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

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Variational Principles. Canonical Transformations

321

because the infinitesimal displacements

123

,,xxxδδδ are independent, it follows that

∂

∑

, 1, 2, 3

= .

(20.3.69)

Now, from (20.3.67) we obtain

tx tx

αα

∂∂

δδ

∂∂

∑∑



;

consequently, the generalized momentum P of components

αα

∂

∑∑



, 1, 2, 3j = ,

(20.3.70)

is conserved if the conditions (20.3.69) are satisfied. The relations (20.3.69) represent

not only a sufficient condition, but also a necessary ones, because taking into account

Lagrange’s equations of motion, the assumption of conservation of the generalized

momentum leads to

11 1

dd d

nn n

jjj

ttx tx x

ααα

αα α

∂∂∂

== =

⎛⎞

== = =

⎜⎟

⎝⎠

∑∑ ∑



LLL

, 1, 2, 3j = .

20.3.3.3 Rotations Group. Conservation of the Generalized Moment of Momentum

Let us consider now a rotation of the system of orthogonal co-ordinates, defined by

the relations

iijj

xax

′

, 1, 2, 3i = ,

(20.3.71)

where

()

SO 3,∈ \a

is an orthogonal matrix. In case of an infinitesimal rotation, this

matrix has the form

δθ=+aEa , (20.3.72)

where

a are the matrices of the generators of the rotations group

()

SO 3, \ ,

δθ

1, 2, 3k = , represent the infinitesimal variations of the rotation parameters and E is the

identity matrix. Hence, we have

()

ij ij ij ij

aaδδθδδ=+ =+a

ij ji

aaδδ=− , ,1,2,3ij= ,

(20.3.72')

δ being the Kronecker symbol. We assume that the transformation (20.3.71) is of the

form (20.3.2), such that, by substituting (20.3.72') into (20.3.71) we have

ijij jji

xxa xaδδ δ==−.

MECHANICAL SYSTEMS, CLASSICAL MODELS

322

It follows that the variations of the particle co-ordinates are of the form

kki

xxa

αα

δδ=− ,

kki

xxa

αα

δδ=−



, 1, 2, 3i = , 1,2,...,nα = ,

(20.3.73)

while the independent variable t remains unchanged, that is

′

= , 0tδ = . Thus, from

the condition (20.3.66), with

0δΩ = , we obtain

xxa

αα

∂∂

⎛⎞

⎜⎟

⎝⎠



The matrix

δa is skew-symmetric and the variations

δθ are independent, such that we

can rewrite this result as a system of equations

xxxx

αααα

∂∂∂∂

−+−=



LLLL

, ,1,2,3jk= .

(20.3.74)

If these equations are satisfied, then the transformation (20.3.71) is a symmetry

transformation and (20.3.67) yields the equation

dd1

dd2

jk k jk

xa x x a

tx t x x

ααα

∂∂∂

δδ

∂∂∂

⎡⎛ ⎞⎤

⎛⎞

=−=

⎜⎟

⎢⎥

⎝⎠

⎣⎝ ⎠⎦



LLL

which is equivalent to the system of equations

jjj

kkk

x x xp xp

α α αα αα

αα

∂∂

−=−



, ,1,2,3jk= .

Therefore, the conserved quantity is the generalized moment of momentum

K, which is

a skew-symmetric tensor of rank two, of components

jk k k

Kxpxp

αα αα

=−, ,1,2,3jk= .

(20.3.75)

Conversely, assuming the conservation of the generalized moment of momentum and

taking into account Lagrange’s equations of motion, one can write

dd d

jjj

kkk

xx xx x

ttx x x tx x

αα αα α

αα α α α

∂∂ ∂ ∂

∂∂ ∂ ∂ ∂

⎛⎞

== − = + −

⎜⎟

⎝⎠

∂



   

LL L L L

kkk

xxxxx

tx xxxx

ααααα

∂∂∂∂∂

⎛⎞

−=−+−

⎜⎟

⎝⎠





LLLLL

, ,1,2,3

k = .

Therefore, the relations (20.3.74), which represent sufficient conditions for the

conservation of the generalized moment of momentum

K, are necessary conditions in

this sense too.

Variational Principles. Canonical Transformations

323

20.3.3.4 Time Translations Group. Conservation of the Generalized Mechanical

Energy

The time translations are defined by the equations

tttδ

′

αα

′

, 0

δ = ,

αα

′



, 0

δ =



1,2, 3k =

1,2,...,nα =

(20.3.76)

and form a one-parameter Abelian group (the time translations group

T ). If we choose

0δΩ = for consttδ = , the condition (20.3.66) is reduced to

∂



(20.3.77)

that is the Lagrangian does not depend explicitly on time. Also, the equation of

conservation takes the form

∂

⎛⎞

−=

⎜⎟

⎝⎠



It follows that the generalized mechanical energy

∂

=−



(20.3.78)

corresponding to the relation of definition (18.2.68), is a conserved scalar quantity

(Jacobi’s first integral or a first integral of Jacobi’s type).

Reciprocally, if the conservation relation (20.3.77) holds, then we can write

dd d d

kkk

xxx

ttx tx x t

ααα

∂∂∂

⎛⎞⎛⎞

== −= + −=−

⎜⎟⎜⎟

⎝⎠⎝⎠







EL L LL

Here, we have used the relation

txx

αα

∂∂

=+ +





LLL

and Lagrange’s equations of motion. Hence, the condition (20.3.77) is not only a

sufficient condition, but also a necessary condition for the conservation of the

mechanical energy.

20.3.3.5 The Galileo Group. Theorem of the Mass Centre Motion

Let be two reference frames: one considered to be a fixed frame of reference and the

other moving uniformly along a straight line relative to the first one, with the velocity

123

(,, )vvv=v . The transformation connecting the co-ordinates of a particle in the

two reference frames is of the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

324

kkk

xxvt

′

=+,

1,2, 3k =

, tt

′

= , 0tδ = ,

(20.3.79)

where the primed co-ordinates correspond to the fixed frame of reference. The operator

associated to this transformation acts in the space of the vectors

()

123

(;) , , ;txxxt=r

and has the associated matrix

100

010

()

001

000 1

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

(20.3.80)

The set of transformations (20.3.79) forms an Abelian group with three parameters

(the components

123

,,vvv of the velocity), called the Galileo group (denoted here by

). The matrices (20.3.80) satisfy the relation

()()()+=tu v tutv,

where

u and v are two arbitrary velocities. Therefore, the set of matrices (20.3.80)

forms a representation of

Γ . The infinitesimal transformation corresponding to

(20.3.79) has the form

′

= ,

kk k

xxtvδ

′

kk k

xx vδ

′



1,2, 3k = ,

(20.3.81)

while the variations of the co-ordinates of the system of particles are given by the

relations

0tδ = ,

xtv

δδ= ,

δδ=



1,2, 3k = , 1,2,...,nα = .

(20.3.81')

With these notations, the condition (20.3.66) becomes

()

xx t

αα

∂∂

δδΩ

∂∂

⎛⎞

+=−

⎜⎟

⎝⎠

∑



;

(20.3.82)

assuming that (20.3.69) is satisfied (i.e., the generalized momentum is conserved), we

have

()

kkk

vPv

∂

δΩ δ δ

∂

=− =−

∑



(20.3.82')

and the right member has to be the total derivative with respect to time of a function of

co-ordinates

x .

Variational Principles. Canonical Transformations

325

The centre of mass of the discrete mechanical system has the co-ordinates

ξ = ,

1,2, 3k =

(20.3.83)

where

∑

is the mass of the mechanical system; therefore, taking into account the definition

(20.3.70), we obtain

kkk

MmxP

ξ ==



1, 2, 3k =

(20.3.83')

Hence, up to an additive constant, we may choose

MvδΩ ξ δ=− , (20.3.82'')

such that the equation of conservation (20.3.67) takes the form

tMv

∂

ξδ

∂

⎡⎛ ⎞ ⎤

−=

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦

∑



and, by the relation (20.3.70), we may write

()

[]

kkk

Pt M v

ξδ−=.

Because the variations

vδ are independent, we get

()

Pt M

ξ−=, 1,2, 3k = ,

const

Pt Mξ−=; the constant is determined by the conditions

()

kk k

Pt M Mξξ

−=−

1,2, 3k =

Finally, we obtain

ξξ=− ,

1,2, 3k =

(20.3.83'')

Taking into account the conservation of the generalized momentum, it follows that the

centre of mass of the considered mechanical system has a uniform rectilinear motion

with respect to both frames of reference. It is a characteristic of classical mechanics

MECHANICAL SYSTEMS, CLASSICAL MODELS

326

that, in case of a Lagrangian invariant with respect to space translations, not only the

generalized momentum is conserved but the equation (20.3.83'') holds too (the law of

motion of the centre of mass).

20.3.3.6 Lagrangians with Certain Symmetry Properties

We will focus now on implications of the reciprocal of Noether’s theorem in

classical mechanics. We restrict our considerations to two-particle interactions, thus

neglecting higher order interactions, and make use of the superposition principle.

Therefore, we assume that the Lagrangian is a sum of Lagrangians, some of them

associated with single particle motions, while the others correspond to two-particle

interactions, that is

()

111

,; ,,,;

nnn

kk k k

xxt xxxxt

ββ

αα α α

αβ

ααβ

===

∑∑∑

LL L ,

(20.3.84)

where

()

123

,,xxxx

ααα

= are the Cartesian co-ordinates of the particle P

, and

similarly for

. On this general form of the Lagrangian we impose the conditions

(20.3.69), (20.3.74) and (20.3.77), that is we want the Lagrangian be invariant with

respect to space translations, space rotations and time translations. In other words, we

consider that the space, in which we study the evolution of our discrete mechanical

system, is homogeneous (in both space and time variables) and isotropic. In fact, these

conditions represent restrictions with respect to the functional form of the Lagrangian in

the variables

xxt

αα



. Firstly, the condition (20.3.77) leads to

()

111

,,,,

nnn

kk k k

xx xxxx

ββ

αα α α

αβ

ααβ

===

∑∑∑

LL L .

(20.3.84')

Then, for every index

1,2, 3k = , the condition (20.3.69) can be regarded as a partial

differential equation, the solutions of this equation being the first integrals of the system

of differential equations (the Lagrange–Charpit sequence)

dd d

... d

11 1

kk k

xx x

λ====,

where

λ is an arbitrary parameter. Thus, we obtain (1)/2nn− first integrals of the

form

const

αβ β

ϕ =−= ,

αβ>

, 1,2,...,nαβ=

, of which only 1n − are

linearly independent. Consequently, if the Lagrangian is of the form (20.3.84'), then

L must be independent of

, while

αβ

L has to depend on

and

only by the

agency of the functions

αβ

ϕ . From the condition (20.3.74), it follows that the

Lagrangian has to depend only on the dot products

αα



()()

xxxx

ββ

αα

−−

()()

xxxx

ββ

αα

−−



()()

xxxx

ββ

αα

−−



, without

summation with respect to

α and β , which are invariant with respect to the space

rotations. Hence, this particular Lagrangian has to be of the form

Variational Principles. Canonical Transformations

327

()

()()()

111

nnn

kk k k kk

kk kk

Txx V x x x x xxxx

ββ ββ

αα α α αα

αβ

ααβ===

=− −−

∑∑∑

  L

(20.3.84'')

where we have introduced the notations

αα

and

αβ αβ

=−L

In classical mechanics, the equations of motion are invariant with respect to the

Galileo group (Galilean invariance of non-relativistic mechanics), such that, by

assuming that

αβ

does not depend on the velocities



and



, the condition

(20.3.82') becomes

()

∂

δΩ δ

∂

=−



Here,

Ω is a function of

, 1,2, 3k = , 1,2,...,nα = , and the right side is a total

derivative of a function of this kind, only if the functions

are proportional to

αα



without summation with respect to α. Finally, the Lagrangian which describes the

motion of a system of n particles, invariant with respect to the group of space

translations, the group of space rotations, the group of time translations and also

conform to the Galilean invariance, has to be of the form

()()()

111

nnn

kk k k

mxx V x x x x

ββ

αα α α

αβ

ααβ

===

=− −−

∑∑∑

L ,

(20.3.85)

where we have introduced the proportionality factors

/2m

, 1,2,...,nα = , to set up

the dependence of the functions T

αα



, without summation with respect to α;

actually,

is the mass of the particle P

, 1,2,...,nα = . In case of such a

Lagrangian, the generalized momentum of this particle has the components

pmx

αα

∂

==



1,2, 3k =

(20.3.86)

while the components of the vector associated to the generalized moment of momentum

of the particle are

ijk jk ijk k k

KKxp

αα

=∈ =∈ , 1,2, 3k = ,

(20.3.87)

both without summation with respect to

α.

20.3.3.7 Application of the Reciprocal of Noether’s Theorem

The example in the previous subsection shows, in a very explicit manner, the close

relation existing between the motion laws in classical mechanics, inferred directly from

the Lagrangian, and the geometrical properties of the space-time, which impose the

analytical form of the Lagrangian. Now, we will use the form (20.3.85) of the

MECHANICAL SYSTEMS, CLASSICAL MODELS

328

Lagrangian to illustrate how to find the symmetry group corresponding to a

conservation law, that is how to apply the reciprocal of Noether’s theorem.

To simplify the formulation, we will consider the motion of a single particle

described by the Lagrangian

()

222

123 123

mx x x Vx x x=++−

L ,

(20.3.88)

where

()Vx is the potential energy of the field of forces acting upon the particle. The

Hessian of this Lagrangian with respect to velocities and its inverse are given by the

identity matrix

E of rank three (abstraction of a multiplicative factor)

100

010

001

⎡⎤

⎢⎥

⎣⎦

100

010

001

−

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

(20.3.88)

Under the assumption that the momentum of the particle of position vector

r is

conserved, we obtain three first integrals (integrals of motion)

()

,const

Dmx==



rr ,

1, 2, 3k =

, (20.3.89)

and from (20.3.27') it follows

kjkkk

∂

δδτδτ

∂

−



]

(20.3.89')

without summation with respect to

1,2, 3k = . Hence, the corresponding infinitesimal

transformation with respect to which the Lagrangian remains invariant is of the form

kk k

xxδτ

′

1,2, 3k =

(20.3.90)

that is an infinitesimal transformation of the form (20.3.68). Consequently, the

conservation of the momentum leads to the invariance of the Lagrangian with respect to

the translations of the Euclidean space

E .

The conservation of the angular momentum provides three first integrals

()

kijk

Dmxx=∈





1,2, 3k =

(20.3.91)

The first integral

()



leads to variations of co-ordinates of the form

123

xxδδτ=−

213

xxδδτ=

0xδ =

and to the infinitesimal transformation

1123

xxxδτ

′

=−

2213

xxxδτ

′

= ,

Variational Principles. Canonical Transformations

329

respectively. In matric notation, the transformation takes the form

22 3

100 0 10

010 1 0 0

000

001

δτ

⎡⎤

′

⎡⎤

−

⎡⎤

⎢⎥

′

⎢⎥

′

⎣⎦

⎢⎥

⎣⎦

(20.3.92)

δτ

′



rErX

(20.3.92')

where

E is the identity matrix, and



is the generator of the group of transformations

associated to the first integral

; by the same method we obtain for the first integrals

()



and

()



the generators

00 0

00 1

01 0

⎡⎤

⎢⎥

=−

⎢⎥

⎣⎦



001

000

100

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

−

⎢

⎥

⎣

⎦



(20.3.93)

The product of the three transformations obtained above leads to an infinitesimal

transformation defined by the matrix

()

123

δτ δτ δτ δτ=+



uEX ,

(20.3.94)

which is a square, orthogonal non-singular matrix

SO(3, )∈ \u . The matrices



1,2, 3i = , satisfy the commutation relations

ij ji

ijk k

−=∈

  

XX XX X

, ,1,2,3ij= ,

(20.3.95)

and can be identified with the matrices of the generators of the

SO(3, )\ group in

exponential parameterization. Consequently, the transformation defined by matrices of

the form (20.3.93) form a group isomorphic to

SO(3, )\ , and the relations expressing

the conservation of the angular momentum lead to the invariance of the Lagrangian

with respect to the proper rotations of the Euclidean space

E . At the same time, the

property of invariance imposes a restriction on the form of the potential energy

()Vx

in (20.3.88). This function has to depend on the variables

,xx and

x in the form

()

222

123

Vxxx++

, the norm of the vector r (i.e.,

()

1/2 222

123

() xxx⋅=++rr )

being invariant with respect to the proper rotations of

E . Indeed, from the condition

(20.3.27) it follows that

()V r is a solution of the partial differential equations

MECHANICAL SYSTEMS, CLASSICAL MODELS

330

11 22 33

kkk

DDD

VVV

xx xx xx

∂∂∂

∂∂ ∂∂ ∂∂

++=



1, 2, 3k =

ijk

∂

∈=, 1,2, 3i = ,

which admits the general solution

()VVϕ= , where

xxϕ = .

The method described above can be generalized to systems of n particles because for

/0t∂∂=



and /0

Vx∂∂=



, in the case TV=−L , the function T is a

homogeneous function of second degree, respectively a positive definite quadratic

form, which can be reduced to its canonical form (i.e., a sum of squares) by means of a

point transformation. Hence, the Hessian of the transformed Lagrangian with respect to

the generalized velocities is a diagonal matrix, and the relations of conservation of the

generalized angular momentum lead to the generators of the

SO(3, )\ group.

We have shown in Sect. 20.3.3.4 that whenever the Lagrangian does not explicitly

depend on time (

/0t∂∂=L ) along a system trajectory (i.e., when Lagrange’s

equations of motion are satisfied), the generalized mechanical energy

(,)

∂

== −

∂



is conserved ( /0t∂∂=E ). In our case, the only independent variable is the time t, and

we consider an infinitesimal transformation of the form

tttδ

′

, where tδ is an

arbitrary infinitesimal function of class

C . If we choose 0δΩ = , then the equation of

conservation (20.3.67) becomes

d( )

ttx

∂

⎛⎞

−+ =

⎜⎟

∂

⎝⎠



which is identically satisfied for

consttδ = , that is the considered transformation is a

time translation. Note that, for time translations

() ( ) () 0

kk k

xt xt xtδ

′′

=−=

, 1, 2, 3k = ,

(20.3.96)

because

()

′′

and ()

xt have to represent the same point in the space of co-

ordinates. Consequently, we also have

xδ =



; it follows that the symmetry

transformation corresponding to the conservation of energy is the time translation for a

one-particle system

tttδ

′

, 0

xδ = ,

′



, 0

xδ =



1,2, 3k = .

(20.3.96')