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

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Lagrangian Mechanics

18.2.3.8 Passing to an Equivalent System of Differential Equation of First Order

In the absence of holonomic constraints, the space of configurations can be even the

representative space

E . Indeed, the relation (18.2.1), written for a fixed t , leads to a

metrics of the form

ddd

sgqq,

(18.2.86)

where the coefficients of the Riemannian space

∂∂

⋅=

∂∂

∑

= , , 1,2,...,

gjks

(18.2.86')

can be obtained from the coefficients (18.2.15') taking masses equal to unity and

maintaining the property

≠det[ ] 0

g ; if the mechanical system is a free one, then

= 3sn, hence an Euclidean space

E , so that – by a linear transformation – we can

obtain the normal form with constant coefficients

()()()

⎡⎤

=++

⎢⎥

⎣⎦

∑

222

() () ()

123

d ddd

iii

s xxx

(18.2.86'')

the square of the element of arc being expressed in the form of a sum of squares. If the

time

t is not fixed,

ds is an arbitrary polynomial of second degree in d

q (corres-

ponding to the analogous form of the kinetic energy); but the character of the kinetic

energy for a mechanical motion leads, by a convenient transformation, to an element of

arc of the form (18.2.86'') too.

But if there exist certain constraints, the space

R becomes a Riemannian space (of

positive curvature), because – reducing the number of the configurative dimensions –

the Euclidean character of the space is destroyed.

In general, the state of a discrete mechanical system

S is determined by its position

and its motion in the neighbourhood of this position, hence by the set

= ()

qqt,

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

jjj

uqutj s . We can thus conceive a representative space with 2s

dimensions, in which the representative point of co-ordinates

=, , 1,2,...,

qu j s, does

represent, at any moment, “the state” of the mechanical system

S ; this space is called

the state space or the phase space. In the case in which

= 3sn, this space is

6n -dimensional and has not the character of a metric manifold; it is only an affine

space and one cannot choose a common unity of measure for both types of co-ordinates,

but it has a particular importance in the analytical representation of the motion.

We have seen in Sect. 18.2.2.4 that one can replace Lagrange’s system of

equations of second order (18.2.45') by a system of

2s differential equations of first

order (18.2.48), written in the normal form, the unknown functions being

= ()

qqt

and

( ), 1,2,...,

uutj s==. This system of equations can be written also in the

form

MECHANICAL SYSTEMS, CLASSICAL MODELS

========

12 12

dd dd

ddd

... ...

qq uu

qut

uu uϕϕ ϕ

(18.2.87)

If we succeed to find

2s first integrals of the form

12 12

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

fqq quu ut

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

Cl s, where

C are arbitrary constants, then we can determine the

unknown functions

12 2

( ; , ,..., )

qqtCC C, =

12 2

( ; , ,..., )

uutCC C, obviously,

the relations

==d d , 1,2,...,

qtuj s, taking place too. The problem of motion of

the mechanical system

S is thus entirely solved.

Let us suppose that the mechanical system

S is catastatic, so that the time t does

not appear explicitly in the expression of the kinetic energy; in this case,, the equations

of motion take the form (18.2.47') (we assume to have only holonomic constraints,

which – in this case – are scleronomic), where Christoffel’s symbols of second kind do

not depend explicitly on time. As well, we suppose that neither the generalized forces

∗

Q do not depend explicitly on time ( 0, 1,2,..,

Qt j s

∗

∂∂= − ). In this case, nor in

the system of equations (18.2.87) the time does not intervene explicitly in the first

ratios; neglecting the ratio

d1t , we will try to integrate the system of equations

=======

12 12

dd dd

... ...

qq uu

uu u

ϕϕ ϕ

(18.2.88)

If we succeed now to find the first integrals

12 12

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

fqq q uu u C,

=−1,2,...,2 1ls

, then we can determinate −21s unknown functions with respect to

one the co-ordinates, e.g.

q , in the form

−

112 21

( ; , ,..., ), 2,3,..., ,

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

qqqCC C k s

uuqCC C j s

Associating the equation

dd1qu t

, we obtain

−

∫

11 1 2 1

(; , ,..., )

uqCC C

τ ,

(18.2.88')

completing thus the integration constants by

τ , the integration of the system of 2s

linear equations (18.2.88) is thus reduced to the integration of a system of

−21s linear

equations and to a quadrature.

Assuming that one can write a Jacobi first integral of the form (18.2.63) too, we can

express then one of the generalized velocities, for instance

q

, in the form

1112

( , ,...,quqq

, , ,..., ; )

quu uh, so that – instead of the system (18.2.88) – we

have to integrate a system of

−22s linear differential equations

Lagrangian Mechanics

=======

12 2 3

dd dd

... ...

qq uu

uu u

ϕϕ ϕ

(18.2.89)

with

12 12

( , ,..., , , ,..., ), 2,3,...,

qq quu u k sϕϕ . Thus, the complete system of

integrals of the equations of motion (18.2.88) is formed by

−2( 1)s integrals of the

system (18.2.89), by Jacobi’s first integral (18.2.63) and by the integral (18.2.88').

We put thus in evidence the trajectory of the representative point in the state space.

Such a study has been made by Hamilton in the phase space and we will present it

systematically in the next chapter.

18.3 Other Problems Concerning Lagrange’s Equations

There are also other problems concerning Lagrange’s equations which present a

particular interest, as the forms taken by these equations in some special cases or

different systems of equations of motion equivalent to them. Besides these results, we

will deal with some significant applications of the analytical methods introduced by

Lagrange.

18.3.1 New Forms of Lagrange’s Equations

In the following, we consider some new forms of Lagrange’s equations, as the

central equation of Lagrange, the equations of Nielsen, the equations of Tsenov and the

Mangeron–Deleanu equations. As well, we will put in evidence the form taken by these

equations in case of continuous mechanical system, in case of a non-inertial frame of

reference and in case of the phenomenon of collision.

18.3.1.1 Lagrange’s Central Equation

Let be a particle

of position vector r

and of mass

m , which belongs to a

discrete mechanical system

S . We can write the obvious relation

() ()

()

⎡⎤

⎡

⎤

⋅δ = ⋅δ +δ + ⋅ δ −δ

⎢⎥

⎢

⎥

⎣

⎦

⎣⎦

rr rr r

rr r

dd dd d

dd d d d d

ii ii i

ii i

tt t t t t

Taking into account the relations (18.2.1), we have

()

∑

12 12

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

Tq q q q q q t m

  ,

wherefrom, using the relations (18.2.28) too, we obtain

()

===

∂

∂∂

δ= ⋅ δ= ⋅ δ= ⋅δ

∂∂ ∂

∑∑∑

rr rr r

111

dd d d

nnn

ii ii i

ji ji jii

jj j

iii

qm qm qm

qtqt tqt 

The principle of virtual work, written in the form (18.2.27), leads to

MECHANICAL SYSTEMS, CLASSICAL MODELS

⋅δ = δ

∑

iijj

mQq

In this case, multiplying the above identity by

m and summing for all the indices i , it

results

()

∂

⎛⎞

⎡

⎤

δ=δ+δ+ ⋅ δ−δ

⎜⎟

⎢

⎥

∂

⎣

⎦

⎝⎠

∑

dddd

jjj i i

qQq m

tq t t t



Observing that

∂∂

δ= δ + δ

∂∂



we can write this relation in the form

()

∂∂ ∂

⎡⎛ ⎞ ⎤

⎡

⎤

−−δ= δ−δ

⎜⎟

⎢⎥

⎢

⎥

∂∂ ∂

⎣

⎦

⎣⎝ ⎠ ⎦

⎡⎤

−⋅δ−δ

⎢⎥

⎣⎦

∑

ddd

jjj

jj j

TT T

Qq q q

tq q q t

ttt





(18.3.1)

We get thus, after Hamel, Lagrange’s central equation (Hamel, G., 1927).

In case of holonomic constraints, the operator relations (18.1.77) take place for the

position vectors in

E ; as well, the operator relations (18.2.13) take always place for

the generalized co-ordinates in

Λ . In these conditions, the equation (18.3.1) leads to

the fourth form of the fundamental equation (the equation (18.2.27')).

18.3.1.2 Mechanical Systems with Separable Variables. Systems of Liouville Type

There are many problems, e.g. in celestial mechanical or in atomic physics, which

lead to a Lagrangian for which the kinetic energy and the simple potential of the

generalized forces are of the form

∑∑

(), ()

jj jj

TVqqUUq ,

(18.3.2)

where

=, , 1,2,...,

UV j s, are functions of only one variable of class

. In this

case, Lagrange’s equations (18.2.29) read

+==

( ) , 1,2,...,

2d d

jjj j

Vqq q j s

 

(18.3.3)

Multiplying by

q and integrating with respect to time, we get

Lagrangian Mechanics

−==

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

jjj jj j

Vqq Uq C j s

(18.3.3')

where

=, 1,2,...,

Cj s, are integration constants. Summing with respect to the index

j . one obtain the first integral of mechanical energy; but we have thus first integrals

corresponding to

s particular mechanical energies (eventually, to the mechanical

energies of

s particles). By s quadratures of the form

+= =

∫

()

d , 1,2,...,

()

jj j

tqjs

Uq C

(18.3.4)

we introduce the integration constants

=, 1,2,...,

sτ

, too. Thus, one obtains the

relations which link the time to each generalized co-ordinate, the problem being thus

entirely solved. The respective mechanical systems are mechanical systems with

separable variables.

A problem similar is that of the discrete mechanical systems, called systems of

Liouville type, for which we have

∑

∑∑

∑

()

() () ,

()

jj j

TuqVqqU



(18.3.5)

and where

and =, , , 1,2,...,

UV u jk s, are functions of only one variable.

Obviously, the case considered above is a particular case of a system of Liouville type

for which

==1const

us . By the substitution

===

d()

( ) , 1,2,...,

jj j

Wq W

Vq j s



we can write

== =

∑∑ ∑

11 1

,(),()

ss s

jjj

kj k

TWUUWuuW



(18.3.6)

Lagrange’s equations relative to the new generalized co-ordinate

W take the form

()

∂∂

−==

∂∂

∑

, 1,2,...,

uW W k s

tWW



Multiplying by



, we get

()

∂∂

−=

∂∂

∑

22 2

kk k

uW uW W uW

tW W

  

MECHANICAL SYSTEMS, CLASSICAL MODELS

On the other hand, the first integral of mechanical energy is

−= =

∑

,const

WUhh



We are thus led to

()

[]

∂∂

=+ +

∂∂

∂

=+=+=

∂

2( ) 2

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

kkk

kkkkk

uW h UW uW

tWW

WhUu huqUqk s





wherefrom one obtains the system of differential equations

++ ++

11 11 1 22 22 2

...

() () () ()

() ()

ss ss s

hu q U q hu q U q

hu q U q

αα

(18.3.7)

Noting that

=++ =

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

kkkk

uW hu U k sα



, summing with respect to all

indices and taking into account the integral of mechanical energy, we find the condition

∑

α ,

(18.3.7')

which must be verified by the constants

=, 1,2,...,

ksα

, thus introduced. The

equations (18.3.7) are with separate variables, the problem being reduced to

quadratures.

18.3.1.3 Case of Continuous Mechanical Systems

Let be a continuous mechanical system with a finite number of degrees of freedom,

which occupies a domain

V in

E and is acted upon by a volumic force Fr() (hence,

by the force

F dV on the element of volume); the torsor of these forces (resultant

force and resultant moment) at the pole

O is given by

==×

∫∫

RFrM rF()d , d

VV.

(18.3.8)

The kinetic energy is expressed in the form

()

∫

() d

(18.3.9)

where

r()μ is the density, so that in the formulae (18.2.15), (18.2.15'), (18.2.15''), and

(18.2.15''') we will have

Lagrangian Mechanics

∂∂

=⋅

∂∂

∂

=⋅==

∂

∫

∫∫

rr rr

() d ,

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

gVjksgV

μμ



(18.3.9')

Analogously, the virtual work of the volumic forces will be given by

∂

δ= ⋅δ =δ ⋅ =δ

∂

∫∫

Fr F

jjj

WVqVQq

so that the generalized forces will be

∂

=⋅

∂

∫

(18.3.10)

The principle of virtual work is expressed in the form

∂

⎡⎤

⋅−δ=

⎢⎥

∂

⎣⎦

∫

() 0



(18.3.11)

By a demonstration analogous to that in Sect. 18.2.2.1, where the sign plus is replaced

by the sign integral, we find again the fourth form of the fundamental equation

(18.2.27'). The results obtained are thus justified for rigid solids too.

18.3.1.4 Electromechanical Analogy

We have seen in Sect. 8.2.2.10 that one can establish an interesting analogy between

an R.L.C. circuit (an ohmic resistance

R , a loading inductance L and a condenser of

capacity C , connected in series with a generator having an electromotive force ()Et)

and a particle

P (of mass →mL, of damping coefficient

′

→

kR and of coefficient

of elasticity

→ 1/kC, acted upon by a given force →() ()Ft Et, where we have put

in evidence the correspondence between various quantities), in a vibrating motion.

Thus, we can introduce quantities analogue to those of mechanical nature in the form

(we use well known notations)

∂

⎛⎞

==− =−=−

⎜⎟

∂

⎝⎠

222

TWqQ RqU q







(18.3.12)

where

(1/2)Rq corresponds to Rayleigh’s function, = ()qqt is the charge, while

==() ()iit qt is the intensity of the current. Lagrange’s equation

TTU

tq q q

∂∂∂

⎛⎞

−=++

⎜⎟

∂∂∂

⎝⎠



leads to the equation (8.2.58).

MECHANICAL SYSTEMS, CLASSICAL MODELS

The above considerations can be applied to complex R.L.C. circuits, corresponding

to discrete mechanical systems with a finite number of degrees of freedom; using the

mentioned electromechanical energy one obtains a system of differential equations

which allows the determination of the electric charges

()qt in each circuit.

18.3.1.5 Nielsen’s Equations

We have established Lagrange’s equations by means of the d’Alembert–Lagrange

principle (the principle of virtual work); analogously, Jourdain’s principle (18.1.57'),

written in the form

⋅δ = ⋅δ

∑∑

av Fv

ii i i i

(18.3.13)

leads to an equivalent system of equations of motion. Starting from the relation

(18.2.14) and taking into account (18.2.4'), we may write

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

ii ii

jj jj

qinjs

qqq qq

∂∂ ∂∂

=+===

∂∂∂ ∂∂



ar rv

(18.3.14)

On the other hand, we notice that Jourdain’s principle can be written for a given

position of the mechanical system, at a fixed moment, varying the velocities; thus, it

results

∂∂

δ= δ= δ

∂∂

ijj





(18.3.15)

where we took into account (18.2.28). In this case, Jourdain’s principle takes the form

∂∂

⎡⎤

−⋅+⋅ δ=

⎢⎥

∂∂

⎣⎦

∑

ii i j



(18.3.13')

Using the expression of the kinetic energy, we get

∂

=⋅=⋅

∂∂

∑∑

va v

ii i ii

tqq

so that

()

11 11

nn nn

ii ii

ii ii ii ii

jj jj

ii ii

mm mm

qt qq qq

== ==

∂∂ ∂∂

∂

=⋅+⋅= ⋅+⋅

∂∂∂∂∂

∑∑ ∑∑

 

av vv

va va

where we have used the relation (18.3.14). Taking into account the expression (18.2.18)

of the generalized forces, the form (18.3.13') of Jourdain’s principle leads to

()

∂∂

⎡⎤

−δ=

⎢⎥

∂∂

⎣⎦

qqt



+ .

(18.3.16)

Lagrangian Mechanics

Noting that the variations

q of the generalized velocities are independent and

arbitrary, in case of holonomic constraints, we can write the equations obtained in 1935

by J. Nielsen in the form (Dolapčev, Bl., 1966)

()

∂∂

−==

∂∂

2 , 1,2,...,

Qj s

qt q



(18.3.17)

These equations are governing the motion of the representative point

P in the space

. In case of the non-holonomic constraints (18.2.10), one must use the method of

Lagrange’s multipliers.

Taking into account the relation (we use the operator relation (18.2.13))

∂∂∂

⎛⎞ ⎛⎞

=== =

⎜⎟ ⎜⎟

∂∂∂

⎝⎠ ⎝⎠

vrv

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

iii

jjj

injs

tq tq q

(18.3.18)

we can write the derivative

===

∂∂ ∂

∂

⎛⎞ ⎛⎞

=⋅=⋅+⋅

⎜⎟ ⎜⎟

∂∂∂∂

⎝⎠ ⎝⎠

∑∑∑

vv v

vav

111

dd d

nnn

ii i

ii ii ii

jj j

iii

mmm

tq t q q tq



in the form

()

∂∂ ∂

⎛⎞

=−=

⎜⎟

∂∂ ∂

⎝⎠

,1,2,...,

jj j

TTT

tq q t q



We see thus that Nielsen’s equations are equivalent to Lagrange’s equations. We

mention also that, in some cases, Nielsen’s equations allow a more simple calculation

that Lagrange’s ones.

We consider also Gauss’s principle (the principle of the least action) (18.1.60') or

(18.1.66), written in the form

⋅δ = ⋅δ

∑∑

aa Fa

ii i i i

(18.3.19)

for a given position and velocity of the mechanical system. at a fixed moment, varying

only the accelerations; in this case, the relation (18.2.14) allows to write

∂∂

qq

(18.3.20)

so that

∂∂

δ= δ= δ

∂∂

ijj

 



(18.3.15')

MECHANICAL SYSTEMS, CLASSICAL MODELS

The relation (18.3.19) takes the form

∂∂

⎡⎤

⋅− ⋅ δ=

⎢⎥

∂∂

⎣⎦

∑

iiij



(18.3.19')

and, taking into account the above results and (18.2.28), we obtain

()

∂∂

⎡⎤

−δ=

⎢⎥

∂∂

⎣⎦

qqt





(18.3.16')

Noting that, in case of holonomic constraints, the variations δ

q of the generalized

accelerations are independent and arbitrary, we find again the equations (18.2.17) of

Nielsen.

The principle of Gauss in finite form (18.2.60) can be written also in the form

⋅Δ − Δ =

∑

ii i j

mQq ,

(18.3.19'')

where we have put in evidence the finite variations of the accelerations

and of the

generalized accelerations

q ; this is the fifth form of the fundamental equation.

18.3.1.6 Tsenov’s Equations

Starting from Gauss’s principle, written in the form (18.3.19) or in the form

(18.3.19'), we can establish also other equations, equivalent to Lagrange’s equations. To

do this, starting from the expression of the derivative

d/dTt, let us calculate

=+⋅

∑∑

ava

(2)

ii ii

where we have introduced the accelerations of second order

(2)

(see Sect. 18.1.2.5

too). Using the equations (18.2.14), it results

∂∂∂ ∂

⎛⎞

=+ + + +

⎜⎟

∂∂∂∂∂

⎝⎠

∂∂∂

⎛⎞

=+ + +

⎜⎟

∂∂∂∂

⎝⎠

rr r r

rrr

(2)

2 2 ...

3 ...,

ii i i

jjjj

jjj j

iii

jj j

qqqqq

qtq qq q

qqq

qqqq



    

  

wherefrom

∂∂

=+ +

∂∂

(2)

3 ...

  ,

(18.3.21)