), 457–462.

[10] W. M. Boothby and H. C. Wang, On contact manifolds. Ann. Math. 2:68 (1958),721–734.

[11] M. Born and L. Infeld, Foundation of a new field theory. Proc. R. Soc. 144 (1934),425–451.

[12] N. E. Byers, Noether’s discovery of the deep connection between symmetriesand conservation laws. In The Heritage of Emmy Noether. (RamatGan, 1996),pp. 67–81.

[13] G. R. Cavalcanti and M. Gualtieri Generalized complex structures on nilmani-folds. J. Symplectic Geom. 2:3 (2004), 393–410.

[14] S. A. Chaplygin, Gas jets. Uch. Zap. Mosk. Univ., Otdel. Fiz.-Mat. Nauk 21 (1904).[15] S. Chynoweth and M. J. Sewell, Dual variables in semi-geostrophic theory. Proc.

R. Soc. Lond. A 424 (1989), 155–186.

487

488 References

[16] S. Chynoweth and M. J. Sewel, A concise derivation of the semi-geostrophicequations. Q. J. R. Meteorol. Soc. 117 (1991), 1109–1128.

[17] M. J. P. Cullen, J. Norbury and R. J. Purser, Generalised Lagrangian solutions foratmospheric and oceanic flows. Siam J. Appl. Math. 51 (1991), 20–31.

[18] M. J. P. Cullen and R. J. Purser, An extended Lagrangian theory of semi-geostrophic frontogenesis. J. Atmos. Sci. 41 (1984), 1477–1497.

[19] G. A. Desroziers, Coordinate change for data assimilation in spherical geometryof frontal structures. Mon. Wea. Rev. 125 (1997), 3030–3038.

[20] V. A. Dorodnitsin, Group properties and invariant solutions of non-linear thermalconductivity equations with source or drain, Preprint No 57 of Applied Mathem-atics Institute of the Academy of Sciences of the USSR (1979).

[21] B. Doubrov and A. Kushner, The Morimoto Problem, Geometry in PartialDifferential Equations. (River Edge, NJ: World Scientific Publishing, 1994),pp. 91–99.

[22] S. V. Duzhin and V. V. Lychagin, Symmetries of distributions and quadrature ofordinary differential equations. Acta Appl. Math. 24:1 (1991), 29–57.

[23] F. J. Ernst, Black holes in a magnetic universe. J. Math. Phys. 17 (1976),54–56.

[24] D. S. Freed, Special Kähler manifolds. Commun. Math. Phys. 203 (1999), 31–52.[25] O.-K. Fossum, On classification of a system of non-linear first-order PDEs of

Jacobi type, Preprint Tromsö University (2002).[26] A. Frölicher and A. Nijenhuis, Theory of vector valued differential forms. Part 1:

derivations in the graded ring of differential forms. Indag. Math. 18 (1956),338–359.

[27] Gibbons J., Tsarev S. P., Reduction of the Benney equations, Phys. Lett. A 211(1996), 19–24.

[28] H. Goldschmidt, Formal integrability criteria for systems of non-linear partialdifferential equations J. Differ. Geom. 1 (1967), 269–307.

[29] E. Goursat, Leçon sur l’intégration des Équations aux Dérivée Partielles duSecond Ordre a Deux Variables Indépendantes, vol. 1 (Paris, 1896).

[30] E. Goursat, Sur le problème de Monge. Bull. Soc. Math. France 33 (1905),201–210.

[31] E. Goursat, Sur les èquations du second ordre à n variables analogues à l’equationde Monge–Ampère. Bull. Soc. Math. France 27 (1899), 1–34.

[32] M. Gross, Special Lagrangian fibrations II: geometry. Surv. Differ. Geom. 5(1999), 341–403.

[33] D. I. Gurevich, V. V. Lychagin and V. N. Roubtsov, Cohomologie de “grandcrochet” et filtration “non-holonome.” C. R. Acad. Sci. Paris Sr. I Math. 313:12(1991), 855–858.

[34] D. I. Gurevich, V. V. Lychagin and Roubtsov V. N., Nonholonomic filtration ofthe cohomology of Lie algebras and “large brackets.” Mat. Zametki 52:1 (1992),36–41.

[35] G. Hart, Geometric Quantization in Action: Applications of Harmonic Analysis inQuantum Statistical Mechanics and Quantum Field Theory. (Dordrecht: ReidelPublishing, 1983).

[36] N. J. Hitchin, Monopoles, Minimal Surfaces and Algebraic Curves, NATOAdvanced Study Institute vol. 105 (1987).

References 489

[37] N. J. Hitchin, The moduli space of special Lagrangian submanifolds. Ann. ScuolaNorm. Sup. Pisa Cl. Sci. 25 (1997), 503–515.

[38] N. J. Hitchin, The geometry of three-forms in six dimensions. Differ. Geom. 55:3(2000), 547–576.

[39] J. K. Hunter and R. Saxton, Dynamics of director fields. SIAM J. Appl. Math. 51:6(1991), 1498–1521.

[40] Igusa, Jun-ichi, A classification of spinors up to dimension twelve. Amer. J. Math.92 (1970), 997–1028.

[41] P. Jakobsen, V. Lychagin and Y. Romanovsky, Symmetries and non-linearphenomena, I. Preprint, Tromsö University (1997).

[42] P. Jakobsen, V. Lychagin and Y. Romanovsky, Symmetries and non linear phe-nomena, II, applications to non-linear acoustics. Preprint, Tromsö University(1998).

[43] K. Jörgens, Differentialgleichung rt − s2 = 1. Math. Ann. 127 (1954), 130–134.[44] T. Karman, The similarity law of transonic flow. Math. Phys. 26 (1947),

182–190.[45] Kazuhiko Aomoto and Toru Tsujishita, Open problems in structure theory of non-

linear integrable differential and difference systems. (Nagoya University, 1984),p. 44.

[46] M. V. Keldysh, Cases of degeneracy of equations of elliptic type on the boundaryof a domain. Dokl. Akad. Nauk SSSR 77:2 (1951), 181–183.

[47] A. A. Kirillov, Geometric Quantization, vol. 4. (2001), 139–176.[48] A. N. Kolmogorov, I. G. Petrovsky and I. S Piskunov. Examining of diffusion

equation with increasing of substance amount, and its application to one biologicalproblem. MSU Bull. A 6 (1937), 1–26.

[49] I. B. Kovalenko and A. G. Kushner, The non-linear diffusion and thermal conduct-ivity equation: group classification and exact solutions. Regular Chaotic Dynam.8:2 (2003), 8–31.

[50] I. S. Krasilshchik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet Spacesand Non-linear Partial Differential Equations. (New York: Gordon and Breach,1986).

[51] I. S. Krasilshchik, Some new cohomological invariants for non-linear differentialequations. Differ. Geom. Appl. 2:4 (1992), 307–350.

[52] B. S. Kruglikov, Nijenhuis tensors and obstructions to the construction ofpseudoholomorphic mappings. Mat. Zametki 63:4 (1998), 541–561.

[53] B. S. Kruglikov, On some classification problems in four-dimensional geometry:distributions, almost complex structures, and the generalized Monge–Ampèreequations. Mat. Sb. 189:11 (1998), 61–74.

[54] B. S. Kruglikov, Symplectic and contact Lie algebras with application to theMonge–Ampère equations. Tr. Mat. Inst. Steklova 221 (1998), 232–246.

[55] B. S. Kruglikov, Classification of Monge–Ampère equations with two vari-ables. CAUSTICS ’98, Warsaw. (Warsaw: Polish Academy of Sciences, 1999),pp. 179–194.

[56] B. S. Kruglikov and V. V. Lychagin, On equivalence of differential equations. ActaComment. Univ. Tartu. Math. (1999), 7–29.

[57] B. S. Kruglikov and V. V. Lychagin, Mayer brackets and solvability of PDEs, I.Differ. Geom. Appl. 17:2–3 (2002), 251–272.

490 References

[58] A. Kushner, Chaplygin and Keldysh normal forms of Monge–Ampère equationsof variable type. Math. Zametki 52:5 (1992), 63–67.

[59] A. Kushner, Classification of mixed type Monge–Ampère equations. Geometryin Partial Differential Equations. (1993) pp. 173–188.

[60] A. Kushner, Symplectic geometry of mixed type equations. Amer. Math. Soc.Transl. Ser. 2 (1995), 131–142.

[61] A. Kushner, Monge–Ampère equations and e-structures. Dokl. Akad. Nauk 361:5(1998), 595–596.

[62] A. Kushner, Contact linearization of nondegenerate Monge–Ampère equations.Natural Sci.

[63] I. B. Kovalenko and A. G. Kushner, Symmetries and exact solutions of non-linear diffusion equation. Math. Model. Comput. Biol. Medicine. July (2002),239.

[64] N. Larkin, B. Novikov and N. Yanenko, Non-Linear Mixed Type Equations.(Novosibirsk: Nauka, 1983).

[65] S. Lie, Classification und integration von gewöhnlichen Differentialgleichungenzwischen x, y, die eine Gruppe von Transformationen gestatten. Math. Ann. 32(1888), 213–281.

[66] S. Lie, Begrundung einer Invarianten-Theorie der Beruhrungs-Transformationen.Math. Ann. 8 (1874), 215–303.

[67] S. Lie Über einige partielle Differential-Gleichungen zweiter Orduung. Math.Ann. 5 (1872), 209–256.

[68] V. Lychagin and O. Lychagina, Finite dimensional dynamics for evolutionaryequations. Nonlin. Dynam. (2006), to appear.

[69] V. V. Lychagin, Contact geometry and second-order non-linear differentialequations. Russian Math. 34 (1979), 137–165.

[70] V. V. Lychagin, Singularities of multivalued solutions of non-linear differentialequations and non-linear phenomena. Acta Appl. Math. 3 (1985), 135–173.

[71] V. V. Lychagin, Differential equations on two-dimensional manifolds. Izv. Vyssh.Uchebn. Zaved. Mat. 5 (1992), 43–57.

[72] V. Lychagin, Lectures on Geometry of Differential Equations, vols 1, 2. (Rome:La Sapienza, 1993).

[73] V. V. Lychagin and V. N. Roubtsov, On Sophus Lie theorems for Monge–Ampèreequations. Dokl. Akad. Nauk BSSR 27:5 (1983), 396–398.

[74] V. V. Lychagin and V. N. Roubtsov, Local classification of Monge–Ampèredifferential equations. Dokl. Akad. Nauk SSSR 272:1 (1983), 34–38.

[75] V. V. Lychagin and V. N. Roubtsov, Non-holonomic filtration: algebraic and geo-metric aspects of non-integrability. In Geometry in Partial Differential Equations.(River Edge, NJ: World Scientific Publishing, 1994).

[76] V. V. Lychagin and R. Romanovsky Yu., Nonholonomic intermediate integralsof partial differential equations. In Differential Geometry and its Applications(Brno, 1989). (Teaneck, NJ: World Scientific Publishing, 1990).

[77] V. V. Lychagin, V. N. Roubtsov and I. V. Chekalov, A classification of Monge–Ampère equations. Ann. Sci. Ecole Norm. Sup.(4) 26:3 (1993), 281–308.

[78] L. J. Mason and N. M. J. Woodhouse, Integrability, Self-duality, and TwistorTheory, London Mathematical Society Monographs, New Series, vol. 15.(New York: Oxford University Press, 1996).

References 491

[79] M. Matsuda, Two methods of integrating Monge–Ampère equations I. Trans.Amer. Math. Soc. 150:1 (1970), 327–343.

[80] J. C. McWilliams and P. R. Gent, Intermediate models of planetary circulationsin the atmosphere and ocean. J. Atmos. Sci. 37 (1980), 1657–1678.

[81] M. E. McIntyre and I. Roulstone, Hamiltonian balanced models: constraints, slowmanifolds and velocity splitting. Forecasting Research Scientific Paper 41 (1996),http://www.atm.damtp.cam.ac.uk/people/mem/

[82] M. E. McIntyre and I. Roulstone, Are there higher-accuracy analogues ofsemi-geostrophic theory? Large Scale Atmosphere–Ocean Dynamics, vol. II,Geometric Methods and Models, ed. I. Roulstone and J. Norbury. (Cambridge:Cambridge University Press), to appear.

[83] T. Morimoto, La géométrie des équations de Monge–Ampére. C. R. Acad. Sci.Paris A-B 289:1 (1979), A25–A28.

[84] T. Morimoto, Open problem in str. theory of non-linear integrable differen-tial and difference systems. The 15 International Symposium, Katata. (1984),pp. 27–29.

[85] O. I. Morozov, Contact equivalence problem for non-linear wave equations.Preprint arXiv: math-ph/0306007 v.1 (2003), pp. 1–13.

[86] O. I. Morozov, Contact equivalence of the generalized Hunter–Saxton equa-tion and the Euler–Poisson equation. Preprint arXiv: math-ph/0406016 (2004),pp. 1–3.

[87] J. D. Murray, Mathematical Biology. (Berlin: Springer-Verag, 1993).[88] A. Newlender and L. Nirenberg, Complex analytic coordinates in almost complex

manifolds. Ann. Math. 65 (1954), 391–404.[89] J. Nielsen, On the intermediate integral for Monge–Ampère equations. Proc.

Amer. Math. Soc. 128:2 (2000), 527–531.[90] V. L. Popov, A classification of spinors of dimension fourteen. Trudy Moskov.

Mat. Obshch. 37 (1978), 173–217.[91] R. J. Purser, Contact transformations and Hamiltonian dynamics in generalized

semi-geostrophic theories. J. Atmos. Sci. 50 (1993), 1449–1468.[92] R. J. Purser and M. J. P. Cullen, A duality principle in semi-geostrophic theory.

J. Atmos. Sci. 44 (1987), 3449–3468.[93] B. L. Rojdestvensky and N. N. Yanenko, Systems of Quasilinear Equations and

their Applications to Gas Dynamics. (Moscow: Nauka, 1968).[94] V. N. Roubtsov and I. Roulstone, Examples of quaternionic and Kähler struc-

tures in Hamiltonian models of nearly geostrophic flow. J. Phys. A 30:4 (1997),L63–L68.

[95] Roubtsov V. N. and I. Roulstone, Holomorphic structures in hydrodynam-ical models of nearly geostrophic flow. R. Soc. Lond. Proc. A 457 (2001),1519–1531.

[96] I. Roulstone and J. Norbury, A Hamiltonian structure with contact geometry forthe semi-geostrophic equations. J. Fluid Mech. 272 (1994), 211–233.

[97] I. Roulstone and M. J. Sewell, The mathematical structure of theories of semi-geostrophic type. Phil. Trans. R. Soc. Lond. 355 (1997), 2489–2517.

[98] O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Non-linear Acous-tics, transl. from Russian by R. T. Beyer, Studies in Soviet Science. (New York:Consultants Bureau, 1997).

http://www.atm.damtp.cam.ac.uk/people/mem/

492 References

[99] A. A. Samarsky, S. P. Kurdyumov, V. A. Galaktionov and Michailov A. P.,Peaking Regimes in Problem for Quasilinear Parabolic Equations (Moscow:Nauka, 1987).

[100] W. Slebodzinski, Exterior Forms and their Applications. (Warsaw: PWN-PolishScientific Publishers, 1970).

[101] R. Salmon, Practical use of Hamilton’s principle. J. Fluid Mech. 132 (1983),431–444.

[102] R. Salmon, New equations for nearly geostrophic flow. J. Fluid Mech. 153 (1985),461–477.

[103] R. Salmon, Semi-geostrophic theory as a Dirac bracket projection. J. Fluid Mech.196 (1988), 345–358.

[104] M. J. Sewell and I. Roulstone, Families of lift and contact transformations. Proc.R. Soc. Lond. 447 (1994), 493–512.

[105] D. V. Tunitskii, Contact equivalence of Monge–Ampère equations with trans-itive symmetries. In Differential Geometry and Applications, Brno. (1995),pp. 479–485.

[106] D. V. Tunitskii, On the contact linearization of Monge–Ampère equations. Izv.Ross. Akad. Nauk Ser. Mat. 60:2 (1996), 195–220.

[107] D. V. Tunitskii, Monge–Ampère equations and characteristic connection functors.Izv. Ross. Akad. Nauk Ser. Mat. 65:6 (2001), 173–222.

[108] F. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo ordine di tipomisto. Rend. Atti Acad. Nazion. Lincei 5:14 (1923), 134–247.

[109] B. Trubnikov and S. Zhdanov, Quasi-Gaseous Unstable Media. (Moscow: Nauka,1991).

[110] C. Udriste and N. Bila, Symmetry group of Titeica surfaces, PDE. Balkan J.Geom. Applic. 4:2 (1999), 123–140.

[111] G. B. Whitham, Linear and Non-linear Waves. (New York: Wiley-Interscience,1974).

[112] V. F. Zaitsev and A. D. Polyanin, Non-Linear Partial Differential Equations,Handbook: Exact Solutions. (Moscow: Nauka, 1996).

[113] L. V. Zilbergleit, Second-order differential equation associated with contact geo-metry: symmetries, conservation laws and shock waves. Acta Appl. Math. 45:1(1996), 51–71.

[114] L. V. Zilbergleit, Geometry of the Hugoniot–Rankine conditions. Russ. J. Math.Phys. 7:2 (2000), 234–249.

Index

algebra of effective forms, 119annihilator, 4

bracketNijenhuis, 425Poisson

of differential 1-forms, 190of functions, 191

Poisson–Lie, 37bundle normal to a distribution, 18

canonical basis, 107Cartan

distribution, 7, 203form, 202ideal, 119

Cauchydata, 307, 313, 334problem, 307, 314

Cauchy–Riemann system, 322caustic, 349characteristic

direction, 301, 303, 312, 331distribution, 301

of a solution, 331number of an exterior

2-form, 148polynomial, 148

codimension of a distribution, 3cohomology

co-effective, 291effective, 291

complex curve, 295connection

in a bundle, 205linear, 206

conservation law, 273of a Jacobi equation, 332

contactdifferential 1-form, 201distribution, 9element, 203equivalent

Monge–Ampére operators, 243Hamiltonian, 216manifold, 201shock wave, 351structure, 201symmetry

of a Monge–Ampére equation, 251transformation, 208vector field, 215

contact structurestandard, 202

curvature of a distribution, 19

derivatives of a distribution, 4derived subalgebra, 31dilatation, 214dimension of a distribution, 3discontinuous solution, 282distribution, 3

Cartan, 7, 203characteristic, 193

of a solution, 301, 303, 312of an equation, 310

characteristic , 298completely integrable, 6, 22, 24complex, 23

493

494 Index

distribution (Continued)contact, 9flat, 19involutive, 24

effective form, 125, 226elliptic

form, 138plane, 143similitudes, 145

equationBianchi, 92Born–Infeld, 440boundary layer, 259Euler, 321Fisher, 262Keldysh, 417Khokhlov–Zabolotskaya, 238, 340Korteweg–de Vries (KdV), 64, 66Laplace, 253Lie, 34, 253Maurer–Cartan, 25Maxwell–Einstein, 420model, 79Monge–Ampére, 238non-linear thermal conductivity, 238Petrovsky–Kolmogorov–Piskunov, 262reaction–diffusion, 238, 357Riccati, 82, 83Schwartzian, 57Sine-Gordon, 276thermal conductivity, 261Titeica–Morimoto, 245, 255Triccomi, 310, 417Von Karman, 237, 242, 264

equation of non-linear thermalconductivity, 357

equivalent forms, 137equivalent Monge–Ampére equations, 422Euler plane, 143

flatconnection, 20distribution, 20

formCartan, 202co-effective, 290divergent, 241dual by Hitchin, 174effective, 125, 226, 229

Killing, 346Lagrange, 74Liouville, 184normed, 298, 310

formulaLagrange, 73

functiongenerating, 33superposition, 90

generating functionfor Lagrangian manifold, 195of a conservation law, 275of a contact vector field, 216of a symmetry, 33of Lagrangian submanifolds, 221

Hamiltonian, 191height, 132Hodge–Lepage

decomposition, 126theorem, 125

horizontalcomponent of a vector field, 216vector, 20

Hugoniot-Rankine condition, 281hyperbolic

form, 138plane, 143similitudes, 145

image of a distribution, 5integral submanifold, 5integrating factor, 29intermediate integral, 304, 313

classical, 304non-holonomic, 305

invariantabsolute, 431relative, 430

involutivesubmanifold, 197

Jacobiequation, 321

elliptic, 325hyperbolic, 325non-degenerate, 325parabolic, 325

Index 495

Jacobi (Continued)plane, 142system, 321

Kahlermanifold, 186potential, 186

Lagrange bracket, 219Lagrangian polarization, 198

manifoldKahler, 186strict contact, 201symplectic, 183

mapP-regular, 5symplectic, 183

maximal integral manifold, 5Maxwell rule, 284Monge–Ampére equation, 238

degenerate, 309elliptic, 297, 309hyperbolic, 297, 310

of divergent type, 241of mixed type, 297on a contact manifold, 258

non-degenerate, 309parabolic, 297, 309

symplecticoperator of divergent type, 241solution regular, classical, 239

multivalued solution, 239, 270, 320

Nijenhuis bracket, 425non-holonomic

almost complex structure, 295almost product structure, 295almost tangent structure, 295

Non-holonomic de Rham complex, 466

operatoradjoint, 54anti-self-adjoint, 73Euler, 234, 292Hamiltonian, 108Keldysh, 417linearization, 34

Monge–Ampére, 224normed, 298, 310of divergent type, 241of non-holonomic differentiation, 466Schrödinger, 56, 74, 75self-adjoint, 54skew-adjoint, 54symplectic, 108Triccomi, 417

oricycle distribution, 9oricycles, 9

P-structure, 94parabolic

form, 138plane, 143

Pfaffian, 136, 147, 296Hitchin, 173

potential, 56integrable, 58

prolongation, 210proposition

Banos, 175Hitchin, 173, 174

quadratures, 31

reduction, 256of the presymplectic manifold, 194

representationcoadjoint, 188

shifts, 209skew-orthogonal

subspaces, 105vectors, 105

solutionautomodel, 257invariant, 257of Monge–Ampére’s equation, 239

solvable Lie algebra, 31space

Hermitian, 117homogeneous, 187Khler, 118

spherization, 204stabilizer of a form, 175strict contact manifold, 201

496 Index

strict diffeomorphism, 208structure

almost complex, 98, 295almost product, 96, 295almost tangent, 295Cauchy–Riemann, 295compatible, 118contact, 201dual, 104presymplectic, 193standard symplectic, 107, 185symplectic, 104

Konstant–Kirillov, 189subalgebra stability, 77submanifold

involutive, 197Lagrangian, 194

subspaceinvolutive, 113isotropic, 88, 113Lagrangian, 88, 113regular, 106symplectic, 106

superposition principle, 90symmetry

characteristic, 16elliptic, 59even, 55hyperbolic, 59infinitesimal

of a distribution, 11infinitesimal

of a Monge–Ampére equation, 251linear, 51odd, 55of a functional, 286of a Monge–Ampére equation, 251of a Monge–Ampére operator, 253of the Jacobi equation, 328parabolic, 59shuffling, 17

symmetry of a distribution, 11symplectic

diffeomorphism, 183form, 104group, 108map, 183operator, 108structure, 104subspace, 106transformation, 108transvection, 109

vector space, 104symplectic equivalent Monge–Ampére

operators, 386symplectomorphism, 183

theoremDarboux, 219Frobenius, 22Frobenius–Nirenberg, 24Hodge–Lepage, 125Kostant–Souriau, 189Lepage, 131Lie–Bianchi, 31Noether, 287Sophus Lie, 81, 244Witt, 115

transformationcanonical, 183contact, 36Euler, 209Legendre, 11, 209, 244optical, 214partial Legendre, 209point, 36, 210, 211polar, 214scale, 36strict symplectic, 210symplectic, 108, 183translation, 36

transformation pedal, 213scale, 209

translation, 209transversal subalgebra, 25traveling wave, 259

universal 1-form, 194

vectorhorizontal, 205primitive, 133vertical, 205

vector fieldcanonical, 189Reeb, 217

vectors fieldHamiltonian, 191

verticalcomponent of a vector field, 216subspace, 20vector, 19

Wronski matrix, 50

Contact Geometry and Non-linear Differential …...61 H. Groemer Geometric Applications of Fourier...

Documents

Transcript of Contact Geometry and Non-linear Differential …...61 H. Groemer Geometric Applications of Fourier...