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

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Chapter 18

Lagrangian Mechanics

The analytical methods of calculation have been introduced to can study in a

systematic and unitary form the discrete mechanical systems (with a finite number of

particles or with a finite number of degrees of freedom); thus, appears the denomination

of “analytical mechanics”, besides that of “theoretical mechanics” (in the English,

German and Russian literature) and that of “rational mechanics” (in the French or

Italian literature). In “physical mechanics” stress is put on the experimental aspect,

while in “technical mechanics” on the possibility to apply in technics the results

obtained. In all these cases, the object of study is the same: mechanics, as a science of

nature.

In this order of ideas, we consider a discrete mechanical system

S , formed by n

particles

P (or a finite number of rigid solids), of position vectors r

, subjected to the

action of given forces

=F , 1, 2, ...,

in (the resultants of all external and internal

given forces which act upon the particle

) and to certain links

=, 1, 2, ...,

Lk n

The basic problem which is put consists in the determination of the motion (position and

velocity) of the mechanical system

S in the time interval

[,]tt (eventually, ∞

[,)t ) if

=FFrr(,;)

iijj



, = rr(,;)

LL t



=rr

()

and

=vv

()

t , = 1, 2, ...,in, = 1, 2, ...,

are given. A distinction between external and internal forces is not necessary; in

exchange, the classification of the constraints and their representation in a finite or

differential form, as well as their elimination from the calculation (when it is possible)

have a particular importance. The elimination of the constraints and of the constraint

forces from the calculation allows to obtain equations of motion which contain only the

position vectors and their derivatives with respect to time. In general, one admits only

frictionless constraints, which – obviously – restrains much the sphere of the problems

which can by treated.

Euler, in 1736, and Maclaurin, in 1742, have written the first treatises of mechanics

in which they pass from purely geometric methods to methods based on differential

calculus (Euler, L., 1736). The brothers Jacob I Bernoulli and Jean I Bernoulli, Daniel

Bernoulli, the son of the latter one, Alexis-Claude Clairaut, Jacob Herman and Pierre-

Louis-Moreau de Maupertuis have made important steps, considering many problems

and enouncing various “principles”, which allowed to put in equation the problems of

mechanics and to give their solution in a unitary form. “A special ability was always

necessary to put in evidence the forces which have to be taken into consideration, so

P.P. Teodorescu, Mechanical Systems, Classical Models,

MECHANICAL SYSTEMS, CLASSICAL MODELS

that each problem become piquant, exciting the emulation; ... the treatise on dynamics

of Jean le Rond d’Alembert, appeared in 1743, has put an end to this kind of

competition, offering a direct and general method to solve or – at least – to put in

equation all the problems of dynamics which one could imagine”. These affirmations

have been made by Joseph-Louis Lagrange in his famous treatise “Mécanique

analytique”, which appeared in Paris, in 1788, in which mechanics is presented as an

abstract science, as a chapter of mathematical analysis, which can be studied without

using any figure (Lagrange,J.L., 1788). Thus, the theory of systems of ordinary

differential equations, the variational calculus, the theory of punctual and canonical

transformations, the theory of Pfaff forms, the theory of integral invariants etc. have

been developed, the impulse given by Lagrange to the study of mechanical systems

having a particular importance for the development of analysis, as well as of other

chapters of mathematics.

We consider, in the frame of this chapter, a discrete mechanical system

S , formed

– in general – by a finite number of particles and by a finite number of rigid solids (as it

has been considered in Sect. 17.1.1.1); hence, the system has a finite number of degrees

of freedom. For the sake of simplicity, we assume – so as it is made in any classical

study, without loss of generality – that the system

S is formed by a finite number of a

particles; when it is necessary or when intervene particular data we refer to rigid solids

too. Thus, after some preliminary results, we will do a detailed study of Lagrange’s

equations of motion, which will be then applied to some important problems (Appell, P.,

1941–1953; Goldstein, H., 1956; Lamb, H., 1929; Landau, L.D. and Lifchitz, E., 1960;

La Valée-Poussin, Ch.J. de, 1925; Lurie, A.I., 2002; Mercier, A., 1955).

18.1 Preliminary Results

First of all, we will present some introductory motions, which allow to pass from the

representation of the discrete mechanical system

S in the space

E to its

representation by means of a representative point

P in the representative space

E .

Passing in review the differential principles of mechanics, we can study the motion of

the mechanical system

S in the new considered space.

18.1.1 Introductory Notions

In what follows we use the notions of displacement for a mathematical representation

of the

n constraints , 1,2,...,

Lk n= ; this allows to introduce the notion of

representative space

E . A particular attention is given to the natural systems and to

the particle subjected to non-holonomic constraints.

18.1.1.1 Displacements. Constraints

One of the most important notions in the frame of the analytical methods is that of

displacement; this notion has been introduced and considered at large in Sect. 3.2.1.1,

putting in evidence the real displacements

, the possible displacements Δr

and the

Lagrangian Mechanics

virtual displacements

δr

of the particle

P , of position vector =r , 1,2,...,

in. Starting

from the displacements, one can define the real velocities

=vrd/d

t , the possible velo-

cities

=Δ Δvr/

t and the virtual velocities

∗

=δ Δvr/

t .

In case of a particle subjected to stay on a fixed curve

C or on a fixed surface S

(steady case), the virtual displacement is directed along the tangent to the curve or is

contained in the plane tangent to the surface at the point

P , respectively; in both cases,

the real displacement belongs to the (finite or infinite) set of virtual displacements. If

the rigid

S , e.g., is movable, then the virtual displacements take place in the tangent

plane to this surface at the moment

t (we consider the surface to be frozen at this

moment), while the real displacement links the particle

P situated at the point P of this

surface at this moment

t to the particle P situated at the point P of the surface at the

moment

+ dtt; hence, in a non-steady case, the real displacement does no more

belong to the set of virtual displacements of the considered particle. This statement can

be made in the case of a deformable surface

S too.

The constraints to which is subjected the discrete mechanical system

S can be

external or internal; in the following, this distinction is not useful. As well, we make

distinction between unilateral (expressed by inequalities) and bilateral (represented in

the form of equalities) constraints; in analytical mechanics, we will consider all the

constraints to be – in general – bilateral. Another distinction is made between

constraints of contact and constraints at distance; both these types of constraints can be

expressed by equalities. A large study of the constraints is made in Chap. 3, Sect. 2.2.

Another classification of the constraints, after Hertz, puts in evidence the holonomic

(finite, of geometric nature) and the non-holonomic constraints (infinitesimal or

differential, of kinematic nature). After Lothar Bolzmann, the constraints which do not

depend explicitly on time are called scleronomic (stationary) constraints, those which

vary in time being rheonomous (non-stationary) constraints. As a matter of fact, taking

into account these classifications, one obtains a mathematical representation of the

constraints. We mention also the distinction between ideal (perfect, smooth) constraints

and constraints with friction (real). Analytical mechanics has been developed for

discrete mechanical systems; some results can be adapted to the case with friction too.

In the absence of the constraints

=, 1, 2, ...,

Lk n, the discrete mechanical system

S is a free mechanical system. The axiom of liberation from constraints states that

there exists always a system of forces

, which is applied upon the mechanical system

S subjected to constraints, so that each particle

P can be treated as a free particle

subjected to the action of the force

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

in; consequently, the

mechanical system

S can be studied as a free one. The forces F

are given (known)

forces, while the forces

are constraint (unknown) forces.

The real elementary work

dW of the given forces F

is given by the relation (3.2.3)

and the corresponding virtual work

δW by the relation (3.2.3'); as well, the real ele-

mentary work d

W of the constraints forces R

is given by (3.2.7), the corresponding

virtual work

W being of the form (3.2.7').

MECHANICAL SYSTEMS, CLASSICAL MODELS

The

p holonomic constraints are expressed in the form (3.2.8) if they are

rheonomous and in the form (3.2.33') if they are scleronomic, the number

=−3snp

representing the number of degrees of geometric freedom of the discrete mechanical

system

S ; the motion takes place only if < 3pn (hence,

> 0s

) . We notice that a

free rigid solid, component of a discrete mechanical system

S , introduces 6 degrees of

freedom. The

non-holonomic constraints are represented – in general – by means of

the Pfaff differential forms in the form (3.2.13) (as well in the form (3.2.13')), with the

aid of the real displacements, or in the form (3.2.15) (analogously, in the form

(3.2.15')), by means of the virtual displacements. In case of scleronomic constraints, the

relations (3.2.32) must also take place; otherwise, the constraints are rheonomous. The

number

=−3rnm represents the number of kinematic degrees of freedom of the

discrete mechanical system

S . In case of holonomic constraints, we have =rs. The

total number of constraints is, in general,

npm, while the number of degrees of

freedom (obviously, kinematic degrees of freedom, because the holonomic constraints

involve the same number of geometric and kinematic degrees of freedom) is

−3

nn.

The non-holonomic constraints of the form (3.2.12), considered above, have been

called non-holonomic constraints of the first kind by G. Hamel; he introduces also

non-holonomic constraints of the second kind, where appear the accelerations of the

particles too, in the form

=rr

(d ,d ;d ) 0

ft,

(18.1.1)

the dimensional homogeneity being ensured.

A constraint relation of the form (3.2.13) corresponds to a holonomic constraint if it

is an integrable differential Pfaff form; otherwise it corresponds to a non-holonomic

constraint. The Theorem 3.2.2 of Frobenius gives the necessary and sufficient

conditions, so that such a differential form of first degree be complete integrable (does

correspond to an exact differential or does admit an integrable factor).

We notice that the two classifications of the constraints (holonomic or

non-holonomic and scleronomic or rheonomous) are independent; we can have, e.g.,

non-holonomic and rheonomous constraints (the most general case) or holonomic and

scleronomic constraints (the most particular case). A mechanical system subjected to

holonomic or non-holonomic constraints is called a holonomic or non-holonomic

mechanical system, respectively; as well, a mechanical system subjected to scleronomic

or rheonomous constraints is called scleronomic or rheonomous mechanical system,

respectively. In case of scleronomic constraints, the real displacements belong to the set

of virtual displacements. As a matter of fact, to have this property it is sufficient that the

conditions

0, 1,2,...,

kmα , be fulfilled in the relations (3.2.13) (it is not

necessary that the constraints by scleronomic); such constraints are called catastatic.

As examples of mechanical systems subjected to holonomic and scleronomic

constraints we mention the particle compelled to stay on a fixed curve or on a fixed

surface, as well as the rigid solid (or a non-deformable discrete mechanical system). If

the curve or the surface are movable (rigid and movable with respect to a fixed frame of

Lagrangian Mechanics

reference or even deformable), then the constraint is holonomic and rheonomous. We

notice that a rigid solid behaves – in a certain sense – as a particle which has not three

but six degrees of freedom. Non-holonomic constraints are, e.g., those corresponding to

the motion of a heavy homogenous sphere (see Sect. 17.1.2.8) or of a heavy circular

disc (see Sect. 17.1.2.9) on a horizontal plane; we mention the motion of a rigid skate

on the ice field too, case considered in Sect. 3.2.2.6.

An important class of constraints, in the frame of which analytical mechanics has

been developed, is the class of ideal constraints. We call ideal constraints those ones

for which the virtual work of the constraint forces vanishes

δ= ⋅δ=

∑R

(18.1.2)

for any system of virtual displacements of the considered mechanical system

S . If the

virtual displacements

δr

are arbitrary (we have not constraints), then the relation

(18.1.2) is fulfilled if and only if

==R0, 1,2, ...,

in. But if p holonomic constraints

of the form (3.2.8) and

m non-holonomic constraints of the form (3.2.13) take place,

then we can use the method of Lagrange’s multipliers, the constraint forces being of the

form (3.2.37). In this case, the real elementary work of the constraint forces is given by

(3.2.37'); this work vanishes if

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

fl p



, and ==

0, 1,2, ...,

kmα

(conditions in which the real displacements belong to the set of virtual displacements,

case of catastatic constraints), in particular in case of scleronomic constraints. Among

the mechanical systems subjected to ideal constraints we mention the particle subjected

to constraints of contact, frictionless (compelled to stay on a fixed or movable curve or

surface), or the discrete mechanical system for which the distance between two particles

is function only on time (constraints at distance). The classical ideal constraints of the

rigid solid are: the simple support, the hinge and the built-in mounting. Ideal constraints

are also the constraints of the rigid solid which can slide without friction along an axis

(or, in general, along a superposable curve), the constraints of a rigid solid which can

slide frictionless on a plane (or on a given surface), as well as the constraints of the

rigid solid which is rolling or pivoting without sliding on a plane (or on a arbitrary

surface); in the first cases, the virtual displacement is normal to the constraint force and

in the last cases the virtual displacement is zero. We mention also the systems of two

rigid solids considered in Sect. 3.2.2.9. As well, a mechanism (a discrete mechanical

system formed of two rigid solids linked rigidly or by hinges or by perfectly smooth or

rough surfaces) can be considered as a mechanical system subjected to ideal constraints.

We mention also the constraints by threads, which are not always bilateral.

Among the constraints with friction, we consider that one which corresponds to the

sliding friction, for which the analytical methods of mechanics could be adapted.

We assume that the virtual displacement

δr takes place in an interval of time δt

(which can be the interval of time

Δt

in which take place the possible displacements,

the difference of which is the virtual displacement). In this case, the virtual

displacement of a point of a free rigid solid with respect to the inertial frame of

reference

′

will be given by

MECHANICAL SYSTEMS, CLASSICAL MODELS

∗∗

′′

δ= δ+ δ×rv r()

ttω ,

(18.1.3)

where

∗

′

δv ()tt

and

∗

δ ()ttω are two arbitrary vectors, which define the virtual roto-

translation at the moment

, r being the position vector with respect to the non-inertial

frame

R ; the position of this solid depends thus on six scalar parameters. We can

express the virtual displacement

′

r in the form

′′

δ=δ +δ×

rr r

(18.1.3')

too, where

represents a virtual rotation of the frame of reference R with respect to

the frame

R (of components δψ , δθ and δϕ , corresponding to Euler’s angles).

In the case in which the rigid solid has a fixed point (e.g., the point

O ), we have

∗

′

, its position depending only on three scalar parameters. As well, if the point

O of the rigid solid has a motion given by

′′

rr()

t , then the real displacement will

be given by

′′

=+ ×

rv rdd(d)

ttω , with

′′

vrd/d

t ; in exchange, for a virtual

displacement (compatible with the constraints at the moment

t ) the point O must have

the position fixed at the moment

t , while

∗

′

δ= δ×

rr()tω , the rigid solid having

three degrees of freedom too.

The virtual work of the forces of inertia is expressed in the form

′′

δ=− ⋅δ

∫∫∫

rar r() (;) d

WtVμ ,

(18.1.4)

where the virtual displacement

′

r is given by (18.1.3),

∗

′

and

∗

ω being arbitrary

functions on time. Noting that

′

ar(;)t

represents the accelerations field in the real

motion of the rigid solid, we can write, with respect to the frame of reference

′

R (we

have

′′

=−rr r

()

[

]

()

{

}

∗∗ ∗

′

′′′ ′

δ=− −×δ⋅+δ⋅vrA D

Wttωω,

(18.1.4')

where

′

is the dynamic resultant of the rigid solid, given by (14.1.27), while

′

the dynamic moment with respect to the pole

′

, given by (14.1.28).

Analogously, in the case of the rigid solid, the virtual work of the constraints forces

of torsor

RM{, }

, will be expressed in the form

()

[

]

()

∗∗ ∗

′′

δ=− −×δ⋅+δ⋅ =

vrR M

RO O O

Wttωω

(18.1.4'')

18.1.1.2 The Representative Space

A basic aspect of the analytical methods of calculation consists in replacing the study

of the motion of a discrete mechanical system

S of geometric support Ω in the space

E by the study of the motion of a single geometric point, called representative point,

Lagrangian Mechanics

in another space, conveniently built up, called representative space. E.g., in Sects.

3.2.2.2 and 11.1.1.5 has been

introduced the representative space

, by passing

from the mechanical system

S of n particles

P , = 1,2, ...,in, to a representative

point

, of generalized co-ordinates

1,2, ...,3kn=

, with the aid of the relations

==−+= =

()

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

Xxk i ji nj ,

(18.1.5)

of the form (3.2.9), where

()i

x is the component along the

Ox -axis of the position

vector

; the real generalized displacements of this point are d

X , the possible

generalized displacements are

X and the virtual generalized displacements are

δ=, 1,2, ...,3

Xk n, the respective properties corresponding to those in the space

E .

In these conditions, the constraints of the mechanical system

S become constraints of

the representative point

P .

We use also the notations

===−+= =

()

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

bb kilinl

αα

(18.1.5')

of the form (3.2.9), where

()i

α are the components along the

Ox -axis of the given

vectors

α , which specify the constraints relations (3.2.13); these relations take the

form

≡+=

∑

d d , 1,2,...,

jk k

bX btj mω ,

(18.1.6)

corresponding to the relations (3.2.13''), where =

1, 2 3

( ,..., ; ),

jk jk

bbXXXt

= 0,1,...,3kn

, or the form

+= =

∑

0, 1,2,...,

jk k

bV b j m,

(18.1.6')

corresponding to the relations (3.2.13'''), where

==d/d

kk k

VX Xt



, = 1,2,...,3kn,

are the components of the generalized velocity of the point

in the representative

space

. Obviously, we assume that the constraints are linearly independent, hence

that the matrices

[]

b are of rank m . If a Pfaff differential form

is not completely

integrable, then the respective constraint is non-holonomic; taking into account the form

(18.1.6') of these relations, then the constraint is called kinematic too. We assume that,

in general, the mechanical system

S has m non-holonomic constraints (the

representative point

P is on a non-holonomic manifold with −3nm dimensions, at

MECHANICAL SYSTEMS, CLASSICAL MODELS

the intersection of

m non-holonomic hypersurfaces in the space

E ). We can express

these constraint relations also in the form

δ= =

∑

0, 1,2,...,

jk k

bX j m,

(18.1.7)

corresponding to the relations (3.2.15''), by means of the virtual displacements, or in the

form

∑

0, 1,2,...,

jk k

bV j m

(18.1.7')

corresponding to the relations (3.2.15'''), where

=δ Δ

VXt are the virtual

generalized velocities (

Δt is the interval of time in which takes place a possible

displacement).

0, 1,2,...,

bj m, then the mechanical system is called catastatic. In this

case, the set of virtual generalized displacements coincides with the set of possible

generalized displacements and the constraints are called catastatic; as well, the set of

virtual generalized velocities (which coincides with the set of possible generalized

velocities) contains also the case in which these velocities vanish, hence the case of rest

with respect to an inertial frame of reference (the denomination used above being thus

justified). A mechanical system which is not catastatic is called non-catastatic; in this

case

≠

b . If the conditions

∂

==== =

∂

0, 0, 1,2,..., , 1,2,...,3

bb j mk n



(18.1.8)

are verified, then the constraints do not depend explicitly on time and are called

scleronomic; otherwise, the constraints are rheonomous. We notice that a scleronomic

mechanical system is catastatic too, but a catastatic one may be rheonomous too; a non-

catastatic mechanical system is always rheonomous.

If a Pfaff differential form is completely integrable (for a system of the form (18.1.6)

we can use the Theorem 3.2.2 of Frobenius), then the corresponding constraint is

holonomic, being of the form

≡==

12 3

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

lk l

fXt fXX X t l p,

(18.1.9)

hence of the form (3.2.8''); such a constraint is also called geometric. In general, we

assume that the mechanical system

S has p holonomic constraints (the representative

point

P is on a manifold with −3np dimensions, at the intersection of p hyper-

surfaces in the space

E ). These finite relations can be written also in the differential

form

∂

+==

∂

∑

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

Xft l p



(18.1.10)

Lagrangian Mechanics

or in the form

∂

+= =

∂

∑

0, 1,2,...,

Vf l p



(18.1.10')

so that

∂

== =

∂

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

bl pk n

Analogously, using the virtual generalized displacements, we can write

∂

δ= =

∂

∑

0, 1,2,...,

Xl p

(18.1.11)

∂

∑

0, 1,2,..., .

Vl p

(18.1.11')

We assume that also these constraints are linearly independent, the matrices

∂∂[/ ]

fX being of rank p .

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

fl p



, the mechanical system S is catastatic; at the same time we

have

∂

⎛⎞

∂

⎜⎟

∂∂

⎝⎠

∂∂



too, so that the respective system is also scleronomic. We notice that these two

properties of the constraints coincide in case of the holonomic systems, for which

≡=

1, 2, 3

( ) ( ..., ) 0

lk l

fX fXX X .

(18.1.12)

Let be two neighbouring, possible, simultaneous (at the moment

t ) positions

()

PX and +δ'( )

PX X , of the same representative point of the holonomic

discrete mechanical system

S . The constraint relations will be of the form

=()0

fX , +δ =()0

lk k

fX X ,

= 1,2,...,lp

. Expanding into a Taylor series, we

can write

∂

+δ+=

∂

∑

( ) ... 0

lk k

fX X

Neglecting the terms of higher order and taking into account the constraint relations,

MECHANICAL SYSTEMS, CLASSICAL MODELS

we find the conditions (18.1.11). One observes thus that

X represent those

differential generalized displacements which must be effected by the representative

point to pass from a position to another one, at the same moment

t , being thus virtual

displacements (corresponding to their definition, as difference of possible

displacements).

=Fi

()i

F represents the resultant of the given forces which act upon the particle

=, 1,2,...,

Pi n

, in the space

, then we can introduce, in the space

, the given

generalized forces (shortly, the generalized forces)

Q , = 1,2,...,3kn, which act upon

the representative point

, in the form

==−+= = =

()

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

QFk i ji nj k n.

(18.1.13)

As well, by using the axiom of liberation from constraints, one introduces the

constraint forces

=Ri

()

R , = 1,2,...,jn, which act upon the particle

P in the

space

; we are thus led to constraint generalized forces

, = 1,2,....3kn, which

act in the space

E upon the representative point P in the form

==−+= =

()

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

RRk i ji nj .

(18.1.13')

In what concerns the mass

m of the particle

P , it is convenient to denote

===

(1) (2) (3)

iii

mm m m,

= 1,2,...,in

so that

==−+= = =

()

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

Mmk i ji nj k n,

(18.1.14)

can be considered as

generalized masses of the representative point

in the space

E .

The real elementary work of the given generalized forces is given by

== =

∑∑ ∑

() ()

11 1

ddd

ij k

WFxQX,

(18.1.15)

while the real elementary work of the constraint generalized forces is expressed in the

form

== =

∑∑ ∑

() ()

11 1

ddd

kkk

ij k

WRxRX

(18.1.15')

Analogously,