,J(;/~,Lecture Notes

in PhysicsEdited by J. Ehlers, MUnchen, K. Hepp, ZUrichR. Kippenhahn, MUnchen, H. A. WeidenmUller, Heidelbergand J. Zittartz, Kain

Managing Editor: W. Beiglback, Heidelberg

107

Jerzy KijowskiWlodzimierz M. Tulczyjew

A Symplectic Framework forField Theories

Springer-Verlag

Berlin Heidelberg New York 1979

Editors

Jerzy KijowskiDepartment ot Mathematical

Methods in Physics

University ot Warsawul. Hoza 74

00-682 WarszawaPoland

Wlodzimierz M. TulczyjewDepartment ot Mathematicsand Statistics

University ot Calgary2920 - 24th Av. NW.

Calgary, Alberta, T2N 1N4Canada

ISBN 3-540-09538-1 Springer-Verlag Berlin Heidelberg New York

ISBN 0-387-09538-1 Springer-Verlag New York Heidelberg Berlin

Library of Congress Cataloging in Publication DataKijowski, J 1943-A symplectic framework for field theories.(Lecture notes in physics; 107)Bibliography: p.Includes index.

1.Symplectic manifolds. 2. Field theory (Physicsl I.Tulczyjew, W. M., 1931- joint authar.II.Title. III. Series.aC174.52.S94K54 530.1'4 79-20519ISBN 0-387-09538-1

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© by Springer-Verlag Berlin Heidelberg 1979Printed in Germany

Printing and bind ing: Beltz Offsetdruck, Hemsbach/Bergstr.2153/3140-543210

CONTENTS

Introduction

I. An intuitive derivation of symplectic concepts

in mechanics and field theory1. Potentiality and reciprocity2. Elastic string3. Elastostatics4. Electrostatics

II. Nonrelativistic particIe dynamics

5. Preliminaries

6. Special symplectic structures. Generating

functions

7. Finite time interval formulation of dynamics

8. Infinitesimal description of dynamics

9. Hamiltonian description of dynamics

10. The Legendre transformation

11. The Cartan form

12. The Poisson algebra

III. Field theory

13. The configuration bundle and the phase bundle

14. The symplectic structure of Cauchy data on a

boundary

15. Finite domain description of dynamics

16. Infinitesimal description of dynamics

17. Hamiltonian description of dynamics

18. The Legendre transformation

19. Partial Legendre transformations. The energy-

-momentum density

20. The Cartan form

21. Conservation laws

22. The Poisson algebra

1

7

7

20

31

35

41

41

42

'1-7

58

69

75

75

79

80

80

85

91

100

116

124

125

143

151

157

IV

23. The field as a mechanical system with an

infinite number of degrees of freedom 160

24. Virtual action and the Hamilton-Jacobi theorem 164

25. Energy-momentum tensors and stress tensors. A

review of different approaches 168

IV. Examples 184

26. Vector field 184

27. The Proca field 191

28. The electromagnetic field 198

29. The gravitational field 209

30. The hydrodynamics 231

Appendices

A. Sections of fibre bundles 2'1-0

B. Tangent mapping 242

c. Pull-back of differential forms 243

D. Jets 245

E. Bundle of vertical vectors 248

F. Tensor product of bundles 250

G. The Lie derivative 250

List of more important symbols 252

References 254

Introduction

These notes contain the formulation of a new conceptual frame-

work for classical field theories. Although the formulation is based

on fairly advanced concepts of symplectic geometry these notes can not

be viewed as a reformulation of known structures in more rigorous and

elegant terms. Our intention is to communicate to theoretical physi-

cists a set of new physical ideas. We have chosen for this purpose the

language of local coordinates which although involved is mor e elemen-

tary and more widely known than the abstract language of modern diffe-

rential geometry. We have given more emphasis to physical intuitions

than to mathematical rigour.

Bince the new framework unifies variational for~llations with ca-

nonical formulations of field theories as different expressions of the

same symplectic structure it is of potential interest to a wid e audi en-

ce of physicists.

Physicists have been interested in variational principles as pro-

viding a method of generating first integrals from symmetry properties

of theories. Powerful methods of solving field equations are based on

variational formulations. We develop a systematic procedure for deri-

ving variational formulations of physical theories. Using this proce-

dure we have succeeded in formulating a number of new variational prin-

ciples such as the formulation af hydrodynamics included in these na-

tes. The usefulness af the procedure is far from being exhausted. Work

is in progress on a variational formulation of hydrodynamics including

thermal processes. A new variational principle for th~ theory of gra-

vity is also included in these notes. This new formulation suggests a

solution of the energy localization problem, provides a basis for ana-

lysing asymptotic behaviaur of gravitational fields at spatial infi­

nity and throws new light on the Oauchy problem for Einstein's equa-

tions and on unified field theories.

Illlll!lj1lIlljlilll1,llll,,1ljl1ljljli1;lll,,;llI;

ljl,,j

~l1

!il!

iili!

llil

ll

2

Quantum physics is one of the main sources of interest in symp­

lectic formulations of physical theories. Physicists interested in ca-

nonical quantization will find in these notes general methods of cons-

tructing symplectic spaces and the associated Poisson algebras. Of spe­

cial importance may be the method of describing local field theories

by finite-dimensional symplectic structures. Lagrangian methods in

quantum field theory such as the method of Feynman integrals may pro-

fit from the more precise formulation of lagrangian systems. Applica-

tions to gauge theories may be of particular interest. The new inter­

pretation of the Legendre transformation throws new light on the re-

lation between lagrangian systems and hamiltonian systems in field

theory. Physicists interested in classical limits of quantum theories

will find the extensive use of lagrangian submaniiolds. Lagrangian

submaniiolds are the objects required by modern W.K.B. theory for cor-

rect formulation oi classical limits and asymptotic expansions.

We give a brief description oi the main features of the new fra-

mework. We consider a class of physical theories which we calI field

theories although particIe mechanics is included as a special case.

Each theory has an underlying manifold M which is the one-dimensional

time manifold in the case of particIe mechanies, it is the physical

three-dimensional space for static field theories and the four-dimen-

sional space-time in the case of dynamical fields. A symplectic "sta­

te space" is associated with the boundary of each Icompactl domain in

M and the dynamics lor staticsl of the field is described by a lagran-

gian subspace of the state space. The lagrangian subspace consists of

states allowed by the physical laws governing the field. This space

can also be described as the set of solutions of the boundary value

problem corresponding to the boundary of the domain. Lagrangian sub-

spaces are usually generated by generating functions, in other words,

they are described by variational principles. Here the generating fun­

ction is the action functional. If the boundary is divided into seve-

-~~._~~-- ....-....--

3

raI components then the associated symplectic space is a product spa-

ce and the lagrangian subspace becomes a symplectic relation. If the

boundary consists of two components thenthe corresponding two-term

symplectic relation may be a symplectic /canonical/ mapping. In non-

relativistic particIe mechanics domains are time-intervals, boundaries

consist of pairs of end points and the corresponding relations turn

out to be mappings. In this way the state at the end of an interval

can be considered as the re suIt of the initial state. Oonsequently

nonrelativistic particIe mechanics can be formulated in terms of hamil-

tonian fields and flows. This situation is exceptional and does not

extend easily to other field theories. Already relativistic mechanics

requires symplectic relations and generalized hamiltonian systems for

its proper formulation /see ref.[37]/. No boundary problems lead to

symplectic mappings in the case of static field theory governed by

elliptic field equations. Only very special boundary problems can be

formulated in hamiltonian terms in the case of a field theory governed

by hyperbolic equations and this formulation is obtained only at the

expense of introducing non-intrinsic elements into the theory and im-

posing conditions excluding physically interesting situations. Such

an element is a one-parameter family of Oauchy hypersurfaces in space­

-time. The Oauchy hypersurfaces must be compact or compactified by im-

posing restrictive asymptotic conditions. Gauge invariance of the theo-

ry must be destroyed by imposing suitable gauge conditions.

Within our framework gauge invariance can be retained by using

symplectic relations and genernlized hamiltonian systems similar to

those appearing in relativistic mechanics. Without compactifying the

Oauchy hypersurfaces dynamics can still be discussed in terms of sym-

plectic relations between initial data, final data and the asymptotic

incoming or out go ing radiation. Boundary problems other than Oauchy

problemscan be discussed within the same framework. Finite-dimensio-

nal formulations are obtained by considering limit s of domains contrac-

lll!il.Illllillililililiil.llilllllllliIllljllllj

IlllIIj

I

ll

4

ting them to points.

We Btt2~h spacial importanPA to generating funetions of lagran-

r:illnsubspgees. IVe mentioned the interp:<:>etation af t11e Bation as a Ge-

ne-""ating f'unction• 'T'hes'~me l ae;J~angi8.nsubspaee can be dese-rihed by

~ations. Eae~ generq~i~s func~ian is assoeiated with Il "specia] sym-

ple,...ticstI'ueture" OT' a "cont-"""Imode" • The La,s;rangian and the Hamil-

tonian af particIe meehanies CBn he snown to he generating funetions

of the same lagrangian ~lhmanifaIrl Isee ref. ~1], ~3J/. Most physical

qllantities Isuc11 as camponents of the energy-momentum tensorl are de-

rived in aur approach as generating functions of lagrangian subspaees

describinc; fi.e]eldyn'lmics.

Charter I is devoted to t11e symplectic analysis of statics. Both

discrete and contirnlous systems are considered on a largeIy intuitive

Ievel. The notion of reciprocity and potentiality of the theory is

diseussed.

Chapter II is a presentation of particIe dynamics together with

mor8 ri[';orousdefinitions of the geometrie structure. Lagrangian sub-

manifolds and their generating funetions are defined in Section 6.

The time evolution of particIe states within a finite time interval

is studied in Seetion 7. It is shown that the Hamiltonian variational

principIe can be derived from composition properties of dynamics. In

Section 8 dynamics is stated in an infinitesimal form in terms of jets

of histories of the particIe system. Section 8 contains also the 1a-

grangian description of dynamics. The hami1tonian description is deri-

ved in Section 9. Seetion 11 contains a formulation of dynamics in

terms af the Cartan form. The Cartan form is an objeet used in the

geometrie formulation of the caIculus of variations developed by Weyl,

Caratheodory, de Donder, Lepage, Dedecker and others Isee ref. [58J,

[6J, [12J, [34-], [9J/.

Chapter III is the main part of these notes. The construction of

i..=

lIIiI

5

canonical moment a of a field is given in Section 13. Field dynamics

is first discussed in finite domains of space-time. This discussion

contained in Section 14 and 15 stays on a heuristic level. A rigorous

formulation of infinitesimal dynamics starts in Section 16. The sym­

plectic structure used here is not a symplectic structure in the strict

sense but a natural generalization of the concept. Section 19 contains

a definition of the energy-momentum density as the potential of the

dynamics associated with a family of control modes. We consider this

definition one of the most important results. Section 19 is technical-

ly the most complex part of this volume. Results of this section are

first stated without proofs and in purely coordinate language. This

is folIowed by a discussion of the intrinsic content and proofs. Dy-

namics is formulated in terms of the Cartan form in Section 20. This

section establishes a relation between our symplectic framework and

the geometric formulation of the calculus of variations of multiple

integrals. The time evolution formulation of dynamics is derived in

Sections 22 and 23. An infinitesimal version of the Hamilton-Jacobi

theorem is proved in Section 24. The last section of the chapter con-

tains R detailed discussion of objects associated with the energy-mo-

mentum of the field. Different definitions of energy-momentum tensors

and stress-tensors are compared and new definitions are proposed.The

results of this section are used in a new formulation of General Re-

lativity given in Chapter IV.

Chapter IV contains examples of field theories selected to illu-

strate various features of the new approach. The simplest example of

a tensor field /the covariant tensor field/ is given in Section 26.

The appearance of constraints in the hamiltonian description is illu­

strated by the example of Proca field in Section 27. An example of a

gauge field /the electromagnetic field/ is discussed in Section 28.

A new formulation of the theory of gravity is given in Section 29.

The new formulation consists in using the affine connection r in spa-

~IIjljlllllllllll,lll~lll,llj

jjllil.jljllilllllll!i!!ji!lil;l!ii1

l!ljllI;il1,liI

llili

J

6

ce-time as the field configuration. The Lagrangian of the theory depends

on the conneetion r and its first derivatives represented by the cur­

vature tensor R. The metric tensor g appears as a component of the mo­

mentum eanonically conjugate to r , and the relation between these two

objects is a part of dynamics together with Einstein's equations. The

momentum conjugate to r has 80 independent components. To obtain the

standard Einstein theory of gravity most of these components are equa-

ted to zero by using a Lagrangian depending only on the symmetric part

of the Ricci tensar instead of the fulI Riemann tensor. Thus the Ein-

stein theory of gravity is a very speeial case of the geometrie field

theory based on an affine eonneetion r. Using the Lagrangian depending

on the eomplete Rieci tensor one can easily reproduce within this fra­

mework one of the versions of Einstein's unified field theories /see

[14J/. Other possibilities of formulating unified field theories are

being investigated.

The last Section contains the formulation of hydrodynamies. An

analysis of hydrodynamics within our framework reveals a formal ana­

logy w:Lth electrodynamics. In both theories a simple variational prin­

cipIe can be formulated only jn terms of potentials which do not appear

directly in ei+:herE'ller' s equ::ltions or Maxwell' s equations. The dis­

cove:ry of potentials for hydrorlynamics and the subsequeut formulation

of a variational principle is au example illustrating the fruitfulness

of the new approach to field theory.

Appendices contain a short review of several geometrie coneepts

frequently used throughout the notes /further details may be found in

references [29J ,[33J, [40], [43J, [46J/.

I. An intuitive derivation of sympleetie eoneepts in meehanies and

field theory.

1. Potentiality and reeiproeity.

In the present ehapter we eonsider a series of examples of statie

physieal systems. Statie theories have the advantage of having a sim-

ple well understood eoneeptual strueture. The aim of this ehapter is

to introduee sympleetie eoneepts as a natural expression of sueh phy-

sieal eharaeteristies of statie systems as reeiproeity and the exis-

tenee of potentials. In this seetion we begin with a very simple exam-

ple with a finite number of degrees of freedom. In subsequent seetions

we use mor e eomplieated examples to gradually develope sympleetie eon-

eepts suitable for deseribing eontinuous systems. The geometrie eon-

eeptsderived in this way are intended for use in primarily dynamie

theories. However the intuitive meaning of these eoneepts is mueh ele-

arer in statie theories. Applying to dynamie theories eoneepts derived

from statie theories agrees with the Minkowskian philosophy of trea­

ting dynamies as staties in spaee-time [3~ •

We eonsider a single material point suspended elastieally in the

three-dimensional physieal spaee Q. The position of the point will be

deseribed by eoordinates (q~, i=1,2,3, belonging in general to a eur­

vilinear system. If an external meehanism is used to eontroI the po-

sition of the point then an infinitesimal displaeement from a position

(qi) to (qi + & qi) requires the meehanism to perform a virtual work

1.1 A

The Einstein summation eonvention is used here and throughout these

notes. The eoefieients fi in the above expression form a eoveetor f

ealled the foree. The foree f is aetually the foree that the eontrol-

8

ling mechanism has to exert to maintain the configuration q. Experi­

ment sh.ows that for each configuration q there is a unique force f ne-

eessary to maintain this configuration. The components f. of this for­J

ee are funetions of the eoordinates (qi)

1.2 f.J

The above formula can be interpreted as defining a differential 1-form

1.3

If the form ~ is evaluated on a virtual displaeement

then the resul t < bq, \fI> is exactly the virtual work 1.1.

We consider nowa finite displacement from a eonfiguration ~ to

a eonfiguration ~ along a path Ó . The total work performed in this

proeess is the integral

1.4

It is an experimentally well verified faet that for elastie suspensions

the work A is the same for all paths joining the eonfigurations ~ and

(il.) C tl' f f f" t' (o) • l .b .q. onsequen y l· a re erenee eon_lgura lon q e.g. an equJ. l rlllm

configuration is ehosen then the formula

1.5 u (q)

defines a function U on Q. We call this funetion the internal energy

of the system. The funetion U and the form f are related by the for­

mula

1.6 dU

9

equivalent to

1.7

If the internal energy is known then form ~ characterizing the sus­

pension system is found from 1.6. The internal energy contains comple­

te information of the behaviour of the system. The internal energy is

a particular example of an object called the potential. The property

of the system leading to the existence of the internal energy is thus

called potentiality. In the language of differential geometry this pro-

perty is referred to as exactness of the form f.An exact form is closed :

1.8 o

or, equivalently,

1.9 o .

ment of reciprocity is that the response of one degree of freedom to

variations in the controI parameter of a second degree of freedom is

1lIillIlllIIIIllI

component fi of the

If the coordinate qi

thenis incremented by

of f .• Formula 1.9 impliesJ

important physical property is called reciprocity. A general state-

tion of this formula. Let the system be displaced in su ch a way as to

increment the coordinate qj by an infinitesimal value E without chan-

ging remaining coordinates. The increment of the

force caused by this displacement isd \f'~ E

d<P" o qJ---1 C; is the corresponding incrementOql

that the two increments are equal. This

For a potential system the work performed in moving the system around

a closed path is zero. Formula 1.9 expresses this property for infi­

nitesimal parallelograms. We give an alternative physical interpreta-

10

the same as the response of the second degree of freedom to variations

in the control parameter of the first.

If Q is simply connected then by Poincare lemma reciprocity im-

plies potentiality. Hence both properties are equivalent.

With no external forces exerted the suspended point will assume

an equilibrium position. I,et (qi) be Cartesian coordinates with the

origin at the equilibrium point. If the suspension is linear, which

is always the case for sufficiently small deflections from equilibrium,

then the comnonents f. of the force are linear functions of the coor-o J

dinates :

1.10 f.J

The linear system is reciprocal if and only if the matrix

metric

k ..lJ

is sym-

1.11 k ..lJ

k ..Jl

The internal energy of a reciprocal linear system is

1.12 U(q)

This function is normalized to vanish at the equilibrium configuration.

Up to now we considered systems whose configuration was controlled.

We may be interested in finding the configuration a system will assume

when a known force is applied to it. In general we may want to find

the response of the system to controlling any combination of position

and force. Equation 1.2 is tOG special to be useful for studying the

behaviour of the system in such situation. We introduce a more symme-

tric description of the elastic properties of the suspension, a des-

cription which does not distinguish any particular method of control.

11

The components f. of the force together with the coordinatesJ

qi of tbe configuration at wbich the force acts can be used as coor-

dinates (qi,fj) of a space F called the phase space of the system.

Points of F represent states of the system. In terms of differential

geometry the space F is the cotangent bundle T*Q of the configuration

space Q. The cotangent bundle T*Q carries acanonical 1-form

1.13

If the form e is evaluated on a virtual change of state Sf from

(qi,f .),J

virtual

to (qi+ tS qi ,f.+ S f.) then the result <Sf, e> is exactly theJ J

work 1.1. We denote by D the submanifold of F composed of sta-

tes with coordinates (qi,f.) satisfying 1.2. The construction of theJ

phase space is the same for alI suspension systems of a point. Each

suspension system is characterized by a submanifold DeF. Points of D

are the states compatible with the elastic properties of the suspen-

sion. In the case of a linear suspension the space F is a vector spa-

ce and D is a linear subspace of F.~ ~

Let the system be moved from a state f to a state f along a path

y . The whole process is compatible with the elastic properties of~

the suspension. This means that the path ~ is contained in D. Pre-

viously we considered processes represented by paths in the configura-~

tion space Q. Since each path Ó in D is completely determined by its

projection t onto Q the two representations of processes are equi­

valent. The work performed in the process previously calculated by in­

tegrating the form ~ along t can now be expressed as the integral

of the canonical 1-form e along y

1.14 A(f) = jet

If the system is potential then this integral is the same for alI paths

12

m «l.) (o)

belonging to D joining states f and f. If a reference state f is cho-

sen e.g. an equilibrium state then the formula

f

1.15 Q(f) = ~8(01f

defines a function U on D. The differential of U is equal to the res-

triction Q/D of the form @ to the submanifold D

1.16 dU G/D.

The above formula means that e/D is exact. The exactness of the form

S/D is thus a criterion for potentiality.

To find the phase space interpretation of reciprocity we define

the canonical 2-form

1.17

in F. The coordinate expression of this form follows from 1.13

The space F together with the 2-form w is an example of what is cal­

led in differential geometry a symplectic manifold([1J,[8J,[57J). Re­

ciprocity means that the form e/D is closed :

o

o.

w

W/D

d (e/D)

This is equivalent to

1.20

1.19

1.18

13

since

1.21 d (e ID) (d 8 )/D c.v ID .

A submarifold D of the symplectic ~8nifold (F,W) satisfying 1.20 is

called an isotropic submanifold. The dimension of D is equal to the

dimension of Q and thus a half of the di~ension of F. If the dimension

of an isotropic submanifold is equal to a half of the dimension of F

then the submanifold is called a lagrangian submanifold (cf. [8] , [55J) .We conclude that an elastic suspension system is reciprocal if the set

D of states compatible with this suspension is a lagrangian submani-

fold of the phase space.

The function U defined on D is related in a simple way to the in-

ternal energy U defined on Q. To each configuration q there corresponds

a unique state f belonging to D. The value of U at q is equal to the00

value of Q at f provided that the reference state f for Q corresponds

to the reference state ~ foi U. The function U is called the proper

vitational field on a spring with spring constant k. The force applied

The submanifold D is the image of this mapping

ded point when the external force instead of the configuration is con-

ll;lllllll;l1

jjilii

Ijiilii

}.D

As an introduction to the analysis of the behaviour of a suspen-

to the system is controlled by plac ing weights on the scale. The weights

dom. The system is a spring balance. A scale is suspended in the gra-

trolled we consider a simple special example with one degree of free-

function of the lagrangian submanifold D. The function U is called the

generating function of D (cf. [45J). The term "generating function" is

justified by the fact that U contains complete information about D.

The differential dU can be considered as a mapping from Q to F = T°j(-Q.

1.22

14

5~o~ed ~t the level of the equilibrium position with no weight.

o o

q

The force and the position are in the relation f = kg reflecting the

elastic characteristic of the spring. If the force is increased from

f to f+ Sf by transfering a weight Sf from the storage to the scale

then the work performed is negative and equal to

1.23 B - g Sf •

If the weight is changed from O to f then the total work is

ff

1.24

H- ~ gdf

1

~ fdf1 f:L==- k =

- 2k.

O

O

This work is spent on the internal energy

1.25

1 k a.,1 a,U

=2 g=2k f

of the spring and on changing the gravitational energy of the weights

by

1.26 - gf - ]. f~ •k

15

Returning to the general example we define in F a 1-form

1.27

The integral

1.28

along a path f contained in D is the work performed in the process

of moving the system along 1 by controlling the force. A mechanism

of such control is ilustrated by the special example of a spring ba­

lance. Since the difference between Q and eH is exact

1.29

where

1.30

(9 -

the wark

the same

B(?) depends only on the end points af the path because

is true for A(t) . The integral

1.31 l!(f)(O)f

defines a function H on D. Obviously

1.32 dH (9H/D •

~If the reference state f is the same as the one used to define the

internal en3rgy then

Q(f) - Y;(f) +

~Usually f is an equilibrium state. In this case

1.33

16

f

~(8 - df)(o)f

lfJ (t)

t (11) •

o and

H u 'fi ID •

Since H is the work performed in the transition from the reference sta-

te it is a form of energy. Formula 1.34 shows that this energy is compo-

sed of the internal energy Q of the suspension system and the term

- ~ which we interprete as the energy stored in the force controlling

mechanism following the example of a spring balance. Assuming that to

each of the controlling forces(fj) there corresponds a unique configu-

ration

1.35i

q

we define a function H by

1.36

Since

1.37

H ef .)J

it follows from 1.32 that

1.38dH~. l

By analogy with thermostatics we calI H the enthalpy of the suspension

system. The function H is a generating function of D since D is com-

17

abstract reciprocity independent of the controI mode. The seeond le-

IIijlllll1llll,llli,lllil!llIllllllllillll

l

1I

llllllllljjlllillll,ljllllljji!lj

o

E is applied to fi then

~i,5 .l

- G) ID(dqj /\ dfj)/D

D

Let the controlling foree be changed by incrementing the compo­

resulting change of the

In above diseussion of the suspension system we used geometrie

concepts whieh belong to two levels. One level is the level of symple-

etic geometry of the phase space F. The sympleetic geometry is repre-

([53J).

sented by 2-form ~ • To this level belong the submanifold D and its

pletely determined by H

of the controI mode. The formula 1.31+ relating the two potentials cor-

is the isotropy of D.

property of being lagrangian. The physieal eharacteristics of the sys-

the other. Hence potentiality is a property of the system independent

energy. The existence of one potential is implied by the existence of

tern which belong to this level are the abstraet potentiality and the

independent of the controI mode. Established in one controI mOde,it

will hold in any other. The geometrie manifestation of this property

nent f, by an infini tesimal amount 6 • Thel 'u, ~X~

d' t l, c -- If' t" teoor "na e q lS c..: a f ,•. :1e J '1~remen ,, J ox,J

the change of qJ is equaJ to c' of,' Sincel

The enthalpy is an example of a potential different from the internal

responding to the two controI modes is called a Legendre transformation

it follows that the response of the system to eontrolling forees is

1.39

reeiproeal. We eonelude that reciproeity is a property of the system

1.1+0

18

vel is the so called level of special symplectic geometries connected

with different specific control modes. To this level belong decompo-

sitions of the phase coordinates into two sets control parameters

and response parameters. With each such decomposition there is asso­

ciated a 1-form on F whose exterior differential is ~ (the complete

definition of a special symplectic structure will be given in Section

6). In the two considered examples these 1-forms are e and QH. The

properties of the system are described on this level by potentials.

In each controI mode the system exhibits reciprocity of the control-

-response relation.

The distinction between these two levels is very useful in dyna­

mical theories. We shall see that the lagrangian and hamiltonian for­

mulation of dynamics (in field theory also the formulation by means

of the energy-momentum tensor) can be treated as different special

symplectic descriptions of the same symplectic objeet.

A similar approach can be used to describe the thermostatics of

1 mole of an ideal gas (cf. ~3]). Let the volume V, the metrical ent­

ropy S, the pressure p and the absolute temperature T be used as coor­

dinates (V,S,P,T) of a manifold F called the phase space. Together

with the 2-form

1.41 w dV A dp + dT A dS

the phase space is a symplectic manifold. The behaviour of the gas

is governed by the two equations of state

1.42pV RT

S

k exp Cv

where R, D and kare constants and

formulae for virtual work in the four modes are :

Reciprocity finds an expression in the Maxwell's identities

energy, the Gibbs function and the enthalpy. These functions are gi-

its thermostatic properties. The submanifold D is usuaIly described

by one of the four thermostatic potentials: internal energy, free

Sexpc

V

R

0'- 1 •

19

k (1-t)'0=1 v

~ 1 ~y:-1 kF P o eXD .Q." c

p

CVT(1 - In T + In k - In R) - RT In V ,

CpT(1 - In T - In R) + cVTln k + RTln p,

f/ =-pdV + TdS ,

8F

=-pdV - SdT ,

eG

=Vdp - SdT

eH

=Vdp + TdS

u(v,S)

F(V,T)

H(S,p)

G(p,T)

It is easy to check that these equations define a Iagrangian submani­

fold D of (F,UJ) • Points of D are the states of the gas allowed by

1.43

ven by the formulae :

where cp = R + cV• These functions correspond to four different cont­

roI modes and are functions of control parameters in each mode. The

20

(~§) V

=(-~~ ) S

- (#) v

=- (~~) T

(~~) p

=- (~~ ) T

(R)

=(~~) SoS p

2. Elastic string

The conceptual framework derived in Section 1 is applied in the

present section to a more complicated example of a static system. The

result is a set of concepts formally identical with those used in Chap-

ter II to describe particIe mechanics.

We consider an elastic string aligned with a coordinate axis.

Each point of the string is labelled with the value of the coordinate

t of its equilibrium position. The configuration of the string stret-

ched .by forces applied in the direction of t will be described by a

function q: R1 ) R1 i.e.

2.1 t q (t) .

The value q(t) of the function denotes the displacement of the point

corresponding to t from its position in the unstretched equilibrium

configuration of the string to the actual position in the described

confi~Jration. External forces applied to the string are described by

a function f

2.2 t , f (t) .

21

are neeessary to maintain the eonfiguration of the string.

work A eorresponding to this displaeement is

JIjj·•·jjll·ljjljjiljljij

I•jijljljj·lljiii

.~il·liijil!lll~lljlil,jl·llilijljlll~

lll,jl

bq(t) = ~t Sq(t). Thus

Sq(t)

[(-:Pet) + fet») ó q(t) - P et) bq(t) l At

6q(t) and its derivative

A

A

A

and replaeing the internal forces by measurable external forces whieh

the parameter smaller than t applies to the remaining portion of the

external forces we introduee the tensio~ u(t) defined as the internql

force whieh the portion0f the string eorresponding to the values of

string. The tension pet) ean be measured by eutting the string at t

2.4-

The foree aeting on an infinitesimal segment of the string eorrespon­

ding to the interval [t,t+L:>tJ is equal to f(t)-ó.t. In addition to

by the value

2.3

represents a virtual displacement of the segment then the virtual

Let us eonsider a segment of the string corresponding to the

interval [t1,t2J.Forces aeting on this segment are: -p(t2) and

P (t1) at the ends and fet) in the interior. If

val [t,t+ L:. t] then the virtual work is

t2

-p(t2) <5q(t2) + p(t1) <5q(t1) + ~ fet) 5q(t)dt

t1If the finite interval [t1,t2J is replaeed by the infinitesimal inter-

2.5

The virtual displacement of the infinitesimal segment is represented

Relations between the forces applied to a segment of the string and

the eonfiguration of the segment depend on the elastie properties of

2.6

22

the string. Specially simple are these relations for an infinitesimal

segment. The equation

- p(t) + fet) o

is the infinitesimal version of the equation

2.8

t2

- p (t2) + p(t1) + ~ f(t)dt

t1

o

expressing the balance of forces applied to the segment. The equation

p (t) - ket) q(t)

is the Hook's lawexpressing the fact that the tension is proportional

to the stretching of the string. Equations 2.7 and 2.9 lead to

2.10 o .

For a finite segment [t1,t2] we obtain thus the system of equations

f (t)

figuration q of the segment. They are thus the analog of equations

since they give the response of the system to the controI of

2.11

These equations give the forces necessary for maintaining a given

tion. In order to verify the potentiality of this response we calcula-

23

te the virtual work 2.4 substituting the values 2.11 of the response

parameters :

ment. Due to equations 2.7 and 2.9 the infinitesimal virtual work 2.5

I!l;!l,

Il!j!l!j!!jl!!!.!jll!l!,;ll,llllljll!ll~lIlIlll

IlllIl]il

o to the given

R1 given by the formula

t2

- ~ ~t [k(t)q(t)] b q(t)dt

t1t2

~ k(t)q(t) & <Ht)dt

t1

A

2.13 U(q)

is equal to

treated as a vector tangent to the space Q([t2,t1]) at q. We may write

t2

~ ket) q(t) & ciCt)dt

t1

2.12

configuration q. A similar analysis applies to an infinitesimal seg-

t2

a ~ k(t)[4(t)]2 dt

t1calculated on the virtual displacement bq. This displacement can be

We consider the space Q([t2,t1]) of alI configurations of the segment.

This space is the space of smooth functions q defined on the interval

[t1,t21.The virtual work 2.12 is the value of the differential of the

2.15

from its unstretched equilibrium configuration q

2.14

It follows that the work is independent of the path between two confi­

gurations and the response of the system is potential. The value U(q)

of the potential is the work necessary to bring the string segment

24

We consider the space Q~ of all configurations of the infinitesimal

segment. As coordinates in Q~ we use q(t) and q(t). The formula 2.15

gives the amount of work performed in the virtual displacement from

the configuration (q(t),q (t») to the configuration (q (t)+ Ó q (t) ,q (t)+

+ bq(t)) • It turns out that this work is the differential of the func­

tion

2.16

evaluated on the virtual displacement (q (t) , li ci et» • The value Ut

u (q)2.17

is the work per unit lenght necessary to bring the infinitesimal

string segment from its unstretched equilibrium configuration (q(t)

= O, q(t) = O) to given configuration (q (t),q(t») • We. see that

t2

~ ut(q(t) ,q(t~ dt

t1

We call U the internal energy of the segment and Ut the internal ener­

gy density. These functions are generating functions of the control-

-response relation. For a finite segment we prove this using formula

t2

- p(t2) b q(t2) + p(t1) b q(t1) + ~ fet) bq(t)dt

t1

2.18 [; u(q)

t2

eS ~ ut(q(t) ,q(t») dt

t1

Leaving aside all mathematical problems connected with the infinite

dimension of the configuration space Q([t2,t1])we can formally write

t2

S~ "t (qCt) ,5.(t») dtt1

25

2.19

The expression

2.20 d- dt

is the sa called Lagrange derivative of the energy density Ut. Compa­

ring 2.18 with 2.19 we obtain equations

2.21

fet)b Ut

&;;- (q(t) ,q(tj)

(lUt

dq\q(t1) ,q(t1))

OUt

_ 2 r;~\q(t2j ,q(t2))

equivalent to 2.11 which proves that the intBrnal energy is indeed the

generating function of the control-response relation. A similar proce-

dure applies to the infinitesimal segment. The virtual work 2.6 isequal to the increment of internal energy. Thus

2.22 (- p (t) + f (t )) b q ( t) - p (t) <5q (t)

This implies equations

- pet) + fet)

2.23

- P (t)

26

equivalent to 2.7 and 2.9.

80 far we treated the string as having its configuration program-

med by an external mechanism providing forces necessary to maintain

a given configuration. The work in changing the configuration was per-

formed by the mechanism and accumulated in the string in the form of

internal energy. Now we consider a segment of the string whose end

points only are attached to a position programming mechanism. The in-

terior of the segment is left free and assumes an equilibrium configu-

ring this displacement is equal to the evaluation of the 1-form

A

controlling mechanism in the virtual displacement

We consider the submanifold D (t.,?,t1) c p (t~ ,tj) which consists of

is equal to

ration compatible with the position of the end points. As control pa­

rameters we take therefore the boundary configurations q(t1) and q(t~ .

They are coordinates in the boundary configuration space Q(t~ ,ti)• The

configuration of the interior of the segment is determined by the boun­

dary configurations. The response of the system on the control of q(t1!

and q(t2) consists of the forces p(t1) and -p(t2) which the position

controlling mechanism has to supply in order to maintain a given con-

figuration. We take these forces together with configurations as coor­

dinates (q(t1) ,q(t2),p(t1},P(t2)) in the space p(t~ ,tj) which we call

the boundary phase space. The virtual work performed by the position

dary values of solutions of equations 2.11 where the force fet) is

equal zero. Points of D(t~ ,~) are boundary states compatible with

elastic properties of the string. The virtual displacement Ci:, q (t1) ,

6 q(t2~ of the boundary position induces the corresponding change of

boundary forces (Sp(t1), bp(t2)) such that the state of the segment

stays on the submanifold D (t",,tj) • The virtual work 2.24 performed du-

2.24

27

2.25

Using equations 2.11 with f = O we may calculate this work in the fol-

lowing way :

t2

~~t[k(tJll(t) 6q(t)] dtt1

2.26

t2

~ k(t)q(t) 6 q(t)~tt1

t2

6 ~ ~ k (t) [q (t)] 2 d tt1

ó Ue q) •

Solving the boundary value problem for equations 2.11 with f = O we

can find the configuration of the entire segment as a function of

0Ct1),q(t2» and thus express the internal energy U as a function of

control parameters. The re suIt is the function W(q(t1) ,q (t2)) equal

to the value of U(q) evaluated on the solution q of 2.11 with the

boundary conditions q(t1) and q(t2). Similarily we can define the fun­

ction VV on D (t~,t.) by setting

2.27

The equation 2.26 shows that

2.28

28

dW

Potentiality of the control-response relation implies abstract reci-

procity which can be expressed by means of the 2-form

2.29

The equation 2.28 implies

2.30 o

1..e. the submanifold D(t~ ,t1) is isotropic. The dimension of DCt,2 ,t1)

is 2 and thus half of the dimensioYl of P(t", , t.) • We conclude that

D(t", , t.) is lagrangian submanifold of the symplectic manifold (p (ta- , t<;

w(t", , t1)) E tl 2?8 1'1 th t 1" t:h T' f t' f D(t", , t1)• _,qua _.OYl.~ meiL.S ..a !!.. J.S .,. e p__oper unc lon o .•

It follows from 2.26 that D(t •. ,t1) is descrihed by the equation

2.31

Thus, similarily as iYl 1.22 we have

2.32

which means that W is generating function for D(t", ,t1) •

Example

Let ket) k be constant. Then

29

3ubstituting this to equation 2.31 we obtain

A similar description applies to the infinitesimal segment [t,

t+ .6t]. The boundary configuration space Q (t~ ,t1) is now replaced by

the infinitesimal configuration space Q~ with coordinates (q(t),q(t» •

The role of the coordinate q(t+6.t) is now played by q(t) + <jJt)·6t.

The role of the tension p(t+~t) is now played by pet) + p(t)".6t. The

response of the system to the controI of q(t) and q(t) consists of the

tension pet) and its derivative pet). We take (q(t) ,q(t) ,p(t),PCt»)

as a coordinate system in the space p~ which we calI the infinitesi­

maI phase space. We consider the submanifold D~ c p~ of states which

are compatible with the elastic properties of the string. It follows

from 2.7 and 2.9 that D~ is a 2-dimensional submanifold described by

equations :

2.33

:P (t)

p (t)

o

- k(t)q(t)

According to 2.24 the virtual work per unit length performed by the

configuration controlling mechanism during the virtual displacement

v = (S q(t) ,6q(t) , bp(t) , bp(t») tangent to D~ is

2.34 Ad

- dt pet) 6 q(t) - Pet) [; q(t) - pet) [;CiCt) •

This work is equal to the evaluation of the form

30

G~ = - p(t)dq(t) - p(t)dq(t)

on the vector v

Ij]~l

Il1

IlJlIIlI!ji]l,l,j~l

lj;

I

a Ut -pet)-,'O q(i)

b Ut(q(t) ,q (t))

A

k(t)q(t) f) q(t)

{(q(t) ,q(t),p(t) ,PCt»)I-p(t)

A

We can also define the function ~t on D~ by setting

2.38

which means that Ut is the proper function of D~. As in the case of

the finite segment the submanifold D~ is a lagrangian submanifold of

the symplectic manifold (p~,G)~) where

2.39

Using equations 2.33 we may calculate this work in the following way

2.36

2.37

The equation 2.35 shows that

2.35

The function Ut is thus a generating function of D~

2.40 -dp(t) 1\ dq(t) - dp(t) 1\ dq(t)

31

3. Elastostatics

We consider in this section an elastic medium under the influen-

ce of fixed external forces. The description of this medium in terms

can be measured by cutting the medium and replacing internal forces

by equivalent external forces.

1

J

!l!l,llllljl~l~jljllljjlljlljlllllll11llIjlli1lijlll~l

1

I

is the force which the

We consider a domain V c M and a piece of the medium which in

an oriented surface element then n~ Pr~"~S

equilibrium fills the domain V. This piece will be deformed by apply-

of symplectic geometry will serve as a model for symplectic formula-

tion of field theory.

Coordinates (x~) will be used in the physical three-dimensional

medium on the negative side of the surface element applies to the posi-

space M. The space M is endowed with a riemannian metric tensor g

whose components are g~v • Covariant derivatives denoted by ~ will

be calculated with respect to the connection r;v ={;v} . The

coordinates (x~) will also label points of the medium in the equili­

brium position with no external forces. The configuration of the me-

dium is the displacement from equilibrium described by a vector field

f whose components are functions ~A(xr). Internal forces are des­

cribed by the stress tensor density Pr ~ (xv) • If n~ "DS represents

tive side. Similarily as a tension of a string in Section 2 stresses

ing a programmed deformation to its boundary av. Deformations of the

boundary form the configuration space Q o V. Elements of Q o V are vec­

tor fields 'P oV = ~, a V tangent to M and defined on the boundary

oV of the domain V. The interior of the piece is under the influence

of a fixed external force field f. For example the medium is placed

in the gravitational field. The medium responds to deformations of the

boundary by forces which must be applied on the boundary by the defor-

mation controlling device in order to maintain a deformation. The sur-

face density of these forces is given by

We describe wi th more rigour an infinitesimal piece 6..V of the medium

liiijiiiIi:&i~:&iiiillliliiljilliI'i

IIIIllIilIlII!IIljllll

I

p/ (x)n~

- ~a,[p,'(X) 1i'P'(X)]dVV

32

applied to the boundary configuration

oV- PI"

6 if av is

lf\ (x)

- ~ Pr~ (x) 6lpl'(x)n~dsav

A

~OV then the virtual work performed is

at!" (x)

A

and PF~(x). Together with the infinitesimal confi.guration they form

a coordinate system (lfl'-(x),lpl'-",(x),P,...?'(x),Gej'-(x)) in the space p~

called the infinitesimal nhase space. Not alI states of pi are allo-~ x

wed by the elastic properties of the medium. The allowed states sa-

As response parameters we choose the coefficients

at a point x eM. The configuration of the piece is described by the

values 'fl'-(x)of the field lf at x and its derivatives at x :

3.2

Infinitesimal configurations form the infinitesimal configuration spa­

ce denoted by Q~. If a virtual change (blf/l'-(x),[)~t<~(x») is applied

to the infini tesimal configuration (lf"'(x), lpf<}\(x»)then according to

3.2 the virtual work performed is

If a virtual change

3.3

33

tisfy the equation

de~ ex)

expressing the balance of forces acting on the infinitesimal element

of the medium. They also satisfy the strain-stress relation

3.7 PI-' ~ (x)

analogous to the Hook's law 2.9. States satisfying the equations 3.6

and 3.7 with given external forces fu(x) form a submanifold Di C pi.r~ x x

A virtual change of configuration induces a corresponding change of

the response parameters SllCh that the state rema in s in Di• The vir­x

tual work A corresponding to this change 'is the value of the 1-form

3.8- [dej'- d lfl-' + Pj<? d'f'"' ~ J. b..V

evaluated on the vector tangent to Di describing this change. Thex

potentiality of the control-response relation means that the form

Di is exact i.e. thatx

3.9 dU

This happens when the matrix k is symmetric

3.10

The function Q is the internal energy of the infinitesimal volume of

the medium. The corresponding generating function jpotentialj U on

Qi is defined byx

34

Substituting 3.6 and 3.7 into 3.11 we obtain

dU ( <.pl", <-pf<-? )

3.12

3.11

3.13

The function

is the internal energy density. The submanifold D~ is completely des-

cribed by the function Ux

3.14 "-Pl'-

The space pi together with the formx

3.15

is a symplectic manifold and Di is a lagrangian submanifold.x

The covariant components

3.16

of the derivative of ~ can be decomposed into the symmetric and anti­

symmetric parts :

3.17

The antisymmetric part describes the rotation of the infinitesimal

element ~V. The symmetric part is related to the Lie derivative of ~:

sjj!IIj;~il

l~;l

Illl~~l~)l~llllllllillljl,lljllj

ijljlll~~

p (v?)

35

2 tf(j<-Y)

P )JA

3.18

The geometrie concepts developed in earlier sections are applied

and describe the change of the shape and the volume of the element.

Usually the internal energy depends only on the symmetric part of

~j<-v • This means that the tensor

3.21

satisfies

in this section to a true field - the electrostatic field.

3.20

It follows that the stres s tensor density is symmetric

In the three-dimensional physical space M with coordinates (xj<-)

we consider a dieletric medium charged with a fixed charge density

g(x). The configuration of the electrostatic field is represented

by the electrostatic potential ~(x). The electrostatic field E? is

defined by

4. Electrostatics

4.1 - O? rp (x)

In addition to the "external charges" represented by ~ there are

36

"internal charges" represented by the electrostatic induction field

p~(x) which is a vector density. If a piece of the dielectric occupy-

ing a domain V is singled out and the rest of the medium is disregar-

ded then the influence of the surroundings on the singled out piece

has to be replaced by a surface charge -n~p~(x)' 6s on each surface

element n ~ ./.'::..s of the boundary a V of V. We will control the bounda­

ry value \.f oV = CP I () V of the potential. The space of boundary va­

lues will be the boundary configuration space Q oV. The response of

the field is the surface charge density _poV = n~p~ I o V which the

potential programming mechanism has to supply. If a virtual change

6roV of the boundary configuration is made then the virtual work is

4.2 A

} p;\(x) [)~ex)n?dsoV

~ d~[ p?(x) Sipex) JdV .V

Passing to an infini tesimal piece ~ V of the medium at a point x E M

we obtain for the virtual work the expression

4.3

The infinitesimal configuration is represented by <.pex) and If,,(x) =

= 0;\ t.p (x) = -E;\(x) and the response is described by de ex) = 2l~p"'(x)

2nd p~(x). The infini.tesimal configuration space is denoted by Q~ and

the infjn:itesimal Dhase space by P~. States compatible with the die­

lectric properties of the medium form a subspace D~ of P~ described

by the field equations. These equations are

4.4 de (x) - 4'Jr· es ex)

•.Ij11

iIIilI,IllIllI,l,l~j,jlj,jj.i

jj

1

)j,jjjjjjj!j,j,jlllj~ljljj·•llljljjl~!l·~

E~I'(x) is a sym-

4;][ ~ (x)

dU

-[dd€ /\d<p - dp:>'/\dE" J. t:>.V

e~ I D~

E'Y'(x)

37

ge. The space pi with the 2-form~ - x

metric tensor density

evaluated on the vector tangent to Di representing the virtual chan­x

and the relation

then the response is reciprocal and

imply the Poisson equation

is a symplectic manifold. If the dielectric constant

stating that the total charge of the infinitesimal element is zero

LJ-.8

A virtual change of infinitesimal configuration induces a correspoding

change of response so that the state of the field remains on D~. The

virtual work A corresponding to this change is the value of the 1-form

where U is the internal energy of the infinitesimal element of the

reflecting the dielectric properties of the medium. These equations

4.10

4.6

38

electrostatic field. The function U is the proper function of Di• Thex

corresponding generating function U is defined on Q~ by

4-.11

Substituting 1:-.4- and 4-.5 into 4-.11 we obtain

4-.12

The function

4-.13 12

is the internal energy density. Complete information on D~ is contained

in the function DX :

1+.14- -de=dUXClE" }.

The space D~ is a lagranr;ian submanifold of the symplectic manifold

(pi W i)x' x •

Let X be a vector field on M whose components are ~(x). Dragging

the electrostatic field along X produces a virtual displacement of its

state. If the electrostatic field is treated as a continuous medium we

can calculate the corresponding virtual work by differentiating the

internal energy. We apply the displacement E-XI"- to the electrostatic

field and the domain AV it occupies but not to the dielectric medium

represented by f,"r and not to the charge density :) • This means that

the boundary potential programming device is displaced with respect to

the charged dielectric. The change of the internal energy due to the

displacement of the domain 6V is equal to the Lie derivative /multi-

plied by 6 / of the i.nternal enecogy treated 8S [) function on M. Since

39

t~e internal energy Ux is q scalar density this change is equaJ. to

f·lu ·.6.VX x

4.15

The change of the electrostatic field due to displacement is equal to

minus the Lie derivative of the field times E • Hence the change of

the intermd. energy due to the change of electrostatic field is equal

to

4.16

The total change of the interna l energy of the piece of the electro-

static field due to the displacement e· xl'- is equal to

4.17

where S~ is the Kronecker's symbol. Due to the potentiality of the

system this change of the internal energy is equal to the virtual wark

performed in the displacement. We compare the result with formula 3.4

of the proceeding section. In this formula we replace S tpl'- and 6<p1"'').

by infini tesimal displacements e Xl"' and c·Xf'..~ and also change ~I'­

and Pl'-? to Tj'-and T?j'-to avoid confusion of notation :

4.18 A

40

The result of this comparison is

4.19

4.20

v/e calI Tj< the force and T\\;<the stress tensor of the electrostatic

field. This quantities measure the real forces which the potential

programming device applies to the piece of the electrostatic field.

We refer to this construction in Section 25 where we define stress

tensors for a general field.

II. Nonrelativistic particIe dynamics

5. Preliminaries

The fundament al geometric space in our considerations is the ma-

nifold Q of alI possible configurations of a particIe system at alI

times. The time manifold M is identified with R1• This identification

corresponds to the choice of a standard clock. The manifold Q is a fib-

re bundle over the time manifold M. The bundle projection Q )M is

denoted by ~ Icf. Appendix Al. Each fibre Qt = 5 -\t) is the con-

figuration space of the system at time t. No standard trivialization

of Q is assumed which means that the rest of the particIe system is

not def:in'Od.This point of v~ew is similar to that of Galilean relati-

vi ty. V/e calI Q the configuration bundle. ·Fibres of Q are assumed to

be simply connected.

We now introduce the notion of the phase bundle of the system.

The phase bundle P is also a fibre bundle over the time manifold. The

bundle projection P---+ IVI is denoted. by ~ . Each fibre Pt = VI, -1C t),

called the phase space at time t, is the cotangent bundle of the con-

figuration space

The cotangent bundle projection Pt----7 Qt is denoted by nt' The bun­

dle structure of P over M can be characterized as that of the adjoint

bundle V*Q of the bundle VQ c TQ of vertical vectors tangent to Q.

tion

5.2

The family of projections ~t defines in an obvious way a fibra­

7C : P --I' Q. The diagram

~p ) Q

~~M

42

is commutative.

The evaluation

< 1!;* u,p ><u, f9 >

where 8(p) is the value of 8 at p and 'll*P denotes the pull-baek of

the coveetor p €i T*Q to T*E lef. Appendix ci.The differential form

6.3

Histories of the particIe system aredescribed by differentiable

is equiv81ent to

In the general ease the eotangent bundle !:. = T*Q of a manifold Q

where Jt::: T*Q --7 Q is the eotangent bundle pro,jeetion and ~*1J is

the tangent mapping from TE to TQ Icf. Appendix B/. The definition 6.1

6.2

is equipped with a eanonical differential 1-form e

metry.

shall ref er to these laws as the dynamics of the system. In subsequent

6.1

sections we give formulatioDs of dynamics in terms of symplectie geo-

sections of the phase bundle P over time. Not alI sections are compa-

tible with the physical laws governing the motion of the system. We

6. Special sympleetie structures. Generating funetions

(u, f9 > of 8 on a veetor u tangent to !:. at a point p €i!:. is given by

I is called the eanonieal 2-form on P. It is a standard re suIt that the

manifold E together with the 2-form w define a sympleetie manifold

43

In the case considered in Section 5, we have a family of canoni­

cal 1-forms @t and canonical 2-forms 0t defined on each phase space

Pt = T~Qt separately.

If a coordinate system (qj), j=1, ••• ,n, is chosen in g then at

8very point of g the differentials dqj form a linear basis for co-

vectors at this point. The components p~ of a covector p with respectu

T.O thi_s basis together ,vith the coordinates qj of the point 2h (p) € g

define coordinates (qj,p.) in the space T*,Q. The local expression forJ

e in this coordinate s;j'stemis

(9

~ne Einst8in summation convention will always be used. The loeal ex-

pression for W is consequently

6.5 w dp. /\ '1qjJ

In the case of the configuration bundle Q we shall use coordina­

te systems (t, qj) whieh are compatihle with the fibration S ' i.e•

.5 (t,qj) = t. The construction introduced above les.ds to the coordi­

nate system (t,qj'Pj) in the phase bundle P, compatible with both fib­

rations ~ and X

t6.6

The fundament al geometrie eoncepts used to deseribe dynamics will

be that of a lagrangian submanifold of a symplectic manifold /see[55] ,

[57J/ .

Definition : A lagrangian submanifold of a symplectic manifold

that there is a function S on N such that

Owj N

wl N = O and dim N = ]. dim P.2 -~ vanishes when evaluated on

GIN

dS

dS

elN of the canonical 1-form El to N is closed since

case the proper function S definies a function S on C which is simply

submanifold N which is simply the image af the section

a projection of § onto C. The function S determines completely the

called isotropie. A simple algebraic argument shows that the dimen-

bivectors tangent to N. A submanifold satisfying this condition is

tion

In a caardinate system (qj,Pj) the abave statement means that the sub­

manifold N is described by equations

44

sion of an isotropie submanifold N of a symplectic manifold ~,

is not higher than ~ dim~. A lagrangian submanifold is thus an iso­

tropie submanifold of maximal dimension.

6.8

This function is called a proper function of N.

Suppose that N is a section of a bundle T*g over C c g. In this

theory will be shown to be generating functions of lacrangian subma­

nifo1ds. Let N be a lagrangian submanifold of (Tj{.g,CJ). The restric-

(;f, w) is a submanifold N c P such that

The condition wiN = O means that

We will always aim at deseribing lagrangian submanifolds in termsQ., 5

of generating functions. Such important objectsAthe Hamiltonian, the

Lagrangian, the action, and also the energy-momentum tensor in field

6.7

We consider only simply connected 1agrangian submanifolds. It fo110ws

45

6.9

'I'hefunetion S is ealled a generating function of N.

Generating functions can be also used in more eomplieated eases,

,,!henN is no longer a section of T*g.

Let C c g be a submanifold of g and let S be a function on C. It

can be easily shown that the set

6.10N

{ P E T*g I 2l:: (p) E C ; <u,p) = <u,dS> for

each veetor u tangent to C at 1rCP)}

is a lagrangian submanifold of (T*g,G)) • The function S is ealled a

generating function of N. Also in this case the generating function

S is a projection onto C of auroper function S given by 6.7. Lagran-

gian submanifolds which can be g8nerated in this way can be 100seJy

charaeteri7,ecJ as those whose proper f.unetions are urojeetible onto

submanifolds of g /cf.[45J/.

Generating functions as well as proper functions are determined

up to an additive eonstant.

If the submanifold C is given by equations

6.11 o 1, ••• ,k ,

in a eoordinate system (qj) and if~ is any continuation of S to g

then the submanifold 6.10 is given by equations /ef. [6]/

Go«(qj)=

O

6.12 aSoGo(= Oqj

+/\", -jp.

Oq

J

p

restricted to a lagrangian submanifold is a differential

o(

and a symplectomorphism

6.13

such that

p

be called special symplectic manifolds. More precisely a special sym­

plectic structure in a symplectic manifold (E,GJ) is a fibration

46

Only the symplectic structure of a manifold is used to define

bundles but are isomorphic to cotangent bundles. Such manifolds will

lagrangian submanifolds. To define proper functions and generating

tangent bundle. In applications of symplectic geometry to dynamics

one encounters symplectic manifolds which are not directly cotangent

functions we needed much more structure namely the structure of a co-

of a function called again a proper function of a manifold. A genera-

The presence of a special symplectic structure in a symplectic

manifold (E,W) makes it possible to describe some lagrangian subma­

nifolds by generating functions defined on submanifolds of g. Genera-

ting functions are constructed as in cotangent bundles. The 1-form

where '!!:: T*g~ g is the cotangent bundle projection. Since O( is

a symplectomorphism the pull-back o(ifw of the canonical 2-form 6J

from T*g to E is equal to the symplectic form w.

ting function is the projection of a proper function to g

Thus we see that the objects of a special symplectic structure

used to def in e generating functions are the projection ot onto a ma-

47

nifold g and the 1-form e sueh that d e = w

One sympleetie manifold may be equipped\ with several speeialsym-

pleetie struetures in whieh ease one lagrangian submanifold may be e;e-

nerated by several different e;enerating funetions. We will find that

the IJ2_grangian and th8 RamiI toniem lin field theQr~T also energy-momen-

tum tensorl are generating functions of the same lagrangian submani-

fold ,,6th respect to different special sympleetie struetures lef. [51]/.

7. Finite time inteTval formulatio~ of dyn2mi~s

As i.2 us';>"li~' ean0ni cs.l formulati0'ls af uarticle dynamics we

a~sume thq existence af a differentiahle two-parameter family of dif-

feomorphisms

7.2

Dynamies is expressed in terms of this family as follows. A section

() : II[ ~ P is dynamie".11 y admissible if and only if

for each (t1,t2). It is assumed that mappings Ret t) are symplecto--- 2' 1morphisms :

7.4

This farmula.tion of dyn9.mics is equivalent to stating a system of first

graph R (t t) c Pt X Pt2' 1 '2 '1

{ (3) f<1) Ct3,t1) I (.1)p,P E P there is a p 6 Pt2

.. et. l' t .) }. (","1'1)(e..) 1+. ] .

such that (p,p) E D -- -, l {, N-1

. (.~ w xthere lS a sequence p, ••• ,p) E PtN-1

7.7

me way we introduce multiple compositions of relations

48

corresponding to divisions of the time interval (t1,tN) into N-1 sub­

intervals. In terms of this definition we have the composition law

Thjs classic formulation of particIe dynamics can be described

position law 7.2 reads

7.6

(t2,t1)The manifold P will be called the boundary phase space corres-

ponding to the time interval [t1,t2] c M. In terms of graphs the com-

in terms suitable for ceneralizations to field theory. We define a

two-parameter family of submanifolds

7.5

order differential eql.18tions. Solutions of the equations are dynami-

cally admissible histories. The family RCt2,t1) is the resolvent of

the sytem of equations. We return to this point in the next section.

•iiiIiiiilI­•iIII!i!!I­•••III-•

49

correspondin[!;to

7.9

(t2,t1)The property 7.4 of the resolvent is equivalent to D bein[!;

a lagrangian submanifold when an appropriate symplectic structure in(t2,t1)

p is chosen. This symplectic structure is given by the 2-form

7.10 [VI

The "minus-sign"jV[

is defined by

7.11

jV[

to Pt and v2, w2 are vectors tangent1is a lagrangian submanifold of mani­(t2,t1)

D -- is the image of the mapping

7.12

and

7.13

A coordinate system (t,qj'Pj) in P gives rise to a coordinate

((•.li (1) (,) j (.)'\. (t2, t1) (-) ..system qu,p.,q' ,p.) ln P for each t2,t1 • In thlS coordlnateJ J

system

7.14

iS

k

o

o

dP' J

o"'kq

O'qoj

",«)kQq~

d'8.~ oJ

o

a'U d~j O(;.lOqjp.p.

__ ,_J

-J

o")oqk o'qk0(1)D·

p.• ll

50

IZ)

CIp·

( .l+ op.l

t~) a (~)j0(»o(~joPoi _q_

p.-~

()",)io'qk Oqko~iq

(~)

óqj O(~lo(~jop·p .

__ J-.J

Cl "l

om ÓPk7p.

Pk p.ll

and

since

(l)j(llojC"j S.)q

=q q,~ J

7.15(l](l) ej ~}.)p.

=Pl q L,JJ

'"

then

(t2,t1)If the mapping Ret t) lor equivalently the submanifold D I2' 1

is described locally by

There is a natural special symplectic

(t2,t1))- ...GJ - deflned by the proJectlon

51

due to 7.15 being acanonical transforrnation /see [6J /.

(t2,t1)We will refer to the lagrangian subrnanifold D as the dyna-

rnics corresponding to the tirne interval [t1,t2] c M.et ,t )

structure in (p 2 1 ,

7.17

qnd the diffpomorphism

7.18

wherp

(t/,t..,) * (t2,t1)p - ~T Q

7.19 ( (~) ('»)p,-p

(t2,t1) _The manifold Q will be cfllled the boundary configuration spa-

ce corresponding to the interval [t1,t2].(t2, t1)

The 2-form 6J is the exterior differential of the 1-form

7.20

defined by

M

7.21

where v1 is a vector tangent to Pt and v2 is a vector tangent to Pt •1 (t2,t1) (t q2

It is easy to see that the form fi is the pull-back by o( ot, 1

of the canonical 1-form in T*Q(t2,t1) • In the coordinate system C~j,

(~) (!lj «) )p.,~ ,p. we haveJ J

(., dWJ'p. q'J

l;t)

O

/t2,t1)p.

=J (~j

oq

,.)

a

/t2,t1)p.

=---J o'qj

'al d,aljp. qJ

The composition law 7.8 for lagrangian submanifolds is reflec-

7.24

equivalent to the familiar formulae

Theorem 1

ted in composition law for their generating functions. We state this

( (t2,t1) (t2,t1)law in the simplest case of no constraints C = Q for

(tN,t1)Let a function W be defined by

52

7.25

7.23

7.22

It will be assumed in the sequel that for each interval [t1,t2](t2, t1)

the lagrangian submanifold D is generated with respect to the(t2,t1)

above special symplectic structure by a function W defined on(t2,t1) (t2,t1)

a submanifold C C Q called the constraint submanifold.

U . d . ('.>.lj ul (., j (., ) d . th thslng coor lnates ,q ,p.,q ,p. an assumlng at ere are no con-

( (t2,t1) (t~,t1))J (t2,t1)straints C = Q we find that the submanifold D

is described by the equation

is a stationary point of

(tN,t1) , ,W lS a generatlng

53

( (/II) (11 ( (11·1) ('3) UJ)where for each q,q) the sequence q ,••• ,q,q E Qt XN-1the right-hand side. The function

(tN, t1)function of D •

We give the proof of this theorem in the case of N

Proof

3.

7.26

where ~ is the statiQnary point of the right-hand side. For given

( (3) (1») (t3 't1) . f') (1) (t3 't1), , , ,q,g E Q denote by (p,p) E D unlgue pOlnt WhlCh proJects

, ",..ct3,t1) t (13)(1)) T k th' , t et) P h th tVla J~ on o g,g. a e e unlque pOln p E t suc aet t) (t t) 2

( f~) (11) 2' 1 / ' ((J) (a») 3' 2 /p,p E D or eql11valently p,p E D -. For the sake of

, l' , d' t d 't' (3)slmp lClty we use coor lna ,e .escrlp lons : p

( (1}j m )g ,p,,1 •

( (3)j (3») (1)q 'Pj' p

It follows from 7.24 and 7.6 that the equation

7.27 o

has the unique solution g (a) (fa» ,q = Je t P for WhlCh2

7.28a ra)p,

J

f«) ( fa)j f«l ) ,(a) ,( f~ (1»,::herep = q' 'Pj . Treatln['; g as a functlon of 'l,g we have

7.29 +

(3)p,

J

+ ~ . ,fklV km' Sln Viii' t

-A ~km' sin {f .t + B cos ~ .t

A cos {}f. t

... (t3,t1)lS a generatlng functlon for D •

pC t)

g(t)

7.32

namically admissible sections

(t2,t1)The manifold D is described by eguations

(>..)

g

7.33

p = -kg

7.31

fJ.)

P

Example 1

Integrating these eguations we obtain the general expression for dy-

54

Under specj.al condi tions which are not stated here the same com-

The configuration bundle of the harmonic oscillator is the tri­

vial bundle Q = M x R1 and the phase bundle P can be identified wi th

?r,,!xR-. In terms of coordinates (t,g,p) the eguations of motion are

position law holds in the presence of constraints. In this case the

sequence (~, ~q~... ,~) must be compatible with the constraints so that..... (tN, t1) (tN,t1)

the rlght-hand slde of 7.25 lS defJ.ned. The constralnt C c Q .

. h .co' ((~) (1)) ••• ( (w·f) (3) (>.))lS t.e set OL palrs g,g for WhlCh statlonary pOlnts g ,••• ,g,g

exist /see[45J/.

7.30

Similarly

55

(t ,t )In order to find the proper function W 2 1

. (r'J 1<»)by coordlnates q,p

7.34-

Hence

1 (1 (,)(1) \r;::-! C" c,') ,fk12 {km p p - vkm q q sin ~m(t2-t1}·7.35

{k1 (t t) ('l C'l • 2. R(t t).cosv m 2- 1 - q p Sln V ~ 2- 1

(t2,t1) (t2,t1)To obtain the generating function W we project W to

(t2,t1)Q • We consider three cases

7.36c.)p

(1)P

'alp

")o

2n~ then equations 7.33 imply

'»q

56

The derivative with respect to ~ is

7.1+0

lustrate the composition law for generating functions we take (t1,t2,

t3) such thot Sin[Ct3-t2).J O and sin{~'ct2-t1).JO. Then

(iii) If {~\t2-t1) =(2n + 1)T

then

'a)

(.) Ul(.)7.39

q=-q ,p=-p

(t2,t1)Hence C is the diagonal of Qt x Qt and it follows from 7.34

. (t2,t1) 2 1that W - = O.

7.38

7.37

57

7.iL2(3.)q

Substitutinc; 7.L~2 into 7.40 we obtain the value

which arcrees with 7.37.

If sina\t3-t1) = O which means that ITCt3-t1)

tionary points exist if and only if

n'jf then sta-

7.43 O

Using the equality

7.44 n'Jt'

the equation 7.43 reads

('3) ()n (.») . J"k\( '\q - -1 q sln~m t2-t1/i.e.

O

which agrees with 7.38 and 7.39. The substitution of the above equa-

lities into 7.40 gives the value o.

58

number of subintervals increasing to infinity and their lengths de-

of P. The term "jet" is taken from modern geometry but it denotes a

p

J~(Q) of 1-jets of sections of the

----;<o> p(t) li: P-r

Definition :.The space Q~

Intnitively the ,jet j1p(t) of a section

There are the canonical jet-target projections

Dividing the time interval [t1,t2] into subintervals we obtained(t ,t )

d . D 2/1 't' f d' d' tynamlcs as a eomposl lon o ynamles eorresponlng o

8. Infinitesimal description of dynamics

creasing to zero. This limit is expressed in terms of jets of sections

the

the subintervals. This procedure can be earried to the limit of the

a!ld

is a limit of the pair (p (t+h) ,pet») E p(t+h,t) when h --> O. In this

sense p; is the limit of pCt+h,t) when h --T O. The same interpretation. i

applJ.es to Q';:;.

finitesimal configuration bundle and the infinitesimal phase bundle,

respectively.

bundle Q at t is called the infinitesimal configuration space at t.

The space P~ = J~(p) is called the infinitesimal phase space at t.The corresponding bundles Qi = J1(Q) and pi = J1(p) are called the in-

geometrie object known already in traditional differential geometry.

Objeets of this type are described for example by Schouten in [43]Isee also Appendix D at the end or our notes/.

8.1

8.2

59

3.3

~here is also the jet prolongation

3.4-1

----+) ,J Q

of X: P---+ Q. We have thus the commutati.ve diagram

lp p

Q

Xi to a fibre p~ will be denotedThe restriction of

We show that e8ch i.nfinitesimal

Ot i . -pi oit ." t ----t'"t •phase space p~ is canonically

diffeomorphic to the cotangent burdle T*Q~. For this purpose we use

the caronical identification

8.6

The s3'mbol VQ denotcs the cuncH", af vertical '.'ectorstanzent to the

'wndle Q---IVJ /Gf. Appendix E/. 3imiL1.rly VeT1(Q) denotes the bundle

of vertical vectors tangent to the bundle J\Q).--t- M. The identifica­

tion 8.6 me ans that each vector tangent to Q~ = J~(Q) may be represen­

ted by the jet of a section

8.7 E TQ-r

for ~ = t. Now let g = j1p(t) e p~ be the jet of the section 8.1. Let

u e TQ~ be a vector attached at the point Xi(g) and represented by

60

the jet j1X(t) E J~(VQ) of the section 8.7. This section is chosen

in such a way that for each 1::E M the vector X ('l;') is attached at

the point X (p(-c) • The evaluation

8.8

defines a covector <. ,gl E T*Q~ associated with g. Using local coor­

dinates we show later that the mapping

8.9

io( t of the ca-

P~ will be de-

is a diffeomorphism. Due to the existence of this diffeomorphism the

infinitesimal phase space P~ is a symplectic manifold with a canoni-

cal special symplectic structure. The pull-backs by

nonical 1-form and the canonical 2-form from T*Q~ to

noted by Ci ~ and GJ~ respectively.

Let v, w be vectors tangent to P~ and attached at the same point.

Using the identification

8.10

similar to 8.6 we represent v and w by jets of sections

8.11

M ::> t -------../ Y (t)

-------->~ Z (1::)

ETP'C

TP'C

We chose these sections in such a way that for each 1::vectors Y (t:) ,

Z ('t) are attached at the same point. It follows from the construction

of o(~ that e~ and w~ satisfy the following equalities

61

8.12

and

<v, @~:>

~~3 ~«Y(t+h) ,Y(t»), @Ct+h,t) >

8.13

Comparing the above formulae with 7.11 and 7.21 we see that in a

tain intuitive sense the forms e~and w~ are limits of ~ (6t+h

and ~ (GJt+h ~ Wt) when h -'>-0. This interpretation justifies

following notation

8.14-

cer-

the

Let (t,qj) be a coordinate system in Q compatible with the bun­

dle str~cture and let (t,qj'Pj) be the corresponding coordinate sys­

tem in P. These systems lead to coordinate systems (t,qj,qj) and

(t,qj,po,qj,po) in Qi and pi respectively. The coordinates qj of theJ J

jet j1q(t) of the section

62

8.15

are calculated from

8.16 •jq

when qj(~) are coordinates of the point q(r) • Since M is identified

with R1 the jet j1q(t) can be interpreted as the vector tangent to

the curve

8.17

The coordinate expression for this vector is

8.18 2>

ot + qj ~oqj

Coordinates Pj are defined in a similar way. Coordinate expressions

for mappings GQ, Gp and xi are

(j •.1L.Q t,q ,q~)

8.19

For a fixed tcM systems (qj,qj) ( j •j • )are coordinate sys-and q ,p.,q ,p. J Jtems in Q~ and p~. The coordinate expressions of forms

et and 6J~ are

et

Md Md .p.dqj + p.dqj

= dt 8t=dt(pjdqJ)=J J8.20 I"ld .

11d .wi =dt Wt=(ff(dPj/\dqJ)=dp.Adqj + dp.Adqjt

J J

63

These expressions follow easily from formulae 8.12 and 8.13.

In Section 6 we described the canonical construction of coordi-

nates in the cotangent bundle from eoordinates of a manifold. Using

this procedure we introduce coordinates (qj,gj,r",s.) of T*Qticorres­J ,]

ponding to coordinates (qj,qj) of Qi. The coordinate expression for

o<~ is thus

8.21 ( j 'j )q ,q ,rj,sj

where r. = p. and s. = p .•J J J J

It is elear from 8.21 that D(~ is a diffeomorphism.

Definition: We say that the /infinitesimal/ dynamics of the sys­

tem is specified if a lagrangian submanifold D~CP~ is chosen at each

time t E M.

If the infinitesimal dynamics is specified then a section

o: M ---p is considered to be dynamieally admissible if and only

if j10" ( t) E D~ for each t E M. Conversely if .the class of dynamically

admissible sections is known then infinitesimal dynamics D~ can be

obtained as the set of alI jets of these sections at t.

We recall that in terms of finite time interval dynamics a sec­

tion O: H-P is dynamicaIIy admissibIe if and only if (t(t2),(t2,t1)

d (t1))E D for each t1,t2 E M. Conversely if the dynamicaIIy

d o obI o (t2,t1) b o d ha mlSSl e sectlons are known then D can e obtalne as t e

set of pairs (o(t2), o(t1)) E p(t2,t1} where Ó is admissibIe section.

We conclude that the finite time interval dynamics can be obtained

from infinitesimaI dynamics:

8.22

64

The above formula can be considered as the limit of formula 7.7 when

the number of subintervals tends to infinity and their lengths to zero.

Conversely the infinitesimal dynamics can be obtained from finite

interval dynamics :

8.23there is a section ({:M.--->- P such that Lg = j1Q'(t) and (6 (t+h), OCt))ED(t+h,t)\.for each h ~

Originally we postulated the existence of a resolvent Ret t) and2' 1

defined the finite interval dynamics in terms of resolvent. The fol-

lowing theorem guaranties the correctness of the above construction

of the infinitesimal dynamics:

Theorem 2

If the resolvent satisfies alI conditions stated in Section 7,

then Dt defined by 8.23 is a lagrangian submanifold of pt.

Proof: Through each point p €. Pt there is a unique dynamically

admissible section

R (z t)(P), ,

This implies that g (P) = j1 O ( t) is the unique element of D~ such

• Since

Using the identification

section 1:: ----* Y (1: )

that Gp(g(P) = p. It follows from the properties of the resolvent

that the mapping p -- g(P) is smooth. The manifold D~ is the image

of this mapping and therefore is smooth. The dimension of D~ is equal

~ dim P~. It remains to prove thatto that of Pt and hence equal to

i I iWt Dt = O. i iLet vETPt be a vector tangent to Dt'

8.10 we represent v by the jet j1Y(t) of a

v is tangent to D~ the section Y may be chosen in such a way that

(Y(~) ,Y(t)) is tangent to D(~ ,t) for each 1:: • Let w be another vec-

65

t:)rtangent to D~ and 1: -----+ Z (t:) the corresponding representant. It

follows from 8.13 that

:)ecauseDCt+h,t) is lagrangian. This completes the proof.

If the dynamics is introduced in terms of a family {D~} then

certain additional conditions have to be satisfied in order that the

formula 8.22 defines a lagrangian submanifold.

We assume that for each teM the infinitesimal dynamics D~ has

a generating function Lt: Q~~ R1• The corresponding proper function

of D~ is denoted by ~t' Both Lt and ~t are defined up to an additive

constant. The family {Lt3 defines a function L: Qi---+R1 which is

called the Lagrangian. The Lagrangian is defined up to an arbitrary

additive function depending only on t. Using coordinates (qj,p.,qj,p.)J J

and the formula 8.20 we find that the generating formula for D~ reads

8.24

equivalent to the familiar formulae

p.=CI

8.25

J

Oqj L(t,qj,qj)

p.

=d .

Joqj L(t,qJ,qj)

It is interesting to note that also the composition law 7.25

for generating functions has its infinitesimal formulation which we

give in the case of no constraints :

Theorem 3Ct2,t1)

Let a function W be defined by

8.26

((;l) U)where for each q,q

66

t,-2 1~ L1;' (j q ('(;'))t1

the section

, q (t) E Q 'C'

is a stationary section lin the sense of the calculus of varia-

tionsl of the right-hand side, such that q(t1) = q , q(t2) = q.(t2,t1) (t2,t1)

The function W is a generating function of D '

Equations 8.25 are obviously equivalent to Euler - Lagrange equ-

ations. This implies that stationary sections are precisely projec-

tions by X Ol dynamically admissible sections.

Due to lack of constraints the Theorem 3 can be stated in terms

of proper functions

Theorem 3'

(t2,t1)Let a function W be defined by

•.:", (1)

where for each (p ,p) E is the

unique dynamically admissible section such that p(t1) = P

't \ r;.) Th f t' .T (t2,t1) . f t' fP( 2) = p. e unc lon ~ lS a proper unc lon o

n (1) W) (t2,t1)Prool: Let u,u E TP be a

(W. M ) .and let A ~ pC?:,),p(?v) be a curve In

(t2,t1)vector tangent to D

(t2 ' t1 ) c;.) (1) •

D· such that (u ,u) lS

its tangent vector at A= O. There is a unique mapping (t ,?c)-7' P(7:' ,?v)

such that for each A., the mapping

P;t. ('l:') p(-;:; ,A.,) E P'C'

67

is a dynamieally admissible seetion and p:\-(t1) = (pO,,), P;>..(t2)Then

(~)P (?-.).

«(al (") (t2,t1»d ,(t2,t1)C,ll f1) ) I

u , u ,d~= d Iv !!. p (~),p (A.-)

iL =0t2t2

~I\, ~ ~'t'(j1p:>-('C')d~ I

=j ~?v ~'C'(j1p:l-('C') I ?\,=Od1:/~=Ot1

t1

!2

t2

~ <V('C'),~'C>d~

=~<v('C'), G~>d'C'

t1

t1

where v ('C')

is the veetor tangent to the eurveA,~j1p:\-(v) E D~ at

?\." = O. The last eguali ty follows from I!1: being a proper funetion

of D~ • Using the identifieation 8.10 we replaee the veetor v('C') by

the jet j1yC'C'")oi the section '7:;'----... YCd E TP't; where Y('C') is the

vector tangent to the eurve A, -P,1, ('C') E P'(;" at A = O. Relations(.1) (.1.) •••

Y(t1) = u , Y(t2) = u are obvlously satlsfled. Now we use the for-mula 8.12

t2

\<ve(;'), 6J~> dl;t1

t2

~ <j1Y(t;) , 8~> d'7:;'t1

This eompletes the proof.

Example 2

Eguations of motion of the harmonie oscillator eonsidered in

Example 1

p = mg

p = -kg

68

define a two-dimensional submanifold D~ of the four-dimensional infi­

nitesimal phase space P~. We restrict the form 8~ = pdq + pdq to D~

and obtain

8.28

It follows that D~ is a lagrangian submanifold generated by the La-

grangian

8.29 L(t,q,q) 1(.2 l 2)2 mq - Kq

8.30

The stationary section of the functional

12

~ L'1;(j1qC't\) d0't1

satisfying the boundary condition q(t1)

from formula 7.32 by

«)

q , q(t2)(J..) ••

q lS glven

8.31

Substituting this expression into 8.29 we obtain

8.32

Integrating this over 't' E [t1,t2] we obtain

8.33

69

which is the same as formula 7.37. This confirms Theorem 3.

9. Hamiltonian description of dynamics

The description of dynamics in terms of a Lagrangian was obtained

by using the canonical special symplectic structure in infinitesimal

phase spaces. The hamiltonian description is associated with different

special symplectic structures which are no longer canonical but depend

on the choice of a trivialization of the configuration bundle Q. Thus

there is a whole family of hamiltonian descriptions associated with

different trivializations of Q. The choice of a trivialization can be

interpreted as the choice of a rest frame. In standard formulations

of particIe mechanics a rest frame is always assumed which leads to

a single hamiltonian description. The generality of our approach is

motivated by later generalizations to field theory.

Each infinitesimal phase space P~ is already fibred over Pt' The

fibration is given by Jt:~ = uplp~. In order to introduce a special

symplectic structure it remainsto construct a diffeomorphism

O(~ : P~ - T*P t' This construction utilizes a choice of a tri viali­

zation of Q. A trivialization in Q induces a trivialization in P. At

each point p E P there is a unique horizontal vector

onto the unit vector in the time manifold M. Sections

a .·at WhlCh

of P are

projects

mappings

from M = R1 to P ; hence jets of sections are vectors tangent to P.

Since a section composed with the projection

ping, vectors tangent to the

g E P; is a jet attached at p

onto zero. It means that g

section project

h " .t. en g - at lSCI

at E TPt ILe.

1is the identity map­

onto the unit vector. If

a vector which projects

is vertical/. The mapping

9.1 g -

is a bundle isomorphism.

Due to the existence of the symplectic form G0t the tangent bun-

70

dle TPt is canonically isomorphic to the cotangent bundle T~Pt' The

cQnonical isomorphism

1ft

is defined by

9 ?.-

The composition

hO\t

is the required diffeomorphism from P~ onto T*Pt•

Let Xt: T>!Pt--+Pt be the canonical cotangent bundle projection.

Let 8~ and W ~ be the canonical 1-form and the canonical 2-form in

T*P t. The condi tion Jl;' ~ = ~ t o oZ~ is obviously satisfied and the equa­

tion w~ = O(~*w~ will be proved using local coordinates. The 1-formA

o(~)('8~ will be denoted by 8~.

Let (t,qj) be a coordinate system of Q and let the trivializa­

tion of Q be the one implied by this system. We use for pi the coor-

g E P~ wi th coor-• (l

+ Pj -- attached~P~J

dinates

dinates

at the

(t j •j • ) . t d d l' mh",q 'Pj ,q' 'P,j ln ro uce ear ler • .L e Jerj .j .. a .j 2(t,q ,p.,q ,p.) lS the vector ot + q .

J J . CqJ

point with coordinates (t,qJ'Pj)' Hence

9.3 g -aot

.jq'O • 3-, + p,-(\qJ J op,J

If coefficients (qj,p.) together with coordinates (qj,p.) are usedJ J

as coordinates in TPt then the coordinate expression for et is the

identity. Coordinates (qj,p~) in P-,-give rise to coordinates (qj,p.,J 0 J

m.,nj)in T*P-,-by a standard procedure described earlier. The canoni-J v

cal forms

9.4

and

9.5

71

"'h "het and GJt are expressed in terms of these coordinates by

If UE; TPt has coordinates (qj'Pj,qj, Pj) then U

Let U>t(u) have coordinates (qj,p.,m., nj). Then1 J J

d 3

qj ~. + Pj ~dqJ PJ

lf\C u)d j jdm.q +n p.

J - J

For each vector w . d daJ .-. + b .-- the equality 9.2 reads

'dqJ J op.,J

9.6

/ • k .k <l d .<--(p.a- - q bij k/\ -,dp./\dqJ >

l - (Iq cp. Jl

Since aj and b. are arbitrary we conclude that m.J J

follows that the coordinate expression for o(~

-qj. It

where

( j j \q ,Pj,mj,n J

9.7 m.J

.p.

J-qj

Substituting 9.7 into 9.4 and 9.5 we obtain the following coordinate

expressions

9.8

72

9.9

The last equality proves that o(~together with ~~ defines a special

symplectic structure in (p~, GV~).

Through each point PEPt there is exactly one dynamieally admis­

sible section of P. The jet of this seetion is the unique element of

D~ attaehed at p. It follows that D~ is the image of a seetion of the

bundle X~ : P;~Pt' Sinee we also assumed that fibres of Q and

consequently fibres of P are simply eonneeted, each lagrangian subma­

nifold D; is generated by a funetion Ft on Pt' Funetions Ht = -Ft de­

fine a funetion on P ealled the Hamiltonian. The Hamiltonian is defi-

ned up to an arbitrary additive funetion depending onI y on t.

Using eoordinates (qj'Pj,qj,pj) we find that D~ is deseribed by

the equation

9.10

analogous to 8.24 and equivalent to the formulae

p.

=a

- - H(t j

9.11

J

Oqj ,q 'Pj)

qtl

=

a

- H(t jop. ,q ,p.)J J I

known as the Hamilton canonieal equations.

Example 3

Equations of motion of the harmonie oseillator ean be written

in the form

• 1q = ID p

p = -kq

Restricting the form e~

73

pdq - qdp to D~ we obtain

It follows that D~ is generated by the Hamiltonian

9.14 H(t,q,p) 1(1 2 2)2 ffiP + kq

In the preceding section we constructed a special symplectic

structure in the symplectic space (p~,GJ~).The fundamental objects

of that structure were the projection Je ~ : p~ ~ Q~ and the 1-form

e~ satisfying d e~ = w ~. In the present section we constructed ano­

ther special symplectic structure in (p~, G.J ~ ) depending on the choice

of a trivialization in Q. The fundamental objects of this structure

are the projection 1t'~: p~~ Pt and the 1-form G~ which again

satisfies d e~ = eJ~ • With respect to the two special symplectic

structures the same objects D~ are described by two sets of genera­

ting functions. The two descrptions are parallel. The formula 9.10

has its counterpart in the formula 8.24.

The difference is a closed 1-form. Due to our topolo-

gical assumptions it is also exact. We define a function ~t on P~ by

9.15

If g has coordinates (qj,p.,qj,p.) thenJ J

9.16

and

qjd

OPj

9.17

Hence

( j .j.)'fit q' 'Pj,q 'Pj

74

a-. + •

oqJ Pj

(7 .

~ 'PjdqJ >p.J

9.18

The function flt on D; defined as

satisfies the equation

Hence -flt is the proper function corresponding to the generating fun­

ction -Ht'Our approach to hamiltonian description of dynamics is different

from, though equivalent to, the standard approach. Since for each point

p EP there is a unique vector /jet/ in Dl) attached at p, the familytt(P

{D;} defines a vector field in P which will be denoted by ~t • The

difference

9.20ddt

is a vertical vector field in P. In coordinates (t,qj'Pj) the fieldsd hdt and X are

9.21

and

~dt

a . 8--:+p.--oqJ J op.J

75

9.22

where qj and p. are funetions on P given by 9.11. Vertieal veetorJ

fields on P are usually called time dependent veetor fields/ef. [1J/.

It ean be shown that the faet that D~ are lagrangian submanifolds is

equivalent to Xh being a loeally hamiltonian veetor field lin the sen­

se of [1J/. In our ease Xh is even globally hamiltonian and it is ele­

ar from equations 9.11 that our Hamiltonian H is the Hamiltonian for

Xh in the usual sense.

10. The Legendre transformation

We assumed that the infinitesimal dynamies D~ is a seetion of the

fibration 'J(.~ : P~----> Q~. It follows that for eaeh element v E Q~ Ive­

loeityl there is a unique element gEP~ sueh that 'Jt;Cg) = v. The map­

ping

is called the Legendre transformation. Since D~ is also a seetion of

the fibration 'X~: P;----;.Pt the Legendre transformation is inver-

tible IcL [9J, [53]/.

11. The Cartan form

The disadvantage of the hamiltonian deseription of dynamics lies

in the dependence on the choice of a trivialization. There is a method

of combining the different Hamiltonians corresponding to different

trivializations into a single object. This objeet is the Cartan form

lef. [7J/. We define this form in terms of the Lagrangian and thus in

a way manifestly independent of trivializations. We show subsequently

how different Hamiltonians are extracted from the Cartan form as com-

ponents with respect to different trivializations.

76

Definition: The Cartan form associated with the dynamics is a

1-form 8 on P such that

11.1 <u,8>

for each vector u E TP tangent to the fibre Pt' and

11.2 i d l)L(X (dti

Theorem 4

For any trivialization of Q the function

H

is a Hamiltonian corresponding to this trivialization.

Proof We eonsider the funetion

11.4

dSinee dt adt + Xh, where Xh is a vertieal veetor field, we may write

11.5

of the funetion (Xh, G t)

formula 9.15/. Thus

lift

leI.

Sinee the mapping 0(~: D~ - Pt is a diffeomorphism we may define

a lift flt of the above function to D~. The lift of the function

Llxi(~t)) gives obviouslY!!t. Sinee xhoX~ = ~ t it follows that the

is equal to <~t' Gt> I D~ = 't'tI D~

11.6 H-t

77

- I:t + o/t I D~

We noted earlier /formula 9.19/ that this function is a lift of a Ha-

miltonian ; thus the proof is complete.

In local coordinates Ct,qj,Pj) the Cartan form is

11.7 8 p.dqj - HdtJ

which follows from 11.1 and 11.3.

The Cartan's form e is independent of trivialization but de-

pends on the choice of the Lagrangian whieh is determined only up to

an additive funetion of time. It is elear from formula 11.2 that if

a funetion fet) is added to the Lagrangian the Cartan form is ehanged

by an additive term f(t)dt. It is interesting to note that the form

Q = d8 , whieh is uniquely determined by dynamics, also eontains

alI information about dynamies.

Theorem 5

A seetion of P

M El t

is dynamieally admissible if and only if for eaeh veetor field

X in P

11.8 (X.JQ)js

where S c P is the image of O •

o

ProofUsing eoordinates (t,qj'Pj) of P we obtain from 11.7 the

78

following expression for ~

11.9 Q. ('OH .

dp. A dqJ _ -.dqJ +J oqJ

oH dpj }A dtOPj

Let the field X be described by

a .2lo11.10

X=A .- + BJ-. + C.-ot

oqJJcp.-J

and let

a be given by

11.11

Then

c et)

'OH .

óH ) .XJQ=A ( - -. dqJ +- dp. - BJdp. +

oqJ'Op. J J

11.12

J

. (. oH Q H )

+ C .dqJ - BJ --o + C. - dtJ oqJ J op.J

the X J Q restricted to S is zero if and

In terms of coordinates j10 is the vector

where ojqdp.(t)

o ~ o HencePj = dt

only if <j1O , X J Q >o . ja.ot + q -. + p.

oqJ J

= o.8-,

op.J

11.13

o H )OPj

o

for all values of A, Bj, C .• This is equivalent toJ

•joHq=

OPj

oH

11.14

=

-oqj

p.J

79

and

11.15'OH

'Op. J

p.Jo

The last equation is a consequence of 11.14 and equations 11.14 are

precisely the conditions for ~ to be dynamically admissible.

Theorem 5 is stated in a form specially suitable for generali-

zation to field theory. Descriptions of dynamics in terms of the Car­

tan form may be found in references [9J, [11j , [16], [17J, [18J, [19J, [2oJ,

[24J, [25J, [26J, [47J, [48J.

12. The Poisson algebra

Using diffeomorphisms <rt introduced in Section 9 /formula 9.2/we can assign to each differentiable function f on P a vertical vec­

tor field Xf defined by

where ft = f Ipt, x~ = Xf I Pt • One-parameter groups of diffeomorphisms

generated by x~ are symplectomorphisms of symplectic spaces (Pt'(~t)'

Differentiable functions on P form a Lie algebra with respect to the

standard Poisson bracket defined by

Theorem 6

The field XH associated with the Hamiltonian H : P ~ R1 is

identical with the field Xh defined by the formula 9.20.

Proof - obvious from 9.22 and 9.11.

III. Field Theory

13. The configuration bundle and the phase bundle

The fundamental geometric space for field theory is a bundle

~ : Q --7 M over a manifold M of dimension m. This bundle will be

called the configuration bundle of the field. The fibre Qx

is called the configuration space at x 6 M. The manifold M is the phy-

sical space in which the field is defined. For static field theories

such as electrostatics or elastostatics M is 3-dimensional physical

space. For dynamical field theories such as electrodynamics and re-

lativistic theories of gravitation M is the 4-dimensional space-time.

The manifold M replaces the 1-dimensional time manifold of particIe

dynamics.

As in particIe dynamics we shall define a phase bundle ~: P ~ M

corresponding to the configuration bundle Q. Histories or states of

the field will be sections of this bundle. Physical laws governing

the behaviour of the field select a class of histories which will be

called dynamically admissible. As in particIe mechanics the laws of

dynamics take form of first order differential equations.

Instead of time segments [t1,t2] used in mechanics we will con­

sider domains /usually compact/ Vc M. Instead of end points (t1,t2)

of segments we will have boundaries oV of domains. In order to desc-

ribe dynamics in terms of symplectic geometry we will associate with

each domain V c M the symplectic space P a V of boundary values on a V.

(t2,t1)This space replaces the space P of mechanics. Boundary values

of dynamically admissible histories define a subspace D aVe P 3V.

In most physical field theories subspaces D oV are lagrangian subma­

nifolds of P a V for a wid e class of domains V.

In the sequel we consider only curvilinear polygons as domains

in M.

81

Definition : A curvilinear jorientedj polygonof dimension k in

M is a diffeomorphic image V = ej) (11) of a k-dimensional oriented po-~ m

lygon V eR.

A polygon V has an orientation which is transferred from V by

the diffeomorphism ~ •

If V is a polygon of dimension m then Q V = ~ (cV) is a sum of

polygons of dimension m-1 • Segments [t1,t2] considered in particIe

mechanics were 1-dimensional polygons and their orientation was deter-

mined by the direction of the time axis. The boundary (t2,t1) of the

segment [t1,t2J is the sum of the positively oriented point t2 and

the negetiveIy oriented point t1• The formulae 7.10 and 7.20 are in­

terpreted as integrals over O-dimensional boundary of the segment

and the negative sign corresponds to the negative orientation of t1•

The generalization of these formulae to more dimensional manifolds M

will require the use of objects which can be integrated over bounda­

ries of m-dimensional polygons. Such objects are differential (m-1)­",-1

forms, which are sections of the bundle I\T*M of (m-1)-covectors in",-1

f1. The vector space of (m-1) -covectors at x E II] is denoted by 1\T~MDifferentiaI (m-1)-forms will be also called vector densities on M

and their values at x E I1 ,viII be called vector densities at x. The

above remarks Iead to the following constructions. Let P denote theq

tensor product

m·f

T* Q ®i\T* Mq x x

~here qE Q and x = 5 (q) E M. The collection of alI spaces Pq form

a vector bundIe n: P --7 Q whose fibres are equal P • The bundIeq

structure of P is that of the fibre tensor product of the bundIe V*Q

(:idjoint to the bundle VQc TQ of vertical vectors tangent to Q) and"'·1

the pull-back f"( 1\ T*M). Because Q is aIready fibred over M we may

82

treat P as a fibre bundle over 1'1with the projection 1= s°9( • Thus

we have the commutative diagram

13.2

p rx ) Q

~A1"1

similar to the diagram 5.2 in Chapter II. As in Chapter II P will be

considered primarily as a fibre bundle over M. The fibre ~-1(x) over

x E 1-1 will be denoted by Px. The restriction of X to Px induces the

mapping

13.3

Definition : The bundle P over 1-1 is called the phase bundle of

the field. The space Px is called the phase space at x.

An element p E Px can be treated as a linear mapping from the11'1··1

tangent space T ()Q to the space /\ Tx* M of vector densities at x.~ (J'C p x

Consequently, we shall calI elements of Px vector-density-valued co-

vectors on Qx. The value of p on a vector u E T:JtCP)Qx is denoted by

The symbol < , /1 can be interpreted as the contraction of the vec­

tor u with the first factor of the tensor product 13.1.

An element p E Px can also be interpreted as a linear mappingm-i

from the space 1\ T r1 of hypersurface volume elements at x E 1"l to thexm-i

cotangent bundle T*Q • The value of p E P on an (m-1) -vector n E 1\ T r1x x - x

jhypersurface volume elementj is denoted by

83

13.5

The symbol < ,> 2 can be interpreted as the contraction of the (m-1)

-vector n with the second factor of the tensor produet 13.1.

In particle dynamics m = 1 and )\T* IV! = R1• Hence the secondx

factor in 13.1 is trivial and

13.6

or

13.7

pq T~ Qx

as in formula 5.1.

Let (x"'), A=1, ••• ,m, be a coordinate system in IV! and let (x;l.,c.pA),

A=1, ••• ,N, be a coordinate system in Q compatible with the bun-

dle structure :

13.8

A vector density in IV! can be expressed as a linear combination of

(m-1)-forms

13.9"1 m

-- -.J dx /\••• 1\ dxoxA.

where the symbol /\ means that the A-th factor has been omitted. It~

follows that each element p E Px is a linear combination p = p~e~ where

13.10

The coefficients p~ together wi th the coordinates (x;L,~ A) of the

84

point X(P) define a coordinate system (x;\.,tpA,p~) in P.There is acanonical vector-density-valued 1-form G on eachx

fibre Px defined by the formula

13.11

analogous to 6.2. Here Jt*p denotes the pull-back of the first fac­x

tor of the tensor produet 13.1 from Qx to Px' If v is a vector tangent

to P at p then the value of e on v is a vector density at x givenx x

by the formula

13.12"'-1

< 'JCx~,v,P)1 E !\ T~ M

analogous to 6.1.

We define the canonical vector-density-valued 2-form on Px

Here the symbol d denotes the exterior differential of the /vector­

-density-valued/ form on P • This means that if n is an (m-1)-vectorx -at x then

13.14

where the differential on the right-hand side is the differential of

the scalar-valued 1-form <n, e ' on P • In the coordinate system- x~ x

13.15

and

13.16 GJ x

85

(';L. A) ('O 11m)dpA /\d t.p \l9 ex:\.-~dx /\•••/\dx •

14. The sympIectic space of Cauchy data on a boundary

Let oV be the boundary

av C\C1V 3V .P ,C' , C-.l correspondlnget , t )

GJ 2 1 of particIe mechanics. At each point we first define objects

oV ?JV C10V oV .. CIQ , p , \J , W derlved from obJects Q , p , <:J, (;.) by pro-x x x x x x x x

jecting the second factor in 13.1 onto the hypersurface oV. This",-1

projection takes eIements of I\T* M /vector densities at x/ intox",,1

eIements of 1\ T~ ( o V) /scaIar densities on av at xl.m-1

The space Qx does not contain the factor /\ T; M. Hence we take

14.1

m-i

The space P = T* Q ® 1\ T* M becomes the space of /scaIar-den-q q x x

sity-on- oV-valued/ covectors on Qx

14.2

and

The projection

14.3

is defined by the formula

p ---+X

14.4

86

for each (m-1)-vector ~ tangent to oV at x Ihypersurface volume ele-

ment on av at x l.If coordinates exA..)of i"l are chosen so that in a neighbourhood

of x the surface oV is described by the equation x1 = const then

12m) .~x , ••• ,x lSpoV E poV is a

14-.4

where

a coordinate system on

linear combination

?tvP

aV. It follows that each element

14-.5 ~Ae A (, 2 m)d 'P Q9 ,dx 1\ ••• 1\ dx

Hence (~A,p~V) is a coordinate system of P~V. It follows from 14.4

that

14.6

~Ae

o for k 2,3, ... ,m.

Hence the coordinate expression of the mapping pr~V is

where

14.7

The form e ~V is a Iscalar-density-on- a V-valuedl differential form

on poV defined by the formulaex

14-.8

or

14.9

where

14.10

G~V(pCV)

<W,G~V(pOV»1

rOVLX

87

is the canonical projection. The form GJoV is defined byx

14.11

The forms G oV and 0JoV are obviously projections of forms f) andx x x

W onto the surface "V. Coordinate expressions of e 'OV and woV arex x x

thus

14.12

14.13 w oVx

1dWA ( 2 mpA I @ dx A ••• 1\ dx )

/We remember that av is described by the equation xi = const./.

The above formulae closely resemble formulae 6.4 and 6.5. The

only difference is the factor dx2 000 dxm which is trivial in ca-

se of m=1.

We now define spaces QOV and poV. The space QÓV is a space of

sections

14.14 av :3 X E

88

and is called the space of Dirichlet data on dV.

The space poV is a space of sections

14.15 oV '3 X

and is called the space of Cauchy data on 3V. These definitions must

be completed by specifying the class of the sections and choosing an

appropriate topology. We do not make such specifications in the pre-

sent paper except in one special case given in the next section.3V . oV

Elements of P are ln a natural way covectors on Q • Let

14.16 oV '3 X sex) E

. oV oV avDe an element of Q • The space T Q of vectors tangent to Q at ssconsists of sections

14.17

Ii

14.18

2V 3 X

CiV '3 X b'(x) E

is an element of poV such that

14.19

then the formula

SeX)

14.20

defines a linear functional

j<X(X),G'(X»1oV

<. ,G' ';> on T QOV /we remember thats

89

(X(x), b (x»1 is a scalar density on oV at x l. If topologies of QOV

and pav are defined in such a way that alI continuous linear functio­

nals on T QOV are of this type thenthe space poV may be identifiedswith the cotangent bundle T~Qav. Assuming that this is the case, we

have the canonical projection

14.21

the canonical 1-form GOV and the canonical 2-form GOoV

formula 14.20 implies the formula

14.22~<yex) ,8 ~v> 1oV

where Y is now a section

14.23 oV :3 X , yex) E

representing a vector tangent to the space of Cauchy data poV at

Similarly

14.24 <y y oV>(1)/\ 'a.)' CU ) av(yex) /\ Y(x) ,6Jx > 1(1.) (,2)

oV

for two such vectors Y and Y. In the coordinate system (xA-, (nA ,PA1)(1) (il) Ibased on a coordinate system (xA-) such that dV is described by the

. 1 hequat~on x = const. we ave

14.25

av

and

14.26 < avY/\Yw =(f) (J.)' >

90

where the following coordinate expressions

sex)=( cpA(x»)

b(X)

=(\fAex) ,pl(x))

14.27 Y(x)

~ A " ~ 1 o= ~ (x) (HpA + PA(x) opl

Y(x)

E A 'O <S 1 a, i=1,2;

=~ (x) ·--A + PAex) -1le) (O o\f U) 'O PA

are used.

The restriction af a section

14.28 M .3 X

to o V is an element of QOV denoted by s I 'O V.

If G' is a section af P :

14.29 M :3 X

then b IdV will denate an element af poV obtained by restricting b

to aV and projecting its values from Px to P~V by means of pr~V. Si­

milarly if

14.30 M 3 X Y(x) E TPx

Is a sectian af the bundle of vertical vectors tangent to P then Y I oV

will denote the section

14.31 oV :3 X --~, (y 12) V) (x' E TpOV, x

91

==~s.ined by restricting Y to cV and projecting its values by means

~.le tangent mapping pr~V* • The section Y I o V is thus the object

~ie type 14.23 and may be treated as an element of TpoV• The form

;:.2;Vevaluated on the vector Y I (7V gives

-.32 <y I av,GoV>j<cy lclV)(x),EJ ~V> 18V

~<y(x), ex> 1av

''--.33 «Y I oV)I\(Z loV),wcV>

5isilarly

\<CY I av) (x)/\(Z I av)(x) ,w~V> 1voV

~ <Y(x)/\ z ex) , wx> 1oV

/"-.34 V :7 X ,.. Z(x) E TPx

~3 a section su ch that Y(X) and Z(x) are vectors attached at the sa­

-" point (;(x') E P for each x E l'.l. Formulae 14.32 and 14.33 are ana-- x

~Jgous to formulae 7.21 and 7.11. In analogy to formulae 7.20 and

-.10 we write

'''':'.35

'...:..36

GdV

woV

l'.l

\ 8x'-?JV

lWxoV

7. Finite domain description of dynamics

Finite domain description of field dynamics can not be presented

92

with a rigour matching the finite time interval description of par-

ticIe dynamics. Field dynamics is based on the theory of partial dif-

ferential equations which is not as well developed as the theory of

ordinary differential equations used in particIe dynamics. Consequent-

ly we give only heuristic considerations as an introduction to a ri-

gorous infinitesimal deseription of field dynamie s given in the next

section.

We begin with the diseussion of eleetrostatics in a 3-dimensio-

nal manifold M which is assumed to be a riemannian manifold with a

metric g. The eonfiguration spaee Qx at eaeh point x E M is the space

of values of the eleetrostatie potential 'P' Hence Qx = R1 and Q is

the trivial bundle Q = M x R1• The value of the electrostatie poten-

tial ~ together with eoordinates (x~) in M define a coordinate systen

(x?v,lD) , A,= 1,2,3 in Q. The first faetor in the formula 13.1 is tri-r ~ ~

vial in this ease. It follows that P = /\ T*M, P = R1)( /\ T*M and:J.., q x x x

p = R1x !\T*M. Elements of Px are thus pairs UP,p) where tpE.R1 is.:l,

the value of the electrostatic potential at x and p E !\ T*M is thex

value of the electrostatic induction field at x. Corresponding to

coordinates (x?v,'P) in Q we have eoordinates (x?v'LP,p~) in P. In term:

of these eoordinates

15.1

and

ex ( 1 2 3 2 3 1 3 1 2)d~Ci!;)p dx/\dx + p dx/\dx + p dx/\dx

15.2LUX = (dP'\dlf)@(dX2/\dx3)+ (dp2/\d\f)®(dX3/\dX1h

+ (dP\ dlf) Ci!;)(dx1/\dx2)

If the coordinates ex?v) are chosen in such a way that the boundary av

of a domain V is described by the equation x1 = const. then (~,p1)

is a coordinate system in the space P~V • Coordinates ~ and p1

93

are interpreted as the value of the potential and the normal component

of the induction on av linterpreted as the surface charge density on

a v l.The equations of electrostatics are

15.3 p = * V~

where V is the exterior differential and * is the Hodge operator

associated with the metric g, and

terms of coordinates the above equations read :

is a 3-form Iscalar densityl r8presenting a fixed charge den-

15·4

where

9sity. In

15.515.6

pA,

() pC\.

dxA.

v p = - 4JtS

\fi! g?-t- a 'PClx;\.

- 4JL ,!g1r

where r is the scalar function defined by the equation r = *' ~ ' or

equivalently, by

15.7 ~ ~ r dx\dx2l'.dx3

Substituting 15.5 into 15.6 we obtain the Poisson equation'

15.8.6 'f - 4']L r ,

where 6 = _1-d;cgA-l'-~ 0l" is the Laplace - Beltrami operator associa-'filted with the metric g. Applying the formula 15.3 to solutions of the

Poisson equation 15.8 we generate dynamically admissible sections of

94

P.

It follows from the formula 15.5 that the normaI eomponent of p

is the normaI derivative of ~. Speeifying the normaI eomponent of p

on 3V determines thus the Neumann data for the Poisson equation. Spe-

eifying the value of ~ on oV determines the Diriehlet data. We eon­

elude that the spaee poV whieh we ealled the spaee of Cauehy data is

the spaee of eombined Diriehlet and Neumann data. In general eombined

Diriehlet and Neumann data on cV do not eorrespond to any solution

of the Poisson equation. Restrietions of dynamieally admissible see-

. 8V avtlons to the boundary of OV form a subspaee D of P • The subspaee

DOV is eomposed of those speeial pairs of Diriehlet and Neumann data

whieh do eorrespond to solutions.

For a suffieiently regular domain V it ean be shown that if the

spaee QOV of Diriehlet data is the Sobolev spaee H1/2 on 'av then DOV

is a lagrangian submanifold of the infinite dimensional sympleetie

( oV aV) .spaee P ,CJ • In thlS ease

15.9 poV T*Q7JV H1/2 x H-1/2

whieh means that the spaee of Neumann data is the Sobolev spaee H-1/2

1/2dual to the spaee H •

A lagrangian submanifold of an infinite dimensional sympleetie

spaee is isotropie and maximaI in a eertain sense /ef. [8] and [57]/.

We use the Green's formula to prove that DaV is isotropie. Let

15.10

and

M 3 X(lp(X),p(x)) li': Px

.J.

R1x /\T* Mx

15.11 M 3 X( (~ cp (x) , ,9) p (x) ) E Px

95

be sections of the bundle P. We denote by Y the vector (5 (n, S p) I av1-1) (1) I (1]

tangent to poV at the point (~,p)lav. If (~,p) satisfies equations

15.3 and 15.4 and if (510, 5p) satisfies corresponding homogenous(l)! Ci}

equations

15.12

15.13

Sp(~)

vC~p)

* v(§ y)

o

then (~+~~ ,p+ sp) satisfies again the inhomogenous equations 15.3and 15.4 which means that the vector Y is tangent to DdV• It is obvious'<j

that alI vectors tangent to DOV can be obtained in this way. Let Y be<~)

a second vector tangent to DOV corresponding to the section

15.14 M " x-~, (6IncX), SP(X») E p'~1 T '0-1 X

AppIying the Stokes formula to 14.26 we obtain

15·15

/ Y Y . av>~ ,,' /\ <1]' LU

H V§)f(X)v

~f§/f(X).§P(X) - (~lf(X).~P(X)}?JV

~ V {,~ 'reX)' §p(X) - (~)f(x),~P(x}v

~ [( '17,2] lp(x») 1\ ,§p(x) + §] tpCX)1\v§,P (x) -V

- (V(~ip(x»)/\ ,2p(X) - ~tp(X)1\ V,§P(X) J

/\ *'VbI0(x) - \7b\l')(x)/\'kV ~\n(x) 1.(" I (1) T (J) T

96

For any pair of funetions f,h on M we have IIfA *Vh = * (vfli.7h)whe-

re the symbol ( l ) denotes the sealar produet of eoveetors defined

by the metrie g. In loeal eoordinates

Vf/\*Vh Ijgi gAf-.2..fd h dx \ dx2 Adx3 •c x:\. ex/'-

The above expression is symmetrie in f and h. This implies that

15.16

Consequently,

V,?)r A* V (~\f - V6lD 1\ * V'b'tDi1) l (l.) Io

15.17 <:: Y/\ Y,c.jW"(~) (al /o

whieh means that D3V is isotropie. Formula 15.15 is the familiar

Green's formula for the Laplaee equation.

Finite dimensional lagrangian submanifolds are defined as iso-

tropie submanifolds of maxi maI dimension. This definition does not

apply to infinite dimensional submanifolds. There is however an equi-

valent eriterion for maximality whieh does apply to infinite dimen-

sional cases : an isotropie submanifold N is lagrangian if and only

if at each point of N the space tangent to N has an isotropie comple­

ment. In the case considered here th~ submanifold DOV is the image of

a seetion of the bundle poV over DCV because given Dirichlet data one

can solve the Poisson equation 15.8 and subsequently ealeulate the

eorresponding Neumann data. It follows that at eaeh point póVeDoV th

spaee tangent to the fibre of poV complements the space tangen

to DOV• Bóth D?V and the fibre are isotropic. We conclude that DCV is

lagrangian.

As a seeond example we shall use the theory of a vibrating strin

The manifold M is nowa 2-dimensional pseudo-riemannian manifold with

97

a metric tensor g of signature (+,-). As in the previous example the

configuration bundle is the trivial bundle Q = Mx R1• The electrosta-

tic potential is replaced by the deflection of the string from its

equilibrium position. It follows that the construction of QOV and poV

are formally the same. We use the same constructions of coordinates

and the same symbols. The coordinates (x1,x2) are interpreted as the

time t and the distance x along the string respectively. The metric

tensor has components

15.18 (g!,-V)[: _~lJ

and g

15.19

det (gf'Y) -1. The dynamical equations are

p = )(, V'f

VP = O

In terms of coordinates

15.20

pi\,

=Rg~F ~

cxFÓ pA.

=O

'ex"

orpt

=olp

15.21

'Ot,pX olf!=

-8X

i)pt

i)pxat +~

=O

It follows that CP(t,X} satisfies the wave equation

15.22a2lp

~t2

98

32lp

Ó x2o

Also for this example one can prove that the space D3V of Cauchy data

compatible with dynamics is a lagrangian submanifold of poV for a wi­

de class of domains V. To prove that DOV is isotropic one uses the

Green's formula as in the previous example. The proof of maximality

is much harder. The argument used for the elliptic case fails because

the Dirichlet problem is not well posed. The general idea of the proof

is however the same. The space pdV is split into two complementary

components and it is shown that the dynamics determines the second

component of the Cauchy data from the first one. In the case of el-

liptic equations of electrostatics we split the Cauchy data into Di-

richlet data and Neumann data. In the case of hyperbolic equations

more general splittings must be used. Such splittings correspond to

different mixed boundary value problems which are well posed and may

be used in the proof of maximality of D~V :

t

x DDDD,N

NDND,N

-<>-D,N

The symbols D - Dirichlet, N - Neumann indicate the data given at

each piece of the boundary of V. Some non-compact domains can be tre-

ated by similar methods. The Cauchy problem

v

D,N

is one example. We return to this example later. The following charac·

99

teristic boundary value problem

t

D D

x

although well posed fails to establish maximality of D2V• In fact it

can be shown that Dav is not lagrangian whenever a piece of oV is

characteristic. We limit alI considerations to domains V for which

DoV. l .lS agranglan.

The lagrangian subspaces DdV exhibit composition properties si­

milar to the composition properties expressed by formulae 7.7 and 7.8.

Let adomain Vo be subdivided into smalIer domains Vi' i=1,2,•••,N :

Vo

We define the following composition law

15.23Noi=1

oV i \ (Wo

D T.?NoP

( CV.) N ?;V. V.1

there is a sequence p l i=1 ' P lEP lsuch that if V· and V. have a common

?Ni l ?N' Jwall then p and p Jare equal on that

common wall for i,j = O,1,2, •••,N .

In the case of M = R1 this law reduces to the formula 7.7. It can be

shown in many cases that

15.24cWD o

Noi=1

oV.D l

100

Lagrangian subspaces DOV are generated very often by generating

functions WoV defined on constraint subspaces COVcQoV. In the case

of electrostatics functions WoV are defined on the full space of Di­

richlet data of class H1/2 /there is no constraints/ and can be exp-

ressed in terms oi Green's functions. The composition property 15.24

implies the composition formula for generating functions :

15.25oV 0(· ?'VO)

W q

N

~?N. C· '(N·L W l q lii=1 '

value

formul~

equal on that commonavo

q on

such that if oV. and cV.l J

wall for i,j=O,1,2, •••,N. /The boundary

oV is thus fixed/. In the case of M = R1 the aboveo

where (qa~ )~-1 is a stationary point of the right-hand side. Thel- V

right-hand side is treated as a function defined on sequences (q~ i~óV. IW·

have a common wall then q l and q Jare

reduces to the formula 7.25.

16. Infinitesimal description of dynamics

8xi, w i, Di which arex x

poV, f93V, w8V, DOV•

pix'

Q3V ,

Considering a domain V shrinking to a point x one obtains infia

in sense limits ofA

corresponding objects

nitesimal objects Qi,x

Definition: The space Q~ = J~CQ) of 1-jets of sections of tl

configuration bundle Q at x is called the infinitesimal configurat:

space at x. The 1-jet bundle Qi = J1(Q) is called the infinitesima:

configuration bundle.

If

(Iv Ax,tp,

(x\ tpA) is a coordinate system of Q then a coordinate sys

er:') of Qi is obtained by taking

16.1\fA ~A(X)

101

and

16.2 \f~A

o ~ Cx);lex

as coordinates of the jet s ,i\.p (X) of the section

16.3 M 3 Y ------70 lp (y), i ;t UJA \\,y , l (y)} ~ Qy

Intuitively the jet j\p (x) is a limit of the boundary values t.p laVE

EQoV when the domain V shrinks to x. In this sense Qi is the limitx

of Q oV.

The canonical identification 8.6 applies also to the present ca­

se. Thus we can represent vectors tangent to Qi as jets of verticalx

vector fields in Q. Let

16.4 M '3 Y X(y) ET lf'CY) Qy

be a section of VQ such that X(y) is a vector tangent to Qy at lp(Y).

The jet j1X(x) represents a vector u tangent to Q~ at the point s

= j\f(X). Intuitively the jet j1X(x) represents the limit of the boun­

dary value X I ,3V which is a vector tangent to QOV at the point ~ I dV.

If in the coordinate system the field X is described by

16.5 X(y) XA(y)a

d\fA

then the vector u is described by

16.6

where

u uA _d_d~A

+ Aud

dCf~

--'-<., .. '''' .••..... ,..... _--.

102

uA = XA(X)

16.7

Au).,

A

l.L(X)Q x?-

It is obvious that each vector tangent to Qi may be represented in thix

way.

The construction of the infinitesimal phase space is more compli-

cated in the general case than in mechanics. One expects intuitively

that the infinitesimal phase space at x is related to the jet space

J1(p). Let there be a sectionx

16.8 ]\1 ~ y p(y) E PY

According to 14.20 the evaluation of the covector plaVET*QoV = poV

on the vector xlav equals

16.9 <xlav,plov>~<x'P) 1

av

The surface integral on the right can be replaced by a volume integra:

using Stokes theorem

16.10 <x I av ,pl av> j div(X,p) 1V

The limit of the right-hand side jper volume elementj is the scalar

density div(X(y) ,P(Y» 1ly=x • We take this expression as the defini­

tion of the evaluation for jets :

16.11 <j1X(x) , j 1p(x» div <X(y) ,p(y» 1 I y=x

The above definition associates with each jet g = j1p(x) a scalar-den

sity-valued covector <. ,g> on Q~. The situation is however differen

103

than in particle mechanics where the evaluation for jets was defined

by the formula 8.8. In the general case many different jets define

the same covector. We illustrate this phenomenon using local coordi-

nates. Let the section 16.8 be described in coordinates by

tfA = Lp A(y)

16.12

?v A-

PA = PACY)

We define coordinates (lfA ,P; , \.f~'P~j<-) of the jet g

\fA=lfA(x)

?v

p~(x)PA

=

~;.

dlpA

=ox:lJx) ,

A.

?v

d PA

PAf-

=--(x)ox!'- .

The formula 16.11 reads

j1p(x) by

16.13

<u,g> . < A a A (Cl 11m ):dlv X ~A' PAd \D ® - -1dx 1\ ••• 1\ dx()lf \ "3x:l.. 1

. f A?v ~ 1 m}dlv LX PA ó x).,JdxA ••• /\ dx

a ( A ?v) 1 m-;:;-:t X PA dx 1\ ••• 1\ dxox

(A/\.u?. PA A 7v \ 1 m+ u PA;l.,) dx 1\••• l\dx

We see that only combinations~

deA = P A A.-Isummation over A II of

components P;~ appear in the result. Two jets having the same value

of

104

~A define the same eoveetor on Q~. We shall eonsider these two

jets equivalent. This equivalenee relation leads to a quotient bundle

J1(p) of the bundle J1(p). The equivalenee elass of a jet j1p(X) will

be denoted by J1P(Xl E J1(p). The "generalized jet" y1p(x) leI. [54-JI

does not eontain the information about alI derivatives of P~ but only

about the divergenee JeA = O~I... p~. It follows that (x;\,,'PA,pi,tp~,deA)

may be used as eoordinates in;r1 (p). The fibre of :r\ p) over the point

x E M is denoted by :r~(p). It ean be parametrized by eoordinates (lfA,I\, A

PA' lpA' ~A)·

Definition : The quotient

nitesimal phase bundle and the

tesimal phase spaee at x.

i~1bundle P = J (p) is ealled the infi-

fibre pi = J1(p) is ealled the infini­x x

The reason why infinitesimal states of the field are equivalenee

classes of jets and not jets themselves is that only the normaI eom-

ponent of P €: Px entered in the eonstruction of the boundary-value spa­

ce poVx •

We denote the canonical jet-target projeetions by ~Q and Gp

respeetively. The jet prolongation of ~ is denoted by 0Ci• We have

thus the eommutative diagram

pi

\'p

.,

p

16.14-tJti1.ix

Ql

lQ

)Q

similar to the diagram 8.5 in Chapter II. In terms of eoordinates

(x'-,fA ,p~, ~~, JeA) the above mappings are described by formulae ana­

logous to 8.19 :

lQ ( x\ lf A, lf~) (x;l,fA)

~6.15

105

(A. A?c Aó-e)l..px , lf ,PA' l[J).,' A

i (A. A /" (()A >.P )0t x ,\f ' PA' T A,' o\- A

(x?c, tpA,p~)

.),., A A)(X , lf ' lf;l

g E J~(P)from pix

~he formula <u,g> = <u,g> , where

:::lassgf J~(p), defines a mapping

-density -valued covectors on Qi :x

is a representant of the

into the space of scalar-

16.16 pi .7 gx o(~Cg). m.

<. ,g> E (T*Q~) @ /\ Ti M

The above formula is anaIogous to 8.9. It follows from the coordinate

expression 16.13 that the mapping CX~ is a diffeomorphism. EIements

OL pi may thus be treated as scaIar-density-vaIued covectors on Qi.x ~ x

Since the space /\T* M of scalar densities at x E M is one-dimensio­x

naI it follows that the space of scaIar-density-vaIued covectors on

Qi has the same dimension as the cotangent bundle T*Qi. Although thex x

space (T*Q~) <&;\ T~ M is not the sympIectic manif'old in the strict

sense there are canonical /scaIar-density-vaIued/ 1-form and 2-form

defined by formulae anaIogous to 6.1 and 6.3. The puII-backs by o(ixof these forms are denoted by ei and GOi. Similarly as in particIex x

~echanics we give an aIternative construction of forms 8~and GJ~

by differentiation of forms e and GJ • Let v, w be vectors tangentx xto pi and attached at the same point. Using the canonical identifica­x

tion simiIar to 8.10 we represent v and w by cIasses of jets of sec-

tions

16.17

M :3 Y

M 3 Y

) Y( y)

Z(y)

E TPY

E TPY

We choose these sections in such a way that for each y vectors y(y)

106

and Z(y) are attached at the same point. It follows from the defini­

tion of C\i that Qi and lvi satisfy the following equalities :x x x

16.18 <v,e~>1 <J1Y(X),G~>1 div( y(y) ,GyA I y=x

16.19

i<vAw'Wx>1 < ~1 ') ~1 () i),j Y(x A j Z\x ,Wx 1

div<Y(YhZ(y) ,Wy> 1 I y=x

We compare these formulae with the integral formulae 14.32 and 14.33

which may be rewritten by means of Stokes theorem in the following

form

16.20 <y I ()V, 8°V>~ div(Y(y)~ 6Y>1

We see that forms

v

«' ) .' , av>,YjClV '/\'(Z]oV),U16.21 j div<Y(y)/\Z(y) ,wY>1V

Si and (,,]i are limits of forms 8°V and WoV whenx x

V shrinks to the point x similarly as v and w are limits of Y!oV and

Z!2V. The above remarks justify the following notation

16.22

16.23

8~

wix

Mdiv 8x

Mdiv wx

similar to 8.14.

Now we find the coordinate exprssions for ei and uJi. If vectorsx x

y(y) and Z(y) are expressed by

Y(y) yACY)o 'A.. ')-+Y() Q

alfA A y (JPA:

107

16.24

Z(Y) a A.) d

ZA(Y)Q~A + ZA(Y oP~

then the corresponding jets are

::I ;> Cl;]yAex) 7J Oy~ ex).~j1y(x)=yA(x) .cA + Yi(xl~ + -o-i" o\f,A + ~ ('p/\' ,

Zllp 2JpA X?v - Af.:>v

16.25

A :l OZA(X);3L a A.. o + oZ (X!_C-_+ ~ •j1ZCX! =

Z ~x) I\'PA + ZA (x)3pA CIxA o'f~ dXt< dPA/<- A /"

Consequently coordinate expressions for equivalence classes v and w

are

o A. (1CyA(x) _0_';)y~cx)a-v = 'J1yCX! = yA(X! Cl\fA+ YA x 3p~ + .-- A+?x:t oce()x;cC'f-?.. J A16.26

azACx) 2

"Z/i-(x)8A a A.) d

A---- +

w = }1Z(x) = Z (x) o~A + ZA(x QP~ + o x?v Glf~Qx?v ad€A

The equation 16.18 reads

(v, 6~>116.27

diV{yA(y)p~(y) ~..JdX~"'l\dXm]1ox y=x

(A /\, A) 1 mY/vPA+Y ~Adx/\ ••• l\dx

Comparing 16.27 with 16.26 we see that

16.28 8~ (A /v. A) 1 mJeAd<p + PAd\f?v ®dXA ••• /\dx

Similarly we show that

16.29i

Wx (..JA I\, A) 1 md d€A1\ ,",,~I + dPA /\dlf/v@dx 1\ ••• 1\ dx •

These expressions are formal derivatives of expressions 13.15 and 13.16.

108

The space (pi, W i·) together wi th the proj ection ;]Ci: pi ----!" Q-'-X X X X x

and the form G~ is a scalar-density-valued special symplectic space.

This means that for each m-vector ~ tangent to M at x Ivolume element

at xl there is in pi a special symplectic structure in the ordinaryx

sense defined by the forms <~,e~>2 and <~'W~>2. In particIe me-

chanics the standard volume element m = :t was used to convert alI

scalar densities to ordinary scalars. The same can be done in any fiel

theory if some standard volume element is present at each point xc M.

In the present approach we do not assume the existence of standard vo­

lume elements because of intended applications to General Relatlvity.

The unit volume element in General Relativity

16.30 m1

R'2

-O /\ex

a--12x

a/\ ~3x

(3

/\ - ~3·dx

where g = det(gFY) can not be used for this purpose since the metric

g is itself a field variable and therefore m is not fixed.

Since volume elements at x form a 1-dimensional space the forms

(~,e~>2 calculated for different volume elements differ from each

other only by a numerical factor. The same applies to the forms<~,w~

It follows that lagrangian submanifolds in p~ defined withx

respect to different symplectic forms <~,w~>2 are the same and wil

be called lagrangian submanifolds of (pi,G0i) • Also proper functionsx x

and generating functions of lagrangian submanifolds can be introduced

These functions will be scalar-density-valued and are defined by for-

mulae 6.7 and 6.10. It must be remembered that the exterior derivati-

ve of a scalar-density-valued function is a scalar-density-valued dij

ferential form. An alternative, equivalent definition of a proper fUl

ction is given below.

Definition : A scalar-density-valued function ~ on N is called

a proper function of the lagrangian submanifold N of (P~tvJ~)if for

109

8ach volume element m at x the function < E!.,§. >2 is a proper function

JI N with respect to the special symplectic structure defined by forms

(E!., e~>2 and < E!.'W~> 2. A scalar-density-valued function S is called

a generating function of N if < E!.,S> 2 is a generating function of N

"..:ithrespect to (E!., G~ >2 and (E!.,W~> 2 •

Definition We say that the infinitesimal dynamics of the field

is specified if a lagrangian submanifold D~ c p~ is chosen at each point

x E M.

The condition 'J1p(x)e D~ for a section M 3 X - p(x) E Px is called

the field equation. The field equation is actually a system of 1-st

order partial differential equations.

Vle assume that for each x E M the infini tesimal dynamics has a ge-

nerating function ci . Thexnoted by ~ • Both L andx x

corresponding proper function of Di is de­x

~x are defined up to an additive constant.

In alI examples of physical field theories considered by us there are

however natural choices of unique generating functions of the dynamies.

There is a unique distinguished state of the field in each theory which

describes the physical vacuum. The distiguished generating function

is the one which vanishes on that state. We shall always use such dis-

tinguished generating function also with respect to other special sym­

plectic structures. The family {l'x} defines the scalar-density-va­

lued function

16.31 L Qi /\ T*M

which is called the Lagrangian. Using coordinates (x;\','fA,p~, 'f~'d€A)

we may write l as

16.32 L L dx \ ••• (\dxm

where L

110

<E:, L> 2 and

16.33 m(3 a

--1 A ••• 1\ -m(Ix dX

The function L is scalar-valued. The generating formula 8.24 assumes

the following form

(, "A ;t A\ 1 m;}€Ad LP + pAd f;t j ® dx 1\ ••• /\ dx

16.34

(" A A) 1 mdLx 'P ' lp::\, ® dx /\••• /\dx

equivalent to the formulae

16.35

'O A,

o x'A..P A df.A

(7

d'-PA L(x~, fA, \f t)'" a

PA = - L Ix" U)A (jlA·)(J~t \. ' T ' I ?v

The formulae 16.35, equivalent to Euler - Lagrange equations, are coor-

dinate expressions of field equations.

It is usual to postulat e field dynamics in infinitesimal form

from which the dynamics for finite domains is derived. The procedure

is similar to that used in particIe dynamics /formula 8.22/. We intro-

duce the class of dynamically admissible sections of P which are so­

lutions of field equations. The Cauchy boundary values of these solu-

. ~V ~Vtlons form subspaces D' c P~

16.36 D8V

{ poV E poV there is a section M" Y ~ PCY)E. P }

I 3V ~1 Yisuch that P 3V :.p and J P(X)E Dxfor x E V

If the composition property 15.24 holds then the above formula

111

2an be considered as the limit of formula 15.23 when the number of

subdomains Vi tends to infinity and their dimensions tend to zero.

In general the proof that the formula 16.36 defines a lagrangian

submanifold of pcV is difficult and has been carried through only in

:ew special cases. However, the proof that DOV is isotropie is rela-

~ively easy and is based on a non-linear version of Green's formula

analogous to the formula 15.15. Let

16.37 ('G",s,x) p_ (X)EPc ,s x

be a two-parameter family of dynamically admissible sections of P, i.e.-1 ij P'"'~ (x)EDx• For each x E!"l denote by Y(x) and Z(x) vectors tangent to

the curve

at '0

't:

o and to the curve

3

Pt: O(X) E p, X

Po ",(x) E Pp x

at S = O respectively. We see that YldV and ZloV are vectors tangent

to the curve

at <::

t;

o and to the curve

~

(P't',O lov) E p3V

'"" (po,S' ?N) E poV

at ~ = O respectively. These vectors are obviQusly tangent to D3V •

Using 16.21 and 16.19 we obtain

112

< CYI oV)/\ (Z I oV), wcV> «r:-1 ( \ ,.,.1 ., i "-'\ J Y X)/\J Z(x/ ,Wx/ 1-.)

V

The integrand vanishes for each x E M since 'J1y(x) and 'J1Z(x) are vec-

16.38

tors tangent to the curve

t ~1 .j p~ O(x~ E Dlv, I X

at L: o and to the curve

5'~ -1

j PO,S (x) EDix

at S o respectively, and D~ is a lagrangian submanifold of P~. Thus

16.38 «y I;W)A (Z I oV) , G}V> o

The formula 16.38 may be considered as a generalized Green's formula

which in the case of electrostatics reduces to the formula 15.17.

If the submanifolds DOV are lagrangian and if there is a unique

solution of field equations for each Cauchy data in D~V then a compo­

sition law for generating functions analogous to 8.26 in particIe me-

chanics can be proved. In the simplest case of no constraints (c8V =

QCV) this law can be formulated as follows.

Theorem 7

Let a function WcV be defined by

wOV (\p0V) Ilx( j1 ~(x))V

where for each \fClV E QOV the section

16.39

16.4-0M :3 X ~ t.p (x) E Qx

113

is a stationary section lin the sense of the calculus of varia-

tionsl of the right-hand side, such that

16.41 lf I~:W 'foV .

The function WOV is a generating function of DOV•

The formula 16.39 can be treated as the limit of the formula

15.25 when the number of subdomains Vi tends to infinity and their

dimensions tend to zero.

Euler -Lagrange equations 16.35 imply that stationary sections

are precisely the projections by X of dynamically admissible sections.

Due to the lack of constraints the above theorem can be stated in terms

of proper functions :

Theorem 7'

Let a function W~V be defined by

!!..OV(pOV)16.42 j Ix (:J1p(x)}V

oV ~v .where for each p ~ D the sectlon

16.43 M '3 X ~ P (X) E Px

is the unique dynamically admissible section of P such that pl3V

av Th f t" Wav" f t" f DOV= p • e unc lon lS proper unc lon o •

The proof is practically the same as in mechanics.

Proof: Let

16.44 re: pClV('t:) E DOV

114

b . D3V Th' ..e a curve ln • ere lS a unlque mapplng

16.45 et ,x) P1;' (x) E Px

such that for each <:: the mapping

16.46 M3 X P" (x) E Px

is the dynamically admissible section satisfying the boundary conditior

16.47 P" I avav

p

Denote by Y(x) the vector tangent to the curve

16.48 't' ~ p.•(x) E Px

at '<:: O. Then J1y(x) is the vector tangent to the curve

1;'..-,..1 ) DiJ P". ( x E x

at 1;' o and YloV is the vector tangent to the curve 16.44 at ~ = O.

< YI dV, tjV;>

Now

16.49

<YlaV,d!{'W>~ WOV(pOVC'(;») Id't' - 't' =0

~~'t'lx(r1p",CX»)!c:=0V

~<}1y(X)' G~>1V

~<Y(X),8X)1oV

= ~i;' ~4:x(J"1P'l:(X») l~ =0

(' Vj<y1y(x) ,dLx> 1V

~div (Y(X), ex? 1v

which completes the proof.

115

Example 1

Equations of eleetrostaties 15.5 and 15.6 define a 4-dimensional

submanifold Di in the 8-dimensional infinitesimal phase spaee pi des-x x

eribed by eoordinates Cr,pA.'lfx-,;}€.) where A= 1,2,3. The submanifoldDi is deseribed by 4 equations :x

pA, = {? g ;1..l'-lf,..

16.50

df, = -4'J"C~r(x)

Restrieting the form 8~ = (oe dlf +p;>"df~) @ dx 11\ dx21\dx3 to D~ we obtain

16.51

SiJDix x rg(-4%r(x)dif+ g'>vf<-Lff'-dtp,J@dx1l\dx2/\dx3

d {~(-4X r(x)\f + ~g;tf<-lfA,lf1,.)&dX\dx2/\dx3 }dL .-x

It follows that Di is a lagrangian submanifold generated by the Lagran­x

gian

16.52 L (x'A, lf ' fA-)

Example 2

~ (~gl\,fllfJv tp~ - 4X rf) dx ~dx2Adx3

The Lagrangian

1(x"', tp , lf/\,) ,..---,( ) 122 1'A, ) O 1 2 3~-g F(lf - 2m t.p + 2g f'~?v \fI'- dx ",dx Adx Adx

",.;here (g;>.,f'-)is a pseudo-riemannian metrie tensor in the four-dimensio­

2al spaee-time M generates the dynamie s

pA, = R gAlL Lfl'-

"-S.53

o:\,p:\' Je R(F/(Lf) - m2yi)

116

which corresponds to the scalar field theory wit h field equations

16.54- (o + m2) Lf F/C,?)

The infinitesimal phase space described by coordinates (tp,pA-,~?v, d-e) ,

Iv = 0,1,2,3, is now 10-dimensional. The submanifold Dl described byx

equations 16.53 is 5-dimensional.

17. Hamiltonian description of dynamics

The description of dynamics in terms of the Lagrangian was obtai­

ned by using canonical /scalar-density-valued/ special symplectic stru

ctures in infinitesimal phase spaces P~. The hamiltonian description

is associated with different special symplectic structures which are

no longer canonical but depend on the choice of a connection in the

configuration bundle Q. A connection in Q enables us to define a ho-

rizontal m-vector at each point of Q, analogous to the Urest frameu ir

mechanics. Thus there is a whole family of hamiltonian descriptions

associated with different connections in Q. This generality is not

necessary in the case of the relativistic field theory in given pseu-

do-riemannian space-time M. In this case Q is a tensor bundle over M.

There is a distinguished connection in Q induced by the given affine

connectio~in space-time M. Thus there is a distinguished hamiltonian

description of such a theory. This is not true in the relativistic

theory of gravitation where the affine connection in space-time is it-

self a field variable and cannot be treated as a fixed structure. Gur

generality is thus motivated by applications to General Relativity.

Before going further into details we have to dra w the attention

of the reader to an ambiguity of the terminology which appears in the

literature. There are two different generating functions which are

called uHamiltoniansu• One of these objects is the Hamiltonian dis-

cussed in the present section. This object is little known by physi-

117

cists. It has been introdueed in geometrical formulations of the eal-

eulus of variations lef. [10J, [17], [241, [261.1. The name "Harniltonian"

is usually given by physicists to the other object which we calI "ener-

gy" and diseuss in the next section. The ambiguity arises because in

the case of dim M = 1 Iparticle mechanicsl the Hamiltonian and the

energy coincide.

Each infinitesimal phase space pi is already fibred over P • Thex x

fibration 0~h : pi~p is given by the jet-target projection 0p•x x x

3ach connection in Q determines a diffeomorphism

17.1 O\hx pix

0't1.

T*P ® i\ T* Mx x

may thus be trea­

ho<.xwe pull-

G:Jh fromx

ted as a Iscalar-density-valuedl covector on Px. UsingAh

forms G andx

which will be constructed later. Each element of pix

-back canonical 1 scalar-density-valuedl

'r*p Q9/\ T* M to pi. We shall prove that in terms of coordinatesx x x,', A! A"lf ,PA' Y'A.' ;}tA)the result are

17.2 G~ o( h*nhx V'x ('lD A A ?v) 1 m= CTC-Adt.p- lp;l,dPA iSldx/\ ••• J\dx

17.3 h!t '" hcXx Cvx (A A Iv) 1 mdeteA/\d'P - drA, dPA @dx/\ ••• /\dx •

The above formulae are analogous to 9.8 and 9.9 in mechanics. The

. h*" h iequallty 17.3 shows that cxx wx = C0x •

Similarly as in particIe mechanics, if the generating function

of the dynamics Di with respect to the above special symplectic struc-x

ture exists then, taken with the opposite sign, it is called the Ha-

miltonian at x and denoted by dlx• The corresponding proper funetions

are denoted by - dtx• The family {~x} defines the sealar-density-

valued funetion

17.4 x prY1.

---~7 /\ T*M

118

which is called the Hamiltonian.

For any coordinate system (x:\'jin M the value of :Je ('Je. and Je. x-x

respectively) is proportional to the scalar density dX~ ••• Adxm. The

corresponding proportionality coefficient is denoted by H (H and Hx -xrespectively)

17.5 'Je H( (DA ;t) d 1 d mx, T ,PA @ X/\ ••• I\ x

The generating formulae for Di are obtained from 17.2~ x

(de Ad fA - lf~dP2') e9dx1/\••• /'\dxm

17.6

-dH (wA ?v)' 1 mx T ' PA ® dx /,••• /\dx

This is equivalent to equations

-dJLX(~A,p~)

(GirA =) 'f~

17.7(?v

0?vPA =) de A - c~A Hex/", 'f'A,pf)

a

c :\,H(x~t, wA A)'PA 1 ,PA

analogous to Hamilton canonical equations in mechanics /see [9J, ~7J,

[24-J , [26) /.

Now we give the construction of the above special symplectic

structure in coordinate-free language. We first define a differential

m-form Gron p depending on the connection in Q. Let ID be an m-vector

tangent to P at p e Px of the form

17.8 ID YA fi

where Y is a vertical vector in P and § is an (m-1)-vector. We define

the value of lY on such m-vectors by

17.9 <§, 19-'/

119

<.!];,<Y, BX>1 > 2

where n = ~~Q is the projection of Q onto M. Let now § be any m-vec­

tor tangent to P such that ~~§= O. Then § can be decomposed into the

sum of m-vectors of the form 17.8. This decomposition is in general

not unique. We use the formula 17.9 to define by linearity the value

af 1r on such sums. We shall show later that this definition does not

depend on the choice of the decomposition of §. This way we defined

the value of ~ an the subspace of m-vectors which have zero projec-

tions onto M. This subspace is of codimension 1. It remains to define

the value of .~ on a single vector which has a non-zero projection on­

to M. This vector will be roughly speaking the horizontal vector. How-

ever, in general the connection in Q does not determine a connection

in P. We adopt the following definition of horizontality :

Definition: An m-vector § in P is horizontal if its projection

~§ onto Q is horizontal with respect to the given connection.

We set the value of tr on horizontal vectors equal zero. Since

horizontal vectors do not form vector subspaces the correctness of

this definition must be demonstrated. We return to this point later.

The above construction of ~ depends on the choice of a connection

in Q. However, the value of VV on m-vectors which have zero projections

anto M does not depend on the connection.

We denote the differential of 1Y by ~

17.10 p dV-

]1he form p is used in the construction of ex:~ •

Let m be an m-vector tangent to M at x / a volume element at x/o

?or each jet g = j1p(X) of a section

120

17.11 M :3 Y , p(y) E PY

we denat e by §g the lift of m to the section. Thus mg is an m-vector

tangent to the section and

17.12 n*mgL -

.!!!

Let g be an element af pix~1Jx P • For each vector Y tangent to Px we

set

17.13 < .!!!, <Y, O<~Cg»1) 2 < YA§g, cP >

where g is any representative of the class g.

We prove the correctness of the above definitions by giving coor·

dinate expressions of li , <i> and o(~. Let (x':\..,lfiA) be a geodesic

be the corresponding coordinate sy

are horizontal with respect to theat points of ~

(;1., A ;l)Let x, Lf ,PAchosen connection.

coordinate system of Q in a neighbourhood of Qx. This means that vec­a

tors oxh attached

tern in P. The reader may easily show that the form

17.14 v- ~ 1 A mPAdx A ••• /\ d\f A ••• J\dxJ\

';l

jthe factor dx?v in the product dx1J\••• /\dxm has been replaced by dtpA

satisfies alI requirements stated in the definition. If, for example,

17.15 yG

o\.fA

and

17.16 fiA--1 Q a a

(-1) --1'1\· •• /\ d).,-1 /\ '1 ).,+11\···/\ox X ux

'O

Oxm

17.17 <Yl\~,v> A,PA

121

< ~,< Y, e x) 1> 2

~s is required in the definition 17.9. One shows in a similar way that

the form 17.14 satisfies the remaining conditions stated in the defi­

tion. It follows from 17.14 that the coordinate expression for ~ in

seodesic coordinates is :

17.18 p A 1A mdpA/\dX A ••• A dr,~ /\••• 1\ dx

:Iv

Let (~A,pl, rt,p;r) be coordinates of the jet g j1p(X) and let

"7.19

then

.!!!

c O

b· a x/I/\••• 1\ a xm

17.20

~~~..l

mg(; o A a A d) (d A Q ?v d \b·-'\1+ \f1 (j,nA+PA1 'O ?v 1\"'/\ -;-m+ lfm QWA+PAm o ?v I .Clx T (I PA ux T PAl

17.21

then

y 'J :\ ()u B _.

AA C'fA + A dP;

17.22 <y 1\ ~g, <P > (A?v :!v A)b' A PA?v - BAlfA,

The above equality proves that the right-hand side of 17.13 does not

depend on the representant g

= (lfA,p~, 1ft, ~A=P}?v)' This

(({)A,p2:"lfA,p~:'r)of the class g =

proves that the deflnition of o(h isx cor-

recto In order to be able to give an explicit coordinate expression"'-

for O(~ we choose coordinates (lfA,p;,rA's~) in T"!fpx® 1\ T~ M cor-

responding to the unique representation

17.23 r

122

(A A ?v, 1 mrAd\f + S;>...dPA/(8) dx /\••• /\dx

of an arbitrary jscalar-density-valuedj covector ~ on Px' Since

17.24<l!!'<Y'~> 1>2 b . (AArA + B~S~)

it follows from 17.22 and 17.13 that

17.25

where

O<~(\fJA,p}, lf~'(J{A) ( A?v A)\ 'f 'PA' rA' S?v

rA =CJeA

17.26

AS?v

A

-lf\

This result is analogous to the formula 9.7. Substituting 17.26 into

17.23 we obtain the formula 17.2 from which 17.3 follows immediately.

Similarly as in mechanics we define the jscalar-density-valuedj

function li) on pi byTx x

17.27 <l!!, ~)x(g) > 2 (.@g, t7/

It is easy to check that also in this case the right-hand side does

not depend on the choice of a representative gE J1(p) of the classx

g E pi. In terms of geodesic coordinates we havex

17.28 rx( 'f A,P~, cpt, 8eA) ?t-lOA 1 mpA T?v dx /'••• 1\ dx

This formula is analogous to 9.17. It follows that

17.29 d 'fx

123

( ~ A A ~.) 1 mpAd lf?v + lf?v dp A 0 dx /,••• /\dx ei _ ehx x

':I'hefunction Je on Di defined as-x x

17.30

satisfies the equation

Xx II) IDi - iIX x -x

17.31 d Je-x d iIJ IDi - d LIX X -x (d ljJx - e~)ID~ - G~ID~

~ence - dex is the proper function corresponding to the generating

function - .'Je •x

Example 1

Field equations 16.50 of electrostatics can be written in the form

17.32

Restricting ehx

olf 1 ~= lff-=-g ;l..P3xf Ijg't-

o pA-

=;re=- 4-7L Rr(x) •

c xA.

(o-edtp - tpl,-dp/<-)0dX1/\••• /\dxm to D~ we obtain

17.33 ehlDix x (- 4-X fi r(x)dtp -

1

fi' gl'\\.PA.dpl')@dx\dx2/\dx3

Hence

- d.f Vf?(4-JC rex)l

- d Je-x

1 ) 1 2 3}+ - gr;\.pl'pC\- dx /\dx /\dx =2g

17.34- 'Rex?, er,pA) {gl (4-01: rex) 'f? + ;g gl';Lpl'p<-) dx \ dx2/\dx3.

124

Example 2

Equations 16.53 of the sealar field theory may be written in the

form

17.35

lf,u-

1Rg,u-?.P'A.

Restrieting

.~ = ~_g' (F/(~) _ m2Lp)

e~to D~ we obtain in this ease

17.36 Je ex"', <f ,p"') 'r-::'(12 2 ( 1 ,-) o~-g 2m 'f - F <f) - 2ggl'-"-pl'pdXA•••Ad

In both examples the equation

17.37 H-x '" I iP lf", Dx - !ox

equivalent to 17.30 is satisfied.

18. The Legendre transformatio~

We define the Legendre transformation assuming that the infinitE

simal dynamies Di is a seetion of the fibration X i : pi. ) Qi.x x x x

This means that for eaeh element V€o Qi l"veloeity"l there is a uniqUEx

element g 6 pi whieh projeets on v :x

Qt~(g) = v.

The mapping

Qi '3 vx J[;~(€~') E Px

is ealled the Legendre transformation lef.

also a seetion of the fibration xhx pix

[9]/. In many eases Di isx

• Px' This implies

125

chat the Legendre transformation is invertible. In terms of coordina-

. A ?c. A ) ices (~J ,PA' lfA-' GeA the above statements mean that Dx may be parame-

crized by coordinates (c.pA, cp~ ) or by coordinates (cpA ,p}). The dyna­

3ically admissible infinitesimal state of the field

g = (lpA,pl,<p~,OfA)

A

is thus uniquely determined by its "position and velocity" /lp and

ft/ or its "posi tion and momentum" / \fA and p~/.

19. Partial Legendre transformations. The energy-momentum density

The complete Legendre transformation replaces the velocities~f

~y momenta p~ as argument s of generating functions. Partial Legendre

cransformations replacing only some components of the velocity by the

~orresponding components of the momentum can also be performed. We

introduce below partial Legendre transformations corresponding to a

,ector field X on M and an (m-1)-vector n in M / a hypersurface volu­

3e element/ transversal to X.

We begin with simple calculations in an adapted coordinate system.

~et n be attached at a point xEM. We choose a coordinate system (x?c.)

in a neighbourhood of x in such a way that the field X is

'9.1 Xa

<h1

~nd the volume element n is

19.2 na

Q x2 /\ ••• /\

d

dxm

~et B(X ,~) be the space parametrized by coordinates (c.pA, 'f;,p1) whe-

~e k = 2,3, ••• ,m, and let

'3.3 0(: pix

"'-

TltB(X,~) @ !\ T~ M

126

be an isomorphism such that the pull-back of the canonical /scalar­n'I_

density-valued/ 1-form in T*B(X,n) ® AT'" M is- x

19.4 G= (A A 1 k A) 1 matAdcp - f1dPA + PAd lfk ®dx /\••• 1\ dx

Since the exterior differential

19.5 de (A A, A) 1 md3eA'df + dPAAdf:t gdx/\ ••• l\dx

is equal to cui, we see that we have defined a special symplectic strx

ture in pi. Comparing 19.4 withx

19.6 e~ ( dwA -::l A) 1 mdeA T' + PAd f:\, @ dx /\••• A dx

A 1we see that ~1 and PA have been exchanged. The complete Legendre

transformation exchanges all velocities and momenta. Here only the

component of the velocity in the direction of X has been exchanged wi

the projection of the momentum onto g. Thus we have performed a par-

tial Legendre transformation.

We defined the scalar-density-valued form & in order to have a

closer analogy with the complete Legendre transformation. However, it

is more convenient to use the scalar-valued form

19.7 8(X,EJ <XA!);, 8)2A A 1 k A

aeAdlf> - ~1dPA + PAdfk

obtained by contracting the form 6 with the volume element X~. The

form e(X,n) can also be obtained as the pull-back of the canonical

1-form in T*B(X,~) by the isomorphism

19.8 o< (X,!);) pix T*B(X,!);)

127

which results from composing the isomorphism ex with the volume ele-

:lent X"n

19.9 «ex,.!!) (g) < x".!!, CX(g) > 2

We introduce in T·JfB(X,.!!)coordinates ((fA, tp~,pl,mA,r~,sA) cano­

nically related to coordinates (lfA, f~,pl) in the base B(X,.!!).The

coordinate expression of CX(X,~) is

19.10

where

. A A, A )O«(X,.!!)(4'7 ,PA' 'fA,' ?fA

mA = Cf.A

(A A1 kA)~ 'c.rk,PA,mA,rA,s

19.11 krA

sA

kPA

-cp~

k)2,

The generating function -E(X,.!!)of the dynamics with respect to

the above special symplectic structure is obtained from the following

generating formula

19.12 -dl;(X,l!) E9(x,l!) ID~

where -l;(X,.!!) is the proper function. In terms of coordinates

19.13 (A A 1,-dE(X,l!) \f ' tflk,PA). A A 1 k A

JeAd <f - f1dPA + PAdl(Jk

This is equivalent to the following form of field equations

OfA

Cl

- 3<.pAE(X,l!)

128

Lft

(3

19.14=--1 E(X,n)

()PA -k

= __ 3_ E(X n) k ?i 2PA04>~ '-

,

The minus sign in front of E(X,Q} appears in order to give E(X,Q) the

interpretation of energy.

The form S(X,n) differs from <XAn, Si>- - x 2

the differential of the function pl ~~. We have

d loA A,d wA ,~A T + PA lA, oy

19.15

Hence

dL3eAd LfA + P~dtft -dE(X,Q) + d(p1~~)

19.16 1;(X,Q)1 A i

P Alf 1 IDx - !! .

Example 1

We calculate the generating function E(X,Q) for the scalar field

theory. The field equations 16.53 can be rewritten in the form

kP ,/:'-kl -kl 1~-g g er l - g g11P k,1=2, ••• ,m

19.17 lf1 1 ( -lk ) 1V -g' g11 - g gk1gl1 P + g11glklpk '

r-----J ( I. 2)a-e = V -g F(t-f) - m \fi

Here we assumed that the matrix gkl is nondegenerate and denoted its

inverse by gkl. This assumption means that Q is not tangent to the

light cone. Restricting the form 19.7 to Di we obtainx

seX,Q) I D~,~( I . 2 ) l j ( -kl ) 1v-g F (\f) - m lf dlf'- Fi g11 - g gk1gl1 P

"9.18

129

-lk ] 1+ g11g lfk dp +

_ d ( 1hV_glg11P1p1

( .C' -ki -ki 1) l~-g g lfl - g g11P dlf

R(F('f) - ~m2lf2) -

__ 1__ -kl(,~ 1)(,~ 1 } .2 R g l-gtpk - gk1P ~-glpl - g11P) = -d~(X,Q).

Ihe corresponding proper function ~(X,Q) expressed in terms of lagran­

sian variabies CLp, Lp:>.Jis:

"9.19

;g; (X,Q ) ,i' ( 1 2 2) 1.1::1 k?v . 1.r--.:> 1q,- V-g F(lp) -2m lf -2 v-g g lf~ lfk+2 V-g g tp;>H1=

(R g1A-Lf~)\f 1- V _g'(F(lf) -~m2lf2+~g AAlf\, 'fI'-)

~his resuit confirms formula 19.16.

Now we construct the space B(X,Q) and the form (9(X,Q)in a coor-

dinate free fashion. It will be shown that in the special coordinate

system the general construction reduces to our previous constructions.

We take the volume element m = XAQ at x. Corresponding to the decompo­

sition of ~ into X and Q there is a decomposition of the space of jets

Qi into two components Qi(X) and Qi(n). The space Qi(n) is the spacex x x- x-

·Jf1-jets at x of sections of Q restricted to an arbitrary submanifold

z=cM tangent to n at x. An element of Qi(n) contains information on- x-

the value of the field t.p and its derivatives in directions tangent

to E /to L, /. We denote by

19.20lQ(E) Q~(E: ) 7 Qx

the canonical jet-target projection. If the coordinate system in M

satisnes 19.2 then a point of Q~(,Q)is described by elfA,lf~) , k;f 2

and the coordinate expression of LQ(,Q)is

19.21

130

A A

lQ(1!) (ep ,Cfk) (~A)

The remaining information about derivatives of srA is eontained in the

Lie derivative "txtp of the physieal field er' wit h respeet to X provi­

ded that the eonfiguration bundle Q eonsists of objeets for whieh the

Lie derivative ean be defined. The value of the Lie derivative is a

veetor tangent to the eonfiguration spaee Qx lef. Appendix Gl. This

justifies the following definition

19.22 Q~(X) TQx

The eanonieal tangent bundle projeetion onto Qx is denoted by

LQ(X) Q~(X) ) Qx

For the sake of simplieity we deal in this seetion only with bundles

of geometrie objeets in M Imore general eases will appear in Chapter

IV l. In this ease the value of the Lie derivative ealeulated in an

adapted eoordinate system 19.1 is equal to

19.23iX CPCx) ~~

8

8'{>A

As eoordinates in Q~(X) we may thus take the system (({JA,<p~) • In

terms of these eoordinates

19.24 c Q(X) (tpA, cp~ ) ( lfA )

The spaee Q~ can be identified with a subset of Q~(X)XQ~(1!)

19.25 Q~ { (sX's1!) E Q~(X)XQ~(1!) IlQ(X) Sx l.,Q(1!)s1!}

131

An element p E P determines a covector Px €i T*Q given by the for-x xmula

19.26 Px <E:,P> 2

",-2,

and a /vector-density-on- L -valued/ covector p E T"*Q ® A T*L:::: de-n x x

fined by the formula

19.27 <~,Pn>2 < -X~,P>2

where k is any (m-2)-vector tangent to ~ at x. We take

19.28

and

/.9.29

p (X)x

Px(E:)

T*Qx

",-2

T*Q ® 1\ T*Z=x x

~anonical projections onto Qx are denoted by

-19.30

s.nd

"'19.31

Xx

Xn

px(X) ~ Qx

px (g)----t Qx

Ii the coordinates OI M satisfy 19.1 and 19.2 we have for an element

p = (~A,p~) the equalities

~9.32

3.nd

Px pld~A

19.33 Pn

132

( k A) a 2 mPA d 'f ® (-,\ k -1 dx A ••• /\ dx )Ox

this m~ans that (~A,pl) may be used as coordinates in px(X) and

p~) as coordinates in Px(~). Obviously

19.34

and

19.35

Je (\1) A 1)X f ,PA

X,g(lf7A,p~)

(lfA)

(~A)

The space P can be identified with a subset of P (XlxP (n)~ x x / x-,

19.36 Px { (PX'P,g) E: Px(X)><Px(,g) I Jt'x(Px) Jr'n(Pn)}

Spaces Q~(~) and Px(~) have been obtained by projecting spaces Q~

Px anto the hypersurface 2:.These spaces can also be obtained ~y 3

plying to the restriction of Q to 2: the procedure that was used ~c

canstruct Ql and P fram Q. If the restriction of Q to L: is den:::~3x x

by QIL then

19.37

and

19.38

Q~(,g)

Px(,g)

J~(QIE)

",-;1,

T*(QII::)x@ !\T;-:Z;

We denote by p(2:) the union of alI such spaces obtained for all

points x E M. The tensar product in the above formula is understo:::c::"

the sense explained in Section 13 and used in formula 13.1.

We may auulv the same procedure / ar rather its simpler ve:!'",:.:z

133

from Chapter II I to the restriction Ol Q to the parametrized integ-

raI curve

19.39 R 3 t ~(t) E M

passing through x and tangent to X at t o. The resuIting spaces are

'19.40

::.nd

"19.41

Q~(X)

px(X)

J6C Q16)

T'* (Q I t )0

~gain we denote by pet) the union Ol alI such spaces obtained lor alI

points Ol 5 . This procedure can be carried one step lurther. We con­

struct spaces pi(X) and piCn) Icorresponding to pil :x x - x

19.42 P~Cg)=J1P(L)x5.nd~9.43

P~(X)=Jil(o) .

~e have canonical projections

19.44

iJCx

iXn

pi(X)x

P~(.Q) ---.

Q~(X)

Q~(.Q)

Gp(X): P~(X) ~ p (X)x

iGp(.Q): Px(.Q) ~ Px(.Q)

The following diagram is commutative

Using coordinates (xk), k = 2, ... ,m, on

o (Ak A-";) .---coordlnates ~ ,PA'LPk'oeA where GIt =

use the coordinate t = x1 on the curve

134

p~(X)

Lp cX)Lpc,g)

P~(,g), PxCX) Px(,g),

19.45

!or~\1x~jLQ(X)

LQ(,g)Q~(X)

)Qx < Q~(,g)

The standard constructions provide also the lagrangian special symplec­

tic structures given by canonical diffeomorphisms :

19.46 o(~(X) : P~(X)}TJf-Qi(X)xand

o(~(,g): P~(,g)

o ",-1

19.47>T*Q~(,g)® /\ T;2:

~ we may construct in picn)x-k I o olPAk. n a Slml ar way we may

O = { xk = const.j . The method

used in Chapter II yields coordinates (lpA,p1,tpA,p1) in P~(X) where

tpA = tp ~ are coordinates of the Lie derivative .txlf and p1 = P11are coordinates of the Lie derivative of the momentum contracted with

n :

19.481

PA1 <.. ,g,lx p> 2

In terms of these coordinates all projections in the diagram 19.45

are effected by omitting apropriate coordinates.

With each pair (gx,gn)€P~(X)XP~C,g) which projects onto the same

element in Qx' i.e.

19.49 Xx" VPCX) (gX) Jrn" VP(,g/~)

we associate an element of p~ denoted by ~(gx,gn). The following for­

mula determines uniquely o(~(4)(gx,gn))

135

< xA~,<u, O(~ ( ~ (gx' g~) ) > 1> 219.50

<ux,O(~(X)(gx» + <~,<U~,o(~(~)(g~»1>2

where u = (UX' un) is a vector tangent to Q~c Q~(X)XQ~(~).

The vector Ux is thus tangent to Q~(X) and un is tangent to Q~~).

The element rp (gx,gn) is completely determined si~ce o(~ is an isomor­

phism.

In order to obtain the coordinate expression for rp we take a vec-

tor

19.51 u A <3 A a

B o lfA + B'A, o lft

It is easily seen that

19.52

and

19.53

If gx

19.54

BAO

Ux =8lpA +

BA

au = d'fA +

n

(A 1 'A '1)~ ' PA' lp , PA and gn

< uX' ()(~(X)(gx»

A (1

B1 ~lf1

A (3

Bk 0lfk

(Ak A ~)~ ,PA' cpk' Jt'A then

BAd BA.1PA + 1PAand

19.55 <~,<u~,CX~(~)( g~)> 1> 2A ~ A k

B :1eA + BkPA

I'he right-hand side of 19.50 is thus equal to BA(P1 + Je A) + B~p~ •

It follows that ~(gx,gn) has coordinates (~A,p1,c.p~,;1eA) , where

19.56 JeA·1PA + JeA

1 kPA1 + PAk

?v

PA';\,

The special symplectic structure in pi which we are going to de­x

136

fine will be a mixture of the lagrangian structure in P~(E) and the

hamiltonian structure in pi(X). In order to define a hamiltonian stru­x

cture in P~(X) we need a connection in the bundle Qlo ' i.e. the no-

tion of the section "constant in the direction of X". This connection

is given by the Lie derivative : the section is constant in the direc-

tion of X if its Lie derivative with respect to X vanishes. We used

this connection already in 19.22 identifying the space of jets Q~(X)

with the tangent bundle TQx'

As the base of the special symplectic structure which we are go­

ing to define we take the space

19.57 B(X,n) f (PX,sn) E Px(X)>(Q~(E) I JrXPXl.. _ l,Q(E)SE}

descri bed in our coordinate system by coordinates (LflA,Lfl~'pl). The

space pi is fibred in an obvious way over B(X,n). Using local coordi-x -

nates one can prove that this fibration is given by

19.58 p~ ~ g --> X (X,E) (g) ( GpeX) gx,Jt'~gE) E B(X,E)

where (gX,gn) is any pair such that 4>Cgx,gn) = g.

We define an isomorphism

19.59

by the formula

c( (X,E) pix ------~) T*B(X,E)

< v, O«X,E)(g) >19.60

<VX,c\~(X)(gX» +

+ <E, <vE' c\~(E)(gE» 1> 2

where v(vx'vn) is a vector tangent to B(X,E) c Px(X)xQ~C~) and

19.61h

CXx P~(X)

137

T*P eX)x

is the isomorphism defining the hamiltonian structure in P~(X) / con­

structed e.g. in Section 9 /. Using local coordinates one can easily

verify the formulae 19.10 and 19.11. This proves that the definition

19.60 does not depend on the choice of the pair (gx,gn).

Formulae 19.50 and 19.60 may be interpreted in the following way.

The element g is decomposed into a pair (gX,gn). The value of the la­

grangian isomorphism o< i on g is obtained by applying the lagrangianx

isomorphisms O(ieX) and O( i(n) to gx and g respectively / formulax x - n

19.50 /. The value of the "mixed isomorphism" o«(X,~} is obtained by

applying the hamiltonian isomorphism c<.~(X)to the componentgx and

the lagrangian isomorphism O(~(~) to the component gn / formula 19.60/.

0orresponding to these constructions also the special symplectic forms

ei and &(X,n) are obtained as combinations of lagrangian and hamil-x -

tonian special symplectic forms in the component spaces P~(X) and

?~(~).We have

/9.62< w, <Xl\~, (9~>2) 1 <Xl\~, <w, e~)1) 2

(wX' G~(x» + (~,<wn' f)~(~»1>2

~~ere Qi(X) and Qi(n) are lagrangian special symplectic forms in piCX)x x- x

~~d P~(~) respectively, w is a vector tangent to p~ and (wX'wn) is a

~ecomposition of w into components tangent to p~cx) and P~(~) respec­

~ively. The pair (wX'wn) is a decomposition of the vector w in the

ssnse that the value of the derivative of 9 is equal to w when cal­

:::ulatedon (WX' wJ • We have also

i9.62' <w, G(X,~» < wx' e~(x» + <~,<w~, 8~C~»1) 2

138

The difference between these two special symplectic forms is

(' W,<XAQ, G~>2 - 8(x,Q» 1. i h

< wx' e/x) - Gx(x) >19.63

<wx,d \jl(x,Q),>,

where, similarly as in the Section 9 / formulae 9.17 and 9.18 /, the

function W(X,n) defined on pieX; is equal toT - x

19.64-- A 1 • A -1~(x,Q)(tp ,PA' \f ,PA)

• A 1'f PA

The function yvex,n) defines a function on p~ by the formula

19.65 If(X,Q)(g) \.fI (X,Q)( gX)

where gx is the first component in a decomposition of g. The coordina­

te expression for ~(X,n) follows from 19.64-

19.66 \f(x,n) (~A,p~, \f~' Ot'A)A 1

\f 1PA

We may rewrite this expression in a coordinate idependent form

\jJ (X,Q) (g)1 A

PA\f1

19.67 . c o < A -:>. ,..,

_ . u A, A o -1 m

< 8x21\'" t\ oxm' ~1 O'fA,PAdt.p ~((lXA,JdXA"'AdX~>2

Obviously

19.68

<n, <ix lf ,P:>1/ 2

<xJ\Q, e~12 - 8(x,Q)

< Q, (ixlf' ' Xh(g) >1> 2

d ~(x,Q) •

139

This implies in the general case the following generating formula

19.69

-d!; (X ,,g)G(X,,g) ID~ = { <XI\,g, 8~ >2 - d r(X,,g1ID~

<XI\,g, e~1D~~ - d{Y (X ,,g) I D~} =

< Xl\,g,d~x ~ - d {f(X,,g)ID~} =

d{(X/I,g, ~x> 2 - \fI(X,,g) ID~}

the

We always assume a normalization of !;(X,,g) such thatAabove formula re-

mains valid when the differentiation sign is omitted :

19.70 !;(X,,g) LJI CX,,g) ID~ - (x",g, Lx)2

We recall that the Lagrangian is always normalized in such a way that

it vanishes on the physical vacuum state. This gives the unique def i­

nition of !;(X,,g). Substituting 19.67 into 19.70 we obtain

"9.71

!;(X ,,g)Cg) <,g, (cixtp , X~(g» </2 - <XI\,g, .lxOO> 2

<,g,(ixlf':iU~(g)1 - X--1L:X(€O)2

~e see that !;(X,,g)(g) depends linearly on the hypersurface volume ele­

=ent n. The / vector-density-valued / function

'3.72!;(X) = <.~\\fl ,p> 11D~ - xJ lx

'",herefor each element g E p~ we denote by p the value of the projec­

~ion of g onto P , is called the energy-moment~m density correspon­x

iing to X.

140

If (xr) are any coordinates in M / not necessarily adapted to X

and n as in 19.1 and 19.2 / the vector density ~(X) may be written in

the form

19.73 ~(x) EA,(X)Cl 1 m

---1dx 1\ ••• A dxd xA-

The values of the components E~(X) follow from 19.72

19.74-

where

and

Elu (X)

x

(lX er A)p~ - XAL

xl\- ~3 x/v

Example 2

(' 1 m~ = L dx A ••• /\ dx

For the scalar field lf the Lie derivative is equal to the ordi­

nary derivative in the direction of X :

19.75

Thus

19.76

ixf

EA-(X)

Xi< lf f'

:Iv

x/yp~ cpf' - 6F!: )

In adapted coordinates 19.1 and 19.2 this expression reduces to 19.16.

The expression tA-r- = pA-tfr - ~~!:is called the energy-momentum ten­

sor density. In this case the energy-momentum density is related to

the energy-momentum tensor density by

19.77 E?v(X)lu

Xf-t r-

141

In the general case the Lie derivative of ~ depends on the value

of X and its derivatives up to some finite order. If Q is a tensor

Dundle over M then the Lie derivative of ~ depends on the first jet

of X. In the case of the gravitational field which will be described

in Chapter IV, the fundamental quantity er is the affine connection

in space-time. The Lie derivative of such a quantity depends on the

second-order jet of the field X.

We discuss in some detail the case when M is equipped with a con-

~ection. This situation occurs when we consider the classical field

~heory in given, fixed gravitational field. In this case the jet of X

is represented by the value of the field X and its covariant deriva­

0ives. In the simplest case when Q is a tensor bundle over M / tensor

~ield theory/ the Lie derivative of ~ contains only the first deri­

,atives of X. The splitting of the jet of X into two parts enables us

~o split the energy-momentum density ~(X)into a part proportional to

~he value of X and a part proportional to its covariant derivative.

-"'-8 have thus the general formula :

~9.78 E:\X) t:\-X;«- + t?uv v.. X f'-f'- f- )/

~here ~X;«- are components of the covariant derivative of X. The ten­

sor density t?u~ is called the energy-momentum tensor. We calI the ten­

sor density t?uv~ the second energy-momentum tensor. A discussion of

~hese objects is given in Section 25. The interpretation of the anti­

symmetric part of t?ujv follows from the following example.

Example ~

We consider a tensor field theory in the fIat spce-time M of Spe­

~ial Relativity. We use an affine coordinate system (x~) in M. With

~espect to this coordinate system the Lorentz metric

~omponents g~?u. Let X be a constant vector field X =

g has constant

xf- _0-. Then-O x'"

\S: Xl'-

19.79

o and

Elv(X)

142

tlv;< Xl-'-

The density ~(X) is in this case the component of the four-momentum

in the direction of X. Let X be a vector field X = ~;<~ g~>xY a:~withconstant skew symmetric coefficients 0~~. This field represents an

infinitesimal rotation of space-time. The corresponding density ~(X)

is given by

EJv(X)

19.80

where the tensor

G);<><g.;e,,(X>t:A,f'- + t?vllj<)

y. ••. (. 'A, SN)W x[>e t f-J + :>J?f'-

19.81~

S ~j<t?V}i O'

e,u. O~]li 1( A.-Y :A,Y)2 t f'-g~)I - t ~gf<)I

is usually called the spin angular momentum tensor. The density 19.80

is the angular momentum density in the direction of GJ~~. It is com-

posed of the orbital angular momentum and the spin angular momentum.

We note that the quantity ~(X,~) defined by the formula 19.71 is

the proper function for the infinitesimal dynamics only if XA~ ~ O.

The formula 19.71 defines ~(X,~) also when XA~ = O. No interpretation

of ~(X,~) in this case is given there. In Section 21 we define conser­

vation laws for the energy-momentum density valid also when XAn = O.

In Section 16 we defined the infinitesimal special symplectic

form G~ as a formal divergence of the form f}x/ formula 16.22 /.

The formal divergence can be decomposed into two parts in analogy with

the general formula

=-.32 X--1divo<'..

143

L O( - div(X.J ex)x

_~~s decomposition is exactly the one given in formula 19.62. The par-

Legendre transformation is a Legendre transformation applied to

-:-~:;first term.

T'he Cartan form

As in particIe mechanics the hamiltonian description of dynamics

~:;~ends on the choice of a connection. The same is true about the des-

_~~ption of dynamics in terms of the energy-momentum tensors. Again as

~~ particIe dynamics there is a method of combining the different Ha-

~~~~onians corresponding to different connections in Q into a single

::~9ct. This object is the Cartan form. We define this form in terms

:~ ~he Lagrangian and thus in a way manifestly independent of connec-

:~:ns. We show subsequently how different Hamiltonians are extracted

:::::-"::::1 the Cartan form as components with respect to different connec-

:~Jns.

We assume that there are no hamiltonian constraints. This means

_:"'~-J,....--.:::-..'-' for each element p E P there is a unique element g E Dix x

= p. Let g E J~(P) be any representative of the class g

such that

and let

- ~~ any m-vector in M at x I volume element at x l. The lift §g of m

:: the jet g is called the m-vector compatible with the dynamics. We

::-:;~allthat the detailed definition of §g was given in Section 17. The

::ordinate description of §g is given by formulae 17.19 and 17.20.

Definition : The Cartan form associated with the dynamics is an

=-:orm e in P such that

- ~_~'. l < §,G»/ A,-!!,v»

:~~ any m-vector ID in P which projects onto zero in M I the form ~

144

was defined for some connection in Q but we remember that the value

ofV on IDdoes not depend on the connection if ~§ = O / and

20.2 < §g,G> <.!!!,l/g»2 < ~§g '~xOO> 2

We show later that the definition correctly defines an m-form Qj

in the sense that 20.2 is fulfilled for any representative g if it is

fulfilled for one representative and if 20.1 holds.

Theorem 8

For any connection of Q let a / scalar-density-valued / function

Je on P be defined byx x

20.3 < .!!!, JexCp» 2 -(§h,e>

where m is a volume element at x and §h is any horizontal m-vec­

tor at pEP which projects onto m. The function Je is the Ha-x - x

miltonian corresponding to the given connection.

Proof

< .!!!,Jex(p) > 2 -(§h' 0) > -<§g,8> +

20.4-

because

+ < §g - §h' (9> -(.!!!,lx(g» 2 + < §g - §h' V/

20.57!-( §g - .@h)

Ag A

V[;.!!!- 1;.!!!hm - m o.

ni --,> pi is a diffeomorphism we may define a liftx x

The last term of 20.4- is equal to < §g ,1-1)­ce the mapping ;n;hx

since < §h' V,> = O. Sin-

145

Jrx of the above function to D~. AppIying the formula 17.27 we obtain

20.6

or

<E!, Jtx(g»2 < E!, Ifx(g) - .lxCg» 2

20.7 1ex lU I Di - 1-1x x -x

This completes the proof.

In coordinates (x~, ~A,p~) which are geodesic at x the formula

20.1 together with 17.14 give

20.8/' a

<--.. --1 .A. •••dx

o o

/\ d"lflA A ••• /\ d xmA,

,9 > (.",

PA

?he formula 20.3 implies

.20.9

',,':lere

o 'O

< --1/\'" /\ ~, 6) >ox dxm- H ,

.2:).10 'Je H dx~ ••• l'\dxm

28 form tv has to vanish on alI other m-products of vectors a ~ '

~ °A and .~ in order to fulfilI the condition 20.1. It fOll~~Sc~ oPA~2at if the form ty exists it must be equal to

.2J.11 El ~ 1 A m 1 mPAdx /\••• A d \fJ A ••• A dx - Hdx /\••• /\dxi\

?v

v- t*Je

146

It remains to be shown that this form satisfies the condition 20.2

for any representative g =(~A,p~,lpt,P~I'-)Of the element g = (lfA,p~,

lfi~' 'JeA • Using formulae 17.19 and 17.20 we obtain

20.12 <§g,8> b(p~lf~ - R(x/\',lfA,p~)) <.!!!, lx(g» 2 •

This proves that the definition was correct.

The Cartan form evaluated on an m-vector §g compatible with the

dynamics gives the value of the Lagrangian. Evaluated on a horizontal

m-vector §h / compatible with the connection / the Cartan form is equ­

al to the Hamiltonian. Also the energy-momentum density can be obtai-

ned by evaluating the Cartan form on an appriopriate m-vector in P.

Let E be a hypersurface volume element in M at x and let g be any re­

presentative of an element ge P • We define the lift fig of n to thex - -

jet g similarly as the lift mg was defined for the volume element m.

If g is the jet of a section

20.13 M :l Y ~ p(y) E PY

at y = x then by ~g we denote the lift of E to the restriction of this

section to the hypersurf'ace L, c M tangent to E. The coordinate des­

cription of ~g is

20.14-

for

fig(dA a )u 'O) (a A a

Ox2 + ~2 o'flA + PA2 2p~ /v··/\ Oxm + ~moLfA +

'Iv Gl)+ PAm o A.,PA

20.15 no

Ox2 /\ ••• /\

adxm

147

Let X be a veetor field in M. In the ease when Q is a bundle of geome­

trie objects in M the field X has unique lift to Q. This lift is asso-

eiated wit h "dragging" the geometrie objeets along the flow of X Isee

Appendix G l. Ii Q is a bundle of geometrie objeets the same is true

about P. It iollows that X ean be lifted to P. The liit oi X to P is

denoted by X. The m-veetor

20.16 .&~ AO"XI\ n°

is "horizontal" in the direetion oi X and eompatible with the dynamies

in the direetion oi n.

Theorem 9

20.17 l: (X,B) Ci!;) -<x~g,e >

where g is any representative oi g E Di•x

Prooi

20.18 <,@, 8) <,@g,8> -<,@g-,@,8>

-<,@g - ,@,-lf/

<XAB, Lx(g» 2

sinee reJ,@g - ,@)side we note that

o. To ealeulate the seeond term oi the right-hand

2:).19 &g Xgl\fl.g

.2ere xg is the liit oi X to the jet g. Thus

20.20 mg - m

148

(xg - x)A.gg

and according to 17.9

< i!lg - i!l,v >20.21

<E:, (xg - X, 8x/1) 2

(E:, <3[* (Xg - X) ,p> 1> 2

where p = Bt~g EPx is the point at which all the vectors used above

are attached. The vector xjxg-X) is difference between the lift of

X to the section

20.22 M '" y '; \f(Y) x (p(y)) EO Qy

and the horizontal lift of X to Q. Hence x*(Xg-X) is the Lie deriva­

tive of this section with respect to X :

20. 23 x*(xg - 1) otxlf

Isee Appendix G l. Thus, according to 19.71 ,

-<'~,8>20.24-

This completes the proof.

<E:,<ixt{J , X~g> 1 - X --1 [x(g»2

.;g; (X ,E:) Ut)

Similarly as in particle mechanics the m+1 -form

20.25 Q d8

149

contains all information about dynamics.

Theorem 10

A section

20.26 M 3 X ) b ex) E Px

is dynamically admissible if and only if for each vector field Y

in P

20.27 (Y--1Q)/S o ,

where S is the image of the section / cf.[9J/.

Proof

Using geodesic coordinates (x~,~A,p~) at x we obtain from 20.11

the following expression for g

20.28

QIv 1 l()A mdPAA dx A ••• A dr A ••• /\ dx -

AI\,

_ ('2 H dWAa~A l

ClH ?v) 1 m+ --Iv dPA /Idx /\•••/\dx'dPA

Let the field Y be vertical and given by

20.29

Then

y BAd ?v a·_-+C -

(l~A A dP~

20.30

LJQA{·· 1 Iv m ClH 1 m}B -dx A ••• /\ ~ A/\••• /\dx - a 'f A dx A ••• 1\ dx +Iv

::\.,( 1 A m oH 1 m 1+ CA ldx A ••• 1\ d \P /\ ••• 1\ dx - -::\n?v dx /\•••A dx •

150

In order to calculate (Y~52) I S we use the coordinate expression for

the section 20.26 :

20.31 (;ex) (A, A A-)x , lfl (x) ,PA (x)

We set

and obtain

dpk I S

d~AI S

A.

OPA dxl'­oxf'

d'tlA dx)'.'Ox!'-

(LJQ) I S20.32

{-a Iv

BA( PAex?v +

OH

Ó~A)+

A. ( A+ C A \f?v- 'OH)1 1Q 'A. 'j dx A ••• A dxmPA

Since equation 20.27 implies 17.7 the section 20.26 is dynamically

admissible. If 20.26 is dynamically admissible then equations 17.7

imply 20.27 for vertical vectors Y. For vectors Y tangent to S 16.27

is trivially satisfied. Hence (Y~~)I S = O for arbitrary vectors.

This completes the proof.

Let a section b: M ----,> P be dynamically admissible / a solution

of field equations / and let X be a vector field in M. The(m-1)-form

(x--18) IS defined on the graph S of the section can be projected onto

M. We obtain this way a vector density bJt'(X~8) in M. It follows

from Theorem 9 / formula 20.17 / that this vector density is equal to

minus energy-momentum density ~(X) :

20.33 ~ (X)( }'16' ) - b*(X --1 G)

The formula 20.33 can be used as a definition of ~(X,~) in the case

151

~hen XAn = O. Many important dynamical variables are also defined by

~he vector density b*(Y~ 8) for fields Y in P whicb are not necessa­

~ily the lifts of vector fields in M.

In this section we assumed that there are no hamiltonian const-

raints. There are, however, important field theories where the projec­

::ionarh of the dynamics Di onto P gives a submanifold Ch c P • Wex x x x x

ienote by eh c P the union of alI these submanifolds and assume that

rh itself is a submanifold. The conditions 20.1 and 20.3 / or equiva­

lently 20.2 / enable us to define the Cartan form at pOln~s belonging

::J eh. Applying this form to m-vectors tangent to Ch we obtain the

~-form 9 on eh. In alI physical field theories whicb we consider in

~hapter IV the field equations for sections of eh over M are equiva­

lent to the condition 20.27 where Y is any vector field in rh. In ge-

~eral, however, this is not necessarily true. The question of this

equivalence is connected with the problem of formal integrability oi

?artial differential equations / cf. [31J/. We do not discuss this

problem in general and limit ourselves to the discussion of examples.

21. Conservation laws

Let b: M ~P be a dynamically admissible section of the phase

:undle / a solution of field equations / and let X be a vector field

~n M. We calculate the divergence of the energy-momentum density

Es (X) ex)

~sing the formula 20.33 we have

~(X)(j1 G' (x))

21.1 div E6'(X) - d{ b'ltCxJG)} - b'l'd(XJ9)

-b'*f l~e" X - XJd@}

152

The second term gives no contribution because of field equations 20.27

fulfilled by the graph S of b . Thus

21.2 div E6'(X) - b;fl~8X

Definition: A vector field X in M is called an infinitesimal

symmetry transformation for the dynamics fDi} if the formL x

is invariant with respect to X :

Q= d8

21.3 i-QX

o

Of special interest are infinitesimal symmetry transformations

which leave also the Cartan form @ invariant :

21.4- t~8X

o

We calI such infinitesimal transformations infinitesimal special sym-

metry transformations.

Theorem 11

The energy-momentum density corresponding to an infinitesimal

special symmetry transformation X is a conserved quantity :

21.5 div E6'{X) o .

The proof follows directly from 21.2.

The field X generates a 1-parameter / local / group {~"} of dif­

feomorphisms of P such that

21.6tt a S~ S'l1 o ~

153

where {~~} is the 1-parameter I local I group oi diiieomorphisms oi

:1generated by X. Diiieomorphisms ~~ enable us to drag along X also

jets and classes oi jets oi sections oi P. It can be easily seen that

equivalence classes oi jets are taken into equivalence classes when

iragged along X. We obtain in this way the liit Xi oi X to pi and cor­

responding I locall group {~~} of diffeomorphisms of pi. The folIo­

wing theorem justifies the definition of symmetry transformations.

Theorem 12

An infinitesimal transformation X of M I a vector field in M I is

an infinitesimal symmetry transformation for the dynamics {D~}

if and only if the group {~t}preserves {D~} in the sense that

the jet ~~(g) is dynamically admissible if gis.

Proof :

The jet g = j1 6' (x) of a section b: I'1 ~ P is dynamically ad­

missible if and only if the equation 20.27 is satisfied at the point

6(X)6 P for any vector Y. This equation is obviously equivalent to

an analogous equation for the jet~i~,,(g)with the form

~*~-?;Q instead

of ~ • The condition 21.3 is equivalent to the equality

21.7 ~* QS-'t w.Q

for alI ~€R1. Thus if X is an infinitesimal symmetry transformation

then the field equations 20.27 for the jet ~~(g) are equivalent to

the field equations for g. Hence ~~Cg) is dynamically admissible if

gis. Conversely, suppose that {~~} preserves the dynamies. We pull

back both sides of the equality

21.8 E)iJDiY Y dl. y

154

from the point y = q~(x)to the point x by means of the adjoint map­

ping ~; *-. The result is

21.9

We not e that

21.10

F.i*-C\iI ri Dia~ LJy /l-t" y

F,l Di6-t y

r.i~dLiJ1: -y

Dix

d (~~*ly)

because the dynamics is preserved. Also

21.11 ri* Si11:.' y 8~

because ~~ preserves the entire internal structure of pi,this struc­

ture being canonical and ~~ being the lift of a diffeomorphism in M.

Thus

On the other hand we have

Gi\Dix x

21.12

Hence

21.13

e~ID~

d ( ~~,. oC y - l x)

d (rY"}, )/l'l:' -y

dl .-x

O

which means that the difference is constant on p~

21.14 '[hi-l01:.' -y -x"'-

dx E /\T; M

155

-~ collection of element s ~ at alI points of M defines an m-formx

X in M. We use the above formula to calculate the pull-back q:~the Cartan form 9 . If the m-vector m attached at the point p €O Px

;:::'o,jectsonto zero in M / i.e. if 11f§ = O / then it follows from 20.1

-::::e.t

2'.15 (§,g;e) < S1;~§'V) /' ~....,,§, )11 V-> <§,17) = < §, 9>.

~~::'ewe used the fact that the form ~ belongs to the canonical struc­

-:-.lI'eof P and this structure is preserved by q: . Let the jet g <= J~(P)

:~ dynamically admissible and let §g be an m-vector compatible with

-::::e dynamies. We denote by f the jet ~~(g). This jet is also dynami­

:~:ly admissible. We have

::-.16 ~H§gAfm

~~i mf is also compatible with the dynamies. It follows from 20.2 and

2- .14 that

< §g, ~~e> <~,;*§g, e> < §f, 8>

< ~*§f,i y (f)/2 < ~*nL*§g, ly (~~(g))h =

~.~.. 17

< ~*§g, (~~'"Ly)(g)/2

< §g, 8 + ~'*o(x>

<1*§g,[x(g) + O<x>2

follows from 21.15 and 21.17 that

_ ~18 r.*J'l;9 8 + Y]Jfo(L

156

where ex is some m-form in M. Thus

~:~~s: de d~:G d(8 +~*'«)

Q + ~>fdCX = Q

which means that .Q is preserved by ~:i.e. 21.3 is satisfie~.

This completes the proof.

An infinitesimal symmetry transformation X of M can thus be c=~-

racterized by the property that any solution of field equations re-

mains a solution when dragged along X. The existence of the vacuum

state enables us to formulate a simple criterion for an infinitesima~

transformation to be an infinitesimal special symmetry transformatic=y

Theorem 13

A vector field X in M is an infinitesimal special symmetry trans­

formation for the dynamics {D~}if and only if both dynamics and

the vacuum state are invariant with respect to X.

Proof :

We have ~y(go) = O, where go denotes the jet of the vacuum sta­

te at the Doint y 6 iVJ. Hence the function r.~i vanishes on ?,i (g '\•• '(1'" y ;]'-1: Ol

The invariance of vacuum means that q~~(go) is the jet of the vacuum

state at x = S-~(y). Since Ix vanishes also on this jet it follows

from 21.14 that o< x O. Using 21.18 we see that the invariance of

the vacuum state is equivalent / for an infinitesimal symmetry trans­

formation X / to the invariance of the Cartan form 8 . This comple-

tes the proof.

The energy-momentum density E(X) is an important dynamical

157

_6 also in the case when it is not conserved. The integral of E(X)

~ver the boundary oV of the domain VcM gives the infinitesimal chan­

~o of the generating function WcV when the domain V is dragged along

~. This integral may be changed into the volume integral of div E(X)

~,er V and calculated with help of the equation 21.2. One gets in this

;';aya field-theoretical analogue of the Hamilton - Jacobi equation.

~~e Hamilton - Jacobi theory is a fascinating and difficult subject.

Same of its consequences are discussed in Section 24. The interested

~eader is referred to the work of Dedecker [9J, [10J, [11J. Conservation

~aws may be considered a part of the Hamilton - Jacobi theory. Usually

~hese laws are derived within the framework of the lagrangian formula­

~ion of field theory by using Noether' s theorems 1L39J , [5J/. Modern

~ormulation of Noether's theorems can be found in ref. [49], ~OJ,~7J

and also [22J •

22. The Poisson algebra

Let ~ c M be a hypersurface in M which is not necessarily the

joundary oV of adomain V. The procedure used in Section 14 can be

applied to obtain spaces Q~, P~ of Cauchy data on L and canonical~ ~

~orms 8, UJ associated with ~ • Under special topological conditions

iiscussed in Section 15, we have pE = T*Q~ which means that (pL ,uJ~)

is a symplectic manifold in the strong sense 1 see[8]/. If (p~,U)~)

is a strongly symplectic manifold then the mapping

22.1 ;h. TL *2:'fE: • p.--. T P

analogous to 9.2 defined by

< X , CPrC~)/ < 3E:.,,3( , Wr: '>(1) I

is an isomorphism. It follows that for each smooth function f on p::::

158

there is a unique vector field xf in P~ such that

22.2 < X,df> <:x 1\ .xf, 6.J~:">

for an arbitrary vector field X. We define the Poisson bracket {f,g}

of two smooth functions f and g on p4 by the standard formula

22.3 {f,g} JSfg

Let (x~) be coordinates of M such that ~ is described by x1

const. and let vectors JE and J[ be represented by sections :u) (~

22.4-

Then

2-:7 X ~ X(x)Ci)

L 3 x __ X(x)<l)

A . Gl +6'P (XI aViA(1) 1

A (1 +b'f ex) etnA(» T

. 1 '3

bp A (x) :;-1(1) u P A

1 . d

bPA(X) ~C~) P A

E: TP~x

E TP~x

22.5 <X A x, wE>= (f cSPA1(x)·SWA(x)(1) (a.) J l(i) (.L)T

:Lr A C1 } 2 m- 0\0 (x). OPA(x) dx 1\ ••• 1\ dx(1) I o.)

exactly as in formula 14-.26.

Let f be a smooth function on P~. The derivative of f in the di­

rection of the vector Jf represented by the section

22.6 L .Ol X---'-'T X ( x) c A a (1 () ,~ar (x)~ + aPA(X) --1 E. TP~ dPA x

is a continuous linear functional on the set of pairs of functions

(SfA, bpl). Hence this derivative can be written in the form of an

integral :

22.7 xf(:X,df; = ~ {BAeX).S'rA(X)+ cAex)OSplCX)}dx2/\••• /\dxm•

159

=2e coefficients BA and CA defined by this formula are usually called

:~ctional derivatives of f and denoted by

::'2.8

BA (x)

CACX)

Sf

8~A(X)

bf

bplCX)

~omparing 22.5 with 22.2 and 22.7 we see that J6f is represented by

:2e section

::'2.9 L ~x --.. X (x)[ifa-1- -- -bp A ex) OlfA

6f ~-- 1b\fA(x) OPA

E TpLX

=2is implies the following coordinate expression for the Poisson brac-

::st :

~2.1 O\[ Sf &g _{f,g} = J bplCX) S\fJACx)

2::E,g Sl" J

1-- 26PA(X) b'f'A(X)dx 1\ ••• l\dxm•

The construction of the Poisson bracket given here applies also

:~ a non-compact hypersurface ~ if the space pL is composed of sec-

:ions vanishing at infinity fast enough to guarantee the convergence

:: alI integrals involved. This construction is specially important

:or hyperbolic field theories where ~ is a Cauchy surface.

Applications require an extension of the above constructions to

'~nctions which are defined and differentiable only on dense subsets

:: pL:. An example is provided by the energy which is defined only on

, dense set of sufficiently smooth Cauchy data.

Definition: The energy / on the hypersurface ~ / associated

,ith the infinitesimal transformation X of M is the function

:2.11 p2: 3 p~ t-cpz:) E R

160

defined by

22.12 t (p:2:) j E(X) (seX)).2::

where for any section

22.13 2....3 X ___ ~) p'" (x) E pEX

and a hypersurface volume element ~ at X on M the element s(x)c B(X,~)c

GPx(X)xQ~(~) is defined by the formula

22.14- sex)(pE(X),j1 gr~(p2':(X»))

The above formula makes sense only for elements p~ such that the

section

22.15 L 3 X ----.. ~~(p~ex») E Qx

equal to X:;(p~)E Q:E.is differentiable. Otherwise the jet in the for­

mula 22.14- is not defined.

23. The field as a mechanical system with an infinite number of deg-

rees of freedom

In this section we consider field theories in a pseudo-riemannias

space-time M. Field equations are assumed to be hyperbolic and the

Cauchy problem is assumed to be well posed on space-like hypersurfa-

ces of H.

Let ~t be a 1-parameter family of space-like hypersurfaces ob­

tained by applying to the hypersurface ~= ~o a 1-parameter group

of transformations generated by a time-like vector field X on H. We

define a bundle fY over R1• For each t '"R1 the fibre ~ over t is

161

~~s space p~t of Cauchy data on ~t' We also introduce the bundle ~

:7er R1 whose fibres are spaces o't = Q:E:.tof Dirichlet data on L t'~~der special topological conditions, which include integrability,

~2.chfibre SUt is the cotangent bundle T*Q,t of the corresponding fib­

:--8 (d,t'For each t ER1 there are canonical forms Gt = ErZt and Wt

= 6V:E:.t.Sections of 9J and ~ are generated by sections of P and Q

~espectively. For example if

M ~ x

_0 a section of P then

R1 3 t •.Pt

pCx) E Px

p ILt:E:.

E P t (pt

~s a section of j). The situation is formally the same as in particIe

=echanics. The difference lies in the fibres ~t and ~t being infi­

=~te dimensional. Since the dimension of ~t is interpreted as the

=~ber of degrees of freedom we can speak of the field as a mechani-

:2.1system with an infinite number of degrees of freedom. The field

~~amics in P induces a dynamics in gJ : the dynamically admissible

sections in jJ are those generated by the dynamically admissible sec­

~~ons of P. Since the Cauchy problem is well posed each element p~ p:E:.t

= jOt determines uniquely a dynamically admissible section of P and

2ence a dynamically admissible section of 9J.

We assume that the bundle Q is composed of objects for which the

~ie derivative can be defined. In this case there is a natural trivia­

~ization of ~. Horizontal vectors are jets of sections with vanishing

~~e derivative with respect to X. We denote by ;t the horizontal vec­

:-orfield in VJ which projects onto the unit vector field in R1• We

ienote by ~t the vector field in 9 compatible with the dynamics which

projects onto the unit vector field in R1• Similarly as in formula

162

9.20 we introduce the vertical vector field

23.1 ?Ehddt

'3at

There is a theorem analogous to Theorem 6 in Section 12

Theorem 14

The field :xE generated by the energy E in the sense of 22.2

is equal to X h.

ProoI :

For the sake OI simplicity we assume the existence of a global

coordinate system ex),,) in M such that E"t is described by x1 t and

the vector field X is equal to d~1 • We define the Iunction E by theequation

23.2 E (X) (s (x)) ~ (A 1 \j)A)· 2 m.t, t.p (X) ,PACx), rk(x) dXI\ ••• l\dx

If a vector y is represented by the section

23.3

then

2:"t .3 X ~ Y(x) C A O C 1· d ~to'P (X)-"lOA + OPA(x) -1 e TP'JT oPA x

23.4

where

<V c> \{ oE ( A dE (A ClE (1 12ma ,d G = l 3 A' 0lp (x) + 3A'Olfk(x) + o 1'OpA(x)J dx 1\••• t\dxLt \(J t.p k PA

23.5 5~~a

dxk '~lfA(x)

Assuming that functions ~A vanish at infinity sufficiently fast we

163

~~~egrate by parts

':::)06

\" ~ QJ O\DA }k"S'fAeX)2: Tk xt-5

Z"t

o oE

S'fAe x) CIxk ()~~

~.J" we use equations 19.14-.The resuIt is

<7.j,dE/ = ~ {-(drA - dOkP~(X»)·StpA(X) +Z:-t x

~:: • 7 + \P~(x) ob pl (X)} dx2;\. ••• ;\. dxm

\' f d \.pA (1 (1pl (A 12mJ L-1(X)oOPA(x)- ~(X)'0'f ex) dx i, ••• ",dxz:- dx xt

..:2lS impIies

~~8::::;.:;.

2-:::lei

23.9

.\'here

otpA(1 t (x)

1dPA (x)Cl t

be:

C1oPA Cx)

8f

S 'fAC x)

M "3 X ----, (lf Aex) ,pkex)) E Px

is the dynamicaIIy admissibIe section determined by the Cauchy data

pEt I the vector y is attached at a point pLtE jOt lo Formulae 22.9,

23.8 and 23.9 impIy that ~e is represented by the section

23.10 XSex)

1()l{JA o OPAO()t ex) 3A + 3""t(x) -1

<p dPA

It is easily seen that xh is represented by the same section. This

completes the proof.

( 'dlptLt ~t''Ot )

164

We see that the part of the Hamiltonian in this time-evolution

picture is played by the energy. Equations 23.8 and 23.9 are formally

analogous to equations 9.11. This is the reason why the energy is usu-

ally called the Hamiltonian of the field. We prefer to reserve this

term for the object defined in Section 17 which in our opinion is a

more natural generalization of the Hamiltonian of particIe dynamics.

The Lagrangian in the time-evolution picture can be shown to be

the function

~ X(x)Jlx(j\.pex))Et

where lft = cP \ Lt' \·Jenote that this Lagrangian as well as the ener-

gy are defined and differentiable only on the dense subsets of smooth

Cauchy data.

This section is meant to give one application of the concept of

energy introduced earlier. The energy appears as the "Hamiltonian" in

the "canonical Hamilton equations" 23.8 and 23.9. These equations are

usually used as one of the standard definitions of energy. We not e that

these equations define only the total energy and not the energy densi-

ty. Moreover, the heavy mathematical assumptions necessary for the for­

mulation of the time evolution picture drastically reduce the appli-

cability of this definition to physical situations.

24. Virtual action and the Hamilton-Jacobi theorem

Let m be a volume element at x E M and let s = j1'f(X) be the jet

of a section

24.1M 3 Y ------- er (y) E Qy

of the configuration bundle at the point x. We consider the rate of

change of the value <~, L (S)~ when both m and s are dragged along the

165

':ector field X in M. For this purpose we assume that Q is a bundle of

seometric objects in M. In this case the group {§~}of diffeomorphisms

Jf M generated by X defines a group of difi'eomorphisms [9-tJ 01' Qi. The

~enerator of this group is the lift X of the field X from M to Qi. Si­

2ilarly the group {9~}can be lifted to the group of transformations

::.L the bundle liT*M of volume elements. We consider the family t.!!! ('l:)}

volume elements and the family {s(~)} of jets obtained by apply­

~~g the appropriate lifts of the group {y~} to ~ and s respectively.

3:;th ~(-C-) and sC-c-)are attached at the point x(~) = §?;; ex). We define

~~e / scalar-density-valued / function YrCX) on Qi by setting

.0.-.2 <~,V(X)(s»2 ~~<~C't') , L (s ('t'») > 2/c; =0

elastostatics / see Section 3 / this quantity / multiplied by 6r /

__ the virtual work which is performed in dragging the piece m of the

=~8stic medium along the the field X. In dynami cal field theories we

=2811 call this quantity the virtual action corresponding to X.

The right-hand side of the formula 24.2 can be treated as a defi­

2~~ion of the Lie derivative of the scalar-density-valued function L

~i wit h respect to the field X. We write

.::-.j w(X) LvLX

__ following theorem shows the relation between the virtual action

the non-conservation of energy.

Theorem 15

_-a -r V'(X) (s) - div .!;(X) (g)

where the value of the right-hand side has been calculated on the

166

jet g of a dynamically admissible section of P determined by the

configuration jet s.

The above theorem is the infinitesimal version of the Hamilton-

Jacobi theorem. A special case of the Hamilton-Jacobi theorem is one

of Noether's theorem which states that if the virtual action vanishes

then the energy E(X) is conserved. Symmetry fields of the theory can

thus be chnracterized as infinitesimal transformations of M which "cost

no virtual action".~~

Proof of Theorem 15

Let

M 3> Y G'CYl € PY

be a dynamically admissible section of P / a solution of the field

equations / such that 24.1 is its projection onto Q, Le. tp='3rb

We denote by g the jet j1 ~(x).According to 20.33

24.5 div ~ eX) (g)=- div G'*(X--18)

where X is the lift of X to P. We decompose the values of X on the

image of

binto part Xlitangent to the image and the vertical rest.

The rest is obviously the

minusLie derivative of6'•Thus

24.6 x Xli - <L bX

Let Eg be an (m-1) -vector tangent to the image of EJ at bex) and let

n be its projection on M. This means that b~~ Eg• Thus

24.7

<~, G'*(Xli -1 6J) >

<~,t\ E,g, Gl >

<b*~,xlI-18>

< XA~, l(j\p» 2

167

:Jecause SC;iA.§gis an m-vector "compatible with the dynamics" / see

20.2/. It follows that

2-'i-.8 G'''' (XII-.J Gl ) x -1 l,(j1'f1

2~~ilarly we obtain from 20.1 and 17.9 the formula

< n, b*(~ bJ 8)>- X <G*E,i b -1 6> >X

.:-.9

~-:7iously

<i bA.§g, 8) = <E,<:L S ,8>1).X X / 2

<E,<X*LXb,6)1)2

::-.10

_ '::-.lS

fK*l S­X

i gr(;X Lxtf

::-."1 5"(1/; -1 (9) = <lx lf' 6')1

divergence of X.J 1(j1 tp) is theLie derivative of the scalar den-

,,[(j1tp) defined in M. This Lie derivative corresponds to drag-

the volume element ~ in 24.2 along X and dragging the jet s along

:~~ ~ift of X to the image of j1tp • We may express this Lie derivativev

_==-=-", the decomposition of the field X similar to 24.6 :

-.'2 vX v J;.1XII - dv J tpX

~=~~e fu is the field tangent to the image of j1~ • Thus

.. - .'"- .'./ div (X -1 L(j1<p)) = .i'6 L

168

The divergence of 24.11 can be calculated from 16.11 and the defini-

tion of the Lagrangian :

div<.t er ' b>1X < '1.,e LOJ XT,.,...16'>, J 1

24.14

< j~ lf ,dL>X= <et j1tp ,dL)

X

The last term is the Lie derivative of ~ with respect to the verti­

cal field Y = L j\'f). ThusX

- div E(X) div G*(JLJ G) 1,.;1XII

-L L.~y

lviX

w(X)

which completes the proof.

25. Energy-momentum tensors and stres s tensor. A review of different

approaches

In Section 19 we introduced the energy density as a generating

function of dynamics. This definition of energy provides not only the

total energy but also its local distribution. In Section 23 we estab­

lished a physical interpretation of the total energy based on the ti­

me-evolution picture. In terms of the local energy density we defined

the energy-momentum tensors. In the simplest case of a tensor field

theory the energy density depends on the first jet of the field X.

Hence there are two energy-momentum tensors t~F and t~~ defined as

coefficients in the expansion formula

25.1 El.. (X) tA, Xl'- + tA.V \7 Xl')L J< v)/

Both energy-momentum tensors must be known in order to know the ener-

gy density. This definition of energy is new. The usual approaches

are based either on conservation laws or on the discussion of sources

169

of the gravitational field. We give a brief review of different defi-

nitions of the energy-momentum tensors and relations between them.

Definitions based on conservation laws (Noether theorem) determi­

ne only the total energy. The localization of this energy and consequ-

ently the energy density are not uniquely determined. This freedom is

used to obtain an expression for the energy density involving only the

~omponents of the field X but not the derivatives of X. This procedure

~esults in a single energy-momentum tensor whose component in the di-

~ection of X is the energy density. In terms of our definition of ener­

gy we may interpret this procedure in the following way. The total en-

e~gy obtained from conservation laws is equal to the integral 22.12

)f our energy density E(X). The formula 25.1 can be rewritten in the

equivalent form

",-:/.2 EA-CX) (tA. _ V. t"'Y )x~ + 17 (t?vY xl')~ y ~ )I ~

=n the simplest case when the second energy-momentum tensor is anti-

.. t"')Isymmetrlc, l.e. F

gence

.:>.,ej'- , the second term is a complete diver-

~:/.3 \7)1 et"';,- Xl'-) (t:\,)I~Xf'),)/

_~e integral of this term over a 3-dimensional surface 2: can thus be

=eplaced by the 2-dimensional integral fiat spatial infinity". This in­

:egral vanishes if appropriate boundary conditions are fulfilled and

:ius gives no contribution to the total energy. In this case one says

:iat the quantity

.:::::;>.4

~?vt j< tlLj< - \ly tA,ij<

_~ the flimproved energy-momentum tensorfl and the quantity

170

25.5

whieh depends on the single tensor t~ is the "true energy-densi ty".

In the ease when t~~ is not antisymmetrie the above eonstruetion fails

beeause the integral of the seeond term of 25.2 over the 3-dimensional

~ does not vanish.

Our definition of energy gives the unique expression 25.1 for

energy density. Subtraeting "eomplete divergenees", even if they do

not eontribute to the total energy is not allowed.

The above proeedure of "improving" the energy-momentum tensor is

used in partieular in eleetrodynamies where a eertain naive approaeh

leads to the so ealled "eanonieal energy-momentum tensor" unaeeeptable

on aeeount of being gauge-dependent. The re suIt of this improvement is

the "symmetrie energy-momentum tensor" whieh turns out to be gauge-in-

variant. A proeedure based on the eorreet geometrie interpretation of

the eleetromagnetie potential leads direetly to the definition of the

symmetrie energy-momentum tensor. This proeedure was given already in

1921 by Bessel-Hagen [5J • We deseribe this proeedure in Seetion 28.

Sinee the energy-momentum tensor t~ obtained in this way is gauge-in-

variant it needs no improvements.

Related to the problem of energy-momentum loealization is the

problem of finding appropriate expressions for sourees of the gravi-

tational field. Sueh sourees are believed to be deseribed by energy

and momentum of matter properly loealized in spaee-time. Construetions

of these sourees have again been attemped by modifieations of the en-

ergy-momentum tensor. We eonsider this problem tOG serious to be sol-

ved by sub/traeting divergences or any other ad hoc modifications. We

find that sources of the gravitational field depend on the response

of matter to space-time deformations and are related to non-conser-

vation of energy rather then to the distribution of conserved energy.

We return to discussion of sourees of the gravitational field in Sec-

171

tion 29. The following construction is used in this discussion. We con­

sider the virtuaI action Yf(X). The eIastostatic analog of vreX) is

the virtual work performed in dragging the eIastic medium along X. In

the case of energy-conservation the virtual action is zero. Using the

HamiIton-Jacobi theorem we obtain

25.6 w(X) - div E(X)

(1mW X)dx/\ ••• A dx

where

25.7 W(X)

The value of W(X) at a point depends for tensor field theories on the

second jet of X at this point since E(X) depends on the first jet. If

M is equipped with an affine connection r then the second jet of X

splits into three parts : the value of X (given by components X~) ,

first covariant derivative ~ X~ and the symmetric part of the se-

cond covariant derivative :

25.8 i7 \1 rvC"- vv) X

AlI these components are independent and may be chosen as coordinates

in the space of second jets. They determine uniquely alI derivatives

part of the second covariant derivative is given by the formula

of the field X up to the second order. For example the antisymmetric

when R is the curvature and Q is the torsion of the connection

25.9

.~~_~m_~ - _._~ _

172

The splitting of the jet of X into three parts enables us to expand

the virtual action V((X) into parts proportional to X~, ~X~ and

~~ ~)X~ with uniquely defined coefficients :

25.10 lV' eX)

The coefficient T :luv,,L<.. is a symmetric tensor density

25.11

The interpretation of the vector density T~ is obtained by setting

\7)J Xi'-= O and 'q;>., \7.) Xl'- = O at the point x E- M. At this point dragging

along X is locally a parallel displacement in the direction of X. The

quantity TI'-measures thus the quantity of action which is necessary

to mak e a local parallel displacement of the physical field ~ • By a­

nalogy with elastostatics we calI TI'-the force. In order to obtain the

interpretation of the remaining terms in 25.10 we put Xl'- = O at XE M.

Dragging along X at XE M is nowa local deformation of the field

without displacement. The deformation is described by derivatives Vvxf'­

and Vc~~)xf'-.By analogy with elastostatics (formula 3.4) we calI

the tensor density TV~ the first stres s tensor (or simply stress ten­

sor) and T~Y~ the second stress tensor.

The detailed analysis of the structure of General Relativity which

we give in Chapter IV suggests that the stress tensors and not ener-

gy-momentum tensors are sources of the gravitational field. Although

these quantities have completely different physical meaning there are

relations between them. It follows from the formulae 25.1 and 25.7

that the stress tensors are completely determined by the energy-momen-

tum tensors. To show this we calculate the covariant divergence of

25.1. For vector densities the covariant and normaI divergences coin-

cide. Thus

--..----

o) the stress tensor

probably the origin of much of the confusion in this domain.

- W(X)

- W(X)

173

25.14 Tf'-\j, A, 1;\'v'= " t f'- + -t I) R f'- ,"". 2

25.15

T"f'-t"t<\7: tA,,,i"'5'] Q"=+

~ t<+ j'-;:Li)

25.16

T '>0)1 j'-2t(~,)I)t',\,}I tY)"= t<= f'- + f'-

It is interesting to note that if the second energy-momentum tensor

t~~ is antisymmetric then the stress tensor 25.17 is equal to the

"improved energy-momentum tensor" 25.4. This complete coincidence is

Using 25.9 for the antisymmetric part we obtain

25.17

For a wide class of field theories dynamics is determined onI y by the

geometry of space-time M, i.e. by the affine connection I~ and the

Comparing 25.13 with 25.10 we obtain the following formulae

In the case of a symmetric connection (Q~o

is equal to

The last term can be decomposed into symmetric and antisymmetric parts.

25.13

25.12

"-~~~~~=======~=~-~-"--"---~--~_._------------

174

pseudo-riemannian metric g. For such theories the Lagrangian i depends

on the values of r and g in addition to the coordinates \{JA and lf~

of the jet of ~

25.18 1(A A j"A,\f ' lf/\,; f'-V , gfv)

We do not assume any relations between and g. Such relations be-

long already to the dynamics of the geometry, i.e. to the General Re­

lativity Theory and will be discussed in Chapter IV. In the present

section we consider a tensor field theory in a given geometry. For the

sake of simplicity we assume that the connection r is symmetric. The

reader may easily generalize alI results to the case of a non-symmet-

ric connection.

We now define an invariance property of the theory usually described

as the invariance of the Lagrangian under coordinate transformations.

This property states that the value of < l!!, 1(s» 2 is invariant under

simultaneous dragging along X the following three objects : 1. physi­

cal field 'fi, 2. the volume element l!! and 3. the geometry of space-ti­

me represented by r and g. The rate of change of <l!!, 1. (s) 2 due

to 1. and 2. with the geometry fixed is just the Lie derivative of the

Lagrangian which appears in the Hamilton-Jacobi theorem. This Lie de­

rivative is equal to the virtual action vv(X). The rate of change with

respect to the deformation 3. of the geometry is equal to

25.19 12

The invariance Gondition thus implies the following

25.2012 G L 1 gf'-Y

d g/"v X

o

~...-~~----------------~-------~--------------.~.••...

175

equivalent to

o12

i gPX

i r"X 1'''

1 oL .. :t,

W(X) -:2 d ~~ txrp

w(x)

25.21

A field which fulfills this property will be called a relativistic

field theory. To calculate the stress tensar for a relativistic field

~heory it remains to substitute in 25.21 the following expression for

the Lie derivative of r and g

25.24

25.25

25.23

25.22

'n'here

:'he result is

',.;here

~ \jV<x~) + 1 ~ \Z \) X'A,

(3g!'Y 2 d I~\ (!' v)

3ymmetrization in the indices ~ and y can be omitted since derivati-

'fes dL

Q g!'''and

w(x)

are symmetric. Thus

1- 2Tp VI{ gP

d La gP'

1(. /"" A, l! ",)- 2 T it R ?-"O' + Tl'v Vi;' gl'

176

Comparing 25.26 with 25.10 we obtain

and

where T~y denotes the symmetrie tensor density

25.26

25.27

25.28

25.29

25.31

25.30

Comparing 25.30 with 25.14 and 25.16 we get the following identity

(ef. [21J where the "hypermomentum" is introdueed by the similar for­

mula). Moreover

Now we use the so called seeond identity for the eurvature tensor

25.32 o

The result is

25.33

177

o •

O •

o .

o

O

sI" g"'" gf'~o(~ •

rf'~= {",:tv} , where {f'~} are the Christoffel

obtain

~= recall that the tensor gs~ t"'\] was denoted by s"'y~ and called

~he spin angular momentum tensor (see formula 19.81 ). Thus

symbols, the covariant derivative of the metric vanishes and we have

.-~ a metric connection

We obtain finally the identity

-o::.on

this identity reduces to

A second identity can be obtained by comparing 25.28 with 25.17. We

valid in the case of metric connection. For the fIat Minkowski space

25.34

25.38

25.36

25.37

:sing the property \Zgb~ = O we mayraise the index b • We use proper­

~ies of the curvature tensor fulfilled in the case of a metric connec-

178

25.39

The antisymmetric part of the stress tensor Trv vanishes. Thus

25.40

lO.

For a metric connection 'i<~ ={:v} , the covariant derivative of the

metric vanishes and the above formula reduces to

25.41 O.

Equations 25.41 are often called the angular momentum conservation

laws. It may be shown that identities 25.33 and 25.40 (or equivalen­

tly 25.34 and 25.41 for the case of a metric connection) together

with 25.30 are necessary and sufficient conditions for the theory to

be relativistic. The formula 25.38 is often called "the General Rela­

tivistic expression of energy-momentum conservation" (see [42]). Sin­

ce in general this formula is not fulfilled in curved space-time at­

tempts were made to "improve" the energy-momentum tensor t?,;<.once mo­

re in order to obtain a "conserved quantity" T~....for which the equa­

tion

25.42 o

holds. This approach is so popular that in some textbooks equations

25.42 are treated as the unique reason for some quantity appearing in

calculations to be called "the energy-momentum tensor" (see e.g. [32J).

In our opinion equations div E(X) = O are conservation laws and not

-- ------------~ ••••L

179

25.42.

In order to establish relations between the Hilbert tensor on one side

bL

12

W(X)

1 dL '1'oL (t= 2 d gf'V dCX gt<"+ -~ g;<v;\?v

2;<''''~ X

25.46

1( a L

oL .+ 2 (~i. ~v),~,2 (1 gj<V

-d~ ~) i:v gl'~

)'<v,;\' X

2 d gf'''X

equality

mation of the geometry :

On the other hand we may rewrite equation 25.21 using the formulae

In traditional variational formulations of gravity the gravita-

and stress tensors and energy-momentum tensors on the other side we

iefined the sources of gravitational field as a symmetric tensor den-

tional field is represented by the metric tensor g. This approach dif-

where L is the matter Lagrangian assumed to depend on ~A, ~~ and ad­

ditionally on the gravitational field g~~ and its derivatives

calculate the derivative of the Lagrangian with respect to the defor-

r

25.45

25.43

25.44

For a relativistic field theory the reasoning used earlier yields the

sity satisfying 25.42. The Hilbert tensor is defined by

fers from ours. The formulation presented in Chapter IV is based on

the connection r treated as the gravitational field. The mathemati­

~al formulation of the traditional approach is due to Hilbert [2~ who

180

----...

25.4-7

and

25.4-8 1, gPX

valid in the case of the metric connection G<~ = f:v} . Using also

25.28 and 25.29 we obtain

W(X) _ 2 T/'V • 2 g~b (\7 r g +2 ?" 2 Vf- <Z, 15>

X

+ iTf<)I(_gl'"gf<f,Lxgo/p)

+

- ~ GTA,Yf-+ T;\'j<)I- Tf<}iA,)~ g".yl:t

The covariant derivative V~ in the last term has been replaced by the

normal derivative since the expression in the bracket is a vector den-

sity. Comparing the last equality with 25.4-6 we obtain an expression

for the Hilbert tensor in terms of stress tensors :

25.50

Consequently

25.51 W(X)

181

T)'-"H

w(X)=- Tf" \J X - [CTN(P) - 2 Tf'~?v) v: X J

H )'- " 2 j'< "" ,N25.53

T'" V XV - [(T?o(f'Y) - 2 Tf'-"A-) V. X ] .H» j< 2' 'I" "j:'\'

~ensors with the help of identities 25.16 and 25.17 :

",).52

~p write 25.51 in terms of derivatives of X

~he Hilbert tensor can be also expressed in terms of energy-momentum

~sing the formula

TI'"=t"'"+\l}" t?vj< >I-V;v (t (:;Vj<).+ t(?,,')j<

_ t (1'>1);1., )H

tf'''

+V-:<, l?vI'J)I-v;, ( t(?,v)j<_ t("'Y)::\')

25.54 t"V

+ 2 V (t?vf'Y_ tf'?vY _ tA-vj<_ t";cj<+ tf'YN + tVI'''')2 ?v

t"'V +

V?Jt?v[i<>i];<[,,?v]

"['-I'J)+ t

- t

?inally

case of ah antisymmetric second energy-momentum tensor (t"'''?"

The above formula is the Belinfante-Rosenfeld theorem (cf. [~ , [4J'

[42J ). We draw the attention of the reader to the fact that in the

25.55

the second stress tensor T""?v vanishes and the formula 25.50 shows

that the Hilbert tensor reduces again to the "improved energy-momen­

tum tensor" 25.4 which turns out to be symmetric :

182

25.56

It follows from 25.29 that this happens when the Lagrangian ~ does

not depend on the connection r but only on the metric. The triple

equality 25.56 seems to be result of pure coincidence.

We calculate the divergence of the Hilbert tensor by first rewritting

the formula 25.55 in the following form:

25.57

Thus

25.58

Hence

25.59 o

follows from the identity 25.37.

__-!!!!!I!----------------"

183

%= jave not found a direct physical interpretation of the Hilbert

~==sor. In Chapter IV we give a formulation of gravity theory signi­

:~~antly different from Hilbert's. Sources of the gravitational field

a~~earing in this formulation are the stress tensors.

IV. Examples

26. Vector field

The configuration bundle Q for the covariant vector field is the

cotangent bundle T*M, where M is the space-time with a pseudo-rieman­

nian metric tensor g and a symmetric affine connection r . No a prio-

ri relations between the metric and the connection is assumed. Su ch

relations belong already to the General Relativity Theory and will be

discussed in Section 29.

We give a local coordinate formulation of the theory. Let (x~)

be a coordinate system in M. In the present Chapter coordinates of

space-time will always be denoted by x~, ~= 0,1,2,3. The system (x~)

induces a coordinate system (x~, ~~) in the configuration bundle Q =

= T*M. The components of the metric tensor will be denoted by g~~ and

gp.~.The connection l' is described by components r;~= I-:~ . The

phase bundle P is the union of tensor products 13.1

l

26.1 pq

where q E Q, X = 5 (q) E M. The space dual to the cotangent space is

the tangent space. Hence

26.2 pq

3T M ® 1\ T* Mxx

It follows that the bundle Px over Qx is trivial

As a bundle over M, P is the Whitney sum of two bundles

26.4 p

__-!IIIIII----------------

185

first bundle is the configuration bundle Q. The second bundle is

tensor product of vectors by vector densities. We calI su ch objects

~:~travariant tensor densities of second ranko Tensors

a (ddx?v ® oxp.JdX0;\••• ;\dx3 )

3

:0rm a basis in the vector space TxM ® /\ T~M. Every element of this

~?ace can be uniquely represented as a linear combination of elements

~ó.~. It follows that the bundle P has coordinates (xfL, ~~,p~~) where

~ie coordinates p~~ are components with respect to the basis 26.5.

:?orms G and 6J at each point x E M are given by formulaex x

26.6

26.7C:Ux

pA,l'-dlf'"0 (a:J< JdxO/\ ••• 1\ dX3)

The infinitesimal configuration bundle is the first jet bundle J1Q.

A coordinate system (xfL,~~ '~~J<) is induced by the coordinate system

(xfL). The coordinates ~~~ represent derivatives

26.8

The infinitesimal phase bundle is the quotient bundle pi = J1p of the

first jet bundle J1p. Coordinates in pi are (xl-'-,t.pA"p?"f'-'Lf~!'-'oe;\,),where

26.9

A t each point x E M the infini tesimal phase space p~ is a symplectic

manifold with the symplectic form

186

26.10

iWx

( ).,::\,u. ) O 3d oe /\ dtf:l.,+ dp r ;\ d lf::t,f' lJ?i dx ;\••• 1\ dx

of the 1-form

The symplectic form is the differential

26.11

26.12

(i :l,..) O 3':1e df)., + p / d lfA-f<- @ dx ;\••• 1\ dx

It is convenient to parametrize jets of sections of P by covariant de­

rivatives of the sections. We thus introduce coordinates (X~'~h'P~~,

cp.lv;< , Je A.) by setting :

lfhJ<- =V;<-lf?v=d~ lf)., - I~;lfo- =lf?v~ - ~'\,~Y;o

26.13

-j\,Vj'- p?v}'-=d p?vl'- +~?v !>'fI..oe~ +n Iv I;'ji-Je ==of<- P'/" "fi.. P

(it must be remembered that p is a tensor density ). The formulae

26.12 and 26.10 can be rewritten in terms of new coordinates. The re-

suIt is

26.14-(-~ ::\, -) O 3ot dLfh + p fLd \.f?vfL @ dx ;\••• 1\ dx ,

26.15i

C0x

The dynamics Di C pi of the field is expressed by field equationsx x

26.16

field equations

187

- A, 2 \ r---:-JgO' Nf'- lOoe = - m V -g co l f'-

n the case when V~gf'-V= O, i.e. when

26.18

are equivalent to the second order system of equations

25.17

26.19

The vacuum state is the zero-section of P

The dynamics and the choice of the vacuum completely determine the

26.20

Lagrangian

26.21

Using the condition L(O,O) = O we obtain

such that

26.22

Since the bundle Q is equipped with the affine connection we may per­

form the Legendre transformation and pass to the hamiltonian descrip­

tion of the theory. If coordinates(xf'-)are geodesic at the point x

188

(i.e. if I-::~vanish at x) then hamiltonian speeial sympleetie form

is

26.23

Sinee i.ngeodesie eoordinate system 9t"" = def'- and ~':\,I'- = ~':\,!'- ' we may

write in general

26.24

This formula is valid in all eoordinate systems, not neeessarily geo-

desie. In order to find the Hamiltonian we rewrite the field equations

in the following form

26.25

Inserting formulae 26.25 into the equation

26.26

we obtain

-dde x

26.27

where

26.28

Let E be a hypersurfaee element (3-veetor) in M and let X be

an infinitesimal transformation of M (veetor field in M) transverse

to n. We use eoordinates (x~) adapted to E and X, i.e. eoordinates in

which

26.29 n

~.) O 3E?uE (X dx 1\ ••• 1\ dx••here

:~ow we pass to a general coordinate system (xl'-) not necessarily adap-

ted to X and n. The Lie derivative of the covector field is given by

26.33

26.35

~b.us

25.17 with respect to the variabIes J{~,p~k and ~AO and insert the

~eBults into formula 26.32. The corresponding proper function ~(X,E)

say be directly obtained by the Legendre transformation 19.72 or 19.74.

order to calculate E we should solve the field equations 26.16 and

26.34

26.36

~he two energy-momentum tensors are

-- z...,__ ii .••••••'

26.37

f = 1,2,3. It is determined by the equation

md

energy E(X,E) is a function of variabIes

x

189

:lence

190

26.38

The stress tensors may be calculated directly by expanding the virtual

action W(X) = - d'A,E?"(X) with respect to X/"-, \J" Xl"- and 'Vc?, \Z) Xl"-. Ins­

tead we use equations 25.17 and 25.16 :

y \7 '1vt I"- + v?J t fi.

26.39

26.4-0

2 \11 "'('LO Vi )- m ~ -g g l'" Tf'

It is easily seen that the first stress tensor is really symmetric as

it should be by virtue of equation 25.28 :

26.4-1

Now we calculate the Cartan form ty in P. The formula 20.11 gives

the Cartan form in terms of coordinates ex/"-)which are geodesic at the

point x

26.4-2 ~ O 3 O 3P f'-dxA ••• 1\ d lf;\, t\ ••• A dx - Hdx /\••• 1\ dx •A

I'-

We derive a formula for ty valid in a general coordinate system. ~Transition to general coordinates requires the replacement of covec­

tors d~~ in the above formula by covectors

191

26.43

The second term is related to the parallel transport of 'f in the di­rection of x -axis. Covectors 26.43 vanish on horizontal vectors in

P. Hence

8 :v OP dx A ••• A

26.44

O 3- H dx ;\••• 1\ dx h O 3P /"-dx;\••• A d lf~ A ••• 1\ dx -;\

JA

where

26.45

Using calculations similar to these used in the proof of Theorem 10

(formula 20.27) we find that field equations may be written in the

form

26.46

26.47

=t is easy to check that the above system is equivalent to 26.16 and

26.17.

27. The Proca field

The Proca field is a covariant vector field with dynamics diffe-

rent from that described in the preceding Section. We have chosen the

example of Proca field to illustrate the appearance of constraints in

the hamiltonian description of dynamies.

192

i iBundles Q, P, Q , P and forms for the Proca

field are the same as for the covector field. The dynamics Di is des­x

cribed by equations

27.2 2 \~g'},)L UJ- m ~ -g 1)L

where

27.3

The derivatives ~~~ and ~~ in the above formulae may be replaced by

covariant derivatives ~~ and ~~ since

27.4

27.5

o ,

, equations 27.4 and 27.5 implyi.e. when

for the antisymmetric tensor density p~~. In the case when

~ J t}[';-ty = U'"

o

27.6

Hence

27.7

This equation is called the Lorentz gauge condition. Using this con-

dition we write

193

?t?v =

27.8\r:g [Dg~?v

!""tlO I\,~-g g \f

because of the equality 25.9. By R~o(= Rf<!\,,I<<< we denote the Ricci cur­

vature tensor. This implies that in the case of metric connection the

system 27.1 and 27.2 is equ/ivalent to the second order system of equ-

ations

27.9

We note that the lagrangian submanifold D~CP~ described by equations

27.1 and 27.2 is bigger than the space of jets of solutions of field

equations. No condition involving the symmetric part of the covariant

derivative ~~~ is contained in the definition of D~. However, such

a condition follows from 27.6. In the case when V~g~~ = O this con­

Qition reads

27.10 O

~nis does not mean tnat tne description of field dynamics in terms of

=~is incomplete since tne additional equation 27.10 is automatically

194

satisfied by any section of P whose jets belong to Di at each XE M.xThe vacuum state is the zero section of P

o

27.11

p?v;<- = O

O } c p

O we obtain

8~ = (&e';l.,d~~- CPVdP~f'-)@dX~'."Adx3

(~';l.,d LO _ CO d [~f<] UJ. d ().;<)) O 3TA, T [:\'.1'] P - ,('?vI') P 0 dx A ••• /\ dx •

27.15

Now we perform the Legendre transformation and pass to the hamilto-

tum is always antisYIDIDetric

The hamiltonian special symplectic structure is given by the form

Using the condition L(O,O)

27.13

such that

27.12

nian description. There are hamiltonian constraints since the momen-

The dynamics and the choice of the vacuumcompletely determine the

Lagrangian

195

The Hamiltonian is defined only on rh. The derivative of .~ withx

respect to p(~~) is thus arbitrary. This implies that ~(~~) is arbitra­

ry on Di• The field equations may be rewritten in the following formx

27.17

or equivalently

27.18

Inserting the equations 27.18 into the formula

27.19 - d Jex = e~ID~we obtain

27.20

where

27.21

Let ~ cM be a Cauchy hypersurface in M. For the sake of simpli­

city we use adapted coordinates (xJ<.) such that ~ is described by

{xO = O} • The space pZ:: of Cauchy data on 2: is described by sections

27.22~ (\fA, ex) ,p;;\'°cx)) E

Not alI element s of P are compatible with dynamics. Cauchy data cor-

responding to solutions of field equations satisfy the following con-

dition implied by the dynamics :

196

27.23

o

Sections 27.22 satisfying 27.23 form a subspace ~:z: OE:l- of P called the

constraint submanifold. The dynamics in the time-evolution picture isL

determined by the energy defined on the constraint submanifold ( •

The construction of the energy density follows the pattern given

in the preceding Section. In adapted coordinates the form 8(x,Q) is

given by the formula

27.24-

Equations 27.1 and 27.2 can be solved with respect to ~~ p~k and

~ kO when I.p:l.,' lf:;t.kand pkO are given. The component CP00 is arbitrary.

This corresponds to the constraint pOO = O. The energy E(X,Q) is de­

termined by the formula 26.31. The corresponding proper function

~(X,Q) may be directly obtained by the Legendre transformation :

27.25X$'(CP;<6' p;<;l,_

The two energy-momentum tensors are

27 .26

and

27.27

- r" L6-

The stress tensors are

+

197

27.28

and

27.29 o •

The 1-st stress tensor is symmetric

27.30

According to the remark given at the end of Section 20 the Cartan

form ~ is defined on the constraint subspace eh c ph consisting of

antisymmetric momenta. In order to calculate the form EV we follow

the same pattern as in the preceding Section. We have

27.31 + H H

since p~~ is antisymmetric and r~~is symmetric. Hence,

27.32

It may be easily checked that equations

27.33 o

for the section of Ch over M are equivalent to field equations 27.1

and 27.2.

The symplectic form in p~

27.34-My

~O 1 2(dp A d\f~) ~ dx Adx Adx3

z:

198

is degenerated when restricted to rE. This means that there

vanishing vectors v tangent to rZ such that the equation

are non-

27.35 o

holds for any vector w tangent to r~. Such vectors v are described

by sections

L 3X27.36 - . C?-O) Z(~lp?v(x),op ex) E TPx

such that the only non-vanishing component is S~O' The equation

27.35 is satisfied since dpOO = o on Cz. The set of all such vectors

v form a distribution in Cz. This distribution is composed of all

vectors tangent to the foliation of C:E by submanifolds obtained by

fixing \fk(x) and pkO(x) and varying lfo (x). The space pZ of leaves

of this foliation is parametrized by 6 functions

27.37 L.... 3 X

We call this space the space of reduced Cauchy data. The form

27.38

M

~ (dpk~ dlfk)® dx1,,\ dx2f\dx3z:::

induced on pZ by w'E is non-degenerate. The pair (pZ::, wZ::) is a symplec­

tic space.

(-r>E . r) C-::c ~~)The proces s of reducing the space ~ ,w to the spa~e P ,w

imitates a well known process of reduc/ing finite dimensional symp­

lectic manifolds (cf. [44], [57]) .

28. The electromagnetic field

The configuration of the electromagnetic field is the electromag-

netic potential. The field itself plays the part of velocity. The elec-

tromagnetic potential is not a covector field in space-time since dif-

199

The representation of the potential as an equivalence class of covec-

II))

Il~llllil

lliilllilljl~;Illllj!j!li!!jlllllljl

Ililjjl!jijilllillllj

ll;I

I

G

rob E. B

M

B

r+ 'teb)

B

B

Gx B 3 (r,b)

~ er- b)

We assume the existence of a class of mappings

Let B be a 5-dimensional differential manifold, let M be the spa-

is a diffeomorphism and

28.4

as a gauge field requires the description of the potential as a con-

ce-time equipped with the pseudo-riemannian metric g and let G denote

the additive group of real numbers. The group G is assumed to act dif-

for rE G.

characterized by conditions

If ly 1 and \f12 belong to this class then

28.3

ferentiably on the manifold B, the manifold B is fibred over M and the

28.1

nection form in a principal fibre bundle associated with the gauge

group esee [40J). The electromagnetic field is then interpreted as the

curvature tensor of the connection.

tor fields related by gauge transformations is also incorrect for se-

28.2

fibres are the orbits of the group action. The fibration is denoted by

veral reasons. The modern interpretation of the electromagnetic field

ferent covector fields correspond to the same physical configuration.

The group action is denoted by

1

< v,c<)' = O ]

o •

1

O

O

do<.

< K,o<.>

d.0b

< K,df/

on B such that

200

28.5

is a mapping constant on fibres of X and can be identified with a

real function A on M

There is a distinguished vertical vector field K on B which is the

image by the group action of the unit vector field in G. The condition

20 implies that

A connection form O( on the principal fibre bundle B is a 1-form

of a connection form c<. is called a connection. Element s of this dis-

cipal fibre bundle with base M and the structural group G. The field

K is called the fundamental vector field and mappings ~ satisfying

conditions 10 and 20 are called trivializations.

28.6

The manifold B with the structure described above is a trivial prin-

28.8

The characteristic distribution

satisfies

28.9

28.7

tribution are called horizontal vectors. The exterior differential

and

28.11

28.10

28.12

201

The condition 28.11 follows from the identity

28.13 LJ d o<.

The condition 28.12 follows from

28.14 i do(K

As a consequence of properties 28.11 and 28.12 there is a 2-form f on

M such that 'f is the pull-back of f by .:t :

28.15 ,/\/' i'

The form ~ is called the curvature form oi' the connection and f is

called the curvature tensar.

Let ~' be a trivialization. Then

28.16

and

28.17

< K, eX - d ~ '/

i (<<K

o

o.

Consequently there is a 1-form (a covector field) A on M such that

28.18

We note that

28.19

Hence

28.20

dO<-

f dA

dX*A X* dA •

If ~ 1 ana ~2 are twa different trivializations of B then by virtue

of 28.5 the corresponding covector fields A1 and A2 differ by the

differential of a function :

202

28.21

Hence,

28.22 - d 1\

-d X/I\

A connection form ~ on B is interpreted as an electromagnetic

potential and the curvature tensor f is the corresponding electromag­

netic field. The field f is an object in space-time. The potential ~

is an object in B. Only when a trivialization ~ of B is chosen can

one associate with ct a covector field A oh M. Bach choice of trivia-

lization is interpreted as a choice of gauge and transition from one

trivialization to another is a gauge transformation.

We have chosen the group of real numbers for the structural group

of B. In applications involving complex wave functions the correct

choice would have be en the multiplicative group U(C,1) of complex num­

bers of modulus 1. The two groups lead to equivalent descriptions of

electromagnetic potentials. In both cases the formula

28.23d f + X* A

establishes a correspondence between connection forms and covector

fields on M if a trivialization yv has been chosen.

At each point x c M we define the configuration space Qx. Elements

of Qx are "values of connection forms at x", i.e. restrictions of con­

nection forms to vectors attached at points b E B =?(, -1 (x) c B. Thex

configuration bundle Q is the union

of configuration spaces. The formula 28.23 proves that elements of Qx

can be parametrized by covectors on M attached at the point x provided

28.24 Q

203

a trivialization .~ has been chosen. Given a trivialization the con­

liguration bundle Q can thus be identified with the cotangent bundle

T*M and each configuration space Q can be identified with the corres-x

ponding covector space T~ M. A coordinate system (x~) in M and a tri-

vialization f of B induce thus a coordinate system (x~,A~) in Q de­

fined by the formula

28.25

Let \jJ1 and ljJ 2 be two trivializations. The identification of Q with

T*M associated with 0/1 can be obtained from the identification asso­

ciated with lf2 by adding to each element of T; M the covector dA (x)•

The linear structure introduced into Q by identification with T* Mx x

changes with a change of trivialization. However, the affine structure

remains the same. We conclude that Qx has a natural affine structure.

Vectors tangent to Q are thus elements of T~ M and the tangent bundlex x

TQ is the trivial bundle Q ~Tx*M. Consequently the cotangent spacex x

for Q is the space T M and the cotangent bundle T*Q is the trivialx x x

bundle Qx~TxM. We conclude that the phase bundle P lor electrodynamics

is the Whitney sum Ol twa bundles

28.26 p

where the secnd term Ol the sum is the same as for the covector field

( Section 26) and for the Proca lield (Section 27 ) • The bundle P

has coordinates (x~,A~,p~~) where p~~ are coordinates Ol the contra­

variant tensor density of second rank and do not depend on the choice

Ol trivialization UJ Ol the bundle B. Forms G and W at each pointT x x

x E M are given by lormulae

28.27

dA. Hence,

( . ';\, . '4f< ) ( O 3 )d ~ {\dA~ + dp A dA~~ 1/9 dx A ••• 1\ dx

iUJx

are components of the electromagnetic field f

204

The quantities

Canonical forms in pi are given by

28.34

The dynamics Di c pi of the electromagnetic field is described byx x

equations

The infinitesimal configuration bundle is the first jet bundle

Qi = J1Q. A coordinate system (x~,A~,A~?) is induced in Qi by a coor­

dinate system Cx~,A~) in Q. The coordinates A~~ represent derivatives

28.33

28.30

28.29

28.32

The affine structure in Qx implies that the differentials dA~ are gau­

ge invariant objects and have tensorial transformation properties.

28.28

28.31

The infinitesimal phase bundle is the quotient bundle pi = J1p of the

first jet bundle J1p. Coordinates in pi are (x~,A~,pk~,A~?,~~) where

is involved in

o •

o

cribed by

205

28.35

28.36

The existence of this constraint is equivalent to the fact that onI y

We recognize that 28.33 are Maxwell equations. Similarly as for the

Proca field equations 28.33 imply a hamiltonian constraint eh des-

The vacuum state is any solution of field equations with vanish­

ing electromagnetic field (a fIat connection in B)

electromagnetic field f (and not alI velocities A~y)

field equations.

28.37

The dynamics and the choice of the vacuum completely determine the

such that

Lagrangian

28.38

28.39

Hence

similar to that used in the preceding section. The re suIt is

28.40

The hamiltonian description of the dynamics may be obtained in a way

206

28.41

defined on constraint submanifold Ch by the formula

28.42

2 p:\'}<. f:l..f'4

Now we are going to calculate the energy of the electromagnetic

field. According to the general discussion given in Section 19 we must

first define the Lie derivative of the electromagnetic potential with

respect to a vector field X in M. The value of the uie derivative at

each point x€M will be a vector tangent to Qx' i.e. an element of the

cotangent space T* M. Since a connection ~ is a differential form inx

B and not in M we have to lift the vector field X from M to B. The na-

tural lift is the horizontal lift associated with ~ • We denote this

lift by X~. The Lie derivative of ~ with respect to x~has the fol-

lowing properties

28.43 iL~ o(=O

Kx'"

and28.44

< K, J.-~ eJ.. > '"O

XoL

and is thus the pull-back of a covector field in M. We calI this co-

vector field the Lie derivative of the electromagnetic potential with

respect to X and denote by i~X

28.45 X/(id-)X

207

In terms of local coordinates (X~,Ah,A~~) induced by a trivialization

f of B the Lie derivative can be calculated as follows. We choose

the coordinate system (x~,y) in B by setting y = y. If

28.46 x

then the horizontal lift of X is

(1

+ c dy

A X,'l..l<-

c

o

where the coefficient c satisfies the horizontality condition

28.49

28.50

28.48

Thus

28.47

( cf. [5]) . The Lie derivative of o( with respect to Xo( is equal to

Thus

28.51 (ci oZ);oX

Consequently

28.52

The two energy-momentum tensors are

28.53

and

28.54

The stress tensors are

t"b

208

o

28.55

and

28.56 o

t?.6"

The covariant stress tensor

28.57

is symmetric and usually is called "the symmetric energy-momentum ten-

sor of the electromagnetic field".

The Cartan form for electromagnetic field can be obtained by si-

milar calculations as for the Proca field. It is defined on the cons­

traint space Ch by the formula

28.58 8 '),14 O 3 . O 3p. dx 1\ ••• /\ ~AA,I\". 1\ dx - Hdx i\ ••• 1\ dx

14

The analysis of the time-evolution picture which we made for the Pro-

ca field applies for the electromagnetic field. If 2: is described in

adapted coordinates by equation xO = O then field equations imply the

constraint ELe P~ for Cauchy data

28.59

28.60

209

The symplectic form

M

L: ?vo 1 2 3j \..dp/\dAj\,)69dx /\dx /\dx2::

is degenerate when restricted to Cl:. The space rE may be reduced

with respect to this degeneracy. The re suIt of this reduction is a

quotient space pL called the space of reduced Cauchy data. This space

is described by sections

28.61

and equipped with the symplectic form &)~.For the detailed descrip­

tion of this reduction see e.g. [26J.

29. The gravitational field

According to General Relativity Theory the gravitational field

is described by the geometry of the space-time (see r3J) . In the

preceding sections we described various matter fields (the scalar

field, the electromagnetic field, the Proca field) in a given geome­

try of space-time. In the present section we describe the dynamics of

the system composed of both matter and geometry interacting with each

other. The geometry of the space-time is described by two structures:

the affine connection r and the metric g. The natural conjecture is

that one of them describes the "configuration" of the gravitational

field and the other describes the "gravitational momentum". In our

formulation of field dynamics momenta always belong to a space equip-

ped with a linear structure and the notion of "vanishing momentum" is

always meaningful. It follows that the connection cannot be interpre-

ted as momentum since there is no vector structure in the space of

connections. No "zero element" is distinguished among different con-

nections. On the other hand the metric tensor belongs to the tensor

space equipped with a linear structure. We conclude that the connection

210

shoul~play the part of eonfiguration and the metrie should be inter-

preted as momentum. The third objeet oeeuring in General Relativity

is the eurvature tensor. The eurvature is not an independent struetu­

re but it is defined by the first jet of the eonneetion (by the eon­

neetion eoeffieients and their first derivatives) • We expeet that the

eurvature tensor will play the part of veloeity. This pieture is ana­

logous to the deseription of eleetrodynamies given in Seetion 28

eonneetion is interpreted as eonfiguration and eurvature as veloeity

( cI. also [28] and [47J ) •

In the present formulation of General Relativity we eonsider on-

ly symmetrie eonneetions (r::~= I'>~) . This is not onI y for teeh-

nieal reasons. Given a symmetric conneetion r in spaee-time M we de-

fine at eaeh point x E M a elass of loeal eoordinate systems. This elass

is composed of these eoordinate systems for whieh the eoeffieients ~~

vanish at x. If ex K) and (yl'-)are two coordinate systems belonging to/

this elass then

29.1 o

A elass of coordinate systems defined by equality 29.1 will be called

a loeal inertial frame at the point x. Conversely, if a loeal inertial

frame at eaeh point x€M is given we may define a symmetrie eonneetion

I' by setting r~~= O in any eoordinate system belonging to the

frame. We see that a symmetrie eonneetion may be treated as a field

of loeal inertial frames defined at eaeh point of spaee-time. The gra-

vitational potential is thus deseribed by the field of loeal inertial

frames in spaee-time. The gravitational field strength is deseribed

by the eurvature tensor whieh gives us the relative aeeeleration of

neighbouring test particIes. This interpretation follows very elosely

Einstein's original ideas ("freely falling elevatorsll) and plays an

important role in understanding the eanonieal strueture of General

211

Relativity.

The configuration bundle Q describing the gravity is thus the

bundle of local inertial frames in M. A coordinate system (XX) indu­

ces coordinates (x~, I~:) in Q. The configuration space Qx composed

OL local inertial frames at x has an affine structure. Subtracting

symmetric tensor in M. It follows that a covec-

two elements

riant,

and I') of Q we obtain a 1-contravariant, 2-cova-et X

In). _ \-oJ..J.{ v .f<-))'1) ('2.1

tor on Qx is a 1-covariant, 2-contravariant, symmetric tensor in M.

Similarly as in the preceding Sections the phase bundle is a Whitney

sum

29.2 p

where S~ is the bundle of 1-covariant, 2-contravariant, symmetric

tensors. The following objects

29.3 os

form a basis in a fibre OL the bundle S~ <;9J\ Tl(-I"l(by Q9s we denote

the symmetric tensor product) • Every element of this fibre may be

uniquely expanded with respect to this basis. The expansion coeffi­

cients Jr,/">f! together wi th coordinates (x'e, I~") in Q form a coor-1,

dinate system (x~, )-01''' '%:1/">2) in P. Of course :

At each point x E. M forms e and 0J are given by the formulaex x

29.5

Gx1

IY f')J'>( r;\.. (Cl O 3)= JL:1, d)l."@

dx~Jdx A ••• II dx2

29.6

Wx1 (dffi:'f'V'J<d r;y )(d O 3)

= 2 J Q., 1\18> dx:jeJdx 1\ ••• 1\ dx •

212

The infi­

J1Q. A

is induced in Qi by the coordi-

I~~ ..~)}~ represent derlvatlvesThe eoeffieient

eoordinate system (x~, r:~( the coefficient a appears because of the symmetry 29.4) •

nitesimal configuration bundle is the first jet bundle Qi

1,71,'), jJ-)}"'-/

29.7

The infinitesimal phase bundle is as usual the quotient bundle pi

= J1p of the first jet bundle J1p. Coordinates in pi are denoted by

Je 1"';(, (;>' /,-"")<. I'?" !'''X , 1"-" ,~v'N ,1'-",.., jA, , where

29.8

The eanonical strueture in pi is given by formsx

The gravitational momentum or~I'-"~ has a priori tOG many eomponents

f'-~>< I"A,) O 3+ dm;'?, /\d t'''''- @dx A ••• 1\ dx •

jJ-~>< n?v ) O 3+ 'JC?v d J f')J ~ ® dx " ••• A dx~(O"f'-Ydr.:\'2)A, 1'-"

N. C\dlV \)"X

and

29.10

for describing the metrie tensor g. However, similarly as in eleetro­

dynamies (formula 28.36) we may expect the existenee of hamiltonian

eonstraints redueing the number of independent eomponents of X • The

29.9

oceurenee of such eonstraints is always equivalent to the faet that

not alI velocities are involved in the field equations. Indeed, Ein-

stein equations eontain only the eurvature tensor

29.11

213

and less than that, namely the Ricci tensor

29.12

Especially important for our purposes is the symmetric part of the

Ricci tensor

29.13

since we don't knowapriori wheather Rrb is symmetric or not C the

symmetry of the Ricci tensor will be a consequence of the dynamics in

the same way as the Lorentz condition is the consequence of the dyna~

mics in the case of the Proca field) • We assume in the sequel thath

the Lagrangian depends onIy on K~~ and not on alI velocities r~~~.We prove that this assumption implies the constraints

29.14

where X~~ is a symmetric tensor density. The number of independent

components of x:v~ is thus equal to the number of independent com­

ponents of ~~; , ie. to the number of independent components of the

metric tensor. We expect that the quantity X~; represents the contra-

variant density of the metric. It will be shown in the sequel thatthe

correct identification of X~; with the metric is :

29.15

where k is the gravitational constant.

Now we prove the formula 29.14. Our assumption about the Lagran-

gian

214

29.16

implies the formula

29.171_ J f'~ dl':\'2 ':\, /"y

where J"t'Y and gr; ~& are arbitrary coefficients satisfying the symmetry

condition

29.18

Using the formula 29.13 we write

29.19

() .,;t)+ rr~d 16'><'

~[J/D' +

On the other han~ the formula 29.9 and the definition af the Lagran-

gian imply

29.20 dLx

Ii

215

Comparing 29.19 with 29.20 we obtain the formula 29.1~. Moreover

29.21

or

29.22 J~){)'

The last term may be rewritten as follows

29.2312 ( S~ 'Jtt>e-

We may add to the right-hand side the expression

29.2~ o

The result is

29.25

Combining 29.25 and 29.22 we obtain

29.26

(we recall that % is a tensor density ) • The coefficients J?/'-v are

216

thus eomponents of the eovariant divergenee of the momentum ~~~y~

and may replace ~~~y as eoordinates in pi.

We use the strueture described above to formulate the dynamics

of the system composed of a matter field ~A and the gravitational-oN i i

field '~y . Bundles Q, P, Q , P are Whitney sums of bundles corres-

ponding to the field ~A and bundles constructed above. The coordina-

~ WA ;e lilA p;\' "" t-,7>< r:cl ~)J CItes are x , I ,PA' T3€. , ?tA' 'I''' , JuCl, , I'y>e and~?v • Forms lJx '

GJ , Si, CVi are eomposed of two parts. One part corresponds to thex x x

matter field and the other to the gravitational field. For example

1+ - Q ~)! d l';\,2 )?v ~v +

29.27

29.28

Field equations for such a system are eomposed of two parts : equations

of the matter field and equations of the gravitational field. The equ-

ations of the matter field are formally the same as in Section 12.

They ean be derived for example from the Lagrangian of the matter

field 25.18 • We denote now this Lagrangian by

29.29 Lmat dXOA ••• A dx3

sine e the symbols L and L are reserved for the Lagrangian of the

fulI system eomposed of the matter and the gravity (this Lagrangian

will be found later) • Thus the matter equations are :

29.30

A,

PA

217

The constrai~t equation 29.14 enables us to treat the above equations

as equations in the space P~. There are also two gravitational equa­

tions. As the first gravitational equation one usually takes

29.31

This is equivalent to

29.32

or

29.33

or

29.34 J fo,?v

o

o

o

Equation 29.31 implies the symmetry of the Ricci tensor similarly as

the Proca field equations imply the Lorentz condition. This enables

us to express the second gravitational equation - the Einstein equa-

tion - in terms of K~v instead of R~y. However, it is not obvious in

the general case which quantity should be used as the right-hand side

of Einstein equations. Let us first consider the simplest case when

the second stress tensor of the matter vanishes i.e. the matter La-

grangian does not depend on r • This is the case of the scalar field,

the Proca field, the electromagnetic field and also hydrodynamics which

is discussed in Section 30. In those cases the first stress tensor T~y

(equal to the Hilbert tensor T~y) is commonly accepted as the right­

hand side of the Einstein equations

29.35

218

kT)'-)J

The coefficient ~ is necessary since in our notation T~> is a ten­

sor density and not a tensor. Contracting both sides with g~> we obtai~

29.36 kT gf.)I!'-)I

This enables us to rewrite the Einstein equations in the following,

equivalent form

29.37

Now we use equation 25.28

29.38

and the identity

29.39k

Rimplied by 29.15. The result is

29.40L (wA A r Iv fA>mat f ' 'P?v ; l p)l , g )

Equation 29.3~ can also be rewritten in a similar way since we assu-

med that Lmat does not depend on

29.41

Equations 29.14, 29.30, 29.40 and 29.41are the complete set of field

equations. They define the dynamics Dicpi. Using the function L tx x -ma

Illii;i!lll~~~~!

Ii1!j!llIllllll~1ll,,~i]i]jjljj~

IlI

+

same class if they have the same value of coordina-

r·;J., 'mi'" f<Y ) i .,~A' f<Y' ,K~y,JN • The form 00x lnduces a sym-

for the matter field and "quasi-hamiltonian" picture for the

1 I-'Y ról,+ -dJ~ /\.d f"Y2

plectic form 6ji on the quotient :x

The proof of this formula follows from the calculations used in 29.19.

The dynamics Di defines a lagrangian submanifold ]icpi described byx x x

equations 29.30, 29.40 and 29.41. The function Lmat is the generating

of D~ with respect to the special symplectic structure 29.43.

gives a mixed picture of the dynamics: the lagrangian

219

The precise definition of pi is the following. We take the subbundlex

Ce pi defined by the constraint equation 29.14. The symplectic forms

wi are degenerat e when restricted to fibres r e pi. We reduce rx x x

with respect to this degeneracy, i.e. we pass to the quotient space

pi. Each element of pi is a class of elements of pi. Two elements of

The above formula can be interpreted as a generating formula for the

lagrangian submanifold in the space p~ described by coordinates (~A,

p~ ' t.p~ ' dfA, rf<~, Jtf'-" ,K)'" ,Ji'Y) . The generation is meant with respect

to the special symplectic structure given by the form

defined on D~ by Lmat we may rewrite the field equations in the fol­

lowing form

29.44

29.42

29.43

iPx belong to the

(A;t. Ates er ,PA ,lf;t,

d (1 r.yf<YK ) IDi2 JiJ )'-" x

dL

d(.f:- .f:mat)

dL

and

220

Hence

Using the identity

29.48

29.47

29.45

which we proved earlier we obtain

gravitational field (we use "quasi" since the genuine hamiltonian

i ~i )picture refers to the space P rather then to the reduced space Px x

Variabies (\fA, ~-t) are the lagrangian variabies for the matter field

and (I:", 'Jtf"") are the hamil tonian variabies for the gravi tational

field. Equations 29.30 are of lagrangian type. Equations 29.40 and

29.41 are of hamiltonian type. The pure lagrangian description of Dix

(or D~)is obtained from the above one by the Legendre transforma-

tion applied to the gravitational degrees of freedom. The Lagrangian

of the fuli system (matter + gravity) is defined by the formula

29.46

29.49 L L 1 . f'~-mat + 2 ar Kf'~

The Lagrangian 29.49 is scaled in such a way that it vanishes for c'~e

O ) • Using 29.15 we write

vacuum. The vacuum is defined

the fIat space-time geometry

221

by the

(CK/,-" =

matter vacuum L-mat = O and

29.50

where R is the scalar curvature. Thus

29.51 L 1l::mat- 2

V _gl Rk

( cf. [23]) • The second term of the right-hand side should not be

interpreted as the gravitational Lagrangian according to the "recipe":

matter Lagrangian + gravitational Lagrangian = the complete Lagrangian.

This term is obtained from the Legendre transformation. It is analo­

gous to the term pq in the formula L = pq - H. Both L t and L are-ma -

generating functions of the dynamics of the fulI system composed of

matter and gravity with respect to two different special symplectic

structures. The notion "matter Lagrangian" is also misleading in this

context. The difference between L and Lmat is analogous to the diffe­

rence between two potentials in two different control modes (e.g. the

internal energy and the enthalpy in thermostatics) • The function L

A A nA, -..,)., A A r::t-depends on lp , ~)., , I f<Y and I f<> ~ or on \.fi ' ~:\, , ,MY and Kl'~. The

function Lmat depends on 'fA, lf~ , l',u~ and gtf'-)1 • In order to calcula­

te L from Lmat it is not sufficient to add the term -~ ~R but it

is necessary to eleiminate the variable Xl'> (the metric) using the

field equations. Similarly the Lagrangian in particle mechanics can

be obtained from the formula L = pq - H if we eliminate the variable

p from the right-hand side.

Example

The Lagrangian for the linear scalar fieldtheory is given by

the formula 16.52 where F = O :

222

29.52

Using the formula

29.53

we write

det 'J1:;t'-V~ 2k4 g det gl'''

29.54

2 ~ '+ k _detJt'l'V

The field equations in this case are 16.53, 29.14, 29.40 and 29.41.

It follows from 19.76 that t~~p = O. Thus T~~ = t~~. We conclude that

29.55

1 2 2)+ -g m \.O2 '),,~ I

Einstein equation 29.40 can thus be rewritten in the following equi-

valent form

29.56

This implies

29.57

and

223

29.58

Combining 29.57 with 29.49 we obtain

29.59

L(~mat + ~ X;1,f'- K;\.,f'-) I D~

1 2 2? V '2 m 'f k- - det'JCJO' 1 2 2.~2 m ~ V -g .

In order to calculate the determinant g we use again the Einstein equ-

ation 29.58

29.60

Thus

(1 2 2)g;V,f'- ;2 m \f .

29.61 g det g:\,f' ( 1 2 2) -4 ( 1. \2 m lf det \f!?v lf/-l. - k K~J'-) •

Inserting 29.61 in 29.59 we obtain the final formula for the Lagrangian

29.62 ( 1 2 2)-\ ( (:2 m 'P V -det lf?v lfr<-

1 .k K?vf')

where Kp is the combination of r~~~~

and If'v~ given by formula 29.13.

The reader may easily che ck that the Euler-Lagrange equations derived

from the variational principle

29.63

where as usual

o

29.64 o 3L dx /\ ••• /\ dx

224

are equivalent to the system composed of equations 15.53, 29.34 and

the Einstein equation 29.55 (eL [28])•

. nA-In the general ease the matter Lagranglan depends on I~v and

the equation 29.34 ean not be written in the form 29.41. It ean be

shown that equation 29.34 is ineompatible with the matter equations

and the Einstein equations in the sense that dynamies defined by them

is not lagrangian. The interaetion between matter and gravity would

not be reeiproeal, so we propose to replace equation 29.34 by another

equation whieh saves the reeiproeity of the interaetion (ef. ~8]).

Our equation follows from the following reeipe: the matter Lagrangian

oC t should always be taken as the generating function of the dynamiesma

in the mixed pieture 29.43, i.e. as the Lagrangian for the matter field

and the (quasi) Hamiltonian for the gravitational field. The dynamies

generated this way is described by equations 29.14, 29.30, 29.40 and

29.41. We proved that the equation 29.40 is the Einstein equation

29.55 k 1V-::g T;<v

It follows from 25.29 that the equation 29.41 is equivalent to

29.55

The eondition ~g~y =0 is no longer satisfied and the eonneetion

r~~is no longer the metric connection {~~} • Hence, the geometry

of the spaee-time differs from the fIat Minkowskian geometry in two

aspeets : the eurvature and the non-metricity of the eonneetion. The

souree of the eurvature is the first stress tensor 29.55. The souree

of the non-metricity is the second stress tensor 29.55. The formula

29.49 deseribing the Legendre transformation to the pure lagrangian

pieture remains valid.

Now we formulate the hamiltonian deseription of the dynamies.

225

The pure hamiltonian picture is not unique since there is no unique

connection in the bundle Q. Each coordinate system (x~) in M defines

its own horizontality : the horizontal sections are defined by the

condition G~(x)= const. The hamiltonian special symplectic struc­

ture defined wit h respect to this horizontality is given by the form

29.67

The Hamiltonian

29.68

is defined on the constraint manifold Ch c P given by equation 29.14-.

The Hamiltonian is a function of variables (\{JA,p~ ,I~ ,m:;P) and

can be obtained by the Legendre transformation from the Lagrangian

HA ?v

1 f')., f'-Y'"= lf>\, PA+ 2 r 1J 3-e m:;?v

- L

A ?v

1 (I'?"/,?v ) )<" 1 Qt'f'-lI'f?v PA + 2 jL" ')"

- jA')"iJ $- L - - K "J-mat 2 1'-

29.69 A ?v_ 2( 1,70 ,--.6" _ r,?v1-'" ) f"- l:mat1?v PA 2 ó;l.,;<)1 o))

t'!\' Je

H 1 ( 1-' ?v ,I,"

_ I'?v ~ 1)) ~)i-ma t - 2 6'?v I ~)iS-1J r?- X

By Bmat we denote the matter hamiltonian

29.70

The Hamiltonian H depends on the choice of the coordinate system (x;<)•. h

However, the Cartan form in C

29.71

226

r.:;\ ?v O A 3CI = PA dx /\•••11 d \(7 A ••• /Idx +A:\,

1 y~ O I"'~ 3 O 3+ - ar; fi- dx A ••• Ad j< y f\ ••• /I dx - Hdx;\ ••• /I dx2 ;<, 1\~

does not depend on the choice of coordinate system as it was proved

in Section 20. The form Ej is composed of two parts. The first part

29.72 8mat ?v d O d VJA d 3 H d O d 3pA x /\ ••• f\ ;\ r /I ••• A X - mat x 1\ ••• /I X?v

corresponds to the matter field and the second part

29.73

corresponds to the gravitational field. The form E9 was first in­gr

troduced by W. Szczyrba (see [47] and ~81) who directly proved its

invariance under general coordinate transformations.

The construction of the energy E(X,~)is similar to the one given

in the preceding Sections. In adapted coordinates the form G(X,~) is

given by the formula

sex,.!:!:)

29.74

227

The corresponding proper function ~(X,Q) may be directly obtained by

the Legendre transformation, i.e. by the formula 19.74 :

(i W A) PA?" +:2 (L r Je ) 0'G fO''''X T 2 X lU 3e

29.75

where

29.76

The Lie derivative of the connection depends on the second jet of the

field X

29.77

Using 25.1 for the matter energy Ei'v t(X) and the formula 29.77 we ob­ma

tain

G"'(. ?v 1 r-:::' (?v 1 p::L ~ )X t (J + 2k ~ -g R o f) - 2 8'G~ R f')) 6' +29.78

We shall prove later that this formula can be transformed to

29.79

where the antisymmetric tensor density H?v~CX) equals

29.80

228

29.81

Formula 29.79 implies the conservation law

29.82 W(x) o

for any vector field X in space-time. This proves that alI transfor-

mations of the space-time are symmetry transformations of the theory.

For physical interpretation of conservation laws 29.82 see [30] , [2~

and ~1J.

The analysis of the time-evolution picture for General Relativity

is similar to that given in Section 28 for electrodynamics. If ~ is

described in adapted coordinates by equation xO = O then the space of

Cauchy data pL is parametrized by sections

29.83

The field equations define a subspace rEcp~ of those data which are

compatible with the dynamies. The symplectic form

is degenerat e when restricted to C~.The space C~may be reduced

with respect to this degeneracy. The result is a quotient space p~ .Elements of pE (reduced Cauchy data) are classes of elements of p~.

To find an appropriate parametrization of reduced Cauchy data is a

serious technical problem (Cf. [2] and ~5J) which we do not dis­

cuss here. A new solution of this problem is proposed in [27].

We prove now the energy formula 29.79 (see [28J). The formula

29.75 can be rewritten in the following form

229

29.85

We use the formula

29.86

\Id... \!~X ex. - V [ol. \j O X Dl.

implied by 25.9 and integrate by parts the last two terms of 29.85.

Then we apply the formula

29.87

The result is

EN(X)

29.88

2 Ro( o( 6'Q2

+ ~ ( V~Jr/~- S~ % ot ('eX.) ~ X~

Now we use equations 25.1, 25.17 and the Einstein equation. We obtain

29.89

230

The non-metricity equation 29.66 implies

and

29.91

The above formulae imply

29.92

+

But

29.93

Using an inertial system

mula

( \',..;\.y = O) one can easily prove the for-

29.94-

The las t two formulae show that the second term in 29.92 vanishes.

This completes the proof since the formula 29.90 implies

29.95 t (J.., 6')b

231

30. The hydrodynamics

To describe the hydrodynamics of a barotropic fluid without vis-

cosity we need a 3-dimensional "material space" Z. Points of Z are in-

terpreted as material points of the fluid. The configuration bundle Q

for this theory is the trivial bundle

30.1 Q fJI x Z

where fJI is a space-time equipped with a fixed geometry (g, r). We will

see in the sequel that the dynamics does not depend on the connection.

A configuration q (x,z) E Q tells us that the point z E Z of the ma-

terial occupies the point x in the space-time. The phase bundle P is

the union of tensor products 13.1

30.2 pq

3

T* Z <81 /\ T* fJIz x

where q (x,z). As a bundle over fJI, P has fibres equal to

px

3T*Z @ 1\ T* Mx

If (XA) is a coordinate system in M and (za) is a coordinate system

in Z, a = 1,2,3, then the tensors

30.4-a

e'{,

form a basis in the vector space P • Every element ~EP of this spa­q

ce can be uniquely represented as a linear combination of tensors 30.4-:

30.5

It follows that the expansion coefficients ~~ together with coordi­a

nates

232

(xA" za) in Q form a coordinate system (x\ za, c;~) in P. The

canonical forms

30.6

G and CJ are given by the formulae:x x

C1 :>., a ( Gl I O 3)C"x = 1: adz @ d x).,- dx t\ ••• 1\ dx

30.7 d Gx ( /v a) ( a I O 3)d 't: a l\. dz @ 'O xA,....J dx J\ ••• 1\ dx .

The infinitesimal configuration bundle Qi is the first jet bundle

J1Q. The coordinate system (x:>",Za)in Q induces a coordinate system

(x~,za,z~) in Qi. The coordinates z~ represent partial derivatives

30.8a

Z;x,

The infinitesimal phase bundle pi = J1p is a quotient bundle of the

first jet bundle J1p. The coordinate system (x ,za, t;) in P induces

a coordinate system (xA..,za, 't;,z~, O€a) in P where

30.9

At each point X€ M the infinitesimal phase space pi is a symplecticx

manifold with the symplectic form

30.10

The lagrangian special symplectic structure is given by the lagran-

gian 1-form

30.11

233

(a A, a) O 3oe dz + 1::dz ~ ~ dx i\ ••• 1\ dx •a a ,,,

The above formalism is common to alI theories ot continuous media.

Each particular theory needs an additional structure of the material

space Z. In the theory of elastic media this additional structure is

a riemannian metric in Z. In the theory ot fluids this structure is

much weaker. It is a scalar density r a differential 3-form) in Z.

This density enables us to measure the quantity of the fluid. For a

given volume CJ c Z the integral of rover LJ tells us how many moles

of the fluid are contained in er •

Given a section

30.12 q(x) (x,z(x») E Q

or, equivalently, a mapping

30.13 x z(x) E Z

we define a matter current j which is a vector density (a 3-form)

in M obtained by taking the pull-back of r through ~

30.14- j

The matter current satisfies the continuity equation

30.15 dj d 'r*r f*( dr ) o

since dr = O as a 4-~form in the 3-dimensional space Z. We consider

the quantity

30.16

234

which describes the density of matter (moles per volume) in the co­

moving reference frame.

Our theory describes a fluid if the Lagrangian L depends on

derivatives z~ only via the matter density :

30.17

We call our fluid homogeneuos if the Lagrangian does not depend on za.

This happens when the dynamical properties of the fluid are the same

for each point ZE Z. The dynamics is given by the generating formula

30.18

equivalent to

30.19

where, as usual,

dL

dL

30.20

The field equations are

30.21

In the simplest case of a homogeneous fluid the above equations re-

duce to

o

30.22

235

u"

where

30.23

is the unit vector in space-time. This vector is the four-velocity of

the matter. We have used the formula

30.24

which the reader may easily check.

In order to give an interpretation of the Lagrangian 1; and to show

the equivalence of the field equations 30.22 with the Navier-Stokes

equations we pass from the lagrangian picture to the energy pic-

ture performing the partial Legendre transformation. Since the bundle

Q is trivial the notion of the Lie derivative for sections of Q coin-

cides with the partial derivative. Dragging the point q = (x,z) E Q

along the field X is simply moving the fixed material point z along

the integral curves of X. Thus

30.25

The formula 19.74 reads

30.26

It follows from 19.78 that the first energy-momentum tensor is equal

to

30.27

236

and the second energy-momentum tensor vanishes

The following equality can be proved

30.28

30.29

o

The simplest way to prove it is to use unimodular coordinates (za)

in Z, i.e. the coordinates for which we have

30.30 r

The corresponding expression for the matter current is

j30.31

where

30.32 (-1) A-

'1 2 3)ó(z ,z ,zd'O 3' (x , •• ,x )

J\I'-

( -1)fLdet

The index ~ means that the ~-th column has be en omitted. Using uni­

modular coordinates the reader may easily verify the following formu-

la :

30.33

Thus

237

30.34

1dL (_ u ." A,R d~ yJ SI' - Uj<-j).,)

and

30.35

On the other hand, the energy-momentum tensor in hydrodynamics is

30.36 +-?v

v )'-

where E is the energy density (per unit volume) in the co-moving

reference frame and p is the pressure.

In order to be able to interpret our theory as hydrodynamics we have

to take

30.37 L

The functions pCg) and E(~) are not independent. We have

RpdL dE:

30.38=

L - dS '3=-Res -Cf3g)

ar . dE)

2 dL30.39 p=- ([ - dg S =~ dS~

The function

quantity v =

is the energy density per one mole. The

is the volume corresponding to 1 mole of the fluid.

We have

238

30.40

and

30.41

We see that the energy of 1 mole of the fluid is

oC

eo + ~ P(~)dv

1Js

equal to the energy

corresponding to the raryfied state (density equal to zero) plus

the amount of work which is necessary to compress our mole from the

raryfied state to the actual density S . Hence, the sa called "con-

stitutive equation" p = p(~)

and the dynamics.

uniquely determines the Lagran~ian

Formula 25.15 implies that the stress tensor is equal to the ener-

gy-momentum tensor,

30.42

~1I

since t J'-

equations

O. Formula 25.34 is thus equivalent to the Navier-Stokes

30.43 o

where the covariant derivative is taken with respect to the metric

connection

There is an interesting analogy between electrodynamics and hyd-

rodynamics. In both cases the Lagrangian depends only on a few combi­

nations of components of the configuration jet (f~lI in electrodyna­

mics and jJ'-in hydrodynamics) • The independence of remaining compo­

nents is connected with a gauge-invariance (in electrodynamics a

"gradient gauge"; in hydrodynamics a change of "names" of material

points without changing the matter density ) •

The first pair of Maxwell equations is automatically satisfied

239

because of the definition of f~v. The continuity equation is automa­

tically satisfied because of the definition of j. The secand pair af

Maxwell equations is thus an analogue of the Navier-Stokes equatians,

and the material variabIes (za) are potentials for the Euler variab­

Ies jY.

Appendices

A. Bections of fibre bundles

:Get

A.1 F B

be a differentiable mapping of a differentiable manifold F onto diffe-

rentiable manifold B. We call X a fibration of F if the inverse ima­

ge Fb = :Jl; -1(b) of each point b E B is a submanifold of F. This subma­

nifold is cal led the fibre over b. An example of a fibration is the

canonical projection

A.2 B

of a product B x F onto the component B. This fibration is called tri-

vial. A fibration A.1 is said to be locally trivial if each point bEB

has a neighbourhood (j such that the set Jl;-1(e) of points of F lying

over er is diffeomorphic to the product (J>< F, in such a way that theo

diffeomorphism

satisfies

A.4- pr(1' o 0\

locally trivial fibration is called a fibre bundle and a diffeomorphism

A.3 is called a local trivialization. If (xA) is a coordinate system

in U and CyA) is a coordinate system in Fb then a local trivialization

241

~ induces a local coordinate system (x~,yA) in F. The coordinate ex­

pression of the mapping ~ is

X (xf-,yA)

The tangent bundle TE of a manifold E is an example of a fibre bundle.

The manifold TE is the collection of alI tangent vectors. The fibra-

tion

A.S TE ~B

assigns to each vector the point at which the vector is attached. ~ach

fibre TbE = 1;'i31(b) is the vector space of vectors tangent to E at b.

Given a coordinate system (x~) in B we assigne to each vector u~ TB

coordinates (X~,UN)consisting of coordinates (x~) of ~E(u) and the

components (u~) of u with repect to the system (x;<-)

A.7 u u::\'

The coordinate system (x;<-,u::\')can be used to define a local triviali­

zation of TE. Another example of a fibre bundle is provided by the

cotangent bundle T*E of covectors in E. The fibration

A.S

assigns to each covector the point at which the covector is attached.

Each fibre T: B = XBtb) is the cotangent space at b. Coordinates (x~

in E induce coordinates (x;<-,p~) in T*E, where p~ are components of

a covector p defined by

A.9 p

242

In general any tensor bundle over a manifold B is an example of a fib-

re bundle. The bundle

A.10 F

of k-covectors (k-covariant, completely antisymmetric tensors) is

an example frequently appearing in the notes.

A section of a fibre bundle A.1 is a mapping

A.11 s B ~ F

such that for each b € B the image s(b) of b belong to the fibre F'b'

A vector field in a manifold B is a section of the tangent bundle TE.

A covector field (a differential 1-~orm) is a section of a cotangentk

bundle T*B. A differential k-form is a section of !\ T:'EB•In a coordi-

nate system (x~,yA) of F a section s is described by sex) = (x~,yA),A A

where y = f (x).

B. Tangent mapping

Let

B.1 M ~ N

be a differentiable mapping af a manifold M on a manifold N. At each

point x (;!VI the derivative of o<.assigns to each vector u tangent to !VI

at x a vector tangent to N at «(x). We denate this vector by ~~u.

The mapping

B.2 TM __ TN

defined this way is called the tangent mapping (induced by 0<.) • The

243

restriction of <x* to T M is thus a linear mapping of T M on T ( ) N.x x C>(X

Let (x~) be a coordinate system in M and let (y~) be a coordinate

system in N. Let

B.3

where

B.4

o( (xl<)

be a coordinate expression for o<. • The value of o<. on a vector A.7 is

equal to

Hence, the coordinate expression for 0\* is

B.G= (o('\x) ,uJ{.

C. Pull-back of differential forms

Let

0.1 o( M

be a differentiable mapping of a manifold M on a manifold N. A covec­

tor p in N attached at a point oZ(x) E; N defines a covector o(*p in M

attached at a point XE- M. The definition of o(*-p is given by the con-

dition

C.2 < O(*u,p>

which we assume to be fulfilled for each vector u E TxM. The mapping

C.3 )(- NT c((x)

244

T* I'1x

defined this way is the adjoint mapping for ~*ITxI'1 (the restriction

of CX* to TxI'1). If

C.4 p PA,dy~

where CyN) is a coordinate system in N and

C.5 u uf<

where Cxf) is a coordinate system in I'1and if B.4 is a coordinate ex­

pression for oZ then the formula C.2 reads

C.6 < uf'-

doZ?

< u,p -dx?-)-?v 8 xl'- /

Hence, the coordinate expression for 0<..* is

C.7

or

C.8

The formula

C.9

enables us to extend the mapping Oi.:* to the space of mul ticovectors

C.10k

1\ T* I'1x

245

Let G be a differential k-form in N

C.11 N :3 Y G(y)

k

E 1\ T* N •Y

We define ak-form 0(*8 in M by setting

C.12 (0<:* e) (x) 0<*8 ( o{ (x))

The form o(~e is ealled the pull-baek of G from N to M. The pull-baek

eommutes with the exterior differentiation :

C.13

D. Jets

dO<!' 8 o(~de

In order to be able to assoeiate intrinsie meaning with partial

differential equations we need geometrie objeets eontaining informa-

tion about partial derivatives of seetions of fibre bundles. The par-

tial derivatives

D.1

of the seetion

D.2 B 3 (x){)~ s(xi<)

deseribed in Appendix A do not define a geometrie objeet sinee know-

ledge of these derivatives in one eoordinate system is insuffieient

to ealeulate sueh derivatives in another eoordinate system. The joint

set (fA,fA~) of values of a seetion together with the derivatives is

already suffieient to define a geometrie objeet. If (fA,f~) are known

in a eoordinate system (x~,yA)

, ~in a eoordinate system (x~,y )

246

I ,

then the eorresponding values (fA, fA,.,)

are given by

D.4

where

x'"

D.5

deseribes the ehange of eoordinates. Formulae D.3 and D.4 express

transformation laws of eomponents (fA,f~) of au objeet ealled the

first jet of a seetion s. Let s1 and s2 be two seetions having the

same jet at a point b E B. Then

D.6

and the two seetions have the same partial derivatives at b with res­

peet to any eoordinate system(xf).The geometrie meanning of this si­

tuation is that the graphs of the two sections are tangent to each

other at the point f = s1(b) = s2(b) E Fb' We eonelude that the class

of seetions of F tangent to each other at a point f EFb is a suitable

abstraet representation of a jet.

As an example we analyse jets of veetor fields. Let

x B TB

be a vector field described locally in a coordinate system (xf,u~)

( see Appendix A ) by

D.8

where

D.9

X (b)

247

We denote by f'\,the partial derivatives

eoordinate system in B and let (XI<',U;l.,I) be

TB. If

a f'" ,--- • Let (x~) be another'(l x~the eorresponding system in

D.10f'=xf'(xJ<)x

then D.11

A.-'

=d x?/ ;\,

u

--udX:A,

Transformation laws for eomponents Cf~,f~~) of jets are

D.12 ?V'f (3 x?v' 'A"--fd x/L

D.13

Aeeording to these transIormation laws partial derivatives I~v alone

do not determine partial derivatives f~,. We see that the partial de-

rivatives separately do not define a geometrie objeet although jointly

with the values I4 they form the set of eomponents OI a geometrie ob­

jeet - a jet. A jet is not a tensor.

The spaee of first jeets of seetions of a fibre bundle F (for­

mula A.1) is denoted by J1F. The jet OI the seetion D.2 at the point

bEB is denoted by j1s(b). Coordinates of j1s(b) are thus (xl'-,IA,I~).

The point seb);;Fb is called thetarget of the jet j1seb). The mapping

whieh assignes to eaeh jet j1SCb) its target seb) is ealled the eano­

nieal jet-target projeetion and is denoted by

D.14- F .

248

We usually treat the space J1F as a fibre bundle over B wit h the pro-

jection

Let

D.15 G .B

be another fibre bundle over the same basis B. We suppose that the

mapping

D.16 F G

preserves the fibre structure, i.e.

D. /17

The mapping X induces a mapping

D.18

defined by the following condition

D.19

The mapping J1X is called the first j et prolongation of X •

E. Bundle of vertical vectors

Let

E.1 F B

be a fibre bundle esee Appendix A) • By VF we denote a submanifold

of the tangent bundle TF composed of vertical vectors, i.e. of vectors

249

tangent to fibres of F. At each point b EB the fibre VbF is equal to

the bundle TFb of vectors tangent to the fibre Fb. We treat the space

VF as a fibre bundle over B

E.2

where

VF B

and '"(FIVF is the restriction to VF c TF 01' acanonical projection '0]'

in the tangent bundle TF (see Appendix A)

If (x!'-,yA) are coordinates in F satisfying A.5 then each verti­

cal vector may be written in the form

E.LJ. u

The coe1'ficients uA together with coordinates (x!'-,yA) form a coordina­

te system (x ,yA,uA) in VF. The coordinate expression for the projec­

tion E.2 is

(xl'-)

There is acanonical identification of the bundle VJ1F and J1VF

which may be described in coordinate language in the following way.

Coordinates (x~,yA,uA) in V]' induce coordinates (x~,yA,uA,yA~,uA~)

in J1VF (see Appendix D) • II' (x~,fA,fA~) are coordinates in J1F then

each vertical vector in J1F may be represented by

E.6 v. 8A _

+ w J'- (J JAt'

250

This means that coordinates in VJ1F are (XJ'-,fA,f~ ,vA,w~).

tification of the two bundles is obtained by identifying fA

fA 'th A A 'th A d A 'th Aj'- Wl Y J'-'V Wl u an w J'-Wl uJ'-'

F. Tensor produet 01 fibre bundles

Let

The iden­

with yA,

F .1

and

F.2

F

G

B

B

be two vector bundles over B. The notion "vector bundle" means that

each fibre Fb and Gb is equipped with a linear structure. The tensor

produet

F.3

of the two bundles is again a vector bundle over B. The tensor produet

is defined by the following formula

F.4

This me ans that fibres of F@ G are tensor products oi fibres of F

and G.

G. The Lie derivative

Assuming that F is a tensor bundle over B we construct the Lie

derivative L s of a section s of F with respect to a vector field XX

in B. A vector field X generates a one-parameter local group of trans-

formations of the manifold B. We denote elements of this group by ~t'

Transformations gt can be applied to tensors. This results in a one­

parameter lo cal group {rt} of transiormations of the bundle F. The

251

generator of the group is a vector field X in F. The field X is a lift

of the field X in the sense that

G.1 X('Jt Ci»

for each f E F. Given a section s of the bundle F we can lift the field

X to a field s*(X) of vectors in F defined on the graph of the section

and tangent to the graph. The difference )( - s*(X) is a field of ver­

tical vectors defined on the graph of section s. The Lie derivative

óE s is a section of the bundle VF defined byX

G.2 (i s) (b)X

Although we assumed that the bundle F is the tensor bundle over B the

same construction will work whenever the action of diffeomorphisms of

B can be lifted to F. In the case when this can be done in a canoni-

cal way we calI the bundle F the bundle of geometrie objects over B.

In the special case of a tensor bundle the construction can be carried

one step further. Since Fb is a vector space we can identify TFb with

FbxFb• Wit h this identification (ix s) (b) is a pair (S(b),f(b))

The section

G.3 B 3 b f (b) E F

is a tensor field of the same type as s. In the case of a tensor bun-

dle it is this field which is usually called the Lie derivative of s.

List of more important symbols

F @ B

F><B

Whitney sum or vectors bundles

exterior produet of differential forms or covectors

interior produet of a vector field X and a differential

form Q

tensor produet

cartesian produet of differentiable manifolds F and B

a v

dD(

'JA,

d;\; =<3

~

ddti

xs

boundary of a domain V

exterior differential of a differential form ~

covariant derivative

~ ...dx~ partlal derlvatl~e

horizontal lift of a unit vector from the basis R1 to a vec­

tor bundle over R1; see Sections 9 and 23

lift of a unit vector to dynamies; see Sections 9 and 23

Lie derivative of a section s with respect to a vector

field X; see Appendix G

D (Dg) wave operator (with respect to a pseudo-riemannian metric g)

TqQ (TQ) space tangent to a manifold Q at a point q E Q (tangent

bundle) ; see Appendix A

T~Q (T*Q) space dual to TqQ (cotangent bundle) ; see Appendix Ak

/\T~M space of k-covectors cotangent to M at x €: MxJ1Q

first jet extension of a fibre bundle Q(space of first

jets of sections of Q) see Appendix D

J1p space of generalized first jets (divergences) of sections

of a fibre bundle P; see Section 16 and Appendix D.1J q(t)...,.1J pet)

VF

first jet of a section q at a point t; see Appendix D

generalized first jet of a section p at a point t; see Sec-

tion 16

bundle of vertical vectors tangent to a fibre bundle F;

see AppendixE

253

the canonical jet-target projection for jets of sections

of bundle F; see Appendix D

x, 'Jf,

'"

X,

'Jl:,i a-aV, vV

0(* p

tA.J'-

t N "f<-

S 'N "f­

T /vf'­

Tk"f­

T;\')<..H

canonical projections from a phase bundle to a configuration

bundle

mapping tangent to a mapping ~ ; see Appendix B

pull-back of a covector p via ~ ; see Appendix C

first energy-momentum tensor; see Section 19

second energy-momentum tensor; see Section 19

spin angular-momentum tensor;.see Section 19

first stress tensor; see Section 25

second stress tensor; see Section 25

the Hilbert tensor; see Section 25

metric tensor in space-time

affine connection in space-time

Christoffel symbols (metric connection)

~ (w,,-I' + w(3o<, )

12 (wO(f' .,.. wl'o()

symmetric part of a tensor W~~

antisymmetric part of a tensor Wo(p

value of covector p on vector u (a contraction of

u and p)

< , >1 contraction of the first factor of the tensor produet 13.1

< ' >2 contraction of the second factor of the tensor produet 13.1

Q IL: restriction of a fibre bundle Q over M to a submanifold

~ c M; see Section 19

restriction of a section s to av;see Section 14

restriction of a form

ei to a lagrangian submanifold Dix x

( dynamies) ;

see Section 8

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