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

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Variational Principles. Canonical Transformations

291

the relations thus obtained corresponding to the considered canonical transformations of

valency

c . The function U is the generating function of the canonical transformation.

Let

U be a function of class

C for which we suppose that

det 0

∂

∂∂

⎡⎤

≠

⎢⎥

⎣⎦

(20.2.53)

where

rR, , 1,2,...,jk s= , represent the variables (20.2.56) in the mentioned order;

we can solve the equations (20.2.52) with respect to

P ,

1,2,...,im=

, and

Q ,

1, 2,...,jm m s=+ + . Introducing in the relations (20.2.52'), we obtain the searched

relations of the canonical transformations (20.2.1).

20.2.2.2 Group Properties. The Symplectic Group

We introduce the 22ss× matrix associated to the Jacobian (20.2.1') in the form

()

∂∂

∂

∂∂

⎡

⎡⎤⎡⎤⎤

⎢

⎢⎥⎢⎥⎥

⎣

⎦⎣ ⎦

⎡⎤

⎢

⎥

⎢⎥

⎣⎦

⎢

⎥

⎡

⎤⎡ ⎤

⎢

⎥

⎢

⎥⎢ ⎥

⎣

⎣⎦⎣⎦⎦

(20.2.54)

where

, 1,2,...,jk s= ; we consider also the matrix

−

⎡

⎤

⎢

⎥

⎣

⎦

(20.2.55)

where

E is the ss× unit matrix.

The matrix

I is called the canonical matrix. We notice that

2 1

−

−−−

⎡⎤⎡⎤

⎡⎤⎡⎤ ⎡⎤

= = =− =− =− =−

⎢⎥⎢⎥

⎢⎥⎢⎥ ⎢⎥

−

⎢⎥⎢⎥

⎣⎦⎣⎦ ⎣⎦

⎣⎦⎣⎦

0E0E E 0 E0 E0

IEII

E0 E0 0E

0E 0E

the last unit matrix having 2s rows and 2s columns; one obtains

1−

=−II, so that

this matrix is non-singular.

Introducing the transpose matrix

J , let us calculate

TT T T

jj j j

kk k k

jj j j

kk k k

QP P Q

qq q q

QP P Q

pp p p

∂∂ ∂ ∂

⎡⎤⎡ ⎤

⎡⎤⎡⎤ ⎡⎤ ⎡⎤

−

⎢⎥⎢ ⎥

⎢⎥⎢⎥ ⎢⎥ ⎢⎥

−

⎡⎤

⎣⎦⎣⎦ ⎣⎦ ⎣⎦

⎢⎥⎢ ⎥

⎢⎥

⎢⎥⎢ ⎥

⎣⎦

⎡⎤⎡⎤ ⎡⎤ ⎡⎤

⎢⎥⎢ ⎥

−

⎢⎥⎢⎥ ⎢⎥ ⎢⎥

⎢⎥⎢ ⎥

⎣⎣ ⎦ ⎣ ⎦ ⎦ ⎣⎣ ⎦ ⎣ ⎦ ⎦

;

then

MECHANICAL SYSTEMS, CLASSICAL MODELS

292

TT TT

TT T

jj jj jj jj

kk kk kk kk

jj jj jj j

kk kk kk k

PQ Q P PQ Q P

qq qq qp qp

PQ Q P PQ Q

pq pq pp p

∂∂ ∂∂ ∂∂ ∂∂

∂∂ ∂∂ ∂∂ ∂

⎡

⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤

−−

⎢

⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣

⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤

−−

⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥

⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦

JIJ

∂

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎡⎤

⎢

⎥

⎢⎥

⎢

⎥

⎣

⎣⎦⎦

[

]

[

]

[

]

[

]

[][][][]

, ,

qq qp c

qp pp

−−

⎡⎤

===

⎢⎥

⎣⎦

where we took into account the necessary and sufficient conditions of canonicity

(20.2.12'). We obtain thus a necessary and sufficient condition of canonicity, equivalent

to the conditions (20.2.12'), and we can state

Theorem 20.2.3 The necessary and sufficient condition for the transformation (20.2.1)

in the space

Γ be canonical consists in the existence of a scalar constant 0c ≠ ,

such that

c=JIJ I.

(20.2.56)

This condition is written in the form

=JIJ I

(20.2.57)

in case of a univalent canonical transformation. A matrix

J which satisfies the

condition (20.2.57) is called a Symplectic matrix. Because

det 1=±I and the

determinant of a product of matrices is equal to the product of the determinants of the

corresponding matrices, we obtain

det 1=±J , hence the Symplectic matrices are non-

singular; as a matter of fact, we have given also another proof in Sect. 20.2.1.2.

A matrix

J which satisfies the relation (20.2.56) will be called a generalized

Symplectic matrix of valency

0c ≠ ; starting from this relation, one can show that

det

c=±J , obtaining again a known result. Hence, the generalized Symplectic

matrices are also non-singular.

Let

J and

J be two 22ss× generalized Symplectic matrices; the matrix

JJ is

a matrix which has the same number of rows and columns. We have

11 1

c=JIJ I,

22 2

c=JIJ I, wherefrom

()

T1T

12221 1

−

=JJIJJI; but

()

12 21

=JJ JJ , so that

()()

21 21 21

cc=JJ I JJ I, the matrix

JJ being also a Symplectic matrix. The

associativity property of the Symplectic matrices follows from the associativity property

of the matrix product. Let be the unit matrix

E , so that ==EJ JE J ,

=EE and

=EIE I; one notices that this matrix is Symplectic, playing the rôle of neutral

element in the set of these matrices (the corresponding canonical transformation is

univalent). Multiplying the relation (20.2.56) at left by the matrix

1−

and at right by

the matrix

1−

and taking into account that

11−−

==II JJ E

, we find the remarkable

relation

-1 1

−

=IJI J

;

(20.2.58)

Variational Principles. Canonical Transformations

293

hence, the inverse matrix

1−

does exist. We notice that det 0≠J , so that

()

−

=JJ; in this case, taking into account the relation (20.2.56) (we assume

that the matrix

J is a generalized Symplectic matrix), we may write

()

() ()( )

11T11TT11

−−

− − −− −−

== = =J IJ J IJ J J IJ J I ,

the matrix

1−

being Symplectic too.

We can thus state that the set of Symplectic matrices forms a group: the Symplectic

group, defined in a real linear space, of even dimension

, and denoted by

()

Sp 2 ,s \ . The essential property of this group consists in making invariant certain

antisymmetric bilinear form defined on

\ . There results that the set of canonical

transformations forms a group too. Analogously, the set of generalized Symplectic

matrices forms the generalized Symplectic group.

We notice also that, if

J is a generalized Symplectic matrix, besides its inverse

1−

J ,

the transpose matrix

J is also a generalized Symplectic matrix; indeed, starting from

the relation

()

111

−−−

=JIJ I and calculating the inverse at right and at left, we

have

() () () ()

T1T 1

11 111 1

−−

−− −−− −

⎡⎤ ⎡⎤

⎡

⎤

==−

⎣

⎦

⎣⎦ ⎣⎦

JIJ J IJ JIJ

()

TT 1

−

=− =− =JIJ I I.

Hence, we may write the necessary and sufficient condition of canonicity (20.2.56) also

in the form

c=JIJ I ;

(20.2.56')

as well, in case of univalent canonical transformations, the condition (20.2.57) becomes

=JIJ I

(20.2.57')

20.2.2.3 Invariance Properties

The univalent canonical transformations admit three basic invariants: the volume

element in the phase space, the Poisson bracket and the Lagrange bracket.

For instance, let us define the univalent canonical transformation by means of the

generating function

()

,;SSPqt= . Before the transformation, the volume element

is given by

()

12 12 12 12

d d ...d d d ...d det d d ...d d d ...d

ss s s

qq qpp p qq qPP P

∂

⎡⎤

⎢⎥

⎣⎦

and after the transformation by

()

1 2 12 12 12

d d ...d d d ...d det d d ...d d d ...d

ss ss

QQ QPP P qq qPP P

∂

⎡⎤

⎢⎥

⎣⎦

MECHANICAL SYSTEMS, CLASSICAL MODELS

294

with

()

det det det det

qP P P q

∂

∂∂

∂∂∂∂

∂∂

⎡⎤

⎡

⎤⎡ ⎤

⎡⎤

⎢⎥

===

⎢

⎥⎢ ⎥

⎡⎤

⎢⎥

⎣⎦

⎣

⎦⎣ ⎦

⎢⎥

⎣⎦

⎣⎣ ⎦ ⎦

()

det det det det

qP q q P

∂∂

∂

∂∂

∂∂∂∂

⎡⎡ ⎤ ⎤

⎡⎤

⎡

⎤

⎡⎤

⎢⎢ ⎥ ⎥

⎢⎥

⎣⎦

===

⎢

⎥

⎢⎥

⎣⎦

⎣

⎦

⎢⎥

⎣⎦

where we took into account the representation (20.2.33). The two determinants at the

right are equal because the corresponding matrices are one the transpose of the other

(or, using Schwartz’s theorem, if

S is a function of class

C ); it follows that

12 12 12 12

d d ...d d d ...d d d ...d d d ...d

ss ss

QQ QPP P qq qpp p=

. (20.2.59)

We can thus state

Theorem 20.2.4 The volume element in the phase space is invariant with respect to a

univalent canonical transformation.

We obtain the same result, observing that

()

12 12 12 12

d d ...d d d ...d det d d ...d d d ...d

ss ss

QQ QPP P qq qpp p

∂

⎡⎤

⎢⎥

⎣⎦

;

in Sect. 20.2.1.2 we have seen that

1J =± , while for the identical transformation

(20.2.39') we have

1J = , so that we find again the relation (20.2.59). In case of a

canonical transformation of valency

c we obtain

12 12 12 12

d d ...d d d ...d d d ...d d d ...d

ss ss

QQ QPP P c qq qpp p= .

(20.2.59')

Concerning the Poisson bracket of the functions

()

,;qptϕϕ= ,

()

,;qptψψ= we

can write

()

ll ll l l k l kl

qp pq Q q Pq Q p Pp

∂∂

∂ϕ ∂ϕ

∂ψ ∂ψ

∂Φ ∂Φ ∂Ψ ∂Ψ

ϕψ

∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛⎞

=−= + +

⎜⎟

⎝⎠

()

jj j

llklkl k

Qp Pp Qq Pq QQ

∂∂

∂Φ ∂Φ ∂Ψ ∂Ψ ∂Φ ∂Ψ

∂∂ ∂∂ ∂∂ ∂∂ ∂∂

⎛⎞

−+ +=

⎜⎟

⎝⎠

() () ()

,,,

jjj

kkk

jj j

kkk

QP PQ PP

∂Φ ∂Ψ ∂Φ ∂Ψ ∂Φ ∂Ψ

∂∂ ∂∂ ∂∂

+++

where

() ()()()

,; ,;, ,; ;QPt q QPt pQPt tΦϕ= ,

() ()()()

,; ,;, ,; ;QPt qQPt pQPt tΨψ=

;

Variational Principles. Canonical Transformations

295

taking into account the sufficient conditions of canonicity (20.2.15), it follows

()

,,ϕψ ΦΨ= ,

(20.2.60)

so that we can state

Theorem 20.2.5 The Poisson bracket is invariant with respect to a univalent canonical

transformation.

In case of a canonical transformation of valency

c , the necessary and sufficient

conditions of canonicity (20.2.15') lead to

()

,,cϕψ ΦΨ= .

(20.2.60')

Let us consider also the Lagrange bracket with respect to the parameters

u and v in

the form

[]

ll ll l l l k l k

qp qp q q pQ pP

uv vu Q u P u Q v P v

∂∂

∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛⎞

=−= + +

⎜⎟

⎝⎠

[]

jjj

lk lk l l k

QPQ

qQ qP p p Q

Qv Pv Qu Pu uv

∂∂∂

∂∂ ∂∂ ∂ ∂ ∂

∂∂ ∂∂ ∂∂ ∂∂ ∂∂

⎛⎞

−+ +=

⎜⎟

⎝⎠

[] [] []

,,,

jj j

kkk

jjj

kkk

QPP

PQ P

QP PQ PP

uv uv uv

∂∂∂

∂∂ ∂∂ ∂∂

+++,

where

() ()()()

;, ;, , ;, ; ,

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

Q tuv Q q tuv p tuv t

P tuv P q tuv p tuv t j s

the sufficient conditions of canonicity (20.2.12'') lead to

[] []

qp QP

uv uv= ,

(20.2.61)

and we may state

Theorem 20.2.6 The Lagrange bracket is invariant with respect to a univalent

canonical transformation.

If the transformation is of valency

c , then the necessary and sufficient conditions of

canonicity (20.2.12''') allow us to write

[] []

qp QP

uv uv

= .

(20.2.61')

20.2.2.4 Functions in involution. Lie’s Theorem

In Sect. 19.1.2.3 we have introduced the canonical conjugate first integrals (the

Poisson bracket of which is constant) and the first integrals in involution (the Poisson

bracket of which is zero). In general, a set of functions of

,qp and t , of class

C , for

which the Poisson bracket of each pair is equal to zero, is a set of functions in

MECHANICAL SYSTEMS, CLASSICAL MODELS

296

involution; obviously, the generalized co-ordinates

()

QQqpt= and the

generalized momenta

()

PPqpt= , 1,2,...,js= , have, everyone, this property.

In general, s arbitrary functions of ,qp and t , of class

C , cannot be the

generalized co-ordinates

, ,...,

QQ Q of a canonical transformation; the following

question arises: is it sufficient that these functions be in involution to have this

property ? More general, we can formulate the question: Let be given

s functions

()

QQqpt= in involution; can one find other s functions

()

PPqpt= ,

1,2,...,js= , so that the transformation

()( )

,,qp QP→ be canonical ?

Obviously, the answer is affirmative in case of a point transformation. In general, let

be the functions in involution

()

qptϕϕ=

, 1,2,...,is= , of class

C , so that

[

]

det / 0

p∂ϕ ∂ ≠ ; because there is not an identical relation connecting the

generalized co-ordinates

and Q and the time

, we have to solve the equations

()

Qqptϕ= with respect to p , obtaining

()

pqQtψ= , ,1,2,...,il s= . We

can write the identity

()

,; 0

qtQϕψ−=, 1,2,...,is= , with respect to the variables

q and Q , where we have denoted

{

}

, ,...,

ψψψ ψ≡ . Calculating the partial

derivatives with respect to

q , there results

klk

qpq

∂ψ

∂ϕ ∂ϕ

∂∂∂

+=,

, 1,2,...,ik s=

;

multiplying by

p∂ϕ ∂ and summing for all the values of k , we find

kk lkk

qp ppq

∂ϕ ∂ϕ

∂ψ

∂ϕ ∂ϕ

∂∂ ∂∂∂

+=, , 1,2,...,ij s= .

Taking the antisymmetric part with respect to the indices

i and j and observing that

()

ϕϕ = , , 1,2,...,ij s= , we may write (we invert the dummy indices k and l in

the first sum)

jj j

iii

ll kl

lkk lkk kl l k

ppq ppq p p q q

∂ϕ ∂ϕ ∂ϕ

∂ψ ∂ψ ∂ψ ∂ψ

∂ϕ ∂ϕ ∂ϕ

∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

⎛⎞

−= −=

⎜⎟

⎝⎠

, , 1, 2,...,ij s= ;

we obtain thus a system of

homogeneous linear equations to determine

unknowns

kl lk

qq∂ψ ∂ ∂ψ ∂− , the determinant of which is equal to

[]

()

det / 0

p∂ϕ ∂ ≠

. It results

∂ψ ∂ψ

∂∂

= ,

,1,2,...,kl s=

;

(20.2.62)

hence, there exists a function

()

,;SSqQt= so that

()

pqQt

∂

==,

1,2,...,ks=

(20.2.62')

Variational Principles. Canonical Transformations

297

with a non-singular matrix

[

]

SQq∂∂∂ (if not, an identical relation connecting

,qp

and

must exist). If we take also

∂

=−

, 1,2,...,js= ,

(20.2.62'')

we find a free canonical transformation, the answer to the above question being, in

general, affirmative.

Let us consider the case in which the

s functions

ϕ , 1,2,...,is= , are first

integrals of the system of canonical equations. The canonical equations in new

canonical co-ordinates derive from Hamilton’s function

()

,;HHSHQPt=+=



;

but the canonical co-ordinates

Q are constant in time along the integral curves of the

canonical system in

,qp and t , while the function

cannot depend on the

generalized momenta

P . Much more, we can choose the canonical transformation so as

to have

0H = , the generalized momenta P being also constant in time along the

trajectory of the representative point of the mechanical system. Hence, the knowledge

s first integrals in involution gives the possibility to determine other s first

integrals, so that the

2s distinct first integrals do specify, completely, the solution of

the problem.

Indeed, if we replace

Q by

{

}

, ,...,

ϕϕϕ ϕ≡ in (20.2.62'), then we obtain an

identity in

,qp and t ; differentiating successively with respect to ,

qp and t , it

follows

kkl

qQq

∂ψ ∂ψ ∂ϕ

∂∂∂

+=,

∂ψ ∂ϕ

∂∂

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

(20.2.63)

∂ψ

ψϕ

∂



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

(20.2.63')

Because

ϕ is a first integral (it verifies a relation of the form (19.1.61)), we can

write

()

kk kl l

kll

jjj

ll l

QQ Qqpqp

∂ψ ∂ψ ∂ψ ∂ϕ ∂ϕ

∂∂

ψϕϕ

∂∂ ∂∂∂∂∂

⎛⎞

=− = = −

⎜⎟

⎝⎠



;

taking into account (20.2.63), we obtain

pq q

∂ψ

∂∂

∂∂ ∂

=− −



1,2,...,ks=

Replacing

p by ψ in

()

,;Hqpt, we obtain

()

,;GQqt , hence

kk k

GHH

qqpq

∂ϕ

∂∂∂

∂∂∂∂

=+ ,

1,2,...,ks=

MECHANICAL SYSTEMS, CLASSICAL MODELS

298

so that

∂



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

(20.2.64)

From (20.2.62) and (20.2.64) it follows that

∂

, GS=−



(20.2.65)

with

()

,;SSQqt= ; if this function corresponds to the canonical transformation

(20.2.62'), (20.2.62''), then the new function of Hamilton is

HGS=+



, expressed by

,QP

and t . Taking into account (20.2.65), we have

0H =

. We obtain thus

()

PPqpt= , 1,2,...,js= , that is s new first integrals of the canonical system in

,qp and t ; these first integrals are in involution, because the Poisson brackets of all

pairs which can be formed vanish. We notice that this result corresponds to Liouville’s

Theorem 19.1.9.

The above considerations can be presented also in a slightly different form. Starting

from

()

qpt aϕ = , const

a = , we may calculate

()

pqatψ= ,

1,2,...,ks=

, with

{

}

, ,...,

aaa a≡ . Replacing the generalized momenta p by

()

,;Hqpt

, we obtain the function

()

,;Gqat

. In this case, ddd

qGt Sψ −=,

with

()

,;SSqat=

, is an exact differential; other s relations /

Sa b∂∂= ,

1,2,...,ks=

, give the generalized co-ordinates. We build up the function S by means

of the

s first integrals in involution, and then find again the Hamilton–Jacobi equation

(19.2.9).

In the particular case in which

ϕ =



, 1,2,...,ks= , and 0H =



, with

()

Hqpϕ = ,

ah= being the energy constant, we have

()

pqaψ=

Sq∂∂= , 1,2,...,ks= , with

()

,SSqa= ; we find again the relations (19.2.15'''),

obtaining thus the solution of the problem.

A system of functions

, ,...,

uu u of class

in a domain D of variables

12 1 2

, ,..., , , ,..., ,

qq qpp pt, so that

()

uu = , , 1,2,...,

ks= , is called a system

in involution. We may state

Theorem 20.2.7 (S. Lie). If ()vvu= , ()wwu= are functions of class

, where

{

}

, ,...,

uuu u≡ is a system in involution defined on the domain D of variables

,qp and t , then

()

,0vw = . (20.2.66)

A direct calculus proves the theorem.

We have seen in Sect. 20.2.1.8 that a function

F is transformed in itself by an

infinitesimal canonical transformation if

()

,0FK = (the formula (20.2.45''')). Let be

such a transformation with

Ku= ; because we have to do with a system in involution,

Variational Principles. Canonical Transformations

299

any function

u is transformed in itself, hence also the function v . We have thus

()

uv= , 1,2,...,

s= . Let us consider now an infinitesimal canonical

transformation with

Kv=

; any of the functions

is transformed in itself, hence also

the function

w . The relation (20.2.66) takes place, the Theorem 20.1.7 being once

more justified.

This theorem can be stated also in a modified form, sometimes useful. Thus, if the

equations

0v = and 0w = are consequences of the equations 0

u = ,

1,2,...,

then

()

,0vw = . In fact, this form of the theorem is known as Lie’s theorem.

As an application, let us consider the functions

, ,...,

ϕϕ ϕ in involution and the

equations

()

,; 0

Qqptϕ−=, 1,2,...,

s= ; solving with respect to the generalized

momenta, we find

()

,; 0

pQqtψ−=, 1,2,...,js= . Equating the generalized co-

ordinates to constants, the last equations are consequences of the first ones. But the first

members of these equations form a system in involution; we can thus say the same thing

about the first members of the second system of equations, so that

()

jj jj

kk kk

ii ii

qp pq

∂ψ ∂ψ

ψψ

∂∂ ∂ ∂

−−

−−= −

ii j

qq qq

∂ψ ∂ψ

δδ

∂∂ ∂∂

=− + =− + = .

We find again the relation (20.2.62) as a consequence of Lie’s theorem.

20.2.2.5 First Integrals Linear with Respect to the Generalized Momenta

As it is known, in case of a dynamical system described by Lagrangian co-ordinates,

one of them (for example,

q ) being cyclic, the corresponding generalized momentum

remains constant during the motion (see Chap.19, Subsec. 1.1.8). One can state an

inverse property, i.e.

Theorem 20.2.8 Any autonomous system which has a first integral linear with respect

to the generalized momenta can be described by means of some generalized co-

ordinates conveniently chosen, so that one of them be cyclic.

Indeed, let be the first integral

pψ , where ()

qψψ= , 1,2,...,js= . We

consider the system of equations

...

ψψ ψ

===,

(20.2.67)

which defines a family of curves in the space

Λ ; the solutions are defined as

intersections of

1s − independent integrals of the partial derivatives equation

∂

= .

(20.2.67')

We denote these integrals by

()

const

qϕ = ,

1,2,..., 1

s=−

; let be now the point

transformation

MECHANICAL SYSTEMS, CLASSICAL MODELS

300

Q ϕ= ,

1,2,...,

(20.2.68)

where

12 1

, ,...,

ϕϕ ϕ

−

are the above obtained functions. We choose then

Q ϕ= , so

that we have

... d

ψψ ψ

====

(20.2.69)

for the previously defined displacements. To obtain

, we integrate, e.g.,

= ,

where

()

11112 1

,,,...,

qQQ QΨψ

−

= . Let us suppose that the transformation (20.2.68)

is reversible, so that we may obtain

()

, ,...,

qFQQ Q= ,

1,2,...,ks=

; for a

displacement along one of the paths defined by (20.2.67) (for which

const

Q = ,

1,2,..., 1js=−) we obtain /

FQ∂∂ ψ= , 1,2,...,ks= . We consider now the

point transformation (see Sect. 20.2.1.4)

qF= ,

∂

1,2,...,ks=

;

we have

const

jjj

Pp p

∂

===

in the new variables, so that

Q is a cyclic co-ordinate, the Theorem 20.2.8 being thus

proved.

20.3 Symmetry Transformations. Noether’s Theorem.

Conservation Laws

In 1918, Emmy Noether proved her famous theorem relating the physical properties

of a mechanical (in general, physical) system and the conservation laws, when the

equations of motion emerge from a variational principle. Thus, she established a general

method to derive conservation laws in classical mechanics; but this method can be

applied equally well in case of relativistic mechanics, quantum mechanics,

electromagnetism etc. This study is, obviously, on the line of the Erlangen programme.

The importance of the results thus obtained is significant because, in general, the

conservation theorems are not deduced systematically, but only in particular cases.

By introducing symmetry transformations, certain properties leading to Noether’s

theory are put into evidence; one obtains thus integration constants (first integrals).

Some considerations concerning Lie groups are made, and then various applications to

discrete mechanical systems, resulting conservation laws, are given.