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

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MECHANICAL SYSTEMS, CLASSICAL MODELS

392

∂∂

⎛⎞

′

δ=−= δ+δ

⎜⎟

′

∂∂

⎝⎠

∫

() () d

CC x xu

αα

ΛΛ Λ

αα

ΛΛ

AA A .

Assuming that

δ=δ=

||0

uu uu

αα

and observing that the variations δx

=+1,2,..., 1sα , are independent, it results that the variational equation

′

δ=δ =

∫

(, )d 0

xx u

ΛA

(21.4.5)

is equivalent to Euler–Lagrange equations

∂∂

⎛⎞

−== +

⎜⎟

′

∂∂

⎝⎠

0, 1,2,..., 1

tx x

αα

ΛΛ

(21.4.6)

Obviously, the principle (21.4.5) is equivalent to Hamilton’s principle (20.1.38'), while

Euler–Lagrange equations (21.4.6) are equivalent to Lagrange’s equations (18.2.38); it

is true that the system of equations (21.4.6) has an equation more than the system of

equations (18.2.38'), but between the first mentioned equations takes place the linear

relation

∂∂ ∂ ∂ ∂

⎡⎛ ⎞ ⎤ ⎛ ⎞

′′′′′

−= − − =−=

⎜⎟ ⎜ ⎟

⎢⎥

′′′

∂∂ ∂ ∂ ∂

⎣⎝ ⎠ ⎦ ⎝ ⎠

dd dd

xxxx

ux x u x x x u u

αααα

αα α α α

ΛΛ Λ Λ ΛΛΛ

which justifies the affirmation thus made. Taking into account that the equations

(18.2.38'), for which the values

q and

q at the moment

t are known, lead to a

unique solution, we can say, corresponding to the affirmation made above, that through

a point

P of the space (,)qt passes only one trajectory of direction specified by

+12 1

d , d ,...,d

xx x.

We call

Λ -dynamic the theory based on the variational equation (21.4.5) and on the

extremal equations (21.4.6); as well, the

α -dynamics will be the theory based on

Hamilton’s principle (20.1.38') and on the extremal equations (18.2.38). These

dynamics are equivalent, the

L -dynamics being a form of the Λ -dynamics, in which

the time

t is at the same time a co-ordinate in the space (,)qt and a parameter on a

curve in this space; as a matter of fact, both dynamics are Lagrangian dynamics.

Instead to impose a Lagrangian

′

(, )xxΛ

, we can impose an equation of energetic

nature

=(,) 0xyΩ ,

(21.4.7)

which links the co-ordinates

of a point in the space (,)qt to the components y

of an

associate

+(1)s -dimensional vector; this equation can be seen – from a geometric point of

view – as associating to each point of

(,)qt a s -dimensional hypersurface in a

+(1)s -dimensional space, tangent to (,)qt , y

being co-ordinates in this tangent space.

It is convenient – sometimes – to solve the equation (21.4.7) with respect to

+1s

y , so that

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

393

112 112

( , ,..., , , ,..., ) 0

yxxxyyyω .

(21.4.7')

If we define by

, 1,2,..., ,

ypj sy H

== =−

(21.4.8)

the generalized momenta and Hamilton’s function, respectively, we find a relation of

the form

12 1 2

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

Hqqqtpppω ,

(21.4.7'')

hence of the form

= (;; )HHqtp,

(21.4.7''')

between the

+22s quantities ,, ,

qtpH. Obviously, one obtains thus the canonical

form of Hamilton’s principle, which reads

δ=δ = =

∫

d0,(,)0yx xy

αα

ΩA ,

(21.4.9)

in the co-ordinates

,xy

, the integration being effected along a curve

with fixed

ends; as well, the extremal curves will be given by the canonical equations

∂∂

==−

∂∂

wyw x

αα

ΩΩ

(21.4.10)

where

dw is an infinitesimal multiplier of Lagrange. A curve = ()xxw

αα

in (,)qt

and the associate vector field

= ()yyw

αα

will form a trajectory.

21.4.1.2 Description of the Motion of a Mechanical System in the

Momentum-Energy Space (, )pH

Being situated in the +(1)s -dimensional space (, )pH , which is called the

momentum-energy space, we use the co-ordinates

, definite by the relations (21.4.8);

the energetic equation (21.4.7) leads us to the energy (21.4.7''). For a curve

C of

equations

==+( ), 1,2,..., 1yyu s

αα

α , in (, )pH , we define a new type of action by

the integral

∫

αα



(21.4.11)

where ()xu

are arbitrary functions, excepting (21.4.7); we can write

()

=−

∫

qp tH



(21.4.11')

MECHANICAL SYSTEMS, CLASSICAL MODELS

394

too. By variation of the curve

C we obtain

()

δ=δ + δ −δ

∫

xy xy yx

αα α α α α



In case of varied curves with fixed ends

(δ = δ =

||0)

uu uu

αα

, we get the

variational equation

δ=δ =

∫

αα



(21.4.12)

which, together with the condition (21.4.7), leads to the same canonical equations

(21.4.10).

It is obvious that the dynamics in the space

(, )pH , based on the equation (21.4.7)

and on the variational equation (21.4.12), is equivalent to the dynamics in the space

(,)qt , based on the same equation (21.4.7) and on Hamilton’s principle (21.4.9) for

varied curves with fixed ends. The two actions which intervene are linked by the

relation

+= + = −

∫∫

ddyx xy xy xy

αα

αα αα αα



(21.4.13)

where

,xy

αα

are referring to the initial moment =

tt. In the space (,)qt , x

are the

co-ordinates of the representative point, while

are the components of an associate

field of vectors; these rôles are inverted in the space

(, )pH . Thus, between the two

formalisms there exists a formal duality. For instance, we could use the technics in the

preceding subsection to define a homogeneous Lagrangian in

(, )pH , as a function of

the arguments

and d / d , 1,2,..., 1yyu s

αα

′

==+, passing then to a classical

Lagrangian, function of

, pH

and d/dpH

In general, the space

(, )pH has a secondary importance. It is useful if, e.g., the

particles of a discrete mechanical system, at the initial moment in free motion, begin to

interact, returning then to a free motion. When the particles are moving freely (before

and after interaction), the representative point

P maintains itself a fixed position in

(, )pH , the interaction having as consequence the motion of this point from a fixed

position to another fixed one.

21.4.2 Formalism in Spaces with +21s or with +22s Dimensions

Starting from the results in the preceding subsection, we consider the

+(2 1)s -dimensional space (,, )qtp and the +(2 2)s -dimensional space (,, , )qtpH ;

by this occasion we put in evidence also the utility of such formalisms.

21.4.2.1 Description of the Motion of a Mechanical System in the Space of States

(,, )qtp

The most used representative space is perhaps, the phase space (, )qp . The totality of the

trajectories, in this space, appears as a congruence of curves, so that, in case of given

conservative forces, through each point passes only one curve; but the problem becomes

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

395

more intricate in case of given non-conservative forces, because through each point passes a

simple infinity of trajectories. In the space

(,, )qtp , which we call the space of states, the

time

is considered the same as the generalized co-ordinates and the generalized momenta;

Hamilton’s function

(,, )Hqtp is a function of position in this space. If the given forces

are non-conservative, then the image of the trajectories is more simple that in the space

(, )qp , because through each point passes one curve of this congruence. We also notice

that, from a mathematical point of view, this

+(2 1)s -dimensional space is the only

representative space which has always an odd dimension.

The canonical equations of motion are (19.1.14); for an equal treatment of all the

co-ordinates, these equations – written in the form (19.1.19) – put in evidence the

natural congruence of the trajectories in the space

(,, )qtp , through each point passing

only one curve. On the other hand, these equations imply the energetic condition

(19.1.25). We define the circulation on a curve

C by

()

κ= δ−δ

∫

()

CpqHt

;

(21.4.14)

giving an infinitesimal displacement (not necessarily along the natural congruence) and

integrating by parts, we can write

()

κ= δ−δ−δ+δ

∂∂∂

⎡⎛ ⎞ ⎛ ⎞ ⎤

=δ + +δ−+ +δ−+

⎜⎟⎜ ⎟

⎢⎥

∂∂∂

⎣⎝ ⎠ ⎝ ⎠ ⎦

∫

d( ) d d d d

dd dd dd.

jj jj

jj j j

CpqqpHttH

HHH

qp t p q t tH t

qpt

If the displacement takes place along the natural congruence, then the equations

(19.1.14), (19.1.25) lead to

=d( ) 0Cκ .

(21.4.15)

On the other hand, if the relation (21.4.15) takes place for an arbitrary curve

C and for

a displacement along an arbitrary congruence, then this congruence must satisfy the

equations (19.1.19), (19.1.25), being thus a natural congruence. As a matter of fact, the

condition (21.4.15), which must be verified by the circulation, is equivalent to the

canonical equations (19.1.14).

Let be the transformation of co-ordinates

→(, ,) ()qpt x , so that

=== =

, , 1,2,..., ,

jjsj j

xqx pj sx t;

(21.4.16)

generically, we denote such a co-ordinate in the form

x , where the capital letters take

values form

+1to2 1s in the frame of a convention of summation. In this case, we

can write

δ− δ= δ(;; )

pq Hqtpt X x

(21.4.17)

where

MECHANICAL SYSTEMS, CLASSICAL MODELS

396

=== =−

, 0, 1,2,..., , ( ; ; )

jjsj

XpX j sX Hqtp;

(21.4.16')

it results

κ= δ

∫

()

CXx

(21.4.17')

wherefrom

()

κ= δ−δ

∫

d( ) d d

AA A A

CXxxX

Denoting

∂∂=

AAB

XxX

, we can also write

()

κ= δ−δ= − δ

∫∫

,,,

d( ) d d d

BB B

AB A A AB BA A

CXxxxx XXxx

;

is along the natural congruence, then the relation (21.4.15) takes place for an

arbitrary curve

C . Hence, the trajectories must satisfy the equations

()

−==+

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

AB BA

XXx A s.

(21.4.18)

Because

−

,,AB BA

XX are the components of an antisymmetric tensor, while d

x are

the components of a vector, we can affirm that the equations (21.4.18) are vector

equations; these equations are satisfied for the co-ordinates (21.4.16), (21.4.16'), so that

they are satisfied in any system of co-ordinates.

One can build up thus a new mathematical formalism of mechanics, starting from the

Pfaff form

; this form is determined, for the same trajectories, excepting an

exact differential.

21.4.2.2 Description of the Motion of a Mechanical System in the Space of States

and Energy (,, , )qtpH

The +(2 2)s -dimensional representative space (,, , )qtpH , which we call the space

of states and energy, allows the most general formalism for the dynamics of a

mechanical system; the time and the energy

H , in this space, are treated as the

generalized co-ordinates and the generalized momenta, obtaining thus formally a

complete symmetry. The

+22s co-ordinates can be divided in two groups: the group

(,)qt and the group (, )pH , corresponding to the space of events and to the

momentum-energy space, respectively, the rôle of which can be – in general – inverted.

It is convenient to use an equation of energy which imply all the

+22s co-ordinates of

the space

(,, , )qtpH to can preserve the symmetry; this equation defines a

+(2 1)s -dimensional hypersurface in the space of states and energy, while the

representative point

must be on this surface. It is often better to use a function of

energy instead of an equation of energy, to can imply the whole space

(,, , )qtpH , not

only the mentioned hypersurface. The space of states and energy corresponds very well

to Hamilton’s optical method; all the theories developed in this representative space can

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

397

be transferred, in an isoenergetical dynamics, in the phase space, by a convenient

reduction of dimensions.

Let be the co-ordinates

and y

, =+1,2,..., 1sα , called conjugate, which have

been previously introduced. To a energy function

(,)xyΩ which we consider, there

corresponds only one energy surface

=(,) 0xyΩ ; to a given energy surface there

corresponds an infinity of energy functions. By means of the variational formalism

(21.4.9) or of the variational formalism (21.4.12), we may obtain canonical equations of

the form (21.4.10). The solutions of the system (21.4.10) fill up the space

(,, , )qtpH

by a natural congruence of trajectories, one for each point. Thus, the totality of the

dynamical trajectories, including those on the energy surface, presents a geometric

image much more simple than that of the space

(,)qt , where there exists only one

trajectory which passes through a point in a given direction.

One takes again – in the representative space

(,, , )qtpH – the whole theory of the

space

(,)qt , but where each quantity and each mathematical or mechanical fact has a

richer significance.

21.4.3 Notions on the Inverse Problem of Mechanics and the

Birkhoffian Formalism

In what follows, we present firstly the inverse problem of the Newtonian mechanics;

is a larger sense, in connection to this problem, we consider some notions concerning

the Birkhoffian formalism too.

21.4.3.1 Inverse Problem of Newtonian Mechanics

In the formulation of the direct problem of Newtonian mechanics, starting from the

motion of a given mechanical system, one obtains a function of Lagrange, which leads

to the equations of motion in one of the representative spaces considered in the previous

sections; these equations are the extremal equations of an action which can be definite

corresponding to the space in which we are situated and to the used formalism.

Starting from these considerations, we can pass to the formulation of the inverse

problem of Newtonian mechanics. Thus, giving the equations of motion, the problem of

existence (including the conditions in which such a thing can take place) of one or

several functions of Lagrange is put, so that the associate Euler–Lagrange equations do

coincide with the given equations of motion or be equivalent to them. Obviously, in this

case, the problem of the effective determination of these Lagrangians is put.

Once, it has been considered that only the natural systems (especially the

conservative system) admit a variational principle, which leads to the associate Euler–

Lagrange equations, equivalent to the Newtonian equations. As well, it has been shown

that, being given a Lagrangian

L , any other Lagrangian =+

′

d/dtϕLL , where

= (;)qtϕϕ (obtained by a gauge transformation), leads to the same dynamical system;

in other words, the Euler–Lagrange systems of equations associated to the two

Lagrangians coincide (see Sect. 18.2.3.2 too). Reciprocally, if two Lagrange’s functions

L and

′

L lead to the same Euler–Lagrange equations, hence if they describe the

same dynamical system, then

′

L is of the form mentioned above. But there exist also

dynamical systems associated to two Lagrangians which have not the mentioned

property; e.g., to the system of equations

MECHANICAL SYSTEMS, CLASSICAL MODELS

398

++= −+= =

111 222

0, 0, , constqaqbq qaqbq ab   

(21.4.19)

one can associate the Lagrangians

()

()()

−

=+ − −

=−+ −

′

11 22

12 12 21 12

22 22

ee,

at at

qq qq qq bqq

qbq qbq

  



(21.4.19')

which cannot be obtained one from the other by a gauge transformation. We mention

also that there are known non-conservative systems which admit a variational principle

and have Lagrangians; among them there is also the system considered above. But there

are also dynamical (non-conservative) systems for which one cannot write Euler–

Lagrange equations, corresponding to a function of Lagrange

= (,;)qqtLL . We are

thus led to a classification of the non-conservative systems after the property of

admitting or not Lagrangians of the above mentioned form. We can consider a class of

dynamical systems (to which belong all the conservative systems) formed by those

systems which have Lagrangians and for which the Euler–Lagrange equations are just

the equations of motion of the system. In case of another class of dynamical systems

one obtains the previous situation after multiplying by an integrant factor.

Hence, the inverse problem is reduced – finally – to the determination of an integrant

factor. Let be thus the equations of motion

−==( , ; ) 0, (!), 1,2,...,

ii j

mq F q q t i s  ,

(21.4.20)

in Newtonian formulation; it is thus put the problem to determine a non-singular matrix

[]

g , for which = (,;)

ij ij

ggqqt , so that

()

∂∂

⎛⎞

−= − =

⎜⎟

∂∂

⎝⎠

∑

, 1,2,...,

ij j j j

gmq F i s

tq q





(21.4.20') ,

for a certain function

= (,;)qqtLL , as well as the problem of the conditions in

which one can find such a matrix.

Starting from the conditions given by H. von Helmholtz in 1887 (for the particular

case

==0

ij j



 and without showing their sufficiency), we write the conditions in

which one can solve the problem put above in the form given by R.M. Santilli in 1984,

that is

∂

∂∂

⎛⎞

−= + −

⎜⎟

∂∂ ∂ ∂∂∂

⎝⎠

∂

∂∂

⎛⎞

+= +

⎜⎟

∂∂ ∂ ∂

⎝⎠

ij ji

ji ji

qq t qqq

qq t q











(21.4.21)

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

399

with summation with respect to

k and where we have denoted =

ij j ij

Amg, without

summation, and

=−

iijj

BgF, with summation with respect to j .

In 1891, G. Darboux showed that any dynamical system with one degree of freedom

admits a variational principle in certain conditions of continuity and regularity, which

are – in general – satisfied; other properties have been put in evidence by D.G. Currie

and E.J. Saletan, in 1966, by J.A. Kobussen, in 1979, and by W.I. Sarlet, in 1982. In

1941, I. Douglas dealt with the case of two degrees of freedom, the problem being

taken again, in 1979–1983 by L.Y. Bahar, H.G.Kwatny and W.I. Sarlet; as principal

mathematical tool, one uses variational methods of self-adjunction. Caviglia showed, in

1896, that – by introducing additional variables – some of Helmholtz’s conditions

become equations of Euler–Lagrange type.

Not all dynamical systems with two degrees of freedom admit an integrant factor

g ,

≠det[ ] 0

g , hence a variational principle; as simple example, we mention the system

−= −=

12 22

0, 0xx xx  ,

(21.4.22)

considered by E.T. Whittaker in 1904, as well as the system

+= +=

12 22

0, 0xx xx  ,

(21.4.22')

considered by S. Hokjman and L.F. Urrutia in 1981. In such a situation, one passed

from a Lagrangian formulation to a Hamiltonian one, by Legendre’s formulation. There

have been searched the conditions in which one can obtain an integrant factor so that

the equation of motion, multiplied by this one, do lead to equations of Hamilton type;

these conditions (necessary and sufficient) are just Helmholtz’s conditions.

21.4.3.2 Notions on the Birkhoffian Formalism

In the case in which one cannot determine (one does not know or there does not

exist) a function of Lagrange or a function of Hamilton such that the dynamical system

become a Lagrangian system or a Hamiltonian system, respectively, one searches a

generalization of the latter ones; thus, the equations of motion in generalized

Hamiltonian form (equivalent to the equations of motion in Newtonian form) must be

obtained by a variational principle too. This represents the Birkhoffian generalization of

the Hamiltonian mechanics (given by G.D. Birkhoff in 1927).

A system of differential equations of first order constitutes Birkhoff’s equations of a

dynamical system if there exist a function

= (;)BBat, ≡

{ , ,..., }

aaa a, called

Birkhoff ’s function (Birkhoffian) of the mechanical system, and an exact Symplectic

2 -form =∧ddaa

μν μ ν

ΩΩ (we use the exterior product and the convention of

summation with respect to Greek indices), so that

= dΩθ, with = dRa

μμ

(

(;)Rat

are the components of Birkhoff’s vector); the system can be written in the

form

∂∂

⎛⎞⎛⎞

−−+=

⎜⎟⎜⎟

∂∂ ∂ ∂

⎝⎠⎝⎠

aa a t

μμ

μν μ

 .

(21.4.23)

MECHANICAL SYSTEMS, CLASSICAL MODELS

400

The tensor

∂

=− =

∂∂

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

at s

μν

Ωμν,

(21.4.24)

is called Birkhoff’s tensor; Birkhoff’s system reads

∂

⎛⎞

−− =

⎜⎟

∂∂

⎝⎠

μν ν

Ω 

(21.4.23')

too, being – in fact – a generalized Hamiltonian system (in particular, Hamilton’s

equations can be written in this form).

The system (21.4.23) is called autonomous if the functions

and B do not depend

explicitly on time (

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

ν== =



); such a system reads

∂

−==

∂

()

() 0, 1,2,...,

aa s

μν ν

Ωμ

(21.4.25)

The system (21.4.23) is called semi-autonomous if only the conditions

= 0R



= 1,2,...,sν , take place; in this case

∂

−==

∂

(;)

( ) 0, 1,2,...,

Bat

aa s

μν ν

Ωμ .

(21.4.25')

In general, the system is non-autonomous. A Birkhoffian system is called regular if

≠det[ ] 0

μν

Ω ; it is singular if =det[ ] 0

μν

Ω .

Let be the equations of motion (21.4.20) in Newtonian formulation; these equations

can be written as a system of first order in the form

−= −= =d d 0, d d 0, (!), 1,2,...,

ii ii i

qqt mqFt i s .

(21.4.26)

Let us consider the application

→=

iiii

qpmq, without summation, by which the

system becomes

−= −==

d d 0, (!), d d 0, , 1,2,...,

ii ii

qpt pFti s

;

(21.4.26')

we can write

−==( ; ) 0, 1,2,...,aAat s

μμ

μ ,

(21.4.27)

too, with the notations

⎡

⎤

⎡

⎤

⎡⎤

⎣

⎦

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎣⎦

⎣

⎦

[]

[] ,[ ]

[]



(21.4.27')

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

401

where

{

}

≡≡

12 12

{ , ,..., }, , ,..., ,

{ , ,..., }, { , ,..., }.

qqq q

mmm m

ppp pFFFF

 

Eliminating the derivatives

 between the equations (21.4.23) and the equations

(21.4.27), we get

∂∂

⎛⎞

−=+=

⎜⎟

∂∂ ∂ ∂

⎝⎠

, 1,2,...,

aa a t

μμ

μν μ

(21.4.28)

hence a system of

2s partial differential equations with + 1s unknown functions. By

integrating this system of equations, one obtains the potential

and Birkhoff’s

function

B , the functions A

(considered to be differentiable) being arbitrary; as in

case of a non-autonomous solution, as well as is case of a semi-autonomous (eventually

autonomous) one, by applying the Cauchy–Kovalevsky theorem of existence and

uniqueness, the obtainment of those solutions is ensured. We can thus state

Theorem 21.4.1 (universal theorem of Birkhoffian representation). Any Newtonian

system admits locally a Birkhoffian representation.

Let be a dynamical system which is neither a Lagrangian, hence nor a Hamiltonian

one; one can put the problem (the inverse problem of Newtonian mechanics in

Birkhoffian formulation) to find a Birkhoff’s function

B , a Symplectic structure

associated to the system and Legendre’s transformations, so that the equations of

motion be obtained in the form of Birkhoff’s equations. The practical method to

determine the Symplectic structure and the Birkhoffian consists in searching an

integrant factor and in imposing some conditions of self-adjunction; one can determine

a Pfaffian from which, according to a variational principle, one obtains Birkhoff’s

equations. This Pfaffian generates a Lagrangian of second order, that is of the form

=+(,,;) (,;) (,;)qqqt Vqqtq W qqt   L ,

(21.4.29)

which leads to the associate equations of Euler–Lagrange type

∂∂∂

⎛⎞ ⎛⎞

+−==

⎜⎟ ⎜⎟

∂∂∂

⎝⎠ ⎝⎠

0, 1,2,...,

jjj

qtqq

 

LLL

;

(21.4.29')

these equations are linear combinations between Newton’s equations and their

derivatives, assuming as solutions the solutions of the given Newtonian system. We can

thus state that to any Newtonian dynamical systems one can associate a functional of

action type, the condition of extremum of which defines the trajectories of the system.