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

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Variational Principles. Canonical Transformations

231

one can obtain in this case a general integral principle, corresponding to asynchronous

varied paths with fixed ends (

() ()

ttΔΔ==rr0,

1,2,...,in=

) in the form

()

2d0

WT PtT tt

ΔΔ Δ Δ

⎡⎤

++ − + =

⎢⎥

⎣⎦

∫

(20.1.35')

As it can be easily seen, in the above calculus an important rôle is played by the

operator

ΔδΔ=+ ,

(20.1.36)

which is applied for the position vector

r and the velocity

v , as well as for all the

quantities of energetical nature or of the nature of an action which occur (even the

relation (20.1.30') can be obtained by applying this operator). As well, the operator can

be used to pass from a synchronous to an asynchronous form of an integral or of a

variational principle. The permutation relation of the form (20.1.5') does no more take

place in this case; one must use the transitivity relation (see relation (20.1.22))

()

dddd

tttt

ΔΔ Δ−= .

(20.1.36')

20.1.3 Hamilton’s Principle

We will present, in what follows, Hamilton’s, Hölder’s and Voss’s principles,

corresponding to synchronous or asynchronous variations of true trajectories,

respectively; some methods of calculus are thus put into evidence.

20.1.3.1 Case of Synchronous Variations

We can write

WU Uδδδ

=∇⋅ =

∑

(20.1.37)

in case of quasi-conservative forces and of synchronous variations; taking into account

the definition relation of the kinetic potential (

TU=+L ), the general integral

principle (20.1.35) leads to

tδ =

∫

L .

(20.1.38)

Using the permutability property (20.1.3'') between the operator

and the integral, we

have

tδδ==

∫

AL.

(20.1.38')

MECHANICAL SYSTEMS, CLASSICAL MODELS

232

We can thus state that along the varied paths of a discrete mechanical system subjected

to ideal constraints and to quasi-conservative given forces, the Lagrangian action

is stationary (necessary condition) in case of synchronous variations; we notice that the

mechanical system must be holonomic, corresponding to this type of variations.

Let us assume that the motion of the mechanical system is described by means of

generalized co-ordinates, so that

()

12 12

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

qq qqq qt=

 

LL ; in this case,

starting from the relation (20.1.38) and using the results of Sect. 20.1.1.2 (Euler–

Lagrange equations (20.1.7'')), we find again Lagrange’s equations of second kind

(18.2.38).

If the given forces do not derive from a quasi-potential, then

∂

⋅=

∂

∑

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

and, starting from (20.1.35), we obtain (we use again the results of Sect. 20.1.1.1 and

the summation convention for dummy indices)

Qqt

qtq

∂∂

⎡⎛⎞⎤

−+ =

⎜⎟

⎢⎥

∂∂

⎣⎝⎠⎦

∫



;

(20.1.39)

applying the basic lemma of Lagrange, we find again the equations (18.2.29). In case of

non-holonomic constraints, we use Lagrange’s multipliers method as in Sect. 20.1.1.4;

there results (we assume to have the constraint relations (18.2.10))

kkj

Qaqt

qtq

λδ

∂∂

⎡⎛⎞ ⎤

−++ =

⎜⎟

⎢⎥

∂∂

⎣⎝⎠ ⎦

∑

∫



(20.1.39')

By the same reasoning as in the above mentioned subsection, we find again Lagrange’s

equations with multipliers (18.2.32), (18.2.33).

Hence, the relation (20.1.38') is sufficient (not only necessary) to specify the motion

in the frame of the mentioned conditions. So, we state

Theorem 20.1.5 (W.R. Hamilton). Among all possible motions of a mechanical system

subjected to holonomic and ideal constraints and acted upon by quasi-conservative

given forces on synchronous varied paths with fixed ends, only and only the motion (of

the representative point on the true curve) for which the Lagrangian action is

stationary takes place.

We mention that the most times – not always – the Lagrangian action realizes a

minimum for the actual motion (on the true curve).

We have obtained this result in the form of a theorem, starting from the principle of

the virtual work; but we can interpret this result also as a variational principle

(Hamilton’s principle), which leads to Lagrange’s equations of motion. This principle

was stated in 1834 by W.R. Hamilton for scleronomic constraints; it was extended by

M.V. Ostrogradskiĭ in 1848 to the case of rheonomous constraints. One observes that,

unlike differential principles (in which, to establish the motion at a given moment, one

considers only the motion in a vicinity of this one), in case of variational principles the

Variational Principles. Canonical Transformations

233

motion of the mechanical system at a given moment is specified by its motion in the

whole (finite) interval of time.

Often, Hamilton’s principle is considered to be the basic principle of mechanics;

indeed, classical mechanics as well as the relativistic one can be deduced starting from

such a principle. As well, the theory of the electromagnetic field can be obtained from

an analogous principle, as other theories of physical nature. Hence, the importance of

those principles consists also in their expression in a unified form, corresponding to the

tendency of extremum of phenomena in the nature. So, the variational principles of

Hamilton’s type lead to “equations of motion” which can be Newton’s, Maxwell’s or

Schrödinger’s equations; using a variational principle as basis of formulation, all these

domains of physical nature have, in a certain measure, a structural analogy. If

experimental results impose the change of physical matter in the theory corresponding

to a certain domain, then this analogy can show similar changes in another domain. For

instance, experiments made at the beginning of this century have put into evidence the

necessity to quantify the electromagnetic radiation as well as the elementary particles;

indeed, starting from the development of quantification methods for the mechanics of

particles, one could construct a quantum electrodynamics. Although its importance

from the point of view of the philosophy of mechanical systems, we must notice that

Hamilton’s principle can – sometimes – lead to more intricate calculations from a

practical point of view as a direct study based on Newton’s equation.

If the given forces are not quasi-conservative, then one cannot introduce the function

L , and if the mechanical system is not holonomic, then the operator δ cannot permute

with the integral. In this case, in general, the varied paths are no more possible curves,

and we must use the relation (20.1.35); the problem is no more a problem of variational

calculus.

We observe that we may distinguish between an integral principle and a variational

one, the first one having a more general character. Indeed, by convention, an integral

principle is deduced from the general integral principle (20.1.35), without the

intervention of the variation of a functional (the variation of an action), this aspect

appearing only under the integral sign; if the variation of a functional occurs, then we

have to do with a variational principle. Perhaps, this distinction is conventional;

however, we must mention that a variational principle is linked to a problem of

extremum of a functional, allowing thus to use a rich methodology of calculus. If the

constraints are non-holonomic but the given forces are quasi-conservative, then

Hamilton’s principle is an integral one and has the form (20.1.38) (it is true that the

virtual displacements are not independent), while if the constraints are holonomic, the

given forces being quasi-conservative or not, then we find again the general integral

principle (20.1.35), which – in fact – represents a more general integral form of

Hamilton’s principle in a synchronous case. Thus, as one can see, Hamilton’s principle

has a sufficient general character, even in the synchronous case.

However, the difference (given by the non-holonomic or holonomic character of the

ideal constraints) between the forms (20.1.38) and (20.1.38') of Hamilton’s principle is

clearly put in evidence.

We notice also that, in case of quasi-conservative forces deriving from a generalized

potential, the virtual work of the given forces can be expressed in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

234

ii jj j

WQq q

qtq

δδδ δ

∂∂

⎡⎛⎞⎤

=⋅= =−

⎜⎟

⎢⎥

∂∂

⎣⎝⎠⎦

∑



ddd

jj j

jjj j

UUU U

qqqUq

qtq qt tq

δδδδδ

∂∂∂ ∂

⎛⎞ ⎛⎞

=− + =−

⎜⎟ ⎜⎟

∂∂∂ ∂

⎝⎠ ⎝⎠

 

;

in case of synchronous variations, for which the operator relation (20.1.5') is valid, the

general integral principle becomes

()

UTt q

δδ δ

∂

∫



Further, assuming variations with fixed ends (

() ()

qt qtδδ==

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

there results that Hamilton’s variational principle maintains its form for any natural

system (we can permute the operator

δ with the integral sign).

The property of extremum expressed by Hamilton’s principle is an intrinsic property;

by way of consequence, it can be expressed in any system of co-ordinates. We will thus

pass from the co-ordinates

q ,

uq=



, 1,2,...,js= , in the space

Λ , to the co-

ordinates

q ,

p , 1,2,...,

s= , in the space

Γ ; a Legendre transformation permits it

(see Sects. 19.1.1.2 and 20.1.1.3). But we must have some care, because the variations

qδ and

uδ are not independent; indeed, to given variations

qδ there correspond

variations of the generalized velocities of the form

δδ= , 1,2,...,

s= .

(20.1.40)

In this case, let us consider the Lagrangian action for which

()

12 1 2

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

qq quu ut=LL ; corresponding to Hamilton’s theorem, this

action must be stationary along the curves satisfying the differential equations

qu=



, 1,2,...,js= . (20.1.40')

The co-ordinates

q are fixed at the ends, while the co-ordinates

u do not possess this

property. Because we deal only with arcs of curve which verify the differential

equations (20.1.40'), the variations being synchronous, it is sure that the conditions

(20.1.40) are verified. Using the method of Lagrange’s multipliers, we obtain

()

[]

12 12

, ,..., , , ,..., ; d 0

jj j

qq quu ut q u tδλ+− =

∫



L ,

where the functions

()

tλλ= , 1,2,...,js= , remain to be determined. The

equations of the extremal curves (the Euler–Lagrange equations) will be given by

∂

−=

∂

−+=

∂

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

Variational Principles. Canonical Transformations

235

obviously, by eliminating the multipliers

λ , we find again Lagrange’s classical

equations.

We denote

()

12 1 2

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

Hqq q p p p t u

∂

=−

∂

where the linear transformation /

pu=∂ ∂L , 1,2,...,js= , allows to pass from the

2s-dimensional space of co-ordinates

q ,

u ,

1,2,...,

, to the phase space

Γ ,

the cap specifying the fact that the transformation has been effected (as in Sect.

19.1.1.3); we assume that the condition

det 0

∂

⎡⎤

≠

⎢⎥

∂∂

⎣⎦

is verified. We can thus state

Theorem 20.1.6 (G.H. Livens). The stationarity condition of Theorem 20.1.5

(Hamilton’s theorem) is written in the form

()

[]

12 1 2

, ,..., , , ,..., ; d 0

pq H q q q p p p t t

δδ=− =

∫



A ,

(20.1.41)

in the phase space.

The form taken by Hamilton’s principle in the phase space (given by Livens in 1919)

is called also the canonical form of Hamilton’s principle.

We notice that

() ()

11 1

00 0

dd d

tt t

jj jj j j j j j j

tt t

pq H t pq H t p q q p q p t

δδ δδδδ

∂∂

⎛⎞

−= −= +− −

⎜⎟

∂∂

⎝⎠

∫∫ ∫



jjjjjj

qppqtpq

δδδ

∂∂

⎡⎛ ⎞ ⎛ ⎞ ⎤

=− −+ +=

⎜⎟⎜⎟

⎢⎥

∂∂

⎣⎝ ⎠ ⎝ ⎠ ⎦

∫



where we took into account the operator relation (20.1.5'). In the hypothesis of fixed

ends (for generalized co-ordinates) the integrated part vanishes. Applying the basic

lemma of the variational calculus and observing that the variations

qδ ,

pδ ,

1,2,...,js= , are arbitrary, we find again Hamilton’s canonical equations (19.1.14);

however, if we write Euler-Lagrange equations for (20.1.41), hence the equations

()

∂

⎛⎞

−− =

⎜⎟

∂

⎝⎠

, 0

∂

⎛⎞

−−− =

⎜⎟

∂

⎝⎠



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

we find again the same equations (19.1.14).

The time and the generalized co-ordinates are fixed at the ends of the interval

(

() ()

qt qtδδ==), but the generalized momenta do not verify a similar

MECHANICAL SYSTEMS, CLASSICAL MODELS

236

property (

()

ptδ ≠ ,

()

ptδ ≠ ); indeed, the conditions at the ends (bilocal) for

p do not furnish any information, taking into account that the derivatives



do not

appear in the integrand expression.

Of course, the two integrals which occur in (20.1.38') and (20.1.41) have the same

value in the actual motion. But Hilbert observed that, in the formulation (20.1.38'), one

can attain a minimum, while in the formulation (20.1.41) – for the same problem – it is

possible to obtain neither a maximum nor a minimum (hence, a saddle point).

We can put in evidence this observation by a simple example: the rectilinear motion

of a particle subjected to its own weight (a field of uniform acceleration

g ), for which

/2xgt vt=+

0t = , (0) 0x = ,

()

11 1

/2x t gt vt=+

. In this case,

/2mx mgx=+



L , while

/2Hp mmgx=−. Let us compare the actual motion

with a motion on a varied path, given by

()

/2xgt vt tα=++,

()

tCα ∈ ,

()

00tαα==; we have

(

)

xmgxtδ+= +

∫



()

vgt mg gt vt tαα

⎡⎤

=+++++

⎢⎥

⎣⎦

∫

 .

There results

()

mv gt mg tδα αα

⎡

⎤

=+++

⎣

⎦

∫



()

tmvgt tααα=++=

∫∫

;

we state that

0δ >

, excepting the case in which 0α =



, hence

()

consttα = , so

that

()

0tα ≡ (taking into account the conditions at the ends of the interval). Hence,

the motion on the true curve corresponds to a minimum of the Lagrangian action. Using

the canonical form of Hamilton’s principle, we can calculate

()

222

p mgx px t p mx x mgx t

⎡

⎤

=− + + =− − + +

⎢

⎥

⎣

⎦

∫∫



A .

For the actual motion we have

/2xgt vt=+,

()

pmvgt=+, while for the varied

motion we can take

()

/2xgt vt tα=++,

()

pmvgt tβ=++,

() ()

,ttCαβ∈ ,

()

00tαα==, without any conditions for

()

tβ at the ends

of the interval. Analogously, we obtain

()

δαα

⎡

⎤

=−−

⎢

⎥

⎣

⎦

∫



A .

We observe that

δ >

A for mβα=



and α non-identical zero, while 0

δ <

for

0α ≡ and β non-identical zero; the functional

is stationary on the true curve

and has neither a maximum, nor a minimum (saddle point).

Variational Principles. Canonical Transformations

237

Let us remember that we have introduced the Lagrangian action in Sect. 19.2.1.6 in

the form of a function

()St which verifies the Hamilton-Jacobi equation, as one can

see by a total differentiation with respect to time; starting from this function, one has

obtained Hamilton’s principal function. So, we can write the relation

()

12 1 2

d d , ,..., , , ,..., ; d

Spq Hqq qpp ptt=−

which corresponds to Hamilton’s Pfaff form (19.1.45).

20.1.3.2 Case of Asynchronous Variations. Hölder’s Principle. Voss’s Principle.

Another Form of Hamilton’s Principle

In case of asynchronous variations, occurs the time variation tΔ as a function of

time. If

0ttΔΔ== it follows that the initial and the final times are the same, the

duration of the motion being also the same on all varied paths; besides, we can make

the latter affirmation also in the more general case in which

0ttΔΔ=≠. We

mention that these conditions are sufficiently restrictive.

Corresponding to the results of Sect. 20.1.1.4, we can write an identity of the form

qtttq

qt q

ΔΔΔΔ

∂∂

⎛⎞

+−=

⎜⎟

∂∂

⎝⎠

∫







qtq

∂∂

⎡⎛⎞⎤

+−

⎜⎟

⎢⎥

∂∂

⎣⎝⎠⎦

∫



(20.1.42)

for Lagrange’s kinetic potential

()

12 12

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

qq qqq qt=

 

LL , where

asynchronous variations are put into evidence. Along the integral curves of Lagrange’s

equations there results

qtttq

qt q

ΔΔΔΔ

∂∂

⎛⎞

+−=

⎜⎟

∂∂

⎝⎠

∫







;

(20.1.42')

in case of varied curves with fixed ends (

() ()

qt qtΔΔ==) and in case of

catastatic constraints (the set of possible displacements coincides with the set of virtual

displacements), we can state

Theorem 20.1.7 (Hölder). Among all possible motions of a mechanical system

subjected to holonomic, catastatic, ideal constraints, and to quasi-conservative given

forces on asynchronous varied paths with fixed ends, only and only the motion (of the

representative point on the true curve) for which one has the relation

qttt

ΔΔΔ

∂

⎛⎞

+−=

⎜⎟

∂

⎝⎠

∫



(20.1.42'')

takes place.

Indeed, in this case, the possible displacements

qΔ

, 1,2,...,js= , are independent,

and we find again Lagrange’s equations; but, in case of non-holonomous constraints

MECHANICAL SYSTEMS, CLASSICAL MODELS

238

one can use the method of multipliers, finding again Lagrange’s equations with

multipliers. The constraints are catastatic, so that



r0,

1,2,...,in=

, for all particles

of the mechanical system; so, a particular configuration of this system is represented by

the same point in the space

Λ , for all values of the time t . As in the case of the

Theorem 20.1.5, the results remain valid for all quasi-conservative given forces which

derive from a generalized quasi-potential. This theorem can be used also as a principle:

Hölder’s principle, stated in 1896; obviously, this principle is an integral one. If

0tΔ = , then we find again Hamilton’s principle.

Using the results of Sect. 20.1.1.4, we can calculate the asynchronous variation of

the Lagrangian action in the form

jjj

tqt qt

qqtq

ΔΔ ΔΔ δ

∂∂∂

⎡⎤⎡⎛⎞⎤

==−+−

⎜⎟

⎢⎥⎢ ⎥

∂∂∂

⎣⎦⎣⎝⎠⎦

∫∫



LLL

AL E

(20.1.43)

where

E is the generalized mechanical energy

∂

=−

∂



(20.1.43')

Assuming that we have to do with varied paths with fixed ends and that

0ttΔΔ==, there results

qtq

Δδ

∂∂

⎡⎛⎞⎤

=−

⎜⎟

⎢⎥

∂∂

⎣⎝⎠⎦

∫



(20.1.43'')

We are thus led to

Theorem 20.1.8 (A. Voss). Among all possible motions of a mechanical system

subjected to ideal constraints, acted upon by quasi-conservative given forces, on

asynchronous varied paths with fixed ends (in space and time), only and only the

motion (of the representative point on the true curve) which makes stationary the

Lagrangian action

tΔΔ==

∫

(20.1.44)

takes place.

This result is valid in case of holonomic constraints (one obtains Lagrange’s

equations), as well as in case of non-holonomic ones (one obtains Lagrange’s equations

with multipliers); it remains valid also in case of quasi-conservative forces which derive

from a generalized quasi-potential.

As above, one can consider this theorem to be a new variational principle: Voss’s

principle (1900). It is an extension of Hamilton’s principle in case of asynchronous

variations; besides, one can easily obtain this result, taking into account the relation

(20.1.30'), which links the asynchronous to the synchronous variation of an action, and

making

0ttΔΔ==.

Variational Principles. Canonical Transformations

239

Along the extremal curves to which lead Lagrange’s equations, the relation

ΔΔΔ

∂

⎡

⎤

=−

⎢

⎥

∂

⎣

⎦



AE,

(20.1.43''')

corresponding to the Poincaré-Cartan integral invariant (see Sect. 21.1.2.4) takes

place. We mention also the connection between this relation and the relation (19.2.28')

concerning Hamilton’s principal function. As well, in case of varied paths with fixed

ends, we can write

tΔΔ+=

AE .

(20.1.43

)

Comparing Hölder’s and Voss’s principles, we see that the second one is a

variational principle (an advantage), and that the mechanical system is not subjected to

catastatic constraints; in compensation, the varied paths must be with fixed ends also for

time, a sufficient severe restriction, not implied by Hölder’s principle. On the other

hand, in the corresponding demonstration the principle of virtual work was not used,

but only the connection with Lagrange’s equations; thus, one must not have

TU=+L (as it was supposed, mentioning the action of quasi-conservative given

forces), but we can consider mechanical systems which admit a Lagrangian

L which

verifies the condition (18.2.34''').

Starting form the general integral principle (20.1.35'), which corresponds to

asynchronous variations with fixed ends and imposing catastatic constraints

(

d/dTtP= ), we can write

()

2d0

WTT tt

ΔΔ Δ++ =

∫

(20.1.45)

If the given forces are quasi-conservative, then we have

WUUtΔΔ Δ=−



, so that we

obtain (

TU=+L )

()

2d0

TtUtt

ΔΔΔ+−=

∫



L .

(20.1.45')

Assuming that the quasi-conservative forces derive from a simple quasi-potential and

observing that, in case of catastatic constraints, we have

qq TT

∂∂

===

∂∂



, 0T =



we find again Hölder’s principle.

If the given forces are quasi-conservative, then the relation (20.1.35') becomes

(

WUUtΔΔ Δ=−



, TU=+L and ETU=−)

()

11 1

00 0

ddd()0

tt t

tPUttEt

ΔΔΔ+−−+ =

∫∫ ∫



L ;

MECHANICAL SYSTEMS, CLASSICAL MODELS

240

integrating by parts, we can write also

()

11 1

00 0

ddd0

tt t

tPttUttEt

ΔΔΔΔ+−−++=

∫∫ ∫



L .

In case of catastatic constraints we have

dddd d dETUWUUt=−= −=−



, so

that

tEtΔΔ+=

∫

L ;

(20.1.46)

we find thus a variational principle of Hamilton type, in the same conditions

0ttΔΔ==. In fact, this is Voss’s principle, stated for TU=+L in the

restricted frame of catastatic constraints, case in which the generalized mechanical

energy is reduced to the mechanical energy (

E=E ).

The condition

0ttΔΔ== is very inconvenient. To eliminate it, we can start

from the relation (20.1.43

), which can be written also in the form

()()

ht htΔΔ++− =

∫

LE ,

where

h is constant with respect to time. If 0=



L , then we obtain Jacobi’s first

integral (or of Jacobi type) h=E ; so we can state

Theorem 20.1.9 Among all possible motions of a generalized conservative mechanical

system, subjected to ideal constraints, on asynchronous varied paths with fixed ends,

only and only the motion (of the representative point on the true curve) for which

()

htΔ +=

∫

L ,

consth =

(20.1.47)

takes place.

Also this result can be used as a principle; in fact, it is a generalization of Hamilton’s

principle for which there are not imposed restrictions concerning time variations (at the

ends of the time interval).

Eh==E in case of catastatic constraints, we can state the same result for a

conservative mechanical system; starting from the relation (20.1.46), one obtains the

same principle. As well, one can obtain the same result using Hölder’s principle.

20.1.3.3 Changes of Independent Variable. Conformal Mappings

As we have seen, one of the important properties of the variational principles is the

possibility to use any system of co-ordinates; the generalized form (20.1.47) of

Hamilton’s principle, corresponding to a generalized conservative mechanical system,

allows to make a step further, i.e. to introduce a change of independent variable. Let be

ddtuθ= , (20.1.48)