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

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Variational Principles. Canonical Transformations

271

dd d d

llll

ll l l k k

jj jj

QQ PP

QP PQ q p

qp pq q p

∂∂∂∂

∂∂

∂∂ ∂∂ ∂ ∂

⎛⎞

−= − +

⎜⎟

⎝⎠



ll ll

jj jj

PPQQ

qp pq q p

∂∂∂∂

∂∂

∂∂ ∂ ∂ ∂ ∂

⎛⎞

−− +

⎜⎟

⎝⎠

[] [] [] []

,,d, ,d

jj j j

kkkkkk

jj j j

HH H H

qp qq q qp pp p

qp pq

∂∂ ∂ ∂

⎛⎞⎛ ⎞

=+ ++

⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

dddd

qpHHt

∂∂

=+=−



;

if HH= , then we are led to the canonical system (20.2.2) (the expression ddHHt−



is invariant by a complete canonical transformation). Thus, we verify once more that

(20.2.9') constitutes a sufficient condition for the considered transformations be

complete canonical, Hamilton’s function preserving its form also in the new canonical

co-ordinates.

Let us represent the Jacobian (20.2.1') as the determinant of a compound matrix

det

∂∂

⎡

⎡⎤⎡⎤⎤

⎢

⎢⎥⎢⎥⎥

⎣

⎦⎣ ⎦

⎢

⎥

⎢

⎥

⎡

⎤⎡ ⎤

⎢

⎥

⎢

⎥⎢ ⎥

⎣

⎣⎦⎣⎦⎦

, 1,2,...,

ks=

the index

j indicating the row and the index k the column in the four submatrices with

ss× elements. We will interchange the first row of submatrices with the second one,

and then the first column of submatrices with the second one; the determinant of the

new matrix will be as well

(with the same sign). We multiply then the submatrices of

the first row, as well as the submatrices of the first column, successively, by

1−

;

transposing this matrix, we obtain the determinant

TTT

det det

jj j j

kk k k

PP P Q

pq p p

∂∂ ∂ ∂

∂∂

∗

⎡

⎤

⎡⎡ ⎤ ⎡ ⎤⎤ ⎡ ⎤ ⎡ ⎤

−−

⎢

⎥

⎢⎢ ⎥ ⎢ ⎥⎥ ⎢ ⎥ ⎢ ⎥

⎣⎦⎣ ⎦ ⎣⎦⎣ ⎦

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎡⎤⎡⎤

−

⎢

⎥

−

⎢⎥

⎢⎥⎢⎥

⎣⎣ ⎦ ⎣ ⎦ ⎦

⎢

⎥

⎣

⎣⎦⎣⎦⎦

We notice that

∗

= , so that

[

]

[

]

[

]

[

]

[][][][]

det 1

qp qq

JJJ

pp qp

∗

⎡⎤

== =

⎢⎥

⎣⎦

(20.2.13)

where we took into account the conditions of complete canonicity (20.2.12); hence

1J =± . In case of a transformation of valency c , we have

Jc=± . Because the

Jacobian

J does not vanish, the condition (20.2.1') is always fulfilled in case of a

complete canonical transformation.

MECHANICAL SYSTEMS, CLASSICAL MODELS

272

Let

()

,FFqp= be a function of class

C . We construct the differential form of

first degree

() ()

,d ,d

jj j

PF Q QF Pϕ =− + ;

(20.2.14)

developing the Poisson brackets

()

QF,

()

PF, 1,2,...,js= , we obtain

[] [] [][]

,,d,,d

jj jj

kkkk kk

jj j j

FF FF

qp qq q pp qp p

qp qp

∂∂ ∂∂

∂∂ ∂ ∂

⎛⎞⎛ ⎞

=+ ++

⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

If the relations (20.2.12) take place, then the transformation

()()

,,QP qp→ is

complete canonical, the differential form being given by

dFω = , where F is an

arbitrary function of class

C . Taking into account (20.2.14), we have

()

∂

()

∂

=−

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

(20.2.14')

In particular, if

FH= , then we find Hamilton’s equations, written by means of the

Poisson brackets (equations of the form (19.1.60'))

()

QQH=



()

PPH=



1,2,...,

(20.2.14'')

If we equate

F to

Q and

P , successively, we obtain

()

QQ = ,

()

PP = ,

()

kjk

QP δ= , , 1,2,...,jk s= ;

(20.2.15)

these relations constitute sufficient conditions of complete canonicity, equivalent to the

conditions (20.2.12). Analogically, the conditions (

1c ≠ )

()

QQ = ,

()

PP = ,

()

kjk

QP cδ= , , 1,2,...,jk s= ,

(20.2.15')

represent necessary and sufficient conditions of complete canonicity.

Obviously, we may write also the sufficient conditions of complete canonicity

()

qq = ,

()

pp = ,

()

kjk

qp δ= , , 1,2,...,

ks= ,

(20.2.15'')

corresponding to the inverse complete canonical transformations. As well, if

1c ≠ ,

then we can write the necessary and sufficient conditions of complete canonicity

()

qq =

()

pp =

()

kjk

δ= ,

, 1,2,...,

ks=

(20.2.15''')

Let be the Pfaffian

Variational Principles. Canonical Transformations

273

dd dd d d

kk kk k kk k

kk ll l l

jj j j

ll ll

pq pq q qp p

qp QP QP

PPPQ P PQ P

∂∂∂∂∂ ∂∂∂

∂∂ ∂∂ ∂ ∂∂ ∂

⎛⎞⎛ ⎞

−= + − +

⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

[] []

,d ,d d d d

jjjj

lll k k

QP Q PP P Q q p

∂∂

=+==+

where we took into account the sufficient conditions of complete canonicity (20.2.12'');

analogically

ddd dd

kk kk

qpP qp

QQ q p

∂∂

∂∂ ∂∂

−=−=−−

By identification, we obtain the relations

∂

∂∂

∂

∂∂

=−

∂

∂∂

=−

∂

∂∂

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

(20.2.16)

In this case, we may write

()

kkllll

ll ll k k

QQpqqp

q p p q PP PP

∂∂

∂∂∂∂∂∂

∂∂ ∂ ∂ ∂∂ ∂∂

=−=−+

as well as other analogous relations; we find thus the relations between Poisson

brackets for the inverse complete canonical transformations (20.2.8) and Lagrange

brackets for the corresponding direct transformations

()

[

]

QQ PP=

()

[

]

PP QQ=

()

[

]

QP QP=

(20.2.16')

1c ≠ , then the relations (20.2.16) are of the form

∂

∂∂

∂

∂∂

=−

∂

∂∂

=−

∂

∂∂

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

(20.2.16'')

while the relations (20.2.16') become

()

[

]

QQ c PP=

()

[

]

PP c QQ=

()

[

]

QP cQP= , , 1,2,...,jk s= .

(20.2.16''')

Analogically, one can establish the relations

()

[]

qq pp

()

[]

pp qq

()

[]

qp qp

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

(20.2.16

)

MECHANICAL SYSTEMS, CLASSICAL MODELS

274

1c = , then we have

()

[

]

qq pp= ,

()

[

]

pp qq= ,

()

[

]

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

(20.2.16

)

Thus, starting from (20.2.12'') and (20.2.12), we find again the sufficient conditions

of complete canonicity (20.2.15) and (20.2.15''), respectively; as well, the conditions

(20.2.12''') and (20.2.12') allow us to write the necessary and sufficient conditions of

complete canonicity (20.2.15') and (20.2.15'''), respectively.

20.2.1.3 Canonical Transformations

In case of an arbitrary transformation (20.2.1), (20.2.1'), in which the time appears

explicitly, we consider the one-form

()

dd d

pq PQ HH tψ =−−−.

(20.2.17)

In this case too, a sufficient condition of canonicity imposes the differential form

(20.2.17) to be an exact differential, hence

dWψ = , (20.2.17')

where

()

,,, ;WWqpQPt=

is a function of class

; taking into account the

canonical form (20.1.41) of Hamilton’s principle, one can easily prove this affirmation.

0W = , then the transformation is homogeneous.

Introducing Pfaff’s forms of Hamilton (20.2.10), the necessary and sufficient

condition of canonicity will be

dcWΩω=− , (20.2.18)

where

c is a non-vanishing constant, while

()

,,, ;WWqpQPt= is a function of

class

C or

()

dd dd

ccpqPQcHHtWωΩ−= − − − = .

(20.2.18')

As a matter of fact, in this case we introduce the differential form

()

dd dd

cp q P Q cH H t Wψ =−−−=.

(20.2.18'')

For

∗

= , arbitrary but fixed, the condition (20.2.18') becomes

()

ddd,,,;

PQ cp q WqpQPt

∗

=− ;

(20.2.19)

this is a necessary and sufficient condition for the transformation

()

QQqpt

∗

= ,

()

PPqpt

∗

= , 1,2,...,

s= , be complete canonical. For the transformation

(20.2.1) be canonical, it is necessary to be canonical for any fixed

(the time is

considered as a parameter), with the same valency

c .

Variational Principles. Canonical Transformations

275

Supposing that the transformation (20.2.1) is canonical for any

t , it follows that we

have (20.2.19) for any

t . After transformation, let us define the Hamiltonian H by

HcHWPQ=++



;

(20.2.19')

we obtain thus

()

dd d

PQ t W t cH H t=− − −



Summing member by member with (20.2.19) (where the differentials

Q and dW

are not extended to the time

, which is fixed), we find again the condition (20.2.18').

Hence, the condition (20.2.18) is a necessary and sufficient condition of canonicity.

On the basis of the above affirmation, it results that also the conditions (20.2.12) are

sufficient conditions of canonicity, while the conditions (20.2.12') are not only sufficient

but also necessary for the canonicity of the transformation. One can then make

analogous affirmations for all conditions established at the previous subsection by

means of Lagrange’s or Poisson’s brackets. As well, the Jacobian (20.2.1') is non-zero

in the general case of canonical transformations too.

Let

()

qqqpt= ,

()

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

(20.2.20)

be the solution of the canonical equations (19.1.14), with the initial conditions

()

qqt= and

()

ppt= . According to Lagrange’s Theorem 19.1.4, the Lagrange

brackets

[

]

[

]

[

]

00 0 0 0 0

,, ,,,

jjj

kkk

qq pp qp

do not depend on time; for

tt= we have

[

]

qq = ,

[

]

pp = ,

[

]

kjk

qp δ= , , 1,2,...,

ks= .

Hence, the transformation

()

,,qp qp→ given by (20.2.20) is a univalent

canonical transformation.

More general, we can state that the values of the canonical co-ordinates at the

moment

t τ+ can be expressed as functions of the values of the same co-ordinates at

the moment

t in the form

() ()()

()

qt qqtptττ+= ,

() ()()

()

pt pqtptττ+= , 1,2,...,

s= ;

, (20.2.20')

hence, the variation of the canonical co-ordinates

and

during the motion can be

considered as a canonical transformation. We can state that the general integral of a

dynamical system, on one hand, and the canonical transformations of this system, on

the other hand, are two problems covering reciprocally themselves from the point of

view of their contents.

20.2.1.4 Point Transformations

A transformation of the form

()

QQq= ,

()

PPqp= ,

1,2,...,

, (20.2.21)

MECHANICAL SYSTEMS, CLASSICAL MODELS

276

with the functional determinant

()

det det det 0

qp q p

∂∂

∂

∂∂∂

⎡⎤⎡⎤

⎡⎤

== ≠

⎢⎥⎢⎥

⎢⎥

⎣⎦

⎣⎦⎣⎦

(20.2.21')

is called a point transformation; in this case, we also have

det 0

∂

⎡⎤

≠

⎢⎥

⎣⎦

det 0

∂

⎡⎤

≠

⎢⎥

⎣⎦

(20.2.21'')

The condition (20.2.9') of complete canonicity can be written in the form

dd d

jj j

kk k

pq PQ P q

∂

in case of a homogeneous transformation; hence, it is sufficient to have

∂

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

for the point transformation (20.2.21), (20.2.21') be completely canonical and

homogeneous. Multiplying both members by

qQ∂∂, ,1,2,...,kl s= , and summing,

we may state that the point transformation

()

QQq= ,

∂

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

(20.2.22)

linear with respect to the generalized momenta, is complete canonical and

homogeneous. We notice that, in the second group of relations, one introduces

()

QQq= ,

1,2,...,

, to have the transformations written in the form (20.2.21).

The condition (20.2.21') is, in this case, automatically fulfilled, as we have seen in Sect.

20.2.1.2.

We can write the relations of inverse transformation

()

qqQ= ,

∂

1,2,...,ks=

(20.2.22')

where we introduce

()

qqQ= in the second group of relations, to have the

transformation written in the form

()

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

Let us introduce also a more general transformation, of the form

(;)

QQqt= , (, ;)

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

(20.2.23)

where the time appears explicitly, and which verifies the condition (20.2.21') (implicitly

the conditions (20.2.21'')). The canonicity condition is

Variational Principles. Canonical Transformations

277

() ()

dd d dd d

jj j

kk k

pq PQ H H t P q Qt H H t

∂

⎛⎞

=+−= ++−

⎜⎟

⎝⎠



;

hence, it follows

∂

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

As in the previous case, we can state that the generalized point transformation

()

;

QQqt= ,

∂

, 1,2,...,

s= ,

(20.2.24)

linear with respect to the generalized momenta, is canonical and homogeneous; we

make

()

;

QQqt= in the second group of relations, to have transformations of the

form (20.2.23). We notice that the Hamiltonian is transformed in the form

HHPQ=+



(20.2.24')

The relations of inverse transformation are

()

;

qqQt= ,

∂

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

(20.2.25)

where we introduce

()

;

qqQt=

, 1,2,...,ks= , in the second group of relations; the

Hamiltonian is given by

HHpq=+



(20.2.25')

20.2.1.5 Free Canonical Transformations. Generating Functions of Canonical

Transformations

Let be a transformation (20.2.1), (20.2.1') for which the supplementary condition

det 0

∂

⎡⎤

≠

⎢⎥

⎣⎦

(20.2.26)

takes place; such a transformation is called a free transformation. In this case, on the

basis of the theorem of implicit functions, the first group of relations (20.2.1) allows us

to calculate

()

ppQqt= , and then

()

PPQqt= , , 1,2,...,

ks= ; as well,

the function

can be expressed in the form

()()

,; ,;Wqpt SQqt= . Thus, the

sufficient condition (20.2.17), (20.2.17') takes the form

()

dd d d dd

jj j

kk k

pq PQ H H t Q q St

∂∂

−−−= + +



;

MECHANICAL SYSTEMS, CLASSICAL MODELS

278

we obtain

∂

= ,

∂

=−

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

(20.2.27)

HHS=+



(20.2.28)

These relations define the canonical transformations (sufficient conditions). Indeed,

differentiating the first relation (20.2.27) with respect to time, we may write

222

kl l

lk lk k

SSS

pqQ

qq Qq tq

∂∂∂

∂∂ ∂ ∂ ∂∂

=+ +





;

inverting the differentiation order and using the relations (20.2.27), (20.2.28), we obtain

()

kll

kk k

pqQ HH

qq q

∂∂

∂

∂∂ ∂

=−+ −





We have

() ()()

,; , ,; ;HQPt HQPQqt t= and

()

,;Hqpt

()()

,,;;HqpQqt t= , so

that

()

klkklk

HHH

qPqqpq

∂∂

∂∂∂∂

∂ ∂∂∂∂∂

−= − −

;

we find the relation

kl l

kk l k l

HH H

pq Q

qq p q P

∂∂

∂∂ ∂

∂∂ ∂ ∂ ∂

⎛⎞

+= − − −

⎜⎟

⎝⎠





Analogically, the second relation (20.2.27) leads to

jj j

HHH

PQq

QQ PQ p

∂∂

∂∂∂

∂∂ ∂∂ ∂

⎛⎞

+=− − − −

⎜⎟

⎝⎠





Assuming that Hamilton’s equations (20.2.2) take place and taking into account

(20.2.26), the second group of relations leads to

qHp∂∂=



; replacing in the first

group of relations, we obtain the second group of Hamilton’s equations (19.1.14).

Hence, the representation (20.2.27) is sufficient to obtain a free canonical

transformation.

If the Hessian of the function

S is non-zero

det 0

∂

∂∂

⎡⎤

≠

⎢⎥

⎣⎦

(20.2.29)

then we obtain the generalized co-ordinates

from the first subsystem (20.2.27); by

replacing in the second subsystem (20.2.27), we are led to the complete determination

Variational Principles. Canonical Transformations

279

of the canonical co-ordinates

,QP

as functions of the co-ordinates ,qp and of the time

t , hence to the transformation (20.2.1). As well, starting from the condition (20.2.29)

we obtain the generalized co-ordinates

q from the second subsystem (20.2.27); by

replacing in the first subsystem (20.2.27), we get the inverse canonical transformation

(20.2.1''). The relation (20.2.28) puts into evidence the connection between Hamilton’s

functions corresponding to the two canonical systems. Hence, we may state

Theorem 20.2.1 (Jacobi). A function

()

,;SQqt

of class

which verifies the

condition (20.2.29) generates a free canonical transformation by means of the relations

(20.2.27).

Indeed, starting from the first subsystem (20.2.27), we obtain

()

QQqpt= ,

1,2,...,

s= ; this subsystem will be identically satisfied if we replace the generalized

co-ordinates

Q thus expressed. Differentiating with respect to

p , we obtain (we

observe that the function

S depends on

p only by means of the generalized co-

ordinates

Q )

kl l

Qq p p

∂

∂∂ ∂ ∂

The Hessian (20.2.29) does not vanish, hence the corresponding matrix is non-singular;

therefore, the matrix

[

]

Qp∂∂, which is the inverse of the previous one, is also non-

singular, the condition (20.2.26) being thus fulfilled.

The function

S is called the generating function of free canonical transformations.

If the Jacobian (20.2.26) is identically zero, then there exists at least one relation

connecting the generalized co-ordinates

,qQ and the time t . Let us assume that the

matrix

[

]

Qp∂∂ is of rank 1s − ; in this case, there is only one relation of the

above mentioned form, let be

()

,; 0QqtΩ = . The relations (20.2.27), (20.2.28) are

replaced by

∂∂Ω

∂∂

∂∂Ω

∂∂

=− −

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

(20.2.27')

HHSλΩ=++



(20.2.28')

where

λ is a non-determinate Lagrange’s multiplier. Analogically, if the matrix

[

]

Qp∂∂

is of rank sr− , then there exist r relations connecting the generalized

co-ordinates

,qQ and the time t , while the formulae of type (20.2.27'), (20.2.28')

contain

r Lagrange’s multipliers.

For example, a point transformation of the form (20.2.21), which is a homogeneous

complete canonical transformation, includes

s relations linking the generalized co-

ordinates

q to the generalized co-ordinates Q ; as a matter of fact, any homogeneous

complete canonical transformation includes at least such a relation. Indeed, the

condition of complete canonicity

pq PQ= implies s homogeneous equations

MECHANICAL SYSTEMS, CLASSICAL MODELS

280

∂

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

hence, to have non-zero solutions

P ,

[

]

det /

Qp∂∂ must vanish, so that

()

,0QqΩ = .

Let us put the problem to determine the generating function

S , which leads to a

special form of the Hamiltonian

H . For instance, if 0H = , then we have 0SH+=



where

()

12 1 2

, ; , ,..., , / , / ,..., / ;

HHqpt Hqq qSqSq Sqt∂∂∂∂ ∂∂== ; we find

thus again the Hamilton-Jacobi equation (19.2.9), the function

S being the

corresponding unknown function. We have seen in Sec. 20.2.1.1 that, in this case,

first integrals of the initial canonical system are determined; the two subsystems

(20.2.27) correspond to the two sequences (19.2.10), which give the solution of

Hamilton’s system in the Hamilton–Jacobi theorem. Hence, the problem to determine

the canonical transformation for which

0H = is equivalent to the integration of

Hamilton’s canonical system.

If we use the necessary and sufficient condition (20.2.18'), then the generating

function

()

,;SSQqt= leads to the representation

∂

=−

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

(20.2.27'')

with

HcHS=+



, 0c ≠ , constc = ,

(20.2.28'')

which, in general, is not univalent.

In case of a complete canonical transformation, the generating function

()

,SSQq=

does not depend explicitly on time (

0S =



) and the Hamiltonian is

transformed by the relation (20.2.10''). We notice that, in such a transformation, the

function

H does not change essentially; to can obtain a canonical transformation

having a simpler Hamiltonian, we must use a generating function

S which contains the

time explicitly.

For instance, let be the affine transformation

jjj

Qqpαβ=+,

jjj

Pqpγδ=+, 0αδ βγ−≠, (20.2.30)

where

,,,αβγδ are constants; one sees easily that, if c αδ βγ=−, then the necessary

and sufficient condition (20.2.11) is fulfilled. In this case,

()

jj jj jj

Vqqqpppαγ βγ βδ=− + +

()() ()

111

222

j j j j jj jj jj

qpqp cqp QPcqpαβγδ=− + + + =− − .

(20.2.30')

We notice that

[

]

det /

Qp∂∂ β= ; hence, if 0β ≠ , then the transformation is free

(as one can directly see from the first relation (20.2.30)). If

0βγ==, then we find