ANNALES DE L’I. H. P., SECTION A

´

JEDRZEJ SNIATYCKI

Dirac brackets in geometric dynamics

Annales de l’I. H. P., section A, tome 20, n^o 4 (1974), p. 365-372

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

Ann. Inst. Henri

Section A :

Poincaré,

Vol.

n°

365

XX,

4, 1974,

Physique théorique.

in

Dirac brackets

geometric dynamics

015ANIATYCKI

Jedrzej

of

The

Canada.

Alberta,

University

Calgary,

of Mathematics

Statistics and

Science

Department

Computing

is formulated in the

of of constraints in

ABSTRACT.

framework of

- Theory

symplectic

dynamics

Geometric

significance

secondary

of Dirac

Global existence

geometry.

is

constraints and of Dirac brackets

given.

brackets is

proved.

1.

INTRODUCTION

The ^successes of the canonical

of

with

quantization

the

dynamical systems

a

finite number of

of

of

freedom,

degrees

necessity

quan-

gravi-

theory

experimental

of

of the

field

tization

and the

that

electrodynamics,

hopes

quantization

could resolve

in

quantum

tational field

difficulties encountered

of the

have

to

rise

canonical structure of

given

thorough investigation

field theories. It has been found that the

formu-

cases

standard Hamiltonian

most

lation of

of

is

in the

dynamics

inadequate

and

physically

interesting

dueto existence of

constraints. Methods

electrodynamics

gravitation

with constraints have been

of

with

several

by

developed

dealing

dynamics

authors and

it

has been realized that the standard Hamiltonian

dynamics

can be formulated in terms of constraints

(1).

has

been

a

mathematical

The aim of

of constraints

Hamiltonian

dynamics

given

very elegant

formulation in the framework of

a

(2).

theory

symplectic geometry

this

in

is to

formulation of the

paper

give

symplectic

As

a

result the

of the classification of

geometric

and of Dirac brackets is

dynamics.

significance

Further,

the

of

constraints

given.

globalization

See refs.

the references

there.

and

[4], [5], [6],

quoted

C)

[2], [3],

[8]

See

refs.

and

[1]

[9].

(~)

eg.

Henri Poincaré -

de l’Institut

Section A -

n° 4 -

1974.

XX,

Annales

vol.

366

JEDRZEJ SNIATYCKI

theresults discussedin theliteraturein termsof local coordinatesis obtained.

the fundamental

with

notion

constraints

is that

As in the case of the Hamiltonian

dynamics,

of

in the

dynamical systems

geometric analysis

of

a

manifold. Basic

of

manifolds are

symplectic

properties

symplectic

reviewed in Section 2.

a

with ^constraints can

be

constraint

A

dynamical system

represented by

a

submanifold of

is described

and

manifold.

constraint

Therefore,

symplectic

dynamics

manifold

a

a

where

is

M,

by

triplet (P,

P,

of canonical

(P, cv)

symplectic

M

is

a

submanifold of

which is called

a

canonical

Ele-

system [12].

their relations to

and the

mentary properties

with

systems,

Lagrangian

a

relation

between

cano-

systems

nical

homogeneous Lagrangians,

and its reduced

phase space

are

in Section 3.

discussed

system

4

Section

contains a

of

constraints. The

discussion

secondary

genera-

lization of Dirac classification of constraints is

in Section 5. Sec-

given

tion

6

is devoted to an

ofthe

of

Dirac brac-

analysis

of

their

geometric significance

a

kets and

existence.

proof

global

2.

SYMPLECTIC MANIFOLDS

P

is

a

manifold

and ^OJ

manifold is

a

where

A

a

(3)

pair (P, OJ)

symplectic symplectic

form on ^P. The most

of

a

is

important example

symplectic

of

is furnished

in this case

the structure of the

manifold in

by

space

dynamics

system,

bracket.

phase

P

the

and ^cv is the

a

represents

phase space

dynamical

Lagrange

be

a

manifold

a

function on ^P. There exists

Let

(P, OJ)

symplectic

and

f

such

form

a

vector field

called the Hamiltonian vector field

unique

that

of f,

J

~ = 2014

d f,

where

and

J

denotes

the left interior

are Hamiltonian vector fields

of a

product

v f

avector field. It

by

correspond-

v f

vg

and

then their Lie brackets

is

to functions

ing

f

g, respectively,

[v f’ ~]

the Hamiltonian vector field

to the Poisson bracket

and the Jacobi

corresponding

( f, g)

for

Further,

( f,

g)

an

identity

of f and ^g.

~(~) = 2014 vg( f ),

the Poisson brackets is

immediate

of

the Jacobi

that ^6D is

consequence

identity

of

fields and the

closed.

for the Lie bracket

vector

assumption

3.

CANONICAL SYSTEMS

- Acanonical

is

a

where

DEFINITION

symplectic

3 . l.

M,

(P, OJ)

(P,

triplet

cv)

system

a

submanifold of P.

M

is

a

manifold and

is

one

from the

if

Canonical

Lagrangian dynamics

and

passes

systems appear

All

manifolds

considered in this

are finite

dimensional,

(3)

paper

paracompact

of class C ".

Annales

de l’Institut Henri

Poincaré - Section

A

DIRAC BRACKETS IN

DYNAMICS

367

GEOMETRIC

with

to the

canonical

of

systems

dynamics

homogeneous Lagrangians.

Let

L

be

a

one

a

on

a

conical

X. function of

defined

homogeneous

degree

domain

We denote

D

of the

TX of

bundle

tangent

space

configuration space

transformation

FL : D - T*X the

by

Legendre

given by

for

the fibre derivative of

L

and

0

the Liouville form on T*X

defined,

is the

by

E

where n : T*X -~

X

a

each v

bundle

by

p(Tn(v))

cotangent

canonical

TpT*X,

The

is

FL,

(T*X,

de)

system

projection.

triplet

range

submanifold ofT*X.

canonical There are two subsets

FL

be

is

a

range

provided

a

Let

(P,

TPM

K

and

N

M,

co)

system.

of

associated to

as

follows:

M,

(P,

co)

and

N=KnTM.

N

is called the characteristic set

for

canonical

of

M. The

The set

w

dynamical signifi-

cance of

N

a

obtained from

a

FL,

(T*X, range

homogeneous Lagrangian

d0)

system

with

on

X

a

L

is

given by

Lagrangian system

the

following.

the

3 . 2.

- A curve

in

X

satisfies

if

PROPOSITION

Lagrange-Euler equa-

where _y denotes

only FL. ,

y

L

and

tions

the

to the

corresponding

prolongation

Lagrangian

^is an

this

curve of N. Proof of

of to

TX,

integral

propo-

sition is

in the reference

and it will be omitted.

given

[11]

system

for

a

of

M

which can be

canonical

Thus,

connected

M,

(P,

points

curves of

N

are related in

a

by integral

be

physically meaningfull

related

time

evolution as

a

in

3.2 or

manner;

they may

by

Proposition

the maximal

can be

a

transformation.

E

by

gauge

Therefore,

M,

given

point

p

N

manifold of

it

exists,

integral

as the

p,

through

provided

interpreted

of

history

p.

if

N

is

a

DEFINITION

3 . 3.

A

canonical

is

M,

(P,

w)

regular

system

of TM.

be

and since cv is closed

subbundle

a

subbundle

Frobenius

a

canonical

Then

N

is

Let

M,

(P,

cv)

regular

system.

is also involutive.

of

N

TM,

theorem

Hence,

unique

by

each

E

M

is contained in

a

maximal

(4),

point

p

integral

manifold of N. Let P~ denote the

set of

M

the

quotient

by

equivalence

relation defined

maximal

manifolds of

that is each element

N,

described

dynamical

system

by

a

integral

of the

in P~

our

cano-

represents

history

by

M ~

Pr denote the canonical

nical

and let

M,

(P,

system

p :

projec-

submersion

M

P~ admits

a

a

tion. If

then there exists

differentiable structure such that

is

p

a

formwr on ^P~ such that

unique symplectic

w

The

manifold

called the

to

of

is

reduced

(Pr,

symplectic

wr)

phase space

on Pr

constants of

Functions

M,

(P,

CD).

correspond

(gauge invariant)

in ^the manner ^described

and

cvr

rise to their Poisson

motion,

algebra

gives

in

2

Section

(5).

All theorems on ^{differentiable} manifolds used in this

can be found

in ref.

(~) (~)

paper

[7].

This Poisson

was ^first studied

in ref.

[2].

algebra

Vol.

n° 4 - 1974.

XX,

JEDRZEJ SNIATYCKI

368

4.

CONSTRAINTS

SECONDARY

on

In

If

is not

points

then

dim

on which dim

e M.

open

(P, M,

regular

p

p

cv)

depends

particular

Np

the ^set S° of

E

M

0

^is an

submanifold

inadmissible

Np

S° as

of

if it is not

this set

M,

Discarding

physically

constraints

here. Consider the class ofall manifolds

empty.

[3]

have lead Dirac

to the notion of

secondary

generalization

M

ofwhich is

given

n
Y

dim

contained in

and

Y’

such that TY

N

is

a

subbundle

of

TY

allow here

(we

order in this class defined as follows

Y

0),

Y

let

denote the

partial

if

and

if

Y

is

a

of Y’.

follows from Zorn’s lemma that

submanifold

It

only

there

to

the

exist maximal manifolds in

class.

we are lead

this

Thus,

following.

DEFINITION

4.1.

A

constraint manifold of

a

canonical

secondary

maximal manifold

is

a
S

contained in

M

such that

N

n

TS

M,

(P,

cv)

system

a

is

subbundle of TS

(6).

S

be

submanifold

cannot use Frobeniustheoremto define the reduced

a

constraint manifold of

If

S

is not an

Let

M,

secondary

of

(P,

be involutive.

open

one

case

M

then

n

TS need not

N

In this

If

N

n

TS

phase space.

relation,

equivalence

maximal

manifolds ^define an

is involutive its

and let

integral

set of

denote the

S

relation.

that the

that there

this

by

such

N

quotient

differentiable structure on

Ps

Suppose

a

exists

tion

canonical

Ps

projec-

S -

is

then

a

submersion.

Since

of_OJ M

is the characteristic set

TS is contained in the characteristic set

ps :

Ps

n

and

N

TS c

TM,

Ns

• of
• defined

there exists

non-

Therefore

The form

the reduced

OJ

unique

S

by

03C9rS

only

constraint

and

a

2-form

on

• such that
• S

is

Ps

N

03C1*S03C9rS.

03C9rS

if

and

if

n

TS

manifold

Thus,

a

degenerate

Ns.

S

phase space

Ps

ofthe

has

natural

structure

secondary

of a symplec-

if

if

TS

n

N

tic manifold

If this condition

and

if

• S
• holds

only

then the structure

Ns.

is

a

submanifold of

Sa

of

as

all

as

constraint

canonical

P,

OJ)

secondary

manifold of

is

the

same

that of

cases

a

M,

(P,

cv).

exactly

regular

This situation

we

in

of

in

interest

S,

system (P,

appears

physics,

regular

therefore in the

shall limit

our considerations to

following

canonical

systems.

5.

A

CLASS OF

CANONICAL

SYSTEM

on

a

first

M

be a

canonical

A

function

P

is

called

Let

M,

(P,

OJ)

system.

f

on

function constant

if its Poisson bracket with

zero on M. This condition

class function

every

can

be reformulated as follows:

is

identically

This definition is

a

of the

in ref.

definition

(~)

globalization

[10].

given

Annales de I’Institut Henri

A

Poincaré - Section

369

DIRAC BRACKETS IN

for each v

GEOMETRIC DYNAMICS

which

P

if

and

E

on

0. Functions

first class

if,

K,

v(f)

only

f is

are

constraint

not first class are called second class functions. Since the

a

M_can

P - dim

be

described

of

system

submanifold

0,

apply

locally

by

equations

the notion ofa ^class can be

extended

f (p)

to

i

M,

1, ..., dim

a

canonical

system. regular

a

canonical

is

the

5.1. - Class of

DEFINITION

of

M,

by

(P,

regular

system

dim K - dim

N,

(dim

N)

_locally M

(’).

integers

pair

If

a

is ofclass

then

...,

can

be described

system

where all

(P, M,cv)

(n, k)

1,

of

i

n and

0,

1, ..., k,

equationsf (p)

0, j

are first class andall

are second

and

class,

it

cannot

functions £.

functions gj

be described

tions. If k

of

with more than n first class func-

by any system

equations

0

we ^call

M

a

first class submanifold of

as ^it can be

(P, OJ)

a

of

i

where all

second

class

0,

1, ..., n,

locally by

system

equations

f (p)

given

if n

M

is called

a

a

are first class.

0,

Similarly,

functions f

of

In

^this case

is

submanifold

manifold.

(P,

(M,

OJM)

symplectic

two theorems which we

hold

shall

later.

need

There

For

PROPOSITION 5.2.

canonical

M,

any regular

(P,

co),

system

N

is an even number.

dim K - dim

M -dim N.

even

P~

dim

K

dim P -dim

M

and dim

dim

forms

dim

Proof.

dim P~ are

and

P

Since both

P

and P~ admit

symplectic

is even.

P - dim P~

numbers.

K - dim

N

dim

dim

Hence,

a

5 . 3.

Let

be

canonical

system

j’ and g

of

PROPOSITION

M,

regular

(P,

class

with

k

&#x3E;

0.

there exist two functions

such

that

Then,

(n, lc)

For each

_E M

there exists

on

a

of in

P

and

0

M’

Proof -

p

neighbourhood

p

Up

two functions

and

such that

n

_gpMn

M

fp

fp

of

Up

gp

Up

Up

and

n

we

1.

associate to the

Further,

M

complement

( fp,

gp)

Up

of the

M

in

P

and

to

closure

functions_{f ’}

covering

zero. This

there

g’ equal identically

we

and since

of

P

have obtained

a

of

is

paracompact,

P,

way

a

this

exists

finite

and

For each

a

of

locally

covering

}.

partition

refinement {

to the

such

we have two

subordinated

on

Ua

Ua

unity {

functions

covering {

that

and g03B1

U (1

^Letf

g

follows,

and

be

P

defined

as

^functions on

for each

Then,

is

to W. M.

due

This definition

(7)

Tulczyjew

(unpublished).

Vol.

n° 4 - 1974.

XX,

JEDRZEJ SNIATYCKI