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

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Hamiltonian Mechanics

131

We introduce Routh’s function

=−

∑

αα







RL,

(19.1.30)

where “the cap” indicates the expression of the generalized velocities



1,2,...,rα =

, by means of Routh’s variables, so that = (, ,,;)

qpqqt

αα

RR ; it

results

==+

∂∂ ∂∂

⎛⎞

δ= δ+ δ + δ+ δ

⎜⎟

∂∂ ∂∂

⎝⎠

∑∑

qp qq

αα



RR RR

Identifying the above relations, where we take into account the notation (19.1.30), we

can write

∂∂ ∂

−= = =

∂∂ ∂

∂∂∂∂

−= −= =++

∂∂∂∂

, , 1,2,..., ,

, , 1, 2,..., .

jjjj

qq p

jr r s

qqqq

αα α

α



LR R

LRLR

Lagrange’s equations (18.2.38) led thus Routh, in 1876, to the equations

∂∂

==−=

∂∂

, , 1,2,...,qp r

αα

α

(19.1.31)

∂∂

⎛⎞

−==++

⎜⎟

∂∂

⎝⎠

0, 1, 2,...,

jr r s

tq q



(19.1.31')

called Routh’s equations (see Sect. 18.2.3.7 too); we remark that a part of these

equations are of Hamilton type (Routh’s function plays the rôle of Hamilton’s function)

and another part of Lagrange type (Routh’s function plays the rôle of Lagrange’s

kinetic potential), which justifies the name of “mixed method” given above.

In the case in which the generalized co-ordinates

=, 1,2,...,qr

, are ignorable we

find again the Routh–Helmholtz theorem (see Sect. 18.2.3.7 too). If the holonomic

constraints to which is subjected the mechanical system

S are also scleronomic, then

the kinetic energy is expressed in the form

∑∑ 12

(,,...,)

jk k

Tgqqqqq .

(19.1.31'')

We also assume that the generalized forces are quasi-conservative, deriving from a

simple quasi-potential

( , ,..., ; )

UUqq qt. In this case,

==+

∂

== + =

∂

∑∑

, 1,2,...,

pgqgqr

αβ β





(19.1.32)

MECHANICAL SYSTEMS, CLASSICAL MODELS

132

Because

[]g

αβ

is a square matrix of rank r and ≠det[ ] 0g

αβ

, corresponding to the

condition (18.2.34'''), we get

==+

=− =

∑∑

, 1,2,...,

qgp hq r

αβ

α,

(19.1.32')

where

αβ

is the normalized algebraic complement of the element

αβ

in det[ ]g

αβ

while

===++

∑

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

hgg rjrrs

αβ

α .

(19.1.32'')

Obviously, the coefficients

αβ

and

are functions only on the palpable co-

ordinates

=+ +, 1, 2,...,

qj r r s

. Observing that

∂∂∂∂∂

⎛⎞

== +

⎜⎟

∂∂ ∂ ∂ ∂ ∂

⎝⎠

∂∂ ∂

⎛⎞

=−

⎜⎟

∂∂ ∂

⎝⎠

∑

kk k

TTTq

qq q q q q

qhp

qq q

αα





    



 

there results

===

=− =− =++

∑∑∑

111

, , 1, 2,...,

rrr

jk jk k jk k

gg ghg gggjkrr s

αβ

αα

αβ

ααβ

. (19.1.33)

Analogically,

∂

∂∂∂ ∂

== =

∂∂ ∂ ∂ ∂ ∂

=δ= =

∑∑

∑

, , 1,2,..., ,

ggp

pp p q p p

gg r

γβ

αβ

ααγ α

ββ

γγ

γβ αβ

γα

αβ





∂

∂∂∂

∂∂ ∂ ∂ ∂

∂

====++

∂

∑

0, 1,2,..., , 1, 2,..., ,

qp q q p

gp rj r r s

αα

βα









so that we can write

= = =+ =+

∑∑ ∑ ∑

11 1 1

rr s s

jr kr

Tgpp gqq

αβ





(19.1.34)

We denote

Hamiltonian Mechanics

133

∗

=+ =+

∑∑

jr kr

Tgqq

(19.1.35)

Let us introduce Routh’s potential

∗

=−

∑∑

UU gpp

αβ

(19.1.36)

and the generalized potential

∗

=+ =

∑∑

Uhpq

Π 

(19.1.36')

Taking into account the relation of definition (19.1.30) and the generalized velocities

(19.1.32'), we obtain Routh’s function in the form

∗

=− +()T ΠR .

(19.1.37)

From (19.1.34), (19.1.35), and (19.1.36), it results

∗∗

−= −TU T U.

(19.1.38)

Lagrange’s equations (19.1.31') can be thus considered as being the equations of motion

of a reduced system, which admits

−R (given by (19.2.37)) as kinetic potential and

which has the same mechanical energy as the initial system.

If, in particular, the kinetic energy of the initial mechanical system does not contain

products of the generalized velocities

 of the ignorable co-ordinates, hence if

== =++0, 1,2,..., , 1, 2,...,

grjrrs

α , then the kinetic energy T is obtained as

a sum of two quadratic forms: a form in the generalized velocities

 and a form in the

generalized velocities

q ; such a mechanical system is a gyroscopic uncoupled one. In

this case,

== =++0, 1,2,..., , 1, 2,...,

hrjrrs

α , while

∗

= UΠ , as well,

∗

=+ =+

∑∑

jk k

jr kr

Tgqq

(19.1.35')

As it can be easily seen, if the initial system is conservative (

= 0U



), then the reduced

system has the same property (

∗

= 0U



If one has

∂∂=/0Uq

too, then it results ∂∂= =/ 0, 1,2,...,qr

αL , so that

the ignorable co-ordinates are hidden ones too. The motions in which vary only the

hidden co-ordinates are called hidden motions; such type of motion occurs, e.g., in case

of mechanical systems which contain gyroscopes. H. Hertz dealt in 1894 with these

problems, considering the potential energy of a conservative system as being the kinetic

energy of some hidden motions.

MECHANICAL SYSTEMS, CLASSICAL MODELS

134

19.1.1.8 Cyclic Co-ordinates

A generalized co-ordinate

which does not intervene explicitly in Hamilton’s

function (

∂∂=/0Hq

) is called, after Helmholtz, cyclic co-ordinate; let us admit

thus that the first

r co-ordinates are cyclic ( = 1,2,...,rα ). Taking into account the

first relation (19.1.12'), we can state that an ignorable co-ordinate (for which

∂∂=/0q

L ) is a cyclic one too. We are thus led to r first integrals

==,constpbb

ααα

, = 1,2,...,rα ; taking into account the results in the preceding

subsection, we can state that the cyclic co-ordinates do not intervene explicitly neither

in Routh’s function, so that the equations (19.1.31) lead to the same results as the

Routh–Helmholtz theorem (see Sect. 18.2.3.7). The Lagrange type equations (19.1.31)

remain to be integrated.

If we deal with the system of canonical equations, after the determination of the

mentioned

r first integrals, we remain with the system of equations (19.1.14) for

=+ +1, 2,...,jr r s (a system of −2( )sr partial differential equations of first order

with

−2( )sr unknown functions). Hamilton’s function will be of the form

++ ++

=≡

12 1 2 12

( , , ; ) ( , ,..., , , ,..., , , ,..., ; )

ssr

rr r r

HHqpbt Hq q qp p pbb bt

;

thus, by integration of the reduced canonical system, we get

=≡

12 12

( ; , , ) ( ; , ,..., , , ,..., )

jj j

qqtabb qtaa abb b

=≡

==++

12 12

( ; , , ) ( ; , ,..., , , ,..., ),

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

jj j

pptabb ptaa abb b

ab jk r r s

Replacing in Hamilton’s function, we obtain

=≡

12 12

( ; , , , ) ( ; , ,..., , , ,..., )

H Hta a b b H ta a a b b b

Introducing in the first

r equations of the first group of equation (19.1.14), we find

also the cyclic co-ordinates

∂

=+=

∂

∫

d , 1,2,...,

qta r

αα

α .

(19.1.39)

Hence,

=≡

12 12

( ; , , ) ( ; , ,..., , , ,..., )

qqtabb qtaaabbb

αααα α

, only the cyclic co-

ordinates depending on all the integration constant.

If, in particular, all the generalized co-ordinates are cyclic (

=rs), then all the

generalized momenta are constant, while

(; , ,..., )

HHtbb b; the cyclic co-ordina-

tes are obtained by

s quadratures, intervening the other s integration constants. If the

mechanical system is also scleronomic, then it results

∂∂= =/,const

jjj

Hb ωω ,

= 1,2,...,js, the generalized co-ordinates being of the form

=+ =,1,2,...,

jjj

qtaj sω ,

(19.1.39')

Hamiltonian Mechanics

135

which justifies the denomination of cyclic co-ordinates.

The problem of integration of Hamilton’s equations (19.1.14) is thus reduced to the

determination of a transformation of generalized co-ordinates to maintain the form of

the canonical equations, the new generalized co-ordinates being cyclic.

As we will see, the intervention of the one or more cyclic co-ordinates may lead to a

simplification of many methods of calculation.

19.1.1.9 The Pfaff Form of Hamilton. The Bilinear Covariant. Whittaker’s Equations

Let be the differential form

∑ 12

( , ,..., )d

Xxx x xω ,

(19.1.40)

where the functions

, 1,2,...,

Xi n= , of class

are definite on a certain domain in

\ ; we assume that ω is not a total differential. An expression of this form is called

Pfaffian (a Pfaff form).

Let us suppose that

( , ), 1,2,...,

xxuuj n, depending on two parameters.

Let be the differential quantities

∂∂

=δ=

∂∂

dd, d

xuxu

Noting that

() ()

∂∂

δ= δ =

∂∂ ∂∂

21 12

12 21

ddd,ddd

xuuxuu

uu uu

and assuming that

x are functions of class

in the two parameters, we have

δ=δd( ) (d )

xx, according to Schwartz’s theorem. If we denote

==δ

∑∑

ii ii

Xx Xxωω

(19.1.41)

and take into account the permutation relation of the differential operators, then we can

write

()

′

=δ=δ−=δ

∂

⎛⎞

=− =

⎜⎟

∂∂

⎝⎠

∑

d, d ,

d , 1,2,..., ,

xj n

ωω ω ω ω

(19.1.41')

obtaining thus the bilinear covariant of the form

ω .

Let us make the change of variables

=∈=

( , ,..., ), , 1,2,...,

jj j

xxyy yxCj n

(19.1.42)

MECHANICAL SYSTEMS, CLASSICAL MODELS

136

so that

∂∂≠det[ / ] 0

xy on a domain in

\ . Denoting

∂

∑

,1,2,...,

YXk n

(19.1.42')

we obtain

∑

Yyω ,

(19.1.43)

∂∂

⎛⎞

′

=δ = − =

⎜⎟

∂∂

⎝⎠

∑∑

, d , 1,2,...,

kk k l

yykn

ωωω .

(19.1.43')

The Pfaff form

ω and its bilinear covariant

′

ω maintain their form.

The necessary and sufficient conditions to have

′

≡ 0

ω for any δ

x , considered

arbitrary, are

==0, 1,2,...,

jnω , being thus led to the differential system associate

to the Pfaff form

∂

⎛⎞

−==

⎜⎟

∂∂

⎝⎠

∑

d0,1,2,...,

xj n

(19.1.44)

The transform of this system, by the change of variables (19.1.42), will – obviously –

∂∂

⎛⎞

−==

⎜⎟

∂∂

⎝⎠

∑

d0,1,2,...,

yk n

(19.1.44')

As it can be easily seen, if

ω is a total differential, then

′

= 0

ω ; consequently, we can

state that the Pfaff forms

ω and

∂

+=+

∂

∑

cFc x

ωω

where

= constc

, and =

( , ,..., )

FFxx x is a function of class

C , have the same

bilinear covariant.

By definition, the Pfaff form of Hamilton is

=−dd

pq Htω ,

(19.1.45)

where we have introduced the canonical variables

qp and the time t (hence,

=+21ns); the bilinear covariant of the form is

()

′

=δ − =δ −δ − δ − δ

∂∂∂

⎛⎞⎛⎞

=− + δ + − δ + − δ

⎜⎟⎜⎟

∂∂∂

⎝⎠⎝⎠

ddddd

dd dd dd.

jj jj

jjjj

pq Ht pq Ht

HHH

ptqqtpHtt

qpt

ωω ω

Hamiltonian Mechanics

137

The differential system of Hamilton associated to the Pfaff form is formed by the

canonical equations (19.1.14) and the differential outcome (19.1.25) of them. By

introducing the notion of exterior differential in the sense of Elie Cartan, the Pfaff form

of Hamilton can lead to a new differential principle of mechanics; choosing

conveniently the Hamiltonian, one can build up new mathematical models (e.g., the

invariantive model of O. Onicescu).

In case of a generalized conservative system (

= 0H



) one obtains the first integral

(19.1.26) of the generalized mechanical energy. Assuming that

∂∂≠

/0Hp (at least

one the components of the generalized momentum must be non-zero, otherwise the

motion of the mechanical system could not take place), the theorem of implicit

functions leads to

=−

11212

( , ,..., , , ,..., ; )

pKqqqppph. The corresponding Pfaff

form will be

=−−

∑

ddd

pq Kq htω ,

(19.1.46)

where

K plays the rôle of Hamilton’s function and

q the rôle of time; indeed,

according to a previous notice, the form

ω and the form + dhtω ( dht is an exact

differential) have the same bilinear covariant.

In this case, the associate differential system will be

∂∂ ∂

′′

==−= =

∂∂ ∂

, , 1,2,..., ,

KK KK

qp js

pq qq

(19.1.47)

where

′′

===

d /d, d /d, 1,2,3,...,

jj jj

qqqppqj s; these −2( 1)s equations of

Hamilton type are known as Whittaker’s equations. One can thus determine the

canonical co-ordinates

123 23

( ; , ,..., , , ,..., , ),

( ; , ,..., , , ,..., , ), 1,2,..., ;

qqqaa abb bh

ppqaa abb bhj s

(19.1.47')

replacing then in the function

K , we obtain =

123 23

( ; , ,..., , , ,..., , )

KKqaa abb bh,

hence the generalized momentum

p too. Finally, Hamilton’s equation =∂ ∂

/qHp

allows to put in evidence the dependence on time of the canonical co-ordinates,

introducing the last integration constant too. Besides, these considerations can be put in

connection to those made in Sect. 19.1.1.6.

Denoting

′

=−

∑

pq KL ,

(19.1.48)

where

′′ ′′′

12 23

( , ,..., , , ,..., )

qq qqq qLL , and effecting a Legendre transformation,

we can replace Whittaker’s equations (of Hamilton type) by Jacobi’s equations (of

Lagrange type) (see Sect. 18.2.3.5).

MECHANICAL SYSTEMS, CLASSICAL MODELS

138

19.1.2 Lagrange’s Brackets. Poisson’s Brackets

In what follows one introduces Lagrange’s and Poisson’s brackets, in connection to

the system of canonical equations; these brackets allow to determine first integrals of

the mentioned equations and put in evidence interesting properties of the integration

constant.

19.1.2.1 Lagrange’s Brackets. Lagrange’s Theorem

Let be the function

12 12

( , ,..., , , ,..., ),

( , ,..., , , ,..., ), 1,2,..., ,

yy i n

ξξ ξ ηη η

(19.1.49)

of class

C . We call Lagrange’s brackets (denoted by square brackets) the sums

[]

∂

∂∂ ∂∂

⎡⎤⎛ ⎞

==−

⎜⎟

⎢⎥

∂∂∂∂∂

⎣⎦⎝ ⎠

∂

∂∂ ∂∂

⎡⎤⎛ ⎞

==−

⎜⎟

⎢⎥

∂∂∂∂∂

⎣⎦⎝ ⎠

∂

∂∂ ∂

⎡⎤

=−

⎢⎥

∂∂∂

⎣⎦

∑∑

∑

(,)

,det ,

(, )

(,)

,det ,

(, )

(,)

det

(, )

ii ii

jjj

kkk

ii ii

jjj

kkk

xy xy

xy x

ξξ

ξξ ξ ξ ξ ξ

ηη

ηη η η η η

ξη ξ η

∂

⎛⎞

⎜⎟

∂∂

⎝⎠

∑

ηξ

(19.1.49')

for

=, 1,2,...,

kn; the denomination of brackets is given because these quantities are

sums of brackets. Lagrange’s brackets are, obviously, anticommutative (skew-

symmetric with respect to the indices

j and

)

[

]

[

]

[

]

[

]

[

]

[

]

=− =− −

,,,,,, ,

jjj j

kk kk

ξξ ξξ ηη ηη ξ ξ

ηη

(19.1.49'')

In quantum mechanics it is used to denote Lagrange’s brackets in the form

{, }

ξξ ,

{, }

ηη

{, }

ξη

Let be

= (; , )

qqtab, ==( ; , ), 1,2,...,

pptabj s, the integrals of the canonical

system (19.2.14), where we have put in evidence the integration constants

a and b ; the

corresponding Lagrange’s bracket is written in the form (the two constants play the rôle

of parameters)

[]

∂∂

∂∂ ∂∂

∂∂

==−

∂∂ ∂∂ ∂∂

∂∂

jjj

qp qp

qp ab ba

(19.1.50)

Let us calculate

Hamiltonian Mechanics

139

[]

∂∂ ∂∂ ∂∂ ∂∂

=+−−

∂∂ ∂∂ ∂∂ ∂∂

jjjjjjj

qp qp qp qp

tababbaba



where we assume that the two parameters are independent on t , so that

∂∂(d/d )( / )tc =∂∂(/ )(d/d)ct, where c can be a or b ; taking into account

Hamilton’s equations, we may write

∂

∂∂

∂∂∂∂∂∂∂

∂

∂∂

=− −

∂∂∂∂∂∂∂

cqpcppc

cqqcpqc



where, analogously, we make – successively –

c equal to a or

. We get

[]

∂∂ ∂∂

∂∂

⎛⎞⎛⎞

=−+−

⎜⎟⎜⎟

∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

⎝⎠⎝⎠

∂∂ ∂∂

∂∂

⎛⎞⎛⎞

+−+−

⎜⎟⎜⎟

∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

⎝⎠⎝⎠

jjj

kk kk

jj jj

kk kk

qq pp

tqqabbappabba

pp qq

qq pq

qp a b b a pq a b b a

Observing that the mixed derivatives of Hamilton’s function do not depend on the order

of differentiation (we assume that

∈

HC), it results that the first two sums vanish

(the derivatives of the function

are symmetric with respect to the indices

and j ,

while the brackets are skew-symmetric with respect to the same indices); if in the last

sums we replace the dummy indices

k and

and k , respectively, then we

remark that the last two sums vanish too. Hence,

=d[ , ]/ d 0ab t , so that

[

]

=,constab

;

(19.1.51)

we can state

Theorem 19.1.4 (Lagrange). Lagrange’s bracket corresponding to two integration

constants is constant along the trajectory of the representative point

P in the phase

space

19.1.2.2 Poisson’s Bracket. Properties

Let be the functions

12 12

( , ,..., , , ,..., ),

( , ,..., , , ,..., ), 1,2,..., ,

xx x yy y

xx x yy y i n

ξξ

ηη

(19.1.52)

of class

C . We call Poisson’s brackets (denoted by usual brackets) the sums

MECHANICAL SYSTEMS, CLASSICAL MODELS

140

()

∂∂∂

∂∂

⎡⎤

⎛⎞

==−

⎜⎟

⎢⎥

∂∂∂∂∂

⎝⎠

⎣⎦

∂∂∂

∂∂

⎡⎤

⎛⎞

==−

⎜⎟

⎢⎥

∂∂∂∂∂

⎝⎠

⎣⎦

∂∂∂

∂

⎡⎤

=−

⎢⎥

∂∂∂

⎣⎦

∑∑

∑

(, )

,det ,

(,)

(, )

,det ,

(,)

(, )

det

(,)

jjj

ii i i i i

jjj

ii i i i i

ii i i

xy x y y x

xy x y

ξξ ξ ξ

ξξ

ηη η η

ηη

ξη ξ ξ

∂

⎛⎞

⎜⎟

∂∂

⎝⎠

∑

(19.1.52')

for

=, 1,2,...,jk n; these brackets are anticommutative too (skew-symmetric with

respect to the indices

and k )

()()()()()()

=− =− −

,,,, ,, ,

jjj j

kk kk

ξξ ξξ ηη ηη ξ ξ

ηη

(19.1.52'')

In quantum mechanics, Poisson’s brackets are usually denoted in the form

[, ]

ξξ,

[, ]

ηη , [, ]

ξη .

Let us put in evidence the connection between Lagrange’s and Poisson ’s brackets in

a unitary form; we will consider

2n functions =

12 12

( , ,..., , , ,..., )

ffxx xyy y,

= 1,2,...,2jn, for which

∂

⎡⎤

≠

⎢⎥

∂

⎣⎦

12 2

12 12

( , ,..., )

det 0

( , ,..., , , ,..., )

ff f

xx x yy y

;

so that, in this case, we can calculate

12 2

( , ,..., )

xxff f, =

12 2

( , ,..., )

yyff f,

= 1,2,...,in. Observing that

∂∂

=δ =

∂∂ ∂∂

∂∂

==δ

∂∂ ∂∂

∂∂

⎛⎞

+=δ

⎜⎟

∂∂ ∂∂

⎝⎠

∑∑

∑

,0,

0, ,

ik ik

ji ji

ik ik

ji ji

xf xf

yf yf

xf yf

for

=, 1,2,...,jk n and =,1,2,...,2pq n, we obtain, easily,

[]

()

=δ =

∑

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

ij i

kjk

ff ff jk n

(19.1.53)

In detail, we can write