Teodorescu P.P. Mechanical Systems, Volume III: Analytical Mechanics

Подождите немного. Документ загружается.

Variational Principles. Canonical Transformations

311

transformations; the relation (20.3.27') shows how to obtain the related infinitesimal

canonical transformations, starting from a given integral of motion. Also, the conditions

(20.3.31), (20.3.31') or the equivalent condition

()

Dqq

qqq

∂∂∂

−

⎛⎞

=−

⎜⎟

⎝⎠





(20.3.31'')

represent a sufficient condition for the case (ii) discussed above.

All these results can be summarized in the form of

Theorem 20.3.1'' (E. Noether; reciprocal). Suppose that

()

,Dqq



is an integral of

motion. To this integral of motion we associate an infinitesimal variation (20.3.27') of

the generalized co-ordinates, which induces a variation

tqq

∂∂

δδτ

∂∂

−

⎡

⎛⎞⎤

=−

⎜⎟

⎢

⎥

⎣

⎝⎠⎦



(20.3.32)

of the Lagrangian. If and only if there is a Lagrangian

′

equivalent to the

Lagrangian

L, as defined in (20.3.24), and

()

,Dqq



is a homogeneous function of the

first degree in the generalized momenta (20.3.31'), then the integral of motion

()

,Dqq



can be correlated with an invariance property of the Lagrangian (

0δ =

′

L ).

20.3.2 Lie Groups

In what follows we present some notions concerning Lie groups, useful in the study

of conservation laws; we consider first the one-parameter Lie groups and then the Lie

groups with m parameters.

20.3.2.1 One-Parameter Lie Groups

We consider the transformations

()

;

xfxa

′

= ,

{

}

12 2

, ,...,

xxx x≡ , 1,2,...,2is= ,

(20.3.33)

which depend on one parameter a, and the identity transformation corresponding to the

zero value of the parameter (

0a = ), that is

()

xfx=

, 1,2,...,2is= . (20.3.33')

We suppose that there exists a value

a of the parameter a, so that

()

;

xfxa

′

{

}

12 2

, ,...,

xxxx

′′′′

≡

, 1,2,...,2is= ;

(20.3.33'')

hence there is an inverse transformation of the transformation (20.3.33). Let

()

;

xfxb

′′ ′

1,2,...,2is=

(20.3.34)

be some other transformation of the form (20.3.33). These transformations constitute a

finite one-parameter topological group if there is a value c of the parameter such that

MECHANICAL SYSTEMS, CLASSICAL MODELS

312

()

;

xfxc

′′

1,2,...,2is=

(20.3.34')

We notice that the parameter c is a function of the parameters a and b

()

,cabϕ= ,

()

,0aaϕ= . (20.3.35)

The group of transformations (20.3.33) is a Lie group if the functions

f are analytic

functions in the parameter a and if the function ϕ is analytic in both arguments. In our

case, 1 is the minimum number of independent parameters necessary to characterize the

elements of the Lie group; this parameter is called essential, while 1 is the order or the

dimension of the group.

In terms of the general method for infinitesimal transformations, we have

() ()

ii ii i ii

xx xfxax axuxa

∂

δδδ

∂

′′ ′ ′ ′ ′ ′ ′

=+ = =+ =+

(20.3.36)

()

(,)

d, ()

ca a aa a aa aa

∂ϕ

ϕδ δ μδ

∂

=+ = =+ =+ ,

(20.3.36')

where c is related to the transformation

()

d;d

iii

xxfxaa

′′

+= +

(20.3.36'')

Indeed, the infinitesimal displacement of the point x can be obtained in two ways: as a

result of successive applications of two transformations or directly. As well, the relation

(20.3.36') is a consequence of the property of associativity. Noting that

()daaaδλ= ,

() 1/()aaλμ= , we can write the system of differential equations of the group in the

form

()()

ux a

′

, 1,2,...,2is= ,

(20.3.37)

where the functions

()

ux form a velocity field. By introducing the parameter

()d

aaαλ=

∫



(20.3.37')

we obtain

()

′

, 1,2,...,2is= .

(20.3.37'')

The new parameter

α is an additive quantity with respect to the composition of

infinitesimal transformations; for an infinitesimal parameter

we have

()()()

;; ;

ffx fxαβ α β=+,

{

}

12 2

, ,...,

fff f≡ , 1,2,...,2is= .

(20.3.38)

Variational Principles. Canonical Transformations

313

Let be a function

()Fx of class

C . By an infinitesimal transformation of the point

x (corresponding to an infinitesimal displacement), the function F has a variation of the

form

d() d () ()

FF F

Fx x u x a Fx a a

xx a

∂∂ ∂

δδδ

∂∂ ∂

== ==

∑∑

X ;

(20.3.39)

the operator

()

∂

∑

(20.3.40)

is called the generator of the transformations group. From (20.3.39) it follows that we

can express this generator also in the form

∂

=X .

(20.3.40')

Hence, the solution of the system of differential equations (20.3.37'') can be expanded

in a power series

...

ii i i

xx x x

′

=+ + +XX, 1,2,...,2is= ,

(20.3.41)

which, for infinitesimal transformations, is reduced to the simple form

()

ii ii i

xx xx uxαα

′

=+ =+X , 1,2,...,2is= .

(20.3.42)

From (20.3.41) it follows

()

;

ii i

xfx ex

′

, 1,2,...,2is= ,

(20.3.43)

and we are led to the equations

∂

∂α

= X

, 1,2,...,2is= .

(20.3.43')

We are now in a position to establish the connection between the canonical

transformations and the one-parameter Lie group, formed by the transformations

(20.3.33). This will allow us to make use of Noether’s theorem, to find the

corresponding conserved quantities.

First of all, it is worthwhile to note that a special form of (20.3.43') is obtained if we

set

(; ) ()

fx xαα= , 1,2,...,2is= , and

tα =

, such that they become the canonical

equations

= X

1,2,...,2is=

(20.3.44)

MECHANICAL SYSTEMS, CLASSICAL MODELS

314

where we have attached the label t to the generator to emphasize the associated

parameter.

Now we assume that the transformations (20.3.33) constitute a Lie group and note

that, in the phase space

Γ , these transformations can be written in a form similar to

(20.2.1), that is

()

iii

qQfqpα

′

≡=

()

iii

pPqqpα

′

≡=

, 1,2,...,is= .

(20.3.45)

We sought the conditions that have to be verified by these transformations (which form

a Lie group) such that they be also canonical transformations. A sufficient condition

consisting in the existence of a function

()

,VVQP= , whose exact differential is a

differential one-form (20.2.9). Thus, if we denote

f∂

∂α

q∂

∂α

, 1,2,...,is= ,

(20.3.46)

we obtain

()

= ,

()

= ,

1,2,...,is=

from (20.3.45), where the dependence of

λ and

μ on Q and P was put in evidence.

Also, the generator (20.3.40) becomes

∂∂

λμ

∂∂

=+X

(20.3.46')

where we have used the summation convention for dummy indices from 1 to s.

By means of the power series (20.3.41) and assuming the series expansion

() ()

VQP V QPα

∞

∑

we obtain the sufficient condition

d ...

ii i i i

pq p

αμ μ

⎛⎞

−++ +

⎜⎟

⎝⎠

dd d ...

ii i

αλ λ

⎛⎞

++ +

⎜⎟

⎝⎠

d(,)

VQPα

∞

∑

which has to be identically satisfied. Hence, equating the coefficients of equal power of

α yields the relations

d0V = ,

()

ddd

ii ii

Vqpμλ=− + ,...

Variational Principles. Canonical Transformations

315

All these relations are simultaneously satisfied if we assume that the function V has the

form

11 1

...

2! 3!

VV V V

αα

α=+ + +XX.

(20.3.47)

To find the functions

λ and

μ , we introduce the function

()

Uqp V pλ=− − , (20.3.47')

such that

ddddd

iiiiii

Uq pqp

μλ

∂∂

=+=−

∂∂

The last equality is identically satisfied in the variables

q and

p if we choose

∂

=−

∂

, 1,2,...,is= .

Thus, we obtain the condition which has to be satisfied such that the transformations

(20.3.45) be canonical transformations, the function U being arbitrary. Consequently, in

terms of the Poisson bracket, the generator (20.3.46') takes the form

()

ii ii

qp pq

∂∂ ∂∂

∂∂ ∂ ∂

=−≡

X .

(20.3.46'')

Finally, by applying this operator to the functions

Uq∂∂ and /

Up∂∂, we obtain

the whole group of canonical transformations, that is

...,

..., 1,2,..., .

Pp i s

∂α∂

∂∂

∂α∂

∂∂

=− + −

=+ + + =

(20.3.48)

Note that we are led to the same result by integrating the system of partial differential

equations

()

∂

∂α

, 1,2,...,is= ,

(20.3.49)

or by integrating the system of differential equations (20.3.46), written in the form

∂

α∂

∂

α∂

=−

, 1,2,...,is= ,

(20.3.50)

MECHANICAL SYSTEMS, CLASSICAL MODELS

316

where

() ()

,,UQP Uqp=− . All these three methods used to determine the Lie group

of canonical transformations are equivalent. The function U is the generating function

of canonical transformations, while the operator

()

, U=X is the generator of the Lie

group. It is also easy to see that, for

tα = and

()

,UHQP= , the system of

differential equations (20.3.50) is identical with Hamilton’s equations.

We denote by x the set of canonical variables

{

}

,qp . Then, a function ()fx is

invariant with respect to the Lie group of canonical transformations if the equation

( ) () ()

fx fx fx

∞

′

∑

(20.3.51)

is identically satisfied. This can be accomplished if and only if

() 0fx =X ,

x∀∈\ .

Explicitly, by using the expression (20.3.46'') for the generator X, we may rewrite this

condition in terms of canonical variables

()

ii ii

Uf Uf

qp pq

∂∂ ∂∂

∂∂ ∂ ∂

−≡=

(20.3.52)

where we have introduced the Poisson bracket. It follows that the conserved mechanical

quantities are invariants of the group of canonical transformations and also solutions of

the equation (20.3.52). If we find m independent invariants of this kind, then we may

choose them as generalized co-ordinates (cyclic co-ordinates in this case), and there

remain to integrate only

2sm− equations of motion.

20.3.2.2 Lie Groups with m Parameters

Let be

(;)

xfxa

′

= ,

{

}

, ,...,

aaa a≡ , 1,2,...,is= ,

(20.3.53)

a set of general transformations, which depend on m independent parameters, real and

continuous, and are such that for

0a = (i.e.

... 0

aa a=== =) we obtain the

identical transformation. We assume that a are essential parameters, and therefore the

transformations (20.3.53) form a Lie group with m parameters. In this case,

(;) (;)

fxa fxa

xaa

∂∂

′

∑∑

, 1,2,...,is= ;

denoting

(;)

()

fxa

∂

= , 1,2,...,is= , 1,2,...,km= ,

(20.3.54)

we may write

Variational Principles. Canonical Transformations

317

d()

xuxaδ

∑

, 1,2,...,is= .

(20.3.54')

As in case of one-parameter Lie groups, the infinitesimal displacement of the point x

will be a differential displacement, and there are two possibilities to obtain it: as a result

of two successive transformations

(;)

xfxa

′

= and

()

iii

xxfxaδ

′′

+= or directly,

()

d;d

iii

xxfxaa

′′

+= +; taking into account the associativity property, the

considered parameters are connected by relations of the form

()

d; ()

jj j j

aa aaa aaϕδ μ δ

+= =+

∑

, 1,2,...,

m= ,

(20.3.55)

where

(,)

()

∂ϕ

∂

= , , 1,2,...,jk m= ,

(20.3.55')

{

}

, ,...,

bbb b≡ being parameters in the transformation

()

xfxb

′′ ′

= ,

1,2,...,is= . The relations (20.3.55) lead to

()d

aaaδλ

∑

1,2,...,km=

(20.3.56)

where

λμ δ

∑

, , 1,2,...,kl m= ,

(20.3.56')

δ being Kronecker’s symbol. From the relations (20.3.54') and (20.3.56) we obtain the

differential equations of the group

() ()

ux a

∂

∑

, 1,2,...,is= , 1,2,...,jm= .

(20.3.57)

If the functions

u are linear independent (what we assume), then we say that the

parameters

a are essential.

Let ()Fx be a function of class

C . This function has a variation of the form

111

d() d ()

ssm

iij

Fx x u x a

∂∂

===

== ≡

∑∑∑

()

jj j

Fx a a

∂

δδ

∂

∑∑

X ;

(20.3.58)

by an infinitesimal transformation of the point x (corresponding to an infinitesimal

displacement); the operators

MECHANICAL SYSTEMS, CLASSICAL MODELS

318

()

∂

∑

X , 1,2,...,jm= ,

(20.3.59)

are called the generators of the transformation group. As one can see from (20.3.58),

the generators can be expressed also in the form

∂

, 1,2,...,

m= .

(20.3.59')

The formalism used in the previous subsection is generalized by introducing m

generating functions

(, )

Uqp, while from (20.3.47') we obtain

()

ii ii

qp pq

∂∂

∂∂ ∂∂

=−=X , 1,2,...,

m= .

(20.3.60)

corresponding to the operator (19.1.58).

Let us consider the product of two generators applied to the function

()Fx

() ()

Fx u u Fx

∂∂

∑∑

11 11

mm s s

iq iq

uuu

xx xx

∂

∂∂

∂∂ ∂∂

== ==

∑∑ ∑∑

, , 1,2,...,pr s= ;

using Schwarz’s theorem (from which, in case of functions of class

C , the mixed

derivatives of second order do not depend on the order of differentiation), we obtain

()

pr rp q

Fx u u

xxx

∂

∂∂∂

⎛⎞

−= −

⎜⎟

⎝⎠

∑∑

XX XX

11 1 1 1

() ()

sm m s m

kk k k k

pr pr pr

ik k i k

cu c u Fx c Fx

∂∂

== = = =

== =

∑∑ ∑ ∑ ∑

;

hence, the generators of the transformations group verify the relation

()

pr r p p r pr

−≡ =

∑

XX XX X X X

, ,1,2,...,pr m= .

(20.3.61)

The Poisson bracket thus obtained is called the commutator of the operators

X and

X . The quantities

are called the structure constants of the group; the property of

antisymmetry

()()

pr rp

=−XX XX is reflected in these constants in the form

pr rp

cc=− , while from the Poisson-Jacobi identity

()()

()

()()()

,, ,, ,, 0

ij j i ij

kkk

++=XXX XXX X XX , , , 1,2,...,ijk m= ,

(20.3.62)

Variational Principles. Canonical Transformations

319

which is easily verified, we obtain the relation

()

sl sl sl

il jk jl ki kl

cc cc cc

++ =

∑

, ,,, 1,2,...,ijks m= .

(20.3.62')

The structure constants of every Lie group are determined, univocally, by the structure

of the group.

The commutator of the generators can be evaluated by applying it to an arbitrary

function

(, )fqp; thus we have

()( )

()

()()()

,,,,,

jjjj j

kkk kk

ffUUfUUf=− = −XX XX XX

;

(20.3.63)

from the above relations of antisymmetry and from the relation (20.3.63) we see that the

Poisson brackets are skew-symmetric (

()()

UU UU=−

) and verify the

Poisson–Jacobi identity

()()

()

()()()

,, ,, ,, 0

ij j i ij

kkk

UUU UUU UUU++=

, , 1,2,...,ijk m=

(20.3.64)

If we set one of the generating functions

UH= we are led to the Jacobi–Poisson

theorem (see Sect. 19.1.2.3) for the case when the time does not appear explicitly. If

and

U are two conserved quantities (

()

,,0

UH UH==

), then

()

is also

a conserved quantity.

Finally, from (20.3.61), (20.3.63) and (20.3.64) we obtain the equation for the

structure constants

c of the group

()

ij ij

UU cU

∑

(20.3.65)

20.3.3 Space-Time Symmetries. Conservation Laws

In what follows we will consider some particular groups (the translations group, the

rotations group, the time translation group and the Galileo group); various conservation

laws (conservation of the generalized linear and angular momenta, conservation of the

mechanical energy and the motion of the mass centre) are thus put in evidence. We

determine then the general form of the Lagrangian for the invariance with respect to

symmetry transformations, introducing thus the Galileo–Newton group.

20.3.3.1 Conservation Laws

Let be a discrete mechanical system, formed by n particles

of masses m

1,2,...,nα = , described by the Lagrangian

()

12 12

, ,..., , , ,..., ;

qq qqq qt L , where

()

qt, 1,2,...,ks= , are the generalized co-ordinates in the configuration space

Note that the only independent variable is the time t, such that the generalized

MECHANICAL SYSTEMS, CLASSICAL MODELS

320

co-ordinates play the rôle of state functions. If

3sn= , then the relation between the

generalized co-ordinates

q and the Cartesian co-ordinates

1, 2, 3

, of the

particle of index

α is given by the relation 3( 1)kjα=−+. In terms of the Cartesian

co-ordinates, the condition (20.3.13) which has to be satisfied by a transformation in

order to be a symmetry transformation has the form

()

tx x t

tx xt t

αα

∂∂ ∂

δδ δ δ δΩ

∂∂ ∂

⎛⎞

+++ =−

⎜⎟

⎝⎠



(20.3.66)

where we have used the summation convention of dummy indices (from 1 to 3 and

from 1 to n, for Latin and Greek indices, respectively). Consequently, the left side in

(20.3.66) has to be the total derivative, with respect to the time variable, of the function

()

;

Ω . Also, the relation of conservation (20.3.15), corresponding to the invariance

of the Lagrangian with respect to a symmetry transformation, becomes

xt x

tx x

αα

∂∂

δδδΩ

∂∂

⎡⎛ ⎞ ⎤

−++=

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦





(20.3.67)

which is equivalent to

const

xt x

αα

∂∂

δδδΩ

∂∂

⎛⎞

−++=

⎜⎟

⎝⎠





(20.3.67')

The equations (20.3.66) and (20.3.67') represent the analytic form of Noether’s

theorem (see the Theorem 20.3.1) in classical mechanics. Thus, to every symmetry

transformation of a mechanical system (that is an infinitesimal transformation which

satisfies the condition (20.3.66)) it corresponds a conservative law expressed by

(20.3.67').

20.3.3.2 Translations Group. Conservation of the Generalized Momentum

The simplest symmetry transformations are the translations of the origin of the frame

of reference (that is of the Cartesian co-ordinates) by a time-independent vector

(

123

,,xxxδδδ). These transformations are defined by the relations (the space

translations group T)

′

, 0tδ = ,

xxx

αα

′

αα

′



1, 2, 3j = , 1,2,...,nα = .

(20.3.68)

If we choose

0δΩ = , then (20.3.66) takes a simpler form

∂

⎛⎞

⎜⎟

⎝⎠

∑∑

;