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

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Hamiltonian Mechanics

151

In this case too, the difficulty consists in expressing the generalized co-ordinates as

functions of the derivatives with respect to time of the generalized momenta, using the

second subsystem (19.1.13).

19.1.3 Applications

We consider, firstly, the motion of a single particle; we pass then to a Hamiltonian

treatment of a system of two or more particles, as well as to the case of a rigid solid or

of a system of rigid solids.

19.1.3.1 Motion of a Particle

In general, the generalized momentum is defined by the relation (18.2.80). In case of

a generalized quasi-potential intervenes the vector part in the form (18.2.81); to

simplify the presentation, we assume, in what follows, that the generalized forces derive

from a simple quasi-potential, the generalized momenta being thus given by the

relations (18.2.80'). We use the results obtained in Sect. 18.3.2.1.

In case of a single particle

123

(,, )Px x x , the representative point in

Γ is expressed

123 1 2 3

(,, , , , ), , 1,2,3

Pq q q p p p q x j , so that the kinetic energy is given by

=++

222

123

(/2)( )Tm xxx

, while the generalized momenta (equal to the usual

components of the momentum) are

==,1,2,3

pmxj

(19.1.79)

Hamilton’s function becomes (we use the formula of definition (19.1.11) with

(18.2.3.4))

=−

123

(,, ;)

HppUxxxt

(19.1.79')

while the canonical equations (19.1.14) read

∂

===

∂

,,1,2,3

jjj

xpp j



(19.1.79'')

In cylindrical co-ordinates (

===

123

,,qrq qzθ ), we obtain

== =

pmrpmrpmz

θ

(19.1.80)

Hamilton’s function being

()

=++−

222

(,,;)

HpppUrzt

θ ,

(19.1.80')

There result the canonical equations

MECHANICAL SYSTEMS, CLASSICAL MODELS

152

== =

∂∂∂

=+==

∂∂∂

111

,,,

,,.

rp pzp

UUU

pppp

θθ





(19.1.80'')

∂∂=∂∂=//0UUzθ , then θ and z are cyclic co-ordinates.

As well, in spherical co-ordinates (

===

123

,,qrq qθϕ) we can write

== =

222

,,sin

pmrpmrpmr

θθϕ



,

(19.1.81)

so that

()

=++ −

22 2

222

11 1

(,, ;)

sin

Hpp pUrt

θϕ

(19.1.81')

We get Hamilton’s equations

()

== =

∂

=+ +

∂

∂∂

=+=

∂∂

222

11 1

,, ,

sin

cot

sin

rp p p

mr mr

ppp

ϕϕ

θϕ

ϕϕ









(19.1.81'')

∂∂=/0U ϕ , then ϕ is a cyclic co-ordinate. We notice that, in case of a simple

potential of the form (18.3.46), we find a system of canonical equations equivalent to

Lagrange’s equations (18.3.47).

In case of a central force of Newtonian attraction we have

/, 0Ukrk=>, so that

∂∂=− ∂∂=∂∂=

//,/ /0Ur krU Uθϕ.

As a matter of fact, in case of a central force the trajectory is a plane curve; if the

intensity of this force is function only on the distance to the fixed pole, Hamilton’s

function will be given by (in polar co-ordinates, in the plane of the motion, the fixed

pole being the origin of the co-ordinate axes)

()

=+−= + −

222 2 2

() ()

HrrUr ppUr



(19.1.82)

There result Hamilton’s equations in

∂

== =+=

∂

11 1

,, ,0

rp pp p p

mr mr

θθθ





(19.1.82')

being a cyclic co-ordinate; we obtain

==,constpmCC

, and then =

rcθ



hence the first integral of areas. Eliminating the generalized momentum

and taking

into account the first integral previously obtain, we can write

Hamiltonian Mechanics

153

∂

1CU



(19.1.82'')

We find thus the equation of motion of the particle along the vector radius (see Sect.

8.1.1.1, formula (8.1.3)); the first integral

=Hh corresponds to the first integral (8.1.6').

In elliptical co-ordinates (see Sect. 18.3.2.1), the formula (18.3.41) leads to

===

111222333

444

mmm

pAqpAqpAq;

Hamilton’s function will be given by

⎛⎞

=++−

⎜⎟

⎝⎠

222

123

ppp

mA A A

(19.1.83)

with

123

(,, ;)UUqqqt, allowing thus to write the canonical equations in the form

== =

∂∂∂

⎛⎞

=++ =

⎜⎟

∂∂∂

⎝⎠

12 3

222

11 22 33

222

123

44 4

,, ,

,1,2,3.

jjj

pp p

qq q

mA mA mA

pA pA pA

mq q q

AAA

 



(19.1.83')

The generalized momenta

p , given by (19.1.79), are the components of the momen-

tum

=Hp; the components of the moment of momentum =×Krp

will be

=∈ = ∈ =,1,2,3

Oi ijk k ijk k

Kxpmxxj .

(19.1.84)

We can thus form Poisson’s brackets

()

∂∂

=−

∂∂ ∂∂

∂∈ ∂∈

=−

∂∂ ∂∂

=∈ ∈ −∈∈ =∈∈ +∈∈

=δδ−δδ =∈∈

∂

∂∈

∂

(, ) ,

() ()

() ( )

(, )

(

(, )

ii ii

nr nm

jlml jlml

knr knr

Oj Ok

ii ii

nm r r

rji

jlm kni jli kir l kli jli kri l

rk jl lk l ijk ilr l

xp px

xp xp

xp px

x p xp xp

xp xp

∂

∂∈

−

∂∂∂

=−δδ ∈ =−∈

)( )

lm l klm l

iii

il klm kjm

xp xp

ppx

MECHANICAL SYSTEMS, CLASSICAL MODELS

154

so that

==∈ =∈=(,)0,(,) ,(,) ,,1,2,3

jji

k Oi Ok ijk Oi Ok ijk

pp K K K pK pjk .

(19.1.85)

If some of the above quantities are conserved (are first integrals), then also the other

quantities have the same property (the Jacobi–Poisson theorem leads to new first

integrals). For instance, if

pK and

K are first integrals, then

,pp and

are first integrals too; other first integrals cannot be obtained, the cycle being closed.

If a particle

P is acted upon by an elastic central force of intensity in direct

proportion with the distance to the pole

O , then the potential U is given by

=− + + =

222

123

(/2)( ), constUkxxxk ; the second group of canonical equations

(19.1.79'') reads

=− =,1,2,3

pkxj .

(19.1.79''')

As it can be easily verified, (19.1.84) are three first integrals of the canonical equations.

The first equation of the first subsystem of equations (19.2.79'') and the first equation

(19.1.79''') (for

= 1j ) give the first integral

111

fmx p

(19.1.86)

As well, Poisson’s bracket

(,)

Kf leads to the first integral

21212

fppmxx

(19.1.86')

One observes easily that one cannot obtain new first integrals starting from the first

integrals (19.1.84), (19.1.86), and (19.1.86') and using the Jacobi–Poisson theorem;

indeed, if one could obtain such a first integral, then the functions

,, 1,2,3

qp j= ,

could be expressed by means of the six integration constants, the generalized momenta

being thus constant (which is not possible, because the motion would be annulled). A

first integral which (unlike the previous ones) contains explicitly the time is

=−=

31 1

cos sin ,

fmx tp t

ωω ωω .

(19.1.86'')

A partial differentiation with respect to time leads to a new first integral

==− −

43 1 1

sin cosff mx tp tωωωω



which is not distinct from the previous ones, because

⎛⎞

⎜⎟

⎝⎠

fkf

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155

As a matter of fact, the maximum number of first integrals in case of a free particle is

six.

19.1.3.2 Problem of Two Particles

Be means of the results in Sect. 18.3.2.3 and of the formulae (18.3.51), (18.3.51') for

the kinetic energy

T and for the simple potential U , respectively, we notice that we

can use a formula of the form (19.1.81'); there results Hamilton’s function

()

=++ −

22 2

222

11 1

sin

fm m

Hpp p

(19.1.87)

Hence, the canonical equations read

()

== =

=+ − = =

22 2

32 2

11 1

,, ,

sin

11 cot

,,0.

sin sin

rp p p

mr mr

fm m

ppp p pp

mr mr

ϕϕϕ

θθ

θϕ

θθ







(19.1.87')

We see that

ϕ is a cyclic co-ordinate, so that

= constp

is a first integral. We remain

only with four canonical equations, determining thus the functions

( ), ( ), ( ), ( )

rrt tp ptp pt

θθ

θθ== = =; we obtain then the generalized angular

velocity

ϕ

and, by a quadrature, we can calculate = ()tϕϕ too.

19.1.3.3 Motion of an Electrized Particle in an Electromagnetic Field

Starting from the results obtained in Sect. 18.3.2.5, we find the generalized momenta

, 1,2,3

iii

pmvqAi=+ = ; the relation of definition (19.1.11) leads to

()()()()

= −− − −− −+

11 1

ii i i i i i ii i

H p p qA p qA p qA qA p qA qA

mm m

where we took account (18.3.59). Finally, we can write Hamilton’s function in the form

()()

()

iiii ii ii ii

H pqApqAqA ppqpAqAAqA

=− −+= −+ +

(19.1.88)

One obtains thus the canonical equations

()

== − = − + =

,,1,2,3

ii i ii j jji

x p qA p p qA A qA i

ν 

(19.1.88')

19.1.3.4 Motion of a Discrete System of Particles

In case of a free discrete system of particles

P of masses =, 1,2,...,

mi n, the

space of configurations is just

E (we have = 3sn). The kinetic energy is given by

(we use the co-ordinates

()

, 1,2,3, 1,2,...,

xj i n== )

MECHANICAL SYSTEMS, CLASSICAL MODELS

156

()()()

⎡

⎤

=++

⎢

⎥

⎣

⎦

∑

222

() () ()

123

iii

Tmx x x.

(19.1.89)

Assuming, for the sake of simplicity, that the given forces derive from a simple

quasi-potential

()

(;)

UUx t

, we obtain Hamilton’s function

=−HTU

Noting that

() ()

, 1,2,3, 1,2,...,

pmxj i n===



, it results

()()()

⎡

⎤

=++

⎢

⎥

⎣

⎦

∑

222

() () ()

123

iii

Hppp

(19.1.89')

The canonical equations are written in the form

∂

====

∂

() () ()

()

, , 1,2,3, 1,2,...,

iii

jjj

xpp j i n



(19.1.90)

Introducing the components of the momentum of the mechanical system

∑

()

,1,2,3

Hpl ,

(19.1.91)

and the components of the moment of momentum of the same system

=∈ =

∑

() ()

,1,2,3

Ol jkl

Kxpl,

(19.1.91')

where we use the summation convention of the dummy indices for

=,1,2,3

k , we

can calculate Poisson’s brackets as in Sect. 19.1.3.1; we obtain thus the relations

()

==∈

=∈ =

,0,, ,

,,,1,2,3,

kOjOkjklOl

Ok jkl l

HH K K K

HK Hjk

(19.1.92)

analogues to the relations (19.1.85). Obviously, we are led to analogous conclusions in

what concerns eventual first integrals too.

In particular, one can obtain all six first integrals previously considered in the case in

which the mechanical system is isolated, hence it is subjected only to the action of

internal forces

, , , 1,2,...,

ijij n≠=F ; the torsor of these forces is, obviously,

equal to zero

∑∑

0 ,

(19.1.93)

=×=

∑∑

MrF

ij ij

0 .

(19.1.93')

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157

Mayer considered, in 1878, internal forces with a more general character that the

Newtonian ones (definite in Sects. 1.1.1.11; see. 2.2.2.8 too), hence forces which verify

only the global conditions (19.1.93), (19.1.93'), hence the conditions of equilibrium of a

non-deformable mechanical system (as the mechanical system would be a rigid one at a

given moment). In this context, we can state (we denote

, , 1,2,...,

ij j i

ij n=− =rrr )

Theorem 19.1.10 (Mayer). The most general quasi-conservative internal forces which

derive from a generalized quasi-potential

U and verify the condition (19.1.93) are

those for which

−−

= rr r rr r

12 13 1, 12 13 1,

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

nn nn

UU t

 

(19.1.94)

Theorem 19.1.10' (Mayer). The most general quasi-conservative internal forces which

derive from a generalized quasi-potential

U and verify the condition (19.1.93) are

those for which

−

= rr r rr r r r r

12 12 1213 1,

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

UU t

 

(19.1.94')

From both theorems taken together, we get

Theorem 19.1.11 (Mayer). The most general quasi-conservative internal forces which

derive from a generalized quasi-potential

and the torsor of which vanishes (verifies

the conditions (19.1.93), (19.1.93')) are those for which the quasi-potential is of the

form (19.1.94).

The motion of the mass centre is rectilinear and uniform and the theorem of areas

can be applied in projection on the three planes of co-ordinates. We can also state

Theorem 19.1.12 (Mayer). The necessary and sufficient condition of existence of the

first integral of the mechanical energy, hence of the existence of a function of co-

ordinates, velocities and time the total derivative with respect to time of which be

⋅

∑∑

ij i

= 0U



; the internal forces must derive from a generalized potential which does not

contain explicitly the time. The first integral is of the form

∂

−+ = =

∂

∑

()

,const

TU v hh

(19.1.95)

If the potential

U verifies the conditions of the Theorem 19.1.10 (is of the form

(19.1.94) with

= 0U



), then the first integral reads

∂

−+ = =

∂

∑∑

,const

TU r hh



(19.1.95')

MECHANICAL SYSTEMS, CLASSICAL MODELS

158

19.1.3.5 Motion of the Rigid Solid

In case of a rigid solid S , free or subjected to constraints, we can start from the

considerations made in Sect. 18.3.2.6. For instance, in case of the motion of a heavy

homogeneous rigid solid of rotation, of mass

M , which slides without friction on a

fixed horizontal plane, we start from the kinetic energy (18.3.62) and from the potential

energy (18.3.62'); the mechanical system being scleronomic and observing that

′′

== =+

′

=++ = +

⎡

⎤

⎣

⎦

1122 3

,,(cos),

sin ( cos )cos , ( ) ,

pMpMp I

pJ I p Mf J

ψθ

ρρ ϕψθ

ψθ ϕψθ θ θ θ



 



we can express Hamilton’s function in the form

()

cos

().

() sin

HTU Mgf

Mf J J

θθ

−

⎤

⎡

′

=−= + + + +

⎥

⎢

′

⎣

⎦

(19.1.96)

Hamilton’s equations read

()

()()

−

′′

===−

−

′

=== =

′′′

′

⎡⎤

⎣⎦

′

+− −−

cos cos

,, ,

sin

cos

sin ( )

0, 0, 0, 0,

() ()

()

cos cos ( ) .

sin

MMI

JMfJ

ppp p

Mf f

Mf J

pp p p Mgf

ϕϕ

ψψ

θθ

ρρϕ

ψθ

θθ

θθ θ





 



(19.1.96')

We notice that

′′

,,ξξϕ and ψ are cyclic co-ordinates, four of the generalized

momenta being thus constant; as well,

=Hh is the fifth first integral of the canonical

system.

In case of the plane motion of a rigid straight bar, we consider the formulation of the

problem in Sect. 18.3.2.7. The kinetic energy is given by (18.3.64) and the potential of

the forces is expressed by (18.3.64'), so that we can define the generalized momenta in

the form

′′

===

1122

,,pMpMpI

ρρθ





, obtaining Hamilton’s function

() ()

⎡⎤

′′

=−= + + + +

⎢⎥

⎣⎦

22 2 2 2

12 2

11 1

sin

HTU p p p kM I

ρθ.

(19.1.97)

We can write the canonical equation in the form

Hamiltonian Mechanics

159

′′

===

′

==− =−

1122

12 2

111

,,,

0, , sin cos .

ppp

MMI

ppkMpkI

ρρθ

ρθθ





 

(19.1.97')

We notice that

′

ρ is a cyclic co-ordinate, the corresponding generalized momentum

being constant; the constraints are scleronomic, so that Hamilton’s function will be a

first integral of this system of equations too.

19.1.3.6 Double Pendulum

The problem of the double pendulum has been treated in Sect. 18.3.2.8. In the frame

of Lagrangian mechanics. The generalized co-ordinates being

θ and

θ , the

corresponding momenta are

=+ + −

11211222 21

22222221 21

() cos(),

() cos().

pIMI MH

θθθθ



(19.1.98)

Starting from these results, we can express Hamilton’s function

′

=−HTU

as a

function on the canonical co-ordinates, obtaining then Hamilton’s equations.

19.1.3.7 Sympathetic Pendulums

Let us take again the case of the sympathetic pendulums, considered in Sect. 18.3.2.9

in the frame of Lagrangian mechanics, Let be two identical physical pendulums, each

one of mass

M , of mass centres

C and

C , respectively, so that ==

11 22

OC OC l,

where

O and

O , are the poles through which pass the respective axes of suspension;

Q and

Q are on

OC and

OC , respectively, with ==

11 22

OQ OQ a, then we

consider that the two pendulum are linked by the elastic spring

QQ of elastic constant

k (see Fig. 18.13 too).

Choosing as generalized co-ordinates the angles

and

made by

and

, respectively, with the vertical line, the formulae (18.3.69) and (18.3.69') lead to

()

()()

=+=−+ ++

22 222 2

12 12 12

TI UMgI MgIka ka

θθ θθ θθ



(19.1.99)

Having to do with a scleronomic system, the generalized momenta will be

pIθ



pIθ



, and Hamilton’s function reads

() ()()

=+−++ +−

22 222 2

12 12 12

HppMgIMgIka ka

θθ θθ.

(19.1.99')

There result Hamilton’s canonical equations

MECHANICAL SYSTEMS, CLASSICAL MODELS

160

=− + − =− + −

1122

11212212

(), ().

p MgI ka p MgI ka

θθ

θθθ θθθ



(19.1.99'')

Eliminating the generalized momenta, we obtain Lagrange’s equations

()()

++ −+ =

−+ ++ =

12 12 2

22 12 2

20,

20.

θωωθωωθ



(19.1.100)

with the notation (18.3.71); we find thus again the solutions (18.3.72).

19.2 The Hamilton–Jacobi method

After some results with a general character, including the Hamilton-Jacobi theorem,

one considers the problem of the systems of equations with separate variables, putting

in evidence Stäckel’s studies; some applications for the motion of a particle or of a

system of two particles are then given.

19.2.1 General Results

To can obtain first integrals of the system of 2s canonical equations, one establishes

the Hamilton–Jacobi theorem, which allows to build up the solution of that system, in

the hypothesis of the determination of the complete integral of the corresponding partial

differential equation; then some properties concerning the integration constants are put

in evidence and one deals with the particular case of the scleronomic constraints, as

well as with the cases which admit cyclic co-ordinates. One makes also other

considerations on the Hamilton–Jacobi method (Hamilton, W.R., 1890; Jacobi, C.G.J.,

1882; Nordheim, L. and Fues, E., 1927; Pars, L., 1965).

19.2.1.1 The Hamilton-Jacobi Partial Differential Equation

Let be a function

()

12 12 1

, ,..., ; ; , ,..., ,

SSqq qtaa aa ,

(19.2.1)

which depends on

s generalized co-ordinates, on time and on

+ 1s

arbitrary constants

(corresponding to the

+ 1s

independent variables); we calculate the partial derivatives

()

∂

≡=

∂

12 12 1

, ,..., ; ; , ,..., ,

Spqqqtaaaa



(19.2.1')

()

∂

12 12 1

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

pqq qtaa aa j s

(19.2.1'')

In this case, + 1s of the + 2s relations (19.2.1), (19.2.1'), and (19.2.1'') allow to

determine the constants

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

aj s , as functions of ,, ,

qtSS



and