LNP107 A Symplectic Framework for Field Theories 0387095381

Lecture Notes in Physics Edited by J. Ehlers, Miinchen, K. Hepp, Ztirich R. Kippenhahn, Miinchen, H. A. Weidenmiiller, and J. Zittat-tz, Kijln Managing Editor: W. Beiglbijck, Heidelberg

Heidelberg

107 Jerzy Kijowski Wlodzimierz M. Tulczyjew

A Symplectic Framework fol Field Theories

Springer-Verlag Berlin Heidelberg

New York 1979

Editors Jerzy Kijowski Department of Mathematical Methods in Physics University of Warsaw ul. Hoza 74 00-682 Warszawa Poland Wlodzimierz M. Tulczyjew Department of Mathematics and Statistics University of Calgary 2920 - 24th Av. N.W. Calgary, Alberta, T2N lN4 Canada

ISBN 3-540-09538-l ISBN O-387-09538-1

Springer-Verlag Springer-Verlag

Berlin Heidelberg New York New York Heidelberg Berlin

Library of Congress Cataloging in Publication Data Kijowski, J 1943A symplectic framework for field theories. (Lecture notes in physics; 107) Bibliography: p. Includes index. 1. Symplectic manifolds. 2. Field theory (Physics) I, Tulczyjew, II. Title. III. Series. QC174.52.894K54 530.1’4 79-20519 ISBN 0-387-09538-l

W. M., 1931-joint

author.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 5 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 0 by Springer-Verlag Printed in Germany

Berlin

Printing and binding: 2153/3140-543210

Beltz

Heidelberg Offsetdruck,

1979 Hemsbach/Bergstr.

CONTENTS Introduction I. An intuitive derivation of symplectic concepts in mechanics and field theory

7

I. Potentiality and reciprocity

7

2. Elastic string

2o

3. Eiastostatics

51

4. Electrostatics

35

II. Nonrelativistic

particle dynamics 4-I

5. Preliminaries 6. Special symplectic structures.

Generating

functions

~2

7. Finite time interval formulation of dynamics 8. Infinitesimal description of dynamics

58

9. Hamiltonian description of dynamics

69

~0. The Legendre transformation

75

11. The Caftan form

75

12. The Peisson algebra

79

III. Field theory

80

15. The configuration bundle and the phase bundle

8o

14. The symplectic structure of Cauchy data on a boundary

85

15. Finite domain description of dynamics

91

16. Infinitesimal description of dynamics

tO0

17. Hamiltonian description of dynamics

116

18. The Legendre transformation

124

19. Partial Legendre transformations. -momentum density

The energy125

20. The Cartan form

143

21. Conservation laws

151

22. The Poisson algebra

157

IV

23. The field infinite 24. Virtual

as a mechanical number

of degrees

tensors

of different

160

of freedom

action and the Hamilton-Jacobi

25. Energy-momentum review

system with an

and stress

approaches

theorem

tensors.

16Zl-

A

168 184

IV. Examples 26. Vector

18zl-

field

191

27. The Proca field 28. The electromagnetic 29. The gravitational

198

field

209

field

231

30. The hydrodynamics Appendices A. Sections B. Tangent

242

mapping

C. Pull-back

of differential

of vertical

F. Tensor product

vectors

of bundles

G. The Lie derivative List of more References

forms

24"3 2#5

D. Jets E. Bundle

24-0

of fibre bundles

important

symbols

2q-8 250 250 252 25zt-

Introduction

~hese notes

contain the formulation

work for classical field theories. on fairly advanced concepts

of a new conceptual frame-

Although the formulation

of symplectic

is based

geometry these notes can not

be viewed as a reformulation

of known structures

elegant terms.

is to communicate to theoretical physi-

Our intention

cists a set of new physical

in more rigorous

and

ideas. We have chosen for this purpose

language of local coordinates

the

which although

involved

is more elemen-

tary and more widely known than the abstract

language

of modern diffe-

rential geometry.

We have given more emphasis to physical

than to mathematical

rigour.

Since the new framework unifies variational nonical formulations same symplectic

intuitions

of field theories

structure

formulations

as different

it is of potential

with ca-

expressions

of the

interest to a wide audien-

ce of physicists. Physicists

have been interested

viding a method of theories. variational

in variational principles

as pro-

of generating first integrals from symmetry properties

Powerful methods formulations.

v i n g variational

of solving field equations

are based on

We develop a systematic procedure

formulations

of physical

theories.

for deri-

Using this proce-

dure we have succeeded in formulating a number of new variational principles such as the formulation tes. ~he usefulness is in progress

of the procedure

on a variational

thermal processes.

of hydrodynamics

is far from being exhausted.

formulation

of hydrodynamics

A new variational principle

vity is also included in these notes.

included in these noWork

including

for ~h~ theory of ~ra-

~his new formulation

suggests

solution of the energy localization problem~

provides

lysing asymptotic

fields at spatial infi-

behaviour

of gravitational

a basis for ana-

nity and throws new light on the Cauchy problem for Einstein's tions and on unified field theories.

a

equa-

Quantum physics lectic

formulations

nonical

is one of the main sources of physical

quantization

tructing

symplectic

cial importance

theories.

and the associated

symplectic

structures.

quantum field theory such as the method fit from the more precise tions to gauge theories

theory.

lagrangian

Physicists

interested

will find the extensive submanifolds

in classical

of classical

We consider

theories

Each theory has an underlying time manifold

interest.

limits

theories

is included

mechanics,

with the boundary

/or statics/

allowed by the physical

The lagrangian

laws governing

problem

to the boundary

they are described

generated

of the new fra-

case.

by variational

the field.

domain in

consists

of

This space

of the boundary value

functions,

principles.

"sta-

by a lagran-

subspace

of the domain.

by generating

ction is the action functional.

A symplectic

of each /compact/

as the set of solutions

are usually

expansions.

and the four-dimen-

fields.

can also be described

spaces

theory for cor-

of the field is described

of the state space.

corresponding

Lagrangian

it is the physical

in the case of dynamical

states

in field

as a special

sional

gian subspace

Applica-

which we call field

space for static field theories

M and the dynamics

may pro-

of quantum theories

submanifolds.

three-dimensional

te space" is associated

in

manifold M which is the one-dimensional

in the case of particle

space-time

Of spe-

The new inter-

systems

of the main features

mechanics

methods

systems.

by modern W.K.B.

a class of physical

although particle

algebras.

Lagrangian

limits and asymptotic

We give a brief description

of cons-

local field theories

and hamiltonian

required

in ca-

throws new light on the re-

use of lagrangian

are the objects

rect formulation

mework.

Poisson

of lagrangian

transformation systems

interested

of Feynman integrals

may be of particular

of the Legendre

lation between

formulation

in symp-

general methods

may be the method of describing

by finite-dimensional

pretation

Physicists

will find in these notes spaces

of interest

Lagrangian

sub-

in other words~

Here the generating

If the boundary

is divided

fun-

into sere-

ral components

then the associated

ce and the lagrangian boundary

consists

symplectic

consist

of end points

out to be mappings. can be considered nonrelativistic tonian fields extend

mechanics

as the result

particle

This

relations

mappings

field

formulated

expense

equations

of introducing

an element

situation

terms

relations

gauge invariance

dynamics

mechanics

systems

for

lead to

governed

problems

by

can be

only at the

into the theory and imsituations.

Gauge invariance

in spaceby im-

of the theo-

gauge conditions. can be retained systems

Without

by using

similar to

compactifying

can still be discussed data,

Such

or compactified

hamiltonian

mechanics.

between initial

the

in terms of sym-

final data and the asymptotic

Boundary problems

other than Cauchy

within the same framework.

are obtained

and does not

is obtained

interesting

suitable

and generalized

be discussed

nal formulations

boundary

elements

conditions.

or outgoing radiation.

problem~can

in terms of hamii-

hamiltonian

must be compact

in relativistic

Cauchy hypersurfaces

incoming

Consequently

family of Cauchy hypersurfaces

by imposing

Within our framework

relations

turn

in the case of a field theory governed

non-intrinsic

asymptotic

ry must be destroyed

plectic

state.

No boundary problems

and this formulation

is a one-parameter

those appearing

boundaries

Already relativistic

Only very special

The Cauohy hypersurfaces

symplectic

In non-

relations

is exceptional

and generalized

excluding physically

posing restrictive

are time-intervals,

in the case of static field theory

equations.

posing conditions

If the

two-term

mapping.

can be formulated

/see ref.[37]/.

in hamiltonian

by hyperbolic

/canonical/

of the initial

mechanics

and flows.

its proper formulation

elliptic

corresponding

spa-

In this way the state at the end of an interval

symplectic

symplectic

relation.

and the corresponding

easily to other field theories.

requires

-time.

domains

space is a product

a symplectic

the~the

may be a symplectic

particle

of pairs

becomes

of two components

relation

relativistic

subspace

symplectic

by considering

limits

Finite-dimensioof domains

contrae-

ting them to points. We attach ~an

special

subspaces.

nerating

We mentioned

function,

many ~enerabinc mations.

importance

furcb~ons

related function

plastic

structure"

ionian

of particle

mechanics

of

sa e ! gr n

an s, hm

rived

/such

describing

level.

can be described

to eao ~ ~th~r by Le~endre is associated

as components

l

ee r e f .

functions

3]I. Most p h y s i c a l

of the enercy-momentum

as generating

sym-

and the Hamil-

can be shown to be generatinz

ifol

by

transfor-

with a "special

~he Lagrangian

functions

tensor/

of lagrangian

are de-

subspaces

fie]@ dynamics.

Chapter I is devoted to the symplectic discrete

of lagran-

of the action as a ge-

subspace

or a "contral mode".

in our approach

functions

the interpretation

mhe same ]agrangian

Each generating

quantities

to generating

and continuous

The notion

systems

analysis

are considered

of reciprocity

of statics.

on a largely

and potentiality

Both

intuitive

of the theory is

discussed. Chapter Ii is a presentation more rigorous manifolds

definitions

is studied principle

of particle

functions

dynamics

structure.

are defined

together with Lagrangian

states within a finite time interval

can be derived from composition

of histories

is stated

of the particle

grangian description

properties

in an infinitesimal system.

of dynamics.

ved in Section 9. Section

sub-

in Section 6.

in Section 7. It is shown that the Hamiltonian

Section 8 dynamics

terms

of the geometric

and their generating

The time evolution

of particle

11 contains

of dynamics.

In

form in terms of jets

Section 8 contains

The hamiltonian

variational

also the la-

description

a formulation

is deri-

of dynamics

in

of the Caftan form. The Caftan form is an object used in the

geometric

formulation

Caratheodory,

of the calculus

de Ponder,

Lepage,

of variations

Dedecker

developed

by Weyl,

and others /see ref. [58],

[~, D2], ~ ] , ~]/" Chapter

ili is the main part of these notes.

The construction

of

canonical momenta of a field is given in Section 13. Field dynamics is first discussed contained

of infinitesimal

dynamics

sense but a natural a definition

generalization

level.

structure

of the concept.

Section 19 contains

associated with a family of control modes.

ly the most complex part of this volume.

is followed by a discussion is formulated

Section 19 is technical-

Results of this section are

section establishes

language.

This

content and proofs.

Dy-

of the Caftan form in Section 20. This

a relation between our symplectic

the geometric formulation

Sections

of the intrinsic

in terms

of the

We consider this

first stated without proofs and in purely coordinate

integrals.

in the strict

of the energy-momentum density as the potential

definition one of the most important results.

namics

A rigorous

starts in Section 16. The sym-

structure used here is not a symplectic

dynamics

This discussion

in Section 14 and 15 stays on a heuristic

formulation plectic

in finite domains of space-time.

of the calculus

The time evolution formulation

22 and 23. An infinitesimal

framework and

of variations of dynamics

of multiple

is derived in

version of the Hamilton-Jacobi

theorem is proved in Section 24. The last section of the chapter contains a detailed discussion

of objects associated with the energy-mo-

mentum of the field. Different definitions and stress-tensors results

of energy-momentum tensors

are compared and new definitions

are proposed.The

of this section are used in a new formulation

of General Re-

lativity given in Chapter IV. Chapter IV contains strate various

features

examples

of field theories

of the new approach.

selected to illu-

The simplest

example

of

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

of constraints

in the h a m i l t o n i a n d e s c r i p t i o n is illu-

strated by the example of Proca field in Section 27. An example of a gauge field /the electromagnetic A new formulation

field/ is discussed

in Section 28.

of the theory of gravity is given in Section 29.

The new formulation

consists

in using the affine connection

~

in spa-

ce-time

as the field configuration.

on the connection vature

~

and its first derivatives

tensor R. The metric

mentum canonically objects

tensor g appears

conjugate

conjugate

standard Einstein

to P

together

by the cur-

as a component

of the mo-

between these two equations.

components.

instead

depending

stein theory of gravity is a very special theory based on an affine

connection

Ricci tensor

mework one of the versions Other possibilities

are equa-

only on the symmetric

of the full Riemann tensor.

r . Using the Lagrangian

of Einstein's of formulating

unified

part

Thus the Ein-

case of the geometric

one can easily reproduce

The

To obtain the

theory of gravity most of these components

of the Ricci tensor

~4]/.

represented

with Einstein's

has 80 independent

ted to zero by using a Lagrangian

on the complete

of the theory depends

to P , and the relation

is a part of dynamics

momentum

The Lagrangian

field

depending

within this fra-

field theories

/see

unified field theories

are

being investigated. The last Section analysis

contains

of hydrodynamics

logy with electrodynamics. ciple can be formulated directly

of a variational

Appendices frequently

In both theories

reveals

equations

a formal

ana-

a simple variational

or Maxwell's

prin-

which do not appear

equations.

and the subsequent

is an example

An

illustrating

The dis-

formulation

the fruitfulness

to field theory.

contain

a short review of several

used throughout

references [29]

our framework

for hydrodynamics

principle

of the new approach

within

of hydrodynamics.

only in terms of potentials

in either Euler's

covery of potentials

the formulation

the notes /further

geometric

details

concepts

may be found in

I. An intuitive

derivation

of symplectic

concepts

in mechanics

and

field theory.

I. Potentiality

and reciprocity.

In the present physical

systems.

Static

ple well understood to introduce

chapter we consider

conceptual

symplectic

sical characteristics

concepts

we use more complicated

cepts derived theories.

examples

described vilinear

to gradually

belonging

mechanism

sition of the point then an infinitesimal (q~

to

~i

+

~ q~

I. I

A

The Einstein notes.

requires

summation

The coeficients

called the force.

=

con-

dynamic

is much cle-

concepts

philosophy

suspended

space Q. The position

If an external

con-

derived

of trea,

[34 .

point

(q~ , i=1,2,3,

sections

The geometric

theories

exam-

symplectic

of these concepts

with the Minkowskian

a single material

by coordinates system.

and the exis-

for use in primarily

to dynamic

in space-time

physical

develope

is

of such phy-

In subsequent

systems.

meaning

Applying

agrees

as statics

three-dimensional

as reciprocity

continuous

the intuitive

from static theories

We consider

expression

of freedom.

in this way are intended

arer in static theories.

ting dynamics

of degrees

for describing

However

of having a sim-

The aim of this chapter

as a natural systems

of static

In this section we begin with a very simple

ple with a finite number

suitable

have the advantage

structure.

of static

tence of potentials.

cepts

theories

a series of examples

the mechanism

elastically

in the

of the point will be in general to a cur-

is used to control the podisplacement to perform

from a position a virtual work

fi ~ qi.

convention

is used here and throughout

fi in the above expression

these

form a covector

f

The force f is actually the force that the control-

ling mechanism

has to exert to maintain

ment shows that for each configuration cessary to maintain ce are functions

fj

1.2

=

?

If the form

~

then the result

is evaluated

of

~

__~ : T~-----+~

the tangent mapping

=

from T~ to ~

@ . ~he evaluation is g~ven by

bundle projection

/cf.

Appendix

and

$~u is

B/. ~he definition

6.fl

to

O(p) Q(p)

~-form

of a manifold

< _z. u,p >

is the cotangent

6.2 where

buudle ~ = T ~

differential



is equivalent

functions

on a vector u tangent to ~ at a point p ~

6.~

where

Generatin~

is the value of

the covector p e T ~

-- _~'(p) 0

at p and

to @*~ /cf. Appendix

The differential

denotes

the pull-back

of

C/.

form

6.3

is called the canonical

gO

2-form

=

d@

on P. It is a standard

manifold P together with the 2-form

(~, ~)/c~. b ] , b l / -

~p

~

define

result that the

a symplectic

manifold

43

In the case considered in Section 5, we have a family of canonical l-forms Pt = ~ Q t

0 t and canonical 2-forms

~ t defined on each phase space

separately.

If a coordinate system

(qJ), j=Q,...,n,

every pQint of ~ the differentials

dq J

is chosen in ~ then at

form a linear basis for co-

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

to this basis together with the coordinates q~ of the point define coordinates

~(p) e

(qJ~pj) in the space T*~. The local expression for

in this coordinate system is

6.~r

~

=

pjdq J

.

9be Einstein summation convention will always be used. The local expression for 60 is consequently

6.5

~

=

dpj A dq J

In the case of the configuration bundle Q we shall use coordinate systems (t,q j) which are compatible with the f~bration

~ ~ i.e.

(t,q j) = t. The construction introduced above leads to the coordinate system (t,qJ,pj) in the phase bundle P, compatible with both fibrations

~

and

~

:

%(t,J,pj)

=

t

6.6

(t,j,pj)

= (t,J)

The fundamental geometric concepts used to describe dynamics will be that of a lagran~ian submanifold of a symplectic manifold /see[55],

[57]/. Definition

: A lagrangian submanifold of a symplectie manifold

44

C~,~)

is

a

submanifold

The condition bivectors

tangent

called isotropic.

N c p such that

CoiN

@ IN = 0 means that to N. A submanifold A simple algebraic

sion of an isotropic

submanifold

@

of generating

argument

Lagrangian,

the action,

nifolds. tion

Let N be a lasrangian

~IN of the canonical

We consider

when evaluated

manifold

submanifold

objects

functions

submanifold

1-form

only simply connected

~

is

is thus an iso-

in terms

the Hamiltonian~

of lagrangian

to N is closed

the

tensor in field

of ( T ~ , a o ) .

lagrangian

on

~,

submanifolds

and also the energy-momentum

theory will be shown to be generating

dim ~.

shows that the dimen-

lagrangian

Such important

I ~

=

this condition

N of a symplectic

aim at describing

functions.

0 and dim N

vanishes

satisfying

is not hisher than ~ dim P. A lagrangian 2 tropic submanifold of maximal dimension. We will always

=

subma-

The restric-

since

submanifolds.

It follows

that there is a function S on N such that

This function Suppose

is called a proper function

that N is a section of a bundle T*~ over C c 2-

case the proper a projection submanifold

function ~ definies

N which

completely

the

is simply the image of the section

dS

In a coordinate

In this

a function S on C which is simply

of ~ onto C. The function S determines

6.8

manifold

of N.

system

N is described

:

C

(qJ,pj)

~ T#~

.

the above statement

by equations

means that the sub-

45

6.9

pj

The f u n c t i o n

S is called

Generating

S

--

% qJ

a generating

functions

function

can be also used

w h e n N is no lon~er a section

of N.

in more

complicated

cases,

of T ~ .

Let C c ~ be a s u b m a n i f o l d

of ~ and let S be a f u n c t i o n

on C. It

I ~ (p) ~ C ; 4 u , p >

for

can be easily shown that the set

N

{

--

6.70

p ~ T*~

each vector

is a l a ~ r a n g i a n ~enerating

submanifold

function

S is a p r o j e c t i o n

of N. Also

u tangent

. The f u n c t i o n

in this

to C at £t(p)}

S is called

case the g e n e r a t i n g

a

function

onto C of a p r o p e r f u n c t i o n S given by 6.9. L a g r a n -

gian submani:Folds which characterized

of (T*~,cO)

=

can be g~erated:~, in this way can be l o o s e l y

as those whose p r o p e r

£unctions

are n r o j e c t i b l e

onto

~ub~ani~olds of 2 /cf.[~@/. Generating

functions

up to an a d d i t i v e

functions

are d e t e r m i n e d

constant.

If the s u b m a n i f o l d

6.77

in a c o o r d i n a t e

as well as p r o p e r

C is ~iven by e q u a t i o n s

G~ 6 q j )

system

t h e n the s u b m a n i f o l d

o ;

:

(qJ)

~ =7,...,k,

and i f ' S is any c o n t i n u a t i o n

6.70 is given by equations

G ~ (qJ)

/cf. [6]/

o

--

6.42

~ Pj

=

%qj

+

2~

~ G - - - ~~

~qJ

of S to :

46

Only the symplec%ic lagrangian submanifolds.

structure

of a manifold

is used to define

To define proper functions

and generating

functions we needed much more structure namely the structure tangent bundle. one encounters

In applications symplectic

bundles but are isomorphic be called special plectie

structure

of symplectic

of a co-

geometry to dynamics

manifolds which are not directly cotangent to cotangent

symplectic

manifolds.

bundles.

Such manifolds will

More precisely

in a symplectic manifold ~ P , ~

a special

sym-

is a £ihration

and a symplectomorphism

such that

where

~

: T*~----~

a symplectomorphism from T ~

is the cotangent bundle projection. the pull-back

~

to ~ is equal to the symplectic

The presence manifold ~ , ~ )

of a special

form

symplec%ic

ring functions

are constructed

~

some lagrangian

on submanifolds

aP

submanifold

subma-

of 2" GeneraThe l-form

is a differential

of a function called again a proper function of a manifold. ting function is the projection

is

in a symplectic

as in cotangent bundles.

restricted to a lagrangian

2-form

~

.

structure

makes it possible to describe

nifolds by generating functions defined

= ~

of the canonical

Since

A genera-

of a proper function to ~ .

Thus we see that the objects of a special symplectic used to define generating functions

are the projection

_~

structure onto a ma-

47

nifold ~ and the q-form

~

such that d ~

= ~

One symplectio manifold may be equipped~ with several special symplectic structures in which case one lagran~ian submanifold may be generated by several different

~enerating functions.

We will find that

the Lagrangian and the Hami!tonian /in field theory also energy-momentum tensor/ ~re generatin~ functions of the same legrangian submanifold with respect to different special symp!ectic structures /cf. ~@/.

7. Finite time interval formulatiou of dynamics

As

is

usual in canonical

formulations

a~sume the existence of a differentiable

Qf particle dynamics we

two-parameter

family of dif-

feomorphisms

g.1

R(t 2,tq)

:

Pt I

~ Pt 2

satisfyin~

7.~

-~(t3,t2) ° ~ 2 , h )

Dynamics : M

--

~t3,h)

is expressed in terms of this family as follows.

A section

~ P is dyn.amic~]!y edmissible if and. only if

7.3

~(t 2)

=

R(t2,tQ ) (~{(tfl) )

for each (tl,t2). It is assumed that mappings R(t2,t~ ) are symplectomorpbisms

:

g •~

This formulation

R ~(t2,tl)

6Or 2

=

6Ot~

of dynamics is equivalent to statin~ a system of first

48

order differential cally admissible

equations.

histories.

the sytem of equations.

Solutions

of particle dynamics can be described to field theory.

We define a

family of submanifolds

D (t2'tq)

graph R(t2,tq ) c

=

p(t2,tl)

The manifold

Pt 2

×

Pt I

=

P (t2'tQ)

law 7.2 reads

D(t3,tl )

•

will be called the boundary phase space corres-

pondins to the time interval [tj,t2] c M. In terms of graphs the

position

of

We return to this point in the next section.

in terms suitable for ~eneralizations

7.5

are dynami-

The family R(t2,tq ~_ is the resolvent

Th~s classic formulation

two-parameter

of the equations

com-

:

=

{

(~3~,p)~ ~4,, P (t3,tl) I there is a ~p @ p t 2

7.6

such that (c~ ,py (~ m D(t3't2 ) , ,(~)(I)\ [p,p) ~

We wil]_ denote the right hand side by O me way we introduce multiple

(t~'tN-~)

.....

(ts,t2)- ° (t2,t~)

compositions

-

(t2'tl]] .

. In the sa-

of relations

D(t3't2 ) o n(t2't! )

there is a sequence

7.7

D

= { ( c ~ $ 0 E p(tN'tl) I

(%'[ ...,p) (~'

~ p

×

...x

iN_ ~

Pt2

such t~at (T',{') ~ D (ti+1'ti) , i .< N-n]

corresponding

to divisions

of the time interval (td,tN)

into N-I sub-

intervals.

In terms of this definition we have the composition

7.8

D

(t N, t~)

=

D

(~N'~N-¢

.....

D

(t2, t~)

law

49

corresponding to

2.9.

R(tN,tl )

=

R(tN,tN_I )

... oR(t2,tl)

o

The property 2.& of the resolvent is equivalent to D (t2~tl) being a lagrangian submanifold when an appropriate symplectic structure in P

(t2,t 1)

2.10

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

(u-) (t2'tl)

~he "minus-sign"

=

M

~

M -

t2

dO

tI

is defined by

j

-

iS a generating function for D (t3'tq).

Under specJ_al conditions which are not stated here the same composition law holds in the presence of constraints.

In this case the

sequence ('~,~4',...,~)must be compatible with the constraints so t~at the right-hand

side of 7.25 is defined.

The constraint

c(tN'tl)cq (tN'tl)

is the set of pairs (~,~) for which stationary points (¢~),...,~{,({) exist /see[g5]/.

Example I The configuration

bundle of the harmonic

oscillator

is the tri-

vial bundle Q = M × R I and the phase bundle P can be identified with M × R 2. In terms of coordinates

mh

(t,q,p)

=

p

=

-kq

the equations of motion are

7.3d

Integrating these equations we obtain the general expression for dynamically admissible

q(t)

sections

A cos~.t

+

B

sin~.t

T~ 7.32

p(t)

-Af~sin~-~.t

+ Bcos~.t

(t2,%) The manifold D

is described by equations

'p) sin ~ (t2-h) 7-33 4)

=

_~

55

(tf,tl)

to

In order

find the p r o p e r

by c o o r d i n a t e s

(q,p)

function

we p a r a m e t r i z e

D

:

dW _ {if' tl)('~,~)

@{t~'t~) I (tf,t 1)

=

aq

-

?. 3~

'"

+ P---- sim

G

(t2-t ~

I}

~-~,

~ p~q

Hence

?.35

.~o~~ (~,_-~ To obtain the g e n e r a t i n g Q

(t2,t 1)

. We c o n s i d e r

(i) I f sin ~ ( [ t 2 - t j )

7.36

Hence

p

C

function

three

~

cases

~, ~

:

c~ cotan ~ t 2 - t l )

= Q

(t2,t I)

and

~_~

we project w (if't1) to

0 then

=

( t 2 , t ~)

W (t2'tl)

_ ~

56

( t 2 ' t q ) r~a~{4, t~,q)

WEt2'tfll

=

_

('~,~'(~,

= -~<x{z) ,p(~)>

I~ m

defines a covector

~t



.im

h-,O

and

~ --t 8.13

=

lim

=

lim h~O

~{ = l i m h~O

{t+h),zCt}),

because D (t+h't) is lagrangian.

If the dynamics certain additional

=

0

This completes the proof.

is introduced

in terms of a family

ID~

then

conditions have to be satisfied in order that the

formula 8.22 defines a lagrangian submanifold. We assume that for each t e M the infinitesimal a generating function L t :

Qit '

dynamics D i has t > Rq" The c o r r e s p o n d i n g proper function

of Dit is denoted by ~t" Both L t and ~t are defined up to an additive constant.

The family

~Lt~

called the Lagrangian.

defines a function L: Qi

The Lagrangian

~R q which is

is defined up to an arbitrary

additive function depending only on t. Using coordinates

(qJ,pj,~J,~j)

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

8.2~

pjdq j + pjd~ j

equivalent

to the familiar formulae

f~j

=

=

dLt(qJ,~J )

~qj

s(t,qJ,~J)

8.25

T,(t,q4,44)

p4

It is interesting to note that also the composition for generating functions has its infinitesimal give in the case of no constraints

:

Theorem 3 Let a function W

(t2,t I)

be defined by

law 7.25

formulation

which we

66

8.26

w(t2'tl)(

)

--

t2

tI where for each

C~' ,~)

the section

M ~ ~

,

q(z)

~

Q~

is a stationary section /in the sense of the calculus of variations/ of~the right-hand side, such that q(tl) = ~ , q(t2) = '~. The function W

(t2,t fl)

is a generating function of D

(t2,t fl) .

Equations 8.25 are obviously equivalent to Euler - Lagrange equations. This implies that stationary sections are precisely projections by ~

of dynamically admissible sections.

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

:

Theorem 3 ~ (t2~t I) be defined by

Let a function W

t2

w(t2, tfl)

8.27

, p,~, )

=

i

,

t~ where for each C~ ~ ,~) e D (t2'~q) the section

~--~pg~)

unique dynamically admissible section such that p(tl)

is the = p ,

P(t2) = ~ . The function ~ (t2'tq) is a proper function of D (t2'tl)

Proof: Let (~ ,u) ~I, e TP (t2' tl) be a vector tangent to D (t2'tl) and let

%

~C~(~),~(~))

its tangent vector at such that for each

~=

be a curve in D (t2'tl) such that (~ ,u) c~), is O. There is a unique mapping

~ the mapping

(u,%]-~p(~,%)

67 l

is a d y n a m i c a l l y

admissible

section

(I)

and p ~ (tl)

= p(%),

p,(ty)

~)

= p(&).

Then



T*Pt

is defined by

9.2

<w, Tt(u)# -- < u ~ w , ~ t >

The composition

=

°

is the required diffeomorphism Let

~t

from Pti onto T~Pt

: T~Pt---+ Pt be the canonical cotangent bundle projection.

Let

@ ~ and ~ht be the canonical Q-form and the canonical 2-form in h ~ h T*P t. The condition ~ t = JUt°O(t is obviously satisfied and the equa^ h will be proved using local coordinates. The q-form tion dO ti = o:

9.6

=

pja j - ~Jbj

Since a j and bj• are arbitrary we conclude

that

follows that the coordinate expression for ~

mj

=

p j,

nJ

=

-~J

.

It

-- 7to~t is

~(qJ,pj,~J,~j) = (qJ,pj,mj,nJ)

where

9.7

Substituting

mj•

=

~j ,

nJ

-q"J

9.7 into 9.$ and 9°5 we obtain the following

expressions 9.8

=

O~

=

pjdq J - ~Jdp j "

coordinate

72

h~ ~ h O(t cOt

9.9

=

d~

j,\dqj

- d

The last equality proves that 04

~JAdpj

=

i cot

"

~h t defines a special

together with

Through each point p ~ Pt there is exactly one dynamically

admis-

sible section of P. The jet of this section is the unique element of D it attached at p. It follows that D ti is the image of a section of the h pi bundle 0"6 i : t ~ Pt" Since we also assumed that fibres of Q and consequently

fibres of P are simply connected,

each lagrangian subma-

nifold Dit is generated by a function F t on Pt" Functions H t = -F t define a function on P called the Hamiltonian.

The Hamiltonian

is defi-

ned up to an arbitrary additive function depending only on t. Using coordinates

• "'[qJ,pj,~J,~j) we find that D ti is described by

the equation

9.10

~jdq j - ~Jdpj

=

- dHt(qJ,pj)

analogous to 8.24 and equivalent

Pj

=

qJ

=

-

to the formulae

_~_~ H(t,qJ,pj ) Dq3 3pj~ H(t'qJ'Pj)

known as the Hamilton canonical equations.

Example 3 Equations

of motion of the harmonic oscillator

in the form I

9.12 =

-kq

can be written

73

Restricting the form

~hl ~t I Dit

~

= ~dq - ~dp to D ti we obtain

=

- kqdq - ~pJdp

:

_

=-d

2~pJfJ2 + kq2~}

9.13 t

(q,p)

It follows that D ti is generated by the Hamiltonian

H (t,q,p)

9.Jz~

=

q#1 2

2[~P

kq2)

+

In the preceding section we constructed a special symplectic structure in the symplectic space (P~,CO~). The fundamental objects i pi i of that structure were the projection ~ t : t ~ Qt and the J-form i satisfying d ~

@t

i . In the present section we constructed ano= cO t

ther special symplectic structure in ( pit , ~ ti) depending on the choice of a trivialization in Q. The fundamental objects of this structure are the projection

~U~ : pit----*Pt and the J-form

~

which again

= @ ti " With respect to the two special symplectic structures the same objects D ti are described by two sets of genera-

satisfies d @

ting functions. The two descrptions are parallel. The formula 9.10 \

has its counterpart in the formula 8.2~. The difference

@i_ ~ h is a closed J-form. Due to our topolot t gical assumptions it is also exact. We define a function ~t on Pti by

9. 5

If

g

9.16

Yt(g)

=

= ~jpj DPj

Hence 9.q8

dVt

=

pjd{J =

~OdPa +

0{ -

O~

The function ~t on Dit defined as

9.q9

tlt

~t l Dti - _Lt

=

satisfies the equation

i t

h

-

=

=

Hence -_Ht is the proper function corresponding

-

i

et

to the generating fun-

ction -H t. Our approach to hamiltonian description

of dynamics

from, though equivalent to, the standard approach.

is different

Since for each point

peP

there is a unique vector /jet/ in D ~p) i

attached at p

the family

ID~

defines a vector field in P which will be denoted by ~ t " The

difference

9.20

xh

-

d

dt

_

~__ at

is a vertical vector field in P. In coordinates dd-~ and X h are :

9.21

and

d

d--t =

a

8--t +

~j

a ~qJ

+ Pj

a ~Pj

"-(t,gJ,pj) the fields

75

9.22

Xh

where

~J _ _~ + ~j ~qJ 3pj

:

qJ and pj are functions

fields

on P are usually

on P given by 9.11.

called time dependent

Vertical

vector fields/cf. [1]/.

i It can be shown that the fact that D t are lagrangian equivalent se of

to X h being a locally hamiltonian

[1]/. In our ease X h is even globally

ar from equations

9.11 that our Hamiltonian

vector

submanifolds

is

vector field /in the senhamiltonian

and it is cle-

H is the Hamiltonian

for

X h in the usual sense.

10. The Legendre

transformation

We assumed that the infinitesimal fibration

i p~____~ i ~t : ~ Qt" It follows

there is a unique

locity/

element

D ti is a section

dynamics

that for each element

g e Pit such that

~(g)

of the

i v ~ Qt / v e -

= v. The map-

ping

i Qt

is called the Legendre the fibration title

/of.

Pt

~ v

Since D ti is also a section

transformation.

~ ht : pit----~Pt

the Legendre

transformation

of

is inver-

[9],[53]/.

11. The Caftan form The disadvantage in the dependence of combining

on the choice

the different

trivializations /cf.

a way manifestly

ponents

object.

corresponding

This object

of dynamics There

Hamiltonians

of trivializations.

are extracted

to different

lies

is a method

to different

is the Cartan form

this form in terms of the Lagrangian

independent

with respect

description

of a trivialization.

Hamiltonians

into a single

[TJ/. We define

how different

of the hamiltonian

and thus in

We show subsequently

from the Caftan form as com-

trivializations.

76

Definition: l-form

~

The C a f t a n form a s s o c i a t e d

w i t h the d y n a m i c s

is a

on P such that

i~.I

--- •

"~ Pt is a d i f f e o m o r p h i s m

a lift --Hi of the above f u n c t i o n

~t~i¢~!~

to this t r i v i a l i z a t i o n .

where X h is a v e r t i c a l

~t : ( - ~ , ~ > the m a p p i n g

of Q the f u n c t i o n

we may define

i to D t. The lift of the f u n c t i o n

gives obviously -~t Since xho~th = ~ t it follows t~at the

lift of t~e funotion <X ~, et> is equal to 2 can be interpreted

n with

the second

In p a r t i c l e factor

in

~ ~Qx

15.1

factor

dynamics

13.6

of the tensor

m = 1 and

is trivial

as the c o n t r a c t i o n product

/°~ T x~ M = R 1

of the

(m-l)

13.1.

Hence

the second

and

Pq

=

T ~q Qx

Px

=

T*Qx

or

15.7

as in formula Let

5.1.

(x~),

%=l,...,m,

A=I,...,,N, dle

structure

be a coordinate

be a coordinate

system

in M and let (x~,~A),

in Q compatible

with the bun-

:

13.8

A)

A vector

system

density

=

(x

in M can be expressed

as a linear

combination

of

~-l)-forms

(-1)k-ldxl^ . . . . . . ^

15.9

where

the symbol

follows

that

A.%

eA

=

that the

coefficients

;9

]dxl~... A d x

~t-th factor

p e Px is a linear

dqOA@( !

The

=

m

~x ~-

means

each element

13.1o

Adx m

p~ t o g e t h e r

~ _Jdx I ...

has been

combination

dx m)

omitted.

p = P A e ~A where

.

~ x~

with the coordinates

It

(X "%, ? A ) o f

the

84

point

g~fp) define a coordinate system (x ~, ~A,pA~) in P. There is a canonical vector-density-valued

l-form

0 x on each

fibre Px defined by the formula

13.11

@x(P~

=

~xP

analogous to 6.2. Here

~ *xp denotes the pull-back of the first fac-

tor of the tensor product 13.1 from Qx to Px" If v is a vector tangent to Px at p then the value of

~x

on v is a vector density at x given

by the formula

I ~v

if

I#.3#

V

is a section

~

x

~ Z(x)

such that Y(x)

and Z(x)

~

TP x

are vectors

attached

at the sa,!

me point

~(x~• ~ Px for each x ~M.

Formulae

I#.32 and 1#.33 are ana-/~J ~ f

logous

to formulae

7.21

and 7 . 1 1 .

In analogy to formulae

7 . 2 0 and

7.10 we write M

14.35

~v 4 ¸

~V M I#. 36

60 8V

= f(~Ox ~V

15. Finite domain description Finite

domain description

of dynamics of field dynamics

can not be presented

92

with a rigour matching the finite time interval description of particle 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 particle dynamics. ly we give only heuristic considerations

Consequent-

as an introduction to a ri-

gorous infinitesimal description of field dynamics given in the next section. We begin with the discussion of electrostatics

in a 3-dimensio-

nal manifold M which is assumed to be a riemannian manifold with a metric g. The configuration space Qx at each point x e M of values of the electrostatic potential

~.

is the space

Hence Qx = R1 and Q is

the trivial bundle Q = M × R 1. The value of the electrostatic potential ~

together with coordinates

(x ~) in M define a coordinate system

[x~,~) , ~ = 1,2,3 in Q. The first factor in the formula 15.1 is tri& $ vial in this case. It fellows that Pq = /~T~M, Px = R ~ / ~ T ~xM and

p : ~1× / k ~ M

~lements of Px are thus pairs (%p) where ? ~

the value of the electrostatic potential at x and p ~ T ~ M value of the electrostatic coordinates ( x ~ , ~ )

i~

is the

induction field at x. Corresponding to

in Q we have coordinates ( x % , ? , p %) in P. In terms

of these coordinates

15.1

@x

: d W ® ( P ld~d~3 + ~2dx3Adxl + p3dx~dx2)

and

dJ x 15.2

+ (d~3AdT) ~(dxIAdx 2) If the coordinates

(x~) are chosen in such a way that the boundary 3V

of a domain V is described by the equation x I = const, then is a coordinate system in the space P~Vx " Coordinates

?

(~,pl)

and pl

93

are interpreted

as the value

of the induction

on

of the potential

8 V /interpreted

and the normal

as the surface

component

charge density

on

~v/. The equations

of electrostatics

~5.3 where

p ~

with the metric

15.4

sity.

=

- ~

is a 3-form /scalar density/

In terms

and

* is the Hodge

operator

g, and

Vp

~

:

= .V T

is the exterior differential

associated

where

are

of coordinates

,

representing

the above equations

a fixed charge denread

:

~X ~

15.6

~P~ ~x m

=

- ~

where r is the scalar function equivalently,

--

~

A?

~=

r dxhdx~dx3

r = , ~

, or

.

15.5 into 15.6 we obtain the Poisson

15.8

where

defined by the equation

by

15.7

Substituting

~Fg-~r

~ g ~ ~ i s

=

- ~

the Laplace

equation'

r ,

- Beltrami

operator

associa-

vo

ted with the metric Poisson

equation

g. Applying

the formula

15.8 we generate

15.3 to solutions

dynamically

admissible

of the

sections

of

94

P.

It follows is the normal on

from the formula

derivative

8 V determines

cifying the value

of ~ .

15.5 that the normal Specifying

the normal

component

of p

component

of p

thus the Neumann data for the Poisson

equation.

of

data.

~

on

8V determines

the Dirichlet

Spe-

We con-

clude that the space pSV which we called the space of Cauchy data is the space of combined Dirichlet Dirichlet

and Neumann data on

of the Poisson

equation.

tions to the boundary D ~V is composed

space Q~V of Dirichlet

to any solution admissible

~ V form a subspace D ~V of pSV.

regular

of Dirichlet

sec-

The subspace

and Neumann data

domain V it can be shown that if the

data is the Sobolev

submanifold

(PaV,co~V).

of dynamically

combined

to solutions.

For a sufficiently

space

Restrictions

of

In general

BV do not correspond

of those special pairs

which do correspond

is a lagrangian

and Neumann data.

~ V then D ~V

space H 1/2 on

of the infinite

dimensional

symplectic

In this case

15.9

paY

=

T~Q~V

=

H 1 / 2 × H-1/2

which means that the space of Neumann data is the Sobolev

space H -1/2

dual to the space H 1/2. A lagrangian

submanifold

of an infinite

and maximal

We use the Green's

formula to prove that D aV is isotropic.

M

~ x

M

x

and

15.11

~ (~(x),p(x))

sense /of.

symplectic

space is isotropic

15.10

in a certain

dimensional

[8] and

~ Px = R 1 × /~ Tx* M

Let

[57]/.

95 be sections of the bundle P. We denote by ,~ the vector tangent to pS~ at the point q5.3 and 15.4 and if

(T,p)J~V. If

((~T,~p)

{~,

~p) [ 8V

(T,p) satisfies equations

satisfies corresponding homogenous

equations

15.~3 then ( ~ + ~

V(,~p)

= o

!,p+ ~ip) satisfies again the inhomogenous equations 15.3

and 15.4 which means that the vector Y is tangent to D 8V. It is obvious that all vectors tangent to D ~V can be obtained in this way. Let Y be a second vector tangent to D 8V corresponding to the section

Applying the Stokes formula to 14.26 we obtain

l!~¢~.~p¢~l ~V

V

15.15

V

V

- ~®¢x).~pcx~I_-

96

For any pair of functions re the symbol by the metric

f,h on M we have

~ i ) denotes

1

"

V We see that forms

0 i and ~P i are limits of forms

0 ~V and CO ~V when

V shrinks to the point x similarly as v and w are limits of YI~V and Z!~V. The above remarks justify the following notation

16.22

0~

: Mdiv @~

16.23

CO i x

=

Mdiv 60

,

x

similar to 8.14. Now we find the coordinate exT~rssions for

~

Y(y) and Z(y) are expressed by

~Cy)

=

yA(y)

~T A

~P~

and

°Oi'x If vectors

107 16.2a

z(y) = zA(~Y) ~-~ + ZA(Y)~PA then the c o r r e s p o n d i n g

jets are

~yA(x )

YA(X)~p~

~ x ~, 3~pA

+

x~

16.25

~ PA#

%

jlZ(x) : Z~(x) 8 • ~?~ -

Consequently

coordinate

-

+

Z~(x) 8 ~p~ ,

expressions

+

8ZA(x) ~ ~.~ ~?~

for equivalence

+

~x ~ 8 P A~~"

classes

v and w

are

~X~(x) +

~x,~ 16.26

~Z~(x) +

•

~x ~ The

equation

~x~

~A

16.18 reads



Qx

If the coordinate system in M

i ) is described by satisfies 19.2 then a point of Qxf~ and the coordinate expression of

C Q ( ~ J is

(~A

~)

, k~2

130

A

A

The remaining information about derivatives of ~ A Lie derivative

~X~

of the physical field

is contained in the

~ with respect to X provi-

ded that the configuration bundle Q consists of objects for which the Lie derivative can be defined. The value of the Lie derivative is a vector tangent to the configuration space Qx /cf. Appendix G/. This justifies the following definition

~9.22

Q~<x) = ~Qx

•

The canonical tangent bundle projection onto Qx is denoted by

~Qcx~

:

Q~(x)

,

%

.

For the sake of simplicity we deal in this section only with bundles of geometric objects in M /more general cases will appear in Chapter IV /. In this case the value of the Lie derivative

calculated in an

adapted coordinate system 19.1 is equal to

~ x T (x) = TA

19.23

~A

•

As coordinates in Qx(X)~ " we may thus take the system rigA ~ ). ~ " terms of these coordinates

The space Qx can be identified with a subset of Q

~9.25

Qxi

=

{

i (sx'Sn)~%(x)

× i n

X)×Q

Qx(-)IL'Q(x)Sx = ~'Q(_n)s_n]

In

131

An element p ~ Px determines a covector PX ~ T~Qx given by the formula

19.26

PX

= ~'P~2

and a /vector-density-on-~-vaiued/

covector pn ~ T*Qx® /~ T ~

de-

fined by the formula

19.27

~ k,Pn ~ 2

=

~ -XAk,p~ 2

where ~ is any (m-2)-vector tangent to ~

19.28

PxCX)

=

,

at x. We take

T~Qx

and

19.29

Px(n)

=

T~Qx @ /~ Tx~~

Canonical projections onto Qx are denoted by

19.3o

~x

" PxCX)

' Qx

gun : Px (n)

~ Qx

and

19.31

m

If the coordinates of M satisfy 19.1 and 19.2 we have for an element p = (~pA p%) the equalities

19.32 and

px

=

p~d~ A

132

Pn

"19.33

this means that

-~d~... (P~d? ~)®¢ ~x

=

(~0A,pZ)

may

be used

~ dx m )

.

as coordinates in Px~X) and

(~A

pk) as coordinates in Px(_n). Obviously

and

~9.35

~nC~A,p~)

= (?A)

The space Px can be identified with a subset of Px(X)×Px(~)

19.]56

Px = ((px,psl~Px(xl~x~-~)

:

~x~p~l-- ~n~Pn~I "

i i Spaces qx(~) and Px(~) have been obtained by projecting spaces Qx and Px onto the hypersurface ~ .

These spaces can also be obtained by ap-

plying to the restriction of Q to ~

the procedure that was used to

i construct Qx and Px from Q. If the restriction of Q to ~ by Q I ~

is denoted

then

i

19.37

Qx(n)

=

J~x(QIz)

and

19.3s

sx(_~)

We denote by P ( ~ )

= e*(Ql~-,)x@ A e * ~ x

the union of all such spaces obtained for all

points x~ M. The tensor product in the above formula is understood in the sense explained in Section 13 and used in formula 13.1. We may apply the same procedure / or rather its simpler version

133

from Chapter II / to the restriction of Q to the parametrized integral curve

19.39

R

~ t

•

~(t) e

M

passing through x and tangent to X at t = O. The resulting spaces are

19.~o

i x) Qx(

=

J (Qf#)

Px (x)

=

T~[QI~)O

and

19.aI

Again we denote by P(~)

the union of all such spaces obtained for all

points of ~ . This procedure can be carried one step further. struct spaces P~(X)

and P~(~) /corresponding to pi

x/

i

1 9.~2

PxCn)

= JxP(~)

and

We have canonical projections

~X

i

~n

:

:

P~(Bt

~

i

Qx (~)

19.~z~

bp(x): pi(x)

~ Px(X)

bp(n) " P~(n)

, Px(n)

The following diagram is commutative

:

.

:

We con-

134

Pxi ( X )

LP(X)

, Px(X)

Px (n)

(

bP(n)

pi(n)

The standard constructions provide also the lagrangian special symplectic structures given by canonical diffeomorphisms

:

and :

,

Using coordinates (xk), k = 2,...,m, on ~ coordinates

( ~ A p~, ~ ,

~ A ) where

~=

use the coordinate t = x ~ on the curve used in Chapter II yields coordinates A =

Qx(_)~/%

x~

we may construct in P~(~)

P~k" In a similar way we may ~ ={ x k = const.3 (~A p],~A

. The method

~]) in P ~ ( X ) w h e r e

are coordinates of the Lie derivative

and PA = PAl

are coordinates of the Lie derivative of the momentum contracted with n

•

I PAl

19.48

= l )

The right-hand It follows that

=

BA~A

+ B AkpAk

A % + B~p A .

side of 19.50 is thus equal to BA(~AI + ~ # ~(gx'%)has

=

The special symplectic

coordinates

=

structure

(~A,pA~,~,~A)

=

PA%

, where

"

in Pxi which we are going to de-

136 4

fine will be a mixture of the lagrangian structure hamiltonian

structure in P~(X).

in P~(~) and the

In order to define a hamiltonian

cture in P~(X) we need a connection in the bundle QI~ tion of the section

stru-

, i.e. the no-

"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 ~ ( 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

i

19.57

described

in our coordinate

system by coordinates

space Pxi is fibred in an obvious way over B(X,~).

(~A

A 1 ). Tk,PA

The

Using local coordi-

nates one can prove that this fibration is given by

Px~

19.58

where

~(X,n)(~)

=

(~x,g~n) is any pair such that

(Op(x)gx,SUngn)eB(X,n)

,

~(~X,~n ) = ~.

We define an isomorphism

~(X,B)

19.59

: piX

> T*B(X,B)

by the formula

( v , (xCX,n) (~) ~ =



-- ~2

%

The formula 20.3 implies

20.9

d_--~A...~,e~

9

-- - H ,

where

=

20.10

The form

a ~A

and

~

has to vanish on all other m-products of vectors

~ 3PA in order to fulfill

that i£ the form

20.11

H dx~... Adx m

0

=

~

the condition

20.1.

It

follows

exists it must be equal to

~ IA . . . A dA~ A^ . . . ^ dx m - H d x ~ . . . A d x PAdX

m

8

~x~'

=

146

It remains to be shown that this form satisfies the condition 20.2 for any representative ~'

~A

2o.~2

g = (~A

p~, ~ ,Ap A ~%) o f

the element ~ = (~A,pA~ ,

" Using formulae 17.19 and 17.20 we obtain



=

b ( pA ~ ~ ,A

- ~(x~, ~A,,~))

_- ~ .

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 Caftan form is equal 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 n be a hypersurface volume element in M at x and let g be any representative

of an element ~

Px" We define the lift ~ g of ~ to the

jet g similarly as the lift ~ g was defined for the volume element ~. If g is the jet of a section

20.13

M

~ y

~ p(y)

e P

Y

at y = x then by ~ g we denote the lift of ~ to the restriction of this section to the hypersurface

~-~c M tangent to ~. The coordinate des-

cription of ~ g is

-~ =C~d+ ~ ~

-~xm+ ~*m WA +

20.1¢

8 9~) PA

+ P~m

for

8 20.15

n

=

8 x 2 A...

A

~X m

.

147

Let X be a vector field in M. In the case when Q is a bundle of geometric objects ciated with

in M the field X has unique lift to Q. This lift is asso"dragging"

the geometric

A p p e n d i x G /,, If Q is a bundle

objects along the flow of X /see

of geometric

objects the same is true

about P. It follows that X can be lifted to P. The lift of X to P is denoted by X. The m-vector

20.16

~

is "horizontal"

=

in the d i r e c t i o n

in the d i r e c t i o n

~Afig

of X and compatible

with the dynamics

of n.

Theorem 9

(X,n)(¢)

20.17

=

-

?IP 2

= ( =

28.25 Let

~I

and ~ 2

+ A~dx ~

be two trivializations.

T~M associated with ciated with

d~

~2

~I

The identification of Q with

can be obtained from the identification asso-

by adding to each element of T * M the covector d A ( x ) X

The linear structure introduced into Qx by identification with T ~ M 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 Qx are thus elements of T~x M and the tangent bundle TQx is the trivial bundle Qx @ T ~ x M. Consequently the cotangent space for Qx is the space TxM and the cotangent bundle T~Qx is the trivial bundle Qx@TxM. We conclude that the phase bundle P for electrodynamics is the Whitney sum of two bundles 3

28.26

P

=

Q

where the secnd term of the sum is the same as for the covector field Section 2 6 )

and for the Proca field

has coordinates

(x~,A~,p~)

where p ~

(Section 27). are coordinates

The bundle P of the contra-

variant tensor density of second rank and do not depend on the choice of trivialization

~

of the bundle B. Forms

~x and

x e M are given by formulae

28.27

0x

=

° \~x~

.^dx 3) ""

ODx at each point

204

28.28

=

o

The affine structure zn Qx implies that the differentials dA~ are gauge invariant objects and have tensorial transformation properties. The infinitesimal configuration bundle is the first jet bundle Qi = jIQ. A coordinate system (x~,A~,A~>) is induced in Qi by a coordinate system (x~,A~) in Q. The coordinates A ~ 28.29

A~

=

~

represent derivatives

A~

The infinitesimal phase bundle is the quotient bundle pi = ~Ip of the first jet bundle jIp. Coordinates in pi are (x~,A~,p ~ , A ~ , ~

~

28.30

=

~p~#

~) where

.

Canonical forms in pi are given by

28.3~

¢x

28.32

CO xi

= ~div e~ =

d 0 xi

=

=

(~dA~ + p~*dA~D~(dxO,...~dx3~

Mdiv dOx

=

The dynamics D xi c P xi of the electromagnetic field is described by equations p~/~

=

~

=

~

g ~ g / ~ (A~

- A~.O)

,

28.53 0

.

The quantities 28.34

f of Qx we obtain a 1-contravariant,

tensor

~

tor on Qx is a 1-covariant,

- !7~

2-cord-

in M. It follows that

2-contravariant,

a

covec-

symmetric tensor in M.

Similarly as in the preceding Sections the phase bundle is a Whitney sum

where S~ is the bundle of 1-covariant, tensors.

The following objects

~9.3

ot~,

=

dx~® ~

~~

~

form a basis in a f i b r e of the bundle the symmetric

tensor product)

2-contravariant,

® ( - - ~~ d x

symmetric

° .. . ~ d x 3 )

~# -#\~'~

(by

~

,o denote

. Every element of this fibre may be

uniquely expanded with respect to this basis. The expansion coefficients

~#~

dinate system

together with coordinates (x ~, I~ ~

,~m~)

29.4-

O-~'~#~'e ,

At each point x 6 M forms

29.5

29.6

Ox

=

_21

(x~,l~)

in P, Of course

=

in Q form a coor:

9"Era~ ' ~

~x and Co x are given by the formulae

~2~.dr~:

® ~~~-2--jdx°^ x" ~

..

. ^dx3)

Jdx0A.

.

, A d x 31

.

212 q

( the coefficient ~ appears because of the symmetry 29.4) . The infinitesimal configuration bundle is the first jet bundle Qi = jIQ. A

coordinate system {x *, r%L

, ~m I~ ~)

is induced in qi by the coordi-

nate system (x ~) in M. The coefficient

I~

29.7

~

represent derivatives

= ~a~ I/.~

The infinitesimal phase bundle is as usual the quotient bundle pi = = ~lp of the first jet bundle JqP. Coordinates in pi are denoted by x ~, ~

,I~ ~

, ~

, ~

29.8

, where

~ =

~ ~ ~-~'~

The canonical structure in pi is given by forms X

=

Ox

=

29.9

2

~dr~ +~'~dr~

and ~x

i

@i x

=

d

--

z~ ( d ~

Mdiv~

=

~dx °...~dx3

[~ ( d ~

x

~ d ~D

]~ dxO''-^dx:

29.1o

A dr~t + d ~ g ~

The gravitational momentum

gU~~

dl ~ @ ® d x °... A dx3

has a priori too many components

for describing the metric tensor g. However, similarly as in electrodynamics

(formula 28.36)

we may expect the existence of hamiltonian

constraints reducing the number of independent components of gg . The occurence of such constraints is always equivalent to the fact that not all velocities are involved in the field equations. Indeed, Einstein equations contain only the curvature tensor 29.1q

R ~

213

and less than that, namely the Ricci tensor 29.12

R~

=

R ~

Especially important for our purposes is the symmetric part of the Ricci tensor

Kfl~

=

RC~ )

=

R~S + R ~ )

=

29.13

since we don't know a priori wheather R~S is symmetric or not

(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 that the Lagrangian depends only on K j~

and not on all velocities ~

.

We prove that this assumption implies the constraints

29.14

where

~U ~

~

=

$~

~

is a symmetric tensor density.

components of

~

The number of independent

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

~e~ represents the contra-

variant density of the metric.

It will be shown in the sequel that the

correct identification of

with the metric is :

29.15

~

~e~

where k is the gravitational

-

1

k ~

g#~

constant.

Now we prove the formula 29.14. Our assumption about the Lagrangian

214

=

29.16

L dxO~... ^ dx 5

implies the formula

~9.~v where J ~ W

~x(~%~

, ~.)

and

=

I- J j ~ d

I~"

2

+ I_ ~

dK,~

2

are arbitrary coefficients satisfying the symmetry

condition j~

=

J ~ wA~

29.q8

Using the formula 29.Q3 we write

=

2-

~-

I/~.

+

29.19

.~

2

+ ~I ( g . ~

+

(S~c ~'~-

- s~)

d r ~% . .

On the other hand the formula 29.9 and the definition of the Lagrangian imply

2

m

215 Comparing 29.19 with 29.20 we obtain the formula 29.q¢. Moreover

or

29.22

Jm#~

_- ~

~m~ ~

- ~ %~ ( ~ )~) ~

_ 0~ ~ ~ ~

The last term may be rewritten as follows

29.23

We may add to the right-hand

side the expression

The result is

~ g~C~C~)~ =

29.25

=

- ~¢F'/~

_ ~

~6"

I~:

=

- ~ - ~

-urns-/~m~je~

Combining 29.25 and 29.22 we obtain

29.26 =

(we recall that ~

~

GC~/~

is a tensor density)

.

The coefficients

J~ ~

ar e

216

thus components of the covariant divergence of the momentum and may replace

~e~

~%~w as coordinates in pi.

We use the structure described above to formulate the dynamics of the system composed of a matter field field

~

~A

and the gravitational

. Bundles Q, P, Qi, pi are Whitney sums of bundles corres-

ponding to the field

~A

tes are x ~, ? A p~ , ? ~

and bundles constructed above. The coordina-

, ~A' ~

'0~u~'

C~

and

~ ~w . Forms

~x '

i

gO x , @~, CO x are composed of two parts. One part corresponds to the matter field and the other to the gravitational field. For example

=

A+

d

29.27 + - ~ 2 29.28

i CO x

-- d 0 i

die~

®dx O... Adx 3

.

Field equations for such a system are composed of two parts

: equations

of the matter field and equations of the gravitational field. The equations of the matter field are formally the same as in Section 12. They can be derived for example from the Lagrangian of the matter field 25.18 . We denote now this Lagrangian by

29.29

since the symbols

~mat

~

=

Lmat dxOA''" ^ dx3

and L are reserved for the Lagrangian of the

full system composed of the matter and the gravity

~this Lagrangian

will be found later) . Thus the matter equations are :

=

Lmat(

29.30

3

217

The constraint as equations

equation 2 9 . ~

in the space pi -

tions.

enables us to treat the above equations There are also two gravitational

equa-

N °

As the first gravitational

equation

one usually

takes

29.31

This is equivalent

to

29.32

~g~

=

0

V~ CU~

=

0

~

=

0

Or

29.53

Or

29.34

J~

Equation

29.31

~

implies

the Proca field us to express

=

the symmetry

equations

imply the Lorentz

the second gravitational

tion - in terms

of K#~ instead

the general

case which quantity

of Einstein

equations.

the second grangian

stress

the Proca field,

(equal

of R ~ .

condition.

~

to the Hilbert

However,

it is not obvious

the simplest i.e.

in side

case when

the matter La-

. This is the case of the scalar field, field and also hydrodynamics

30. In those cases the first

tensor T ~ )

hand side of the Einstein

equa-

should be used as the right-hand

the electromagnetic

in Section

as

This enables

- the Einstein

tensor of the matter vanishes on

similarly

equation

Let us first consider

does not depend

is discussed

of the Ricci tensor

equations

is commonly :

which

stress tensor T ~

accepted

as the right-

218

4 29.35

The coefficient

V~g 7 is necessary

sor density and not a tensor.

29.36

-

form

in our notation

Contracting

T#~ is a ten-

both sides with g ~

we obtain

=

This enables us to rewrite equivalent

since

the Einstein

equations

in the following,

:

k 29.37

Now we use equation

29.38

25.28

T#~

=

~ g ~ Lma t

and the identity

~g#~

implied by 29.45.

The result

29.~0

Ke~

Equation

9

k

29.39

-

m~~

med that Lma t does not depend

Equations

equations.

j~

29.44,

29.30,

=

~ g~

is

29.34 can also be rewritten

29.41

2

on

&~

in a similar way since we assu:

a

29.40 and 29.44

~

are the complete

set of field

They d e f i n e t h e dynamics DXi ~ P ~ . Using t h e f u n c t i o n ~mat

219

i defined on D x by Lma t we may rewrite the field equations in the following form

29.42

d~_mat = (~Ad ~A+p~ d~ A +~J~/~ d,~%/~-I-K2~ d 0 ~ l I Dix

The above formula can be interpreted as a generating formula for the lagrangian submanifold in the space ~ix described by coordinates p~ , ~ A

, ~@A' ~

(~A,

' gg~ ,K~ ,J~Y) . The generation is meant with respect

to the special symplectic structure given by the form

29.43

~ mat x

The precise definition of ~i Px is the following.

We take the subbundle

a pi defined by the constraint equation 29.14. The symplectic forms COx are degenerate when restricted to fibres With respect to this degeneracy,

cPi.x We reduce

i.e. we pass to the quotient space

~i. Each element of ~i is a class of elements of p i

Two elements of

pi belong to the same class if they have the same value of coordinaX

tes (~A p~ , ~

, ~ A , ~ ,~6~y ,K~,J~ ~) . The

plectic form ~

on the quotient

CO x~i

=

d@matx

=

form CO ix induces a

sym-

:

(d~AAd~ A +

dp~Ad~

A

+

29.44 + 1 - d J ~ A d J-I~~

+ ~-d~A

2

d K ~ ) ~ dx ° ... ^ dx 3

2

The proof of this formula follows from the calculations used in 29.19. The dynamics D xi defines a lagrangian submanifold D~ c

x described by

equations 29.30, 29.40 and 29.41. The function Lma t is the generating ~i function of D x with respect to the special symplectic structure 29.43. This structure gives a mixed picture of the dynamics: picture for the matter field and

I!

quasi-hamiltonian

11

the lagrangian picture for the

220

gravitational

field

(we use "quasi" since the genuine hamiltenian i picture refers to the space Px rather then to the reduced space ~i ) X

Variables and

(~A

~)

"

are the lagrangian variables for the matter field

(P~v ,gg~w) are the hamiltonian variables for the gravitational

field. Equations 29.30 are of lagrangian type. Equations 29.40 and 29.44 are of hamiltonian type. The pure lagrangian description of D i x ( or Dx) is obtained from the above one by the Legendre transformation applied to the gravitational degrees of freedom• The Lagrangian of the full system

(matter + gravity)

is defined by the formula

--

2

X

"

Using the identity

which we proved earlier we obtain

--

2

x

Hence

29.

- _La t )

= d(~i gg~ K#~) Dxi and

29•¢9

L

=

_Lmat + ~ ~ # ' K ~

Di X

•

The Lagrangian 29.49 is scaler in such a way that it vanishes for the

221 vacuum. The vacuum is defined by the matter vacuum the flat space-time geometry

29.50

~K~

=