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

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Variational Principles. Canonical Transformations

301

20.3.1 Symmetry Transformations. Noether’s Theorem

In what follows, we will present the symmetry transformations which allow to state

Noether’s theorem; an important particular case is that of the Lagrangian which does

not depend explicitly on time. We mention also the rôle played by the reciprocal of

Noether’s theorem.

20.3.1.1 Equations of Motion

Let be a physical system (for instance, the mechanical system considered in Sect.

20.1.5.1), the states of which are determined by means of the independent variables

{

}

, 1,2,...,

xxi n≡= and the functions of state (dependent variables) u

1,2,...,mα =

; the derivatives of the functions of state are denoted by

uux

αα

∂∂= , 1,2,...,in= . The time t has not a privileged position and can be

anyone of the variables

x . We assume that the equations of motion derive from the

functional

()

,, d

xu u

αα

Ω

Ω=

∫

AL ,

(20.3.1)

where

d d d ...d

xx xΩ = is the volume element of the domain Ω , and the

Lagrangian density

L (corresponding to the unit volume) is of class

C . To obtain the

equations of motion, we introduce the variations of the independent variables

()

ii ii i

xx xx xδεη

′

=+ =+

, 1,2,...,in= ,

(20.3.2)

where

ε is a small parameter, while

η and

xδ are arbitrary functions and

infinitesimal functions of class

C , respectively. The relations (20.3.2) define an

infinitesimal transformation of the domain

Ω into Ω

′

. Consequently, the state

functions and their derivatives are transformed according to the relations (analogously,

we take

()ux

αα

δεξ= , where

ξ are arbitrary functions of class

C )

() ( ) ()ux u x ux

ααα

′′

=−

, () ()

ii i

uuxux

αα α

′′

=−

1,2,...,in=

1,2,...,mα =

(20.3.3)

where we used similar notations, while the variation of the functional (20.3.1) is given

() ()

,, d ,, d

ii ii

xu u xu u

αα αα

ΩΩ

δΩΩ

′

′′ ′ ′

=−

∫∫

AL L ,

(20.3.1')

where

d d d ...d

xx xΩ

′′′′

. The Jacobian of the transformation (20.3.2) is

()

det det det

jjj

kk k

xxx

xx x

∂∂δ

∂

δδ

∂∂ ∂

′

⎡⎤

⎡

⎤

== =+

⎢⎥

⎢

⎥

⎣

⎦

⎣⎦

MECHANICAL SYSTEMS, CLASSICAL MODELS

302

hence

()

∂

δε

∂

=+ +O

(20.3.4)

where we have used Einstein’s summation convention for dummy Latin indices from 1

to n. The volume elements associated with the domains Ω and

Ω

′

are thus related by

ddJΩΩ

′

= , and for the inverse transformation we will have

()

∂

δε

∂

−

=− +O .

(20.3.4')

With these definitions and using Einstein’s summation convention for dummy Greek

indices from 1 to m, we may write the Taylor series expansion

()( )

,, , ,

ii ii ii

xu u x xu u u u

αα α αα α

δδ δ

′′ ′

=+ + +LL

()

ii i i

xu u x u u

xu u

αα α α

αα

∂∂ ∂

δδ δε

∂∂ ∂

=++++

LL L

Hence, taking into account (20.3.4), the variation (20.3.1') of the functional

A can

be written in the form

ii i

xx u u

xxu u

αα

Ω

∂∂ ∂ ∂

δδδδδΩ

∂∂∂ ∂

⎛⎞

=+++

⎜⎟

⎝⎠

∫

LL L

(20.3.1'')

As, in general,

()

uux

αα

δ∂δ∂≠ , we introduce the variations

() () ()

ux u x ux

ααα

∗

′′′ ′

=−

() () ()

ii i

ux u x ux

ααα

∗

′′′ ′

=−

(20.3.5)

to integrate by parts the last term in (20.3.1''). Thus, we have the Taylor series

()

ux ux x ux u x

αα α α

∂

δδδδ δδε

∂

∗∗ ∗ ∗

′

=+= + +

O ,

where

∗

and

xδ are variations of order ε; by neglecting

()

εO

, we have

()

()ux ux

αα

δδ

∗∗

′

. Therefore, from (20.3.3) we obtain

() ()

[

]

()

[

]

() ()ux u x ux ux ux

ααα αα

′′ ′ ′

=−+−,

() ()

[

]

()

[

]

() ()

ii i ii

ux u x ux ux ux

ααα αα

′′ ′ ′

=−+−,

while the notations (20.3.5) and new expansions into Taylor series lead to

() ()

ux ux ux

ααα

δδ δ

∗

=+, () ()

i i ij j

ux ux ux

αα

δδ δ

∗

=+,

(20.3.5')

Variational Principles. Canonical Transformations

303

where the mixed derivatives of second order

ij i j

uuxx

αα

∂∂∂= , , 1,2,...,ij n= , have

been also introduced. The second relation (20.3.5) may be written also in the form

() () ()

[]

()

ux u x ux ux

ααα α

∂∂

δδ

∂∂

∗∗

′′′′ ′

=−=

′′

(20.3.5'')

where we took into consideration the first relation (20.3.5); analogically,

() ()

ux ux

αα

∂

δδ

∂

∗∗

= , () ()

iijj

ux ux ux

ααα

∂

δδδ

∂

∗

=+.

(20.3.5''')

Introducing the operator of total differentiation (

L depends on

x explicitly or by

the agency of the functions

and

)

iji

ii j

xx u u

αα

∂∂ ∂

=+ +

, 1,2,...,in= ,

(20.3.6)

we observe that

()

i i ii iij j

iii i

x x xu xu x

xxxuu

αα

∂∂ ∂ ∂

δδδδδ

∂∂ ∂ ∂

=++ +

LLL

;

taking into account also the relations (20.3.5'), the variation (20.3.1'') of the functional

A becomes

()

xuu

αα

Ω

∂∂

δδδδΩ

∂∂

∗∗

⎡⎤

=++

⎢⎥

⎣⎦

∫

(20.3.7)

The last term in

δA can be written in the form

iii ii

uu u

uxu xu

αα α

∂∂ ∂

δδ δ

∂∂ ∂

∗∗ ∗

⎛⎞⎛⎞

=−

⎜⎟⎜⎟

⎝⎠⎝⎠

LL L

so that

[]

xu u

αα

Ω

∂

δδδδΩ

∂

∗∗

⎡⎛ ⎞ ⎤

=++

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦

∫

AL L

(20.3.7')

where

[]

uxu

αα

∂∂

⎛⎞

=−

⎜⎟

⎝⎠

, 1,2,...,mα = ,

(20.3.8)

is the Euler–Lagrange derivative (corresponding to the functional derivative (20.1.79)).

Finally, by means of the equations (20.3.5') we obtain the variation of the action

A in

the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

304

[]

()

ijjjj

iii

xuux uux

xuu

αα αα

αα

Ω

∂∂

δδδδδδΩ

∂∂

⎡⎛ ⎞ ⎤

=+−+−

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦

∫

AL L

(20.3.7'')

To obtain the equations of motion, according to the conditions of Hamilton’s

principle, we assume that the variations

xδ

vanish identically inside and on the

boundary of the domain

Ω (i.e. 0

xδ ≡ in Ω and on ∂Ω, and the integration domain

remains unchanged), and the variations of the functions of state

are identically zero

on the boundary of

Ω (

∂Ω

δ ≡

). By way of consequence, applying the flux-

divergence formula (A.2.67'), we show that the first term in (20.3.7'') vanishes and we

obtain

[]

Ω

δδΩ=

∫

AL ;

(20.3.9)

equating to zero the variation

δA and using the second basic lemma of the variational

calculus, we obtain the equation of motion in the differential form

[

]

1,2,...,mα =

(20.3.10)

The function

L which leads to a certain type of equations of motion is not uniquely

determined. Indeed, these equations are invariant if we replace the function

L by λL,

where

λ is an arbitrary non-zero constant (scale transformation) or by d/d

fx+L ,

where

()

fxu

1,2,...,

, are arbitrary functions of class

(divergence

transformation). Indeed, observing that

jjj

fff

xxu

∂∂

=+ ,

ux u

αα

∂

∂∂

⎛⎞

⎜⎟

⎝⎠

jjj j j j

jjj

fff f f f

xxux

uuxuu uu u

αα

βββα β β β

∂∂ ∂ ∂ ∂

∂∂∂

∂∂

∂∂∂∂∂ ∂ ∂ ∂

⎛⎞

⎛⎞ ⎛⎞ ⎛⎞

=+ = + =

⎜⎟ ⎜⎟ ⎜⎟

⎜⎟

⎝⎠ ⎝⎠ ⎝⎠

⎝⎠

it is easy to show that

[

]

d/d 0

. In particular, when the only independent

variable is the time t we obtain the gauge transformations discussed in Sect. 18.2.3.2.

20.3.1.2 Symmetry Transformations. Noether’s Theorem

By symmetry transformations we denote the transformations of independent and

dependent variables under which the form of the equations of motion remains invariant.

Thus, the general form of a transformation of independent variables is

()

xxϕ

′

= ,

1,2,...,in=

(20.3.11)

and it changes the functions of state into

Variational Principles. Canonical Transformations

305

()

() ,()ux xux

αα

′′

= ,

1,2,...,mα =

(20.3.11')

where

{

}

, ,...,

uuu u≡ . The functional (20.3.1) is invariant with respect to the

transformation (20.3.11) if the relation

()()

,, d ,, d

ii ii

xu u xu u

αα αα

ΩΩ

′′ ′ ′

′

(20.3.12)

holds. On the other hand, the equations of motion (20.3.10) are invariant if

()()

,, ,,

ii ii

xu u xu u

αα αα

′′ ′ ′′ ′

′

LL .

(20.3.12')

Thus, the transformations (20.3.11) are symmetry transformations for the considered

mechanical system if and only if both conditions (20.3.12) and (20.3.12') are satisfied.

In the case of the infinitesimal transformations (20.3.2), (20.3.3), these conditions

become

()()

,, d ,,d

ii i i i i

xxuuuu xuu

αααα αα

δδ δΩ Ω

′

++ + =

′

LL,

()

,, d

ii i i

xxuuuu

αααα

δδ δΩ

′

++ +

′

()

,, d d

ii i i j

xxu uu u f

αααα

δδ δΩδΩ=+ + + +L

;

in the last term we have introduced the variation

fδ (a quantity of order ε) and we

have used the relation

()

fx fxδδ

′

= (we neglect

()

εO with respect to ()εO ),

observing that

′

and dΩ

′

differ from d

x and dΩ, respectively, by ()εO (which

can be neglected, because

fδ is of the order of ε).

Eliminating

()

,, d

ii ii

xxuuuu

αααα

δδδΩ

′

+++

′

L , we obtain

()()

()

,, ,,

ii i i i i j

xxuuuu xuu fJ

αααα αα

δδ δ δ

−

⎡⎤

++ += −

⎢⎥

⎣⎦

;

finally, substituting (20.3.4') and neglecting

()

εO , we obtain (expansion in a Taylor

series)

()

iiiji

iiij

xu u x fxu

xu ux x

αα α

αα

∂∂ ∂∂

δδ δ δ δ

∂∂ ∂∂

⎛⎞

+++ =−

⎜⎟

⎝⎠

L .

(20.3.13)

Hence, an infinitesimal transformation of the form (20.3.2), (20.3.3) is a symmetry

transformation if, for a given L, there exist functions

()

fxu

of class

C , so that

the equation (20.3.13) be satisfied. It can be shown that, for a given

L, the symmetry

transformations form a group, which represents the symmetry group of the considered

mechanical system.

MECHANICAL SYSTEMS, CLASSICAL MODELS

306

If, in particular, together with E.L. Hill, the left member of the equation (20.3.13) is

identically zero, and if the Jacobian of the transformation is equal to unity, then we say

that the density

L has an invariant form.

From (20.3.1') and (20.3.13) we have (we integrate on the arbitrary domain

Ω and on

the domain

Ω

′

, respectively,

()

Ω

δδΩ+=

∫

A ,

(20.3.13')

so that (20.3.7'') can be written in the form

[]

()

ijjijj

iii

x u ux f u ux

xuu

αα αα

αα

Ω

∂∂

δδ δδ δδΩ

∂∂

⎡⎛ ⎞ ⎤

+− ++ − =

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦

∫

(20.3.14)

This integral has to vanish on any domain

Ω; if we assume that the integrand is a

continuous function, then it must vanish at any point

x Ω∈

. Consequently, when the

equations of motion (20.3.10) are satisfied, then we obtain an equation of conservation

of the form

ijji

iii

xuuxf

xuu

αα

∂∂

δδ δδ

∂∂

⎛⎞

+− +=

⎜⎟

⎝⎠

(20.3.15)

It follows that an infinitesimal symmetry transformation of the mechanical system is

associated with a certain equation of conservation; as the symmetry transformations

form a group, the relation (20.3.15) establishes the connection between the symmetry

group of a mechanical (physical) system and a certain conservation law. The general

connection between the symmetry properties of a mechanical system and the

conservation laws is given by

Theorem 20.3.1 (E. Noether). If the Lagrangian of a physical (mechanical) system is

invariant with respect to a continuous group of transformations with m parameters,

then there exist m quantities which are conserved during the evolution of the system.

20.3.1.3 Case of a Generalized Conservative System

Let be a mechanical system described by a Lagrangian

()

,qq= LL ,

{

}

, ,...,

qqq q≡ ,

{

}

, ,...,

qqq q≡

 

, which does not depend explicitly on time

(



ubjected to catastatic constraints and

conservative given forces; see Sect. 18.2.3.4). We also assume that neither the

Hamiltonian

()

,HHqp= ,

{

}

, ,...,

ppp p≡ (which is a consequence of the

considered constraints), nor the integrals of motion of the system do not depend

explicitly on time.

For an infinitesimal transformation of the generalized co-ordinates

), for instance a mechanical system s

Variational Principles. Canonical Transformations

307

kk k

qq qδ

′

1,2,...,ks=

(20.3.16)

we have

δδ=



(20.3.17)

We obtain thus the relation between the Lagrangian

()

,qqL

and the Hamiltonian

()

,Hqp by means of the generalized momenta given by (18.2.80); to pass from H to

L we use Hamilton’s first equation (19.1.14). If

()

,Cqp is an integral of motion in

the Hamiltonian formalism, then in the Lagrangian formalism this integral of motion

will take the form

() ()

,,Dqq Cqp=



; also, along the trajectory of the representative

point, this constant of motion will satisfy the relations

() ()

,,0

Cqp Dqq



(20.3.18)

The variation of the Lagrangian, induced by the transformation (20.3.16), can be

written in the form (we use the summation convention of dummy indices from 1 to s;

see Sect. 20.1.1.2)

[]

∂

δδδ

∂

⎛⎞

=−

⎜⎟

⎝⎠



(20.3.19)

which, along the trajectory of the representative point in

, becomes

∂

δδ

∂

⎛⎞

⎜⎟

⎝⎠



(20.3.19')

We consider three cases when the relation (20.3.19') leads to conserved quantities

(integrals of motion), i.e.:

(i) If

0δ =L , then the corresponding integral of motion is

()

kkk

Dqq q p q

∂

δτ δ δ

∂

==



(20.3.20)

where

p are the generalized momenta in the Hamiltonian formulation and δτ is a

constant factor.

(ii) If there exists a function

()fq such that

()

δδ=

(20.3.21)

then we obtain the integral of motion

()

Dqq q

∂∂

δτ δ

∂∂

⎛⎞

=−

⎜⎟

⎝⎠



(20.3.21')

MECHANICAL SYSTEMS, CLASSICAL MODELS

308

In this case, we may assume that the Lagrangian of the system has the form

d/dft=−

′

LL , so that 0δ =

′

L . Let us notice that the two Lagrangians (L and

′

) are related by a divergence transformation; consequently, they are equivalent in

the sense that the form of the equations is preserved.

(iii) If

()

gqq

δ =

L ,

(20.3.22)

then the associated integral of motion is

() ()

Dqq q gqq

∂

δτ δ

∂

=−



(20.3.22')

In this case, there is no one Lagrangian

′

to be equivalent to L and to satisfy the

relation

0δ =

′

L .

It is evident that the first and the second case are particular cases of the third case.

Actually, there is no other case (i.e.

()

d,/dgqq tδ ≠ L ) which could lead to an

integral of motion associated with the infinitesimal transformation (20.3.16). Note that

not all the three cases considered above correspond to symmetry transformations.

Indeed, the only independent variable is the time t, while the Lagrangian L does not

explicitly depend on t. Hence, the defining relation (20.3.13) takes the form

()

;

qq fqt

qq t

∂∂

δδ δ δ

∂∂

⎛⎞

≡+ =−

⎜⎟

⎝⎠



LL;

(20.3.23)

it follows that only the first case and the second one correspond to symmetry

transformations.

We may state

Theorem 20.3.1' (E. Noether). To every infinitesimal transformation of the form

(20.3.16), inducing a variation of the Lagrangian of the form (20.3.22), it corresponds

a conserved quantity defined by (20.3.22').

If the infinitesimal transformation (20.3.16) is a symmetry transformation (i.e.,

() ()

,gqq fqδ=



), then there exists an invariant Lagrangian

()

=−

′

(20.3.24)

20.3.1.4 The Reciprocal of Noether’s Theorem

The Hessian of a given Lagrangian with respect to the generalized velocities has the

elements

jk kj

qq q q

∂

∂∂ ∂ ∂

== ==

  

, , 1,2,...,jk s= ,

(20.3.25)

Variational Principles. Canonical Transformations

309

and det 0

≠

⎡⎤

⎣⎦

H is the necessary and sufficient condition for the integrability of

Lagrange’s equations. The inverse of the matrix

⎡

⎤

⎣

⎦

has the elements

jk kj

∂

∂∂

−−

===



HH , , 1,2,...,

ks= .

(20.3.25')

The relation (20.3.22') represents the condition which must be satisfied by the most

general integral of motion of a generalized conservative mechanical system whose

evolution is described by Lagrange’s equations. An equivalent form of this condition is

given by (20.3.18); thus we have

()

Dqq q q

tqq

∂∂

=+=





(20.3.18')

In our case

()

,qq= LL ,

()

,HHqp= and

()

qqqp=



, 1,2,...,ks= , so that

(we use the relations (19.1.13), (19.1.14))

k l

jj j j j

kk k lk

qqp pq pq qqp

∂∂

∂∂ ∂∂ ∂∂ ∂

∂∂∂ ∂∂ ∂∂ ∂∂∂

⎛⎞ ⎛⎞

⎛⎞

===−=−

⎜⎟

⎜⎟ ⎜⎟

⎝⎠

⎝⎠ ⎝⎠





;

we obtain the generalized accelerations in the form (we use the relations (20.3.25'))

kk l k

jj j

jjjj

qq qq

qqqp q

tq p qqp pq

∂∂ ∂∂

∂∂

∂∂ ∂∂∂∂∂

== + =− +

 

    



qqq p

∂

∂∂

∂∂∂ ∂

⎛⎞

=−

⎜⎟

⎝⎠



1,2,...,ks=

(20.3.26)

In terms of the elements of the inverse Hessian (20.3.25'), and by means of (20.3.26),

the condition (20.3.18') becomes

klkj

kkl k

qqqq q

∂∂ ∂ ∂

∂∂∂∂ ∂

−

⎛⎞

=−

⎜⎟

⎝⎠



(20.3.27)

where

()

,Dqq



is an integral of motion determined by (20.3.22'). The condition

(20.3.27) can also be expressed by means of the Poisson bracket (note that

()

,/ 0Cqp t∂∂=)

()()

,, 0Cqp H = , (20.3.28)

this being a necessary and sufficient condition for

()

,Cqp be a first integral of the

canonical system.

For

KC=

and

εδτ=

in (20.2.46) and by means of (20.3.25'), we may rewrite the

variations of the generalized co-ordinates in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

310

()

kk jk

CD D

qqC

pqpq

∂

∂∂ ∂

δδτδτδτδτ

∂∂∂∂

−

=== =





H , 1,2,...,ks= ; (20.3.27')

the conditions (20.3.27) and (20.3.27') are equivalent to (20.3.22'). It follows that an

integral of motion

()

,Dqq



is associated with an infinitesimal transformation defined

by (20.3.16) and (20.3.27'), such that the condition (20.3.22') be satisfied, and the

function

()

,gqq



is determined by

()

gqq D

∂∂

δτ

∂∂

−

⎛⎞

=−

⎜⎟

⎝⎠





(20.3.27'')

The variation of the Lagrangian, induced by this infinitesimal transformation, is given

by (20.3.22). The condition (20.3.27) is a necessary condition which has to be satisfied

by the most general integral of motion given by Noether’s theorem. Because this

condition is a homogeneous partial differential equation of first order, in 2s independent

variables

()

,qq



, it has

21s −

functional independent solutions

()

,Dqq



. Hence, the

equation (20.3.27) yields all the constants of motion

()

,Dqq



, which do not explicitly

depend on time. As well, the equation (20.3.27) is also a sufficient condition for the

integrals of motion; we may rewrite it in the form

()

Dqq

ttqqq

∂∂∂

−

⎡⎛ ⎞ ⎤

=−

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦





(20.3.29)

as one can easily prove. We are thus led to

()

d,/d0Dqq t=



along the integral curves

of the system.

In case of integrals of motion corresponding to symmetry transformations, we have

() ()

()

gqq fq f q qδ∂∂δ==



, hence

()

gqq

∂∂

δτ

∂∂

−

=



;

(20.3.30)

by means of the relations (20.3.22), (20.3.27'') we obtain

()

Dqq Eqp p

∂

′′

′



(20.3.31)

where we denoted

{

}

, ,...,

pppp

′′′′

≡ , with

kk k

qq q

∂∂∂

∂∂ ∂

′

=−=

′



, 1,2,...,ks= .

(20.3.31')

Thus, the integral of motion

()

,Dqq



has to be a homogeneous function of the

generalized momenta (20.3.31'). This condition is equivalent to the condition which has

to be satisfied by the generating function of infinitesimal, homogeneous, canonical