Teodorescu P.P. Mechanical Systems, Classical Models Volume II: Mechanics of Discrete and Continuous Systems

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MECHANICAL SYSTEMS, CLASSICAL MODELS

== =

∑∑ ∑

HH v r

11 1

nn n

iiiii

ii i



(11.1.1')

for a discrete mechanical system

S of n particles

P of masses

m and of position

vectors

, 1,2,...,in= (the momentum of a single particle

iii

m=Hv



The moment of the momentum

with respect to the pole O of the considered

frame is the moment of momentum of the particle

P , with respect to that pole, and is

given by

ii iii iii i

Oi Oi

mm m=×=×=×=



ΩKrH rv rr

where we have introduced the areal velocity (5.1.16) too. The moment of momentum

(angular momentum) of the mechanical system

S with respect to the pole O (which

can be a given fixed point, immovable with respect to an inertial frame) is given by

dd2d

mmm

ΩΩ Ω

=× =× =

∫∫ ∫



ΩKrv rr

;

(11.1.2)

in case of a discrete mechanical system, we obtain

11 1 1 1

() 2

nn n n n

ii i ii iii i

OOi Oi

ii i i i

mm m

== = = =

==×=×=×=

∑∑ ∑ ∑ ∑



ΩKK rHrv rr

. (11.1.2')

In components, we have

()

Hmx

∑



() () ()

ii i

Oj jkl

kl Oj

KmxxmΩ

= ∈ =

∑∑



, 1, 2, 3j = .

(11.1.3)

The notion of torsor, introduced in Chap. 6, Sect. 1.1.1 for a single particle, allows

us to write

{

}

{

}

τ=HHK;

(11.1.4)

hence, the set formed by the linear and the angular momenta of a mechanical system

represents the torsor of the system

{

}

H , formed by the momenta of the component

particles of the mechanical system with respect to the considered pole. For the problems

of dynamics, the torsor plays thus a rôle analogous to that played in statics.

11.1.1.2 Work. Kinetic and Potential Energy. Conservative Forces

The notion of work has been introduced in Chap. 3, Sect. 2.1.2, in the form of

elementary work (3.2.3) (we omit the adjective “real”) of a system of given forces

F ,

applied at the points

P of position vectors

r , 1,2,...,in= . In general, we denote by

F the given external forces; the given internal forces

F , which verify a relation of the

11 Dynamics of Discrete Mechanical Systems

form (1.1.81) (inclusive a relation of the form (2.2.50)), put in evidence the influence of

the particle (point)

P upon the particle (point)

P . The elementary work

int

dW of the

given internal forces is expressed in the form (3.2.4); we notice that

int

dW is a non-

negative quantity, vanishing if and only if the mechanical system

S is non-deformable.

Analogously, the elementary work of the constraint external forces

R ,

1,2,...,in=

, is given by

=⋅

∑

Fr,

(11.1.5)

while the elementary work of the constraint internal forces

ik ik

R=Ru (

ik ki

RR= are

positive quantities in case of repulsive constraint forces and negative ones in case of

attractive such forces) has the remarkable expression

()

int

11 11

dddd

nn nn

Rikkikikik

ik k ik k

WRr

−−

=+ = =+ =

=⋅+⋅=

∑∑ ∑∑

RrRr

ik ik

∑∑

, ik≠ ,

(11.1.5')

where

ik ik k

r==−rurr, vers

=ur, is the vector which links two particles

(points) of the mechanical system

We can introduce the kinetic energy

/2 /2

iii ii

Tmv m==



r , which is a quantity

of state of the particle

P . The kinetic energy of the mechanical system S is, in this

case, given by

Tvm m

ΩΩ

∫∫



(11.1.6)

and if this system is discrete, then it results

ii ii

Tmv m

∑∑



r .

(11.1.6')

If there exists a function

U so that dd

ik ik ik

UFr= , then we obtain a potential U

of the form (3.2.6'); in this case, the internal forces are conservative forces (the

mechanical system is conservative) and we can introduce the potential energy

VU=−

, being thus led to the mechanical energy (6.1.15).

Immaterial whether the forces are external or internal ones, we introduce also the

virtual work of the given forces in the form (3.2.3'), while the virtual work of the

constraint forces is given by (3.2.7').

As in Chap. 6, Sect. 1.1.3, one can introduce the notions of power and mechanical

efficiency for a mechanical system

S. Thus, starting from the elementary work (3.2.3),

we get the power of the given external forces in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

iii

== ⋅= ⋅

∑∑

FFv

;

(11.1.7)

as well, the power of the internal forces is

−

== =+= ==

== ⋅= =

∑∑ ∑ ∑ ∑∑

int

11 11 11

dd2d

nn nn nn

ik ik

ik ik ik

ik ikk ik

PFF

ttt

, ik≠ .

(11.1.7')

In case of conservative internal forces, which derive from a potential

U, we obtain

int

;

(11.1.7'')

we can make analogous observations concerning the power

P of the external forces.

Analogously, we may introduce the power of the constraint forces too.

11.1.1.3 Formulation of Problems of Mechanical Systems in Motion

Let us consider a free discrete mechanical system

S, formed by n particles

P , of

masses

m and position vectors

r , acted upon by the given external forces

(eventually, resultants of given forces, applied at the points

) and by the given

internal forces

ik≠

, , 1,2,...,ik n= . According to the principle of action of

forces, Euler showed that the motion of the mechanical system

S is governed by the

system of vector differential equations (to simplify the notation, the sums of terms with

two indices will be denoted by “prime” if the case of equal indices is excluded)

ii i

∑



rF F, 1,2,...,in= ;

(11.1.8)

in components, we may write

() () ( )

ii ik

jj j

mx F F

∑

 , 1,2,...,in= , 1, 2, 3j = .

(11.1.8')

The position of the free discrete mechanical system

S at a given moment t can be

determined if one knows the functions

() ()

()

xxt=

, 1,2,...,in= , 1, 2, 3j = ;

hence, this mechanical system has 3

n degrees of freedom. In general,

12 12

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

ii i

tt=≡

  

FFrr rrr r Frr, (,;)

ik ik l l



FFrr,

ik≠

, , , 1,2,...,ikl n= ,

the motion of each particle depending on the motions of all other particles. As well, the

given internal forces can be determined (that is, may be given) only taking into account

some hypotheses concerning the structure of the mechanical system (rigidity,

deformability – elasticity, plasticity, viscosity etc.), hence if a mathematical model of it

is set up; in fact, in case of deformable mechanical systems, the motion of the particles

and of the internal forces may be determined only simultaneously.

In the first fundamental problem (the direct problem) which is put the forces

F and

F , ik≠ ,

, 1,2,...,

ik n=

, are given, and one must determine the trajectories of the

particles, hence the vector functions

()

t=rr, and the velocities () ()

ii i

tt==



vv r.

The problem is solved by integrating the system of

n vector equations (11.1.8) or the

system of 3

n scalar equations of second order (11.1.8') with certain boundary

conditions. For the most part, initial conditions (at the initial moment

tt= ) of the

form

()

t =rr

()

t =vv

1,2,...,in=

(11.1.9)

are put; one may put also other boundary conditions (e.g., bilocal conditions at the

moments

tt= and

tt= ). In certain conditions, sufficiently large, the solution of the

problem is unique.

In the second fundamental problem (the inverse problem) the motion of the particles

P is known (hence, the position vectors

r are given) and the forces

F and

F ,

ik≠ , , 1,2,...,ik n= , which provoke this motion, have to be determined. In general,

the solution of the problem is not unique.

We mention also the mixed fundamental problem in which some elements which

characterize the motion and some elements which characterize the forces are given; one

must determine the other elements remained unknown, to can specify entirely the

motion of the particles and the forces applied upon them. This problem has also not a

unique solution, in general, but only in certain conditions.

As in the case of a single particle (see Chap. 6, Sect. 1.1.4 too), we must impose

other conditions to the law which expresses the forces, so that the two last problems be

determined.

Using the results in Chap. 6, Sect. 1.1.5, we may express the equations of motion in

other systems of co-ordinates, as it is more convenient from the point of view of the

computation.

The system of equations of motion (11.1.8) of the discrete mechanical system

S is

written in an inertial frame of reference

R, with respect to which the principles which

are at the basis of the Newtonian model of mechanics are supposed to be verified. They

remain, further, in the same form with respect to any other inertial frame which is

deduced from the first one by a rectilinear and uniform motion of translation, hence by

a transformation of co-ordinates belonging to the Galileo-Newton group (see Chap. 6,

Sect. 1.2.3 too).

By means of the notations (see Chap. 3, Sect. 2.2.2 too)

()

3( 1)

−+

()

3( 1)

−+

, 1,2,...,in= , 1, 2, 3j = ,

(11.1.10)

11 Dynamics of Discrete Mechanical Systems

11.1.1.4 Theorems of Existence and Uniqueness

MECHANICAL SYSTEMS, CLASSICAL MODELS

we can pass from the geometric support

Ω of the discrete mechanical system S in the

space

E (formed by the geometric points

P ) to a representative geometric point P (of

co-ordinates

, 1,2,...,3kn= ) in the representative space

; we also introduce

the notations

() ( )

3( 1)

iik

QFF

−+

⎛⎞

⎜⎟

⎝⎠

∑

, 1, 2,...,in= , 1,2, 3j = .

(11.1.10')

Therefore, we replace the study of the motion of the discrete mechanical system

S by

the study of the motion of the representative point

P in the representative space

E ;

the motion of the point

P is governed by the system of differential equations



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

(11.1.11)

We replace this system of

3n differential equations of second order by a system of

6n differential equations of first order, written in the normal form

XV=



, =



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

(11.1.11')

where

()

;

kkl

VVXt= ,

()

kkll

QQXVt= , , 1,2,..., 3kl n= . Such a system is non-

autonomous; if the time does not intervene explicitly in

V and

Q , then the system is

autonomous (or dynamic). We associate the initial conditions (corresponding to the

conditions (11.1.9))

()

Xt X= ,

()

Vt V= , 1,2,..., 3kn= ,

(11.1.11'')

to this system, so that the boundary value problem (11.1.11'), (11.1.11'') becomes a

problem of Cauchy type. The boundary value problem (11.1.8), (11.1.9) is equivalent to

the boundary value problem (11.1.11'), (11.1.11''); for the latter problem, one may state

Theorem 11.1.1 (of existence and uniqueness; Cauchy-Lipschitz). If the functions

and

Q , 1,2,...,3kn= , are continuous on the interval (6n+1) – dimensional D,

specified by

00kk kkk

XX XXX−≤≤+,

00kk kkk

VV VVV−≤≤+,

tt ttt−≤≤+,

,, const

XVt= , 1,2,...,3kn= , and defined in the space

Cartesian product of the phase space (of canonical co-ordinates

12 3

, ,...,

XX X

12 3

, ,...,

VV V ) by the time space (of co-ordinate t), and if the Lipschitz conditions

()

;;

kl kl l l

VXt VXt X X

−≤−

∑

()

,; ,;

kll kll l l l l

QXVt QXVt X X V V

−≤−+−

∑

for

1,2,...,3kn= , where T is an independent of

V and t time constant, while τ is a

time constant equal to unity, are satisfied, then it exists a unique solution

()

XXt= ,

()

VVt= of the system (11.1.11'), which satisfies the initial conditions (11.1.11'')

and is defined on the interval

tTttT−≤≤+, where

min , , ,

⎛⎞

≤

⎜⎟

⎝⎠

()

max ,

VQτ=V in D.

According to Peano’s theorem, the existence of the solution is ensured by the

continuity of the functions

and

on the interval D. For the uniqueness of the

solution, Lipschitz’s conditions must be fulfilled too; the latter conditions can be

replaced by other more restrictive ones according to which the partial derivatives of

first order of the functions

and

, 1,2,...,3kn= , must exist and be bounded in

absolute value on the interval

D, as it was shown by Picard, using a method of

successive approximations. As a matter of fact, the Theorem 11.1.1 can be

demonstrated by an analogous method. We must mention that the conditions in

Theorem 11.1.1 are sufficient conditions of existence and uniqueness, which are not

necessary too.

The existence and the uniqueness of the solution have been put in evidence only on

the interval

[

]

,tTtT−+, in the vicinity of the initial moment

(in fact,

can be

an arbitrary chosen moment, not necessarily the initial one); taking, for instance,

tT+

as initial moment, by repeating the above reasoning, it is possible to extend the

solution on an interval

TT>

(obviously, if the sufficient conditions of

existence and uniqueness of the Theorem 11.1.1 are fulfilled in the vicinity of this new

initial moment). We can obtain thus a prolongation of the solution for

[

]

,ttt∈ ,

corresponding to an arbitrary interval of time in which the considered mechanical

phenomenon takes place or even for

(,)t ∈−∞+∞.

As in the case of a single particle (see Chap. 6, Sect. 1.2.1), we can put in evidence

some important properties of this solution; we thus state:

Theorem 11.1.2 (on the continuous dependence of the solution on a parameter). If the

functions

()

VXtμ ,

()

,;,

kll

QXVtμ are continuous with respect to the parameter

[

]

,μμμ∈ and satisfy the conditions of the theorem of existence and uniqueness,

and if the constant

T of Lipschitz does not depend on μ, then the solution (, )

Xtμ ,

(, )

Vtμ , 1,2,...,3kn= , of the system (11.1.11') which satisfies the conditions

(11.1.11'') depends continuously on

μ.

Theorem 11.1.3 (on the analytical dependence of the solution on a parameter;

Poincaré). The solution

(, )

Xtμ , (, )

Vtμ , 1,2,...,3kn= , of the system (11.1.11')

which satisfies the conditions (11.1.11''), depends analytically on the parameter

[

]

,μμμ∈ in the neighbourhood of the value

μμ= if, in the interval

[

]

,μμ×D , the functions

V and

Q are continuous with respect to t and analytic

with respect to

X ,

V , 1,2,...,3kn= , and μ.

11 Dynamics of Discrete Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

Theorem 11.1.4 (on the differentiability of the solutions). If in the vicinity of a point

()

XV tP the functions

()

;

VXt and

()

kll

QXVt, 1,2,...,3kn= , are of

class

C , then the solutions ()

Xt and ()

Vt of the system (11.1.11'), which satisfy

the initial conditions (11.1.11''), are of class

in that vicinity.

One can state theorems (analogue to the Theorem 11.1.2) concerning the continuous

dependence of the solution on the initial conditions or on several parameters.

The points

P in the vicinity of which the boundary value problem (11.1.11'),

(11.1.11'') has not solution or, even if the solution exists, that one is not unique, are

called singular points; the integral curves (in the space

) formed only of singular

points are called singular curves, the corresponding solution being a singular solution.

For the singular points there are necessary supplementary conditions, which can lead to

the choice of one of the branches of the multiple solution.

11.1.1.5 First Integrals. General Integral. Constants of Integration

An integrable combination of the system (11.1.11') may be, e.g.,

()

12 3 12 3

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

XX X VV V t= ,

(11.1.12)

obtaining thus a finite relation of the form

()

12 3 12 3

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

XX X VV V t C= , constC = ,

(11.1.12')

hence a link between the co-ordinates

X and the components

V of the velocity at the

time

t; the function f, which is reduced to a constant along the integral curves, is called

first integral of the system (11.1.11').

If we determine

6hn≤ first integrals for which

()

XVt C=

const

C = , 1,2,...,jh= ,

(11.1.13)

the matrix

()

12 3 12 3

, ,...,

, ,..., , , ,...,

XX X VV V

∂

⎡⎤

≡

⎢⎥

∂

⎣⎦

(11.1.13')

being of rank

h, then all these first integrals are independent (for the sake of simplicity

we say not “functional” independent) and we may express

h unknown functions of the

system (11.1.13) in terms of the other ones; replacing in (11.1.11'), the problem is

reduced to the integration of a system of equations with only

6nh− unknowns. If

6hn= , then all the first integrals are independent and the system (11.1.13) of first

integrals determines all the unknown functions. For

6hn> the first integrals (11.1.13)

are no more independent, so that we cannot set up more than

6n independent first

integrals.

Assuming that

6hn=

and solving the system (11.1.13), we get (the matrix M is a

square matrix of order

6n for which det 0≠M )

()

12 6

; , ,...,

XXtCC C= ,

()

12 6

; , ,...,

VVtCC C= , 1,2,...,3kn= ,

(11.1.14)

where

= const

C , = 1,2,...,6ln, hence the general integral of the system of

equations (11.1.11'). The general integral of the system of vector equations (11.1.8) is,

analogously,

()

12 6

;,,...,

tC C C=rr , 1,2,...,in= ;

(11.1.15)

eventually, we have

()

12 2

; , ,...,

t=rr KK K , 1,2,...,in= ,

(11.1.15')

where

const

JJJJG

, 1,2,...,2ln= . Thus, 6n scalar constants or 2n vector constants of

integration are put into evidence. Because the vector functions (11.1.15) or (11.1.15')

verify the equations (11.1.8) for any constants of integration, we can state that the same

mechanical system

S, acted upon by the same system of forces, may have different

motions. Imposing the initial conditions (11.1.9), written in the form (11.1.11''), we

obtain

()

12 6

;,,...,

XtCC C X= ,

()

12 6

; , ,...,

VtCC C V= , 1,2,...,3kn= ;

the conditions (11.1.11'') being independent, we may write

()

00 0 00 0

12 3 12 3

12 6

, ,..., , , ,...,

det 0

, ,...,

XX X VV V

CC C

∂

⎡⎤

≠

⎢⎥

∂

⎣⎦

and, according to the theorem of implicit functions, we deduce

()

00 0 00 0

01 2 3 12 3

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

CCtXX XVV V= , 1,2,...,6jn= .

Thus, we obtain, finally,

()

;, ,

kk ll

XXttXV=

()

;, ,

kk ll

VVttXV=

1,2,...,3kn=

(11.1.16)

()

;, ,

ii jj

tt=rr rv,

()

;, ,

ii jj

tt=vv rv, 1,2,...,in= .

(11.1.16')

In the frame of the conditions of the theorem of existence and uniqueness, the

principle of the action of forces and the principle of the initial conditions determine,

univocally, the motion of the mechanical system

S in a finite interval of time; by

11 Dynamics of Discrete Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

prolongation, this statement may become valid for any

t. The deterministic aspect of

Newtonian mechanics is thus put in evidence.

Sometimes, a calculation in two steps can be made, as in the case of only one particle

(see Chap. 6, Sect. 1.2.2). Starting from the system of differential equations of second

order (11.1.11), we can find integrable combinations, leading to

3n first integrals of the

form

()

12 3 12 3

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

XX X XX X t Cϕ =

 

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

(11.1.17)

if, starting from these relations, in a second stage, we build up other

3n integrable

combinations, leading to the first integrals

()

12 3 12 3

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

knk

XX X tCC C Cψ

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

(11.1.17')

then the problem is solved. Indeed, we notice that

()

12 3

, ,...,

det 0

, ,...,

XX X

ψψ ψ∂

⎡⎤

≠

⎢⎥

∂

⎣⎦

finding thus the first group of relations (11.1.14).

11.1.2 General Theorems. Conservation Theorems

We present in what follows the general theorems of mechanics (the theorem of

momentum, the theorem of moment of momentum and the theorem of kinetic energy)

with respect to an inertial frame of reference (for the sake of simplicity, we do not

mention it in the following) in case of a free discrete mechanical system and in case of a

discrete mechanical system subjected to constraints, as well as the corresponding

conservation theorems. Among the applications of these results, we mention the

problem of

n particles. We notice that, in fact, the general theorems are differential

consequences of the system of equations of motion (11.1.8); we may also say that these

theorems represent necessary conditions which must be verified in the motion of the

mechanical system

S with respect to an inertial frame of reference.

11.1.2.1 Theorem of Momentum. Theorem of Motion of the Centre of Mass

Summing the equations of motion with respect to an inertial frame of reference (11.1.8)

for all the particles of the mechanical system

S and taking into account the relation

(2.2.50) verified by the internal forces, we obtain

ii i

∑



where the internal forces disappear. Introducing the momentum (11.1.1'), it results,

finally,

== =

∑



HFR

()

HFR

∑



1, 2, 3j =

(11.1.18)

and we may state

Theorem 11.1.5 (theorem of momentum). The derivative with respect to time of the

momentum of a free discrete mechanical system is equal to the resultant of the given

external forces which act upon this system.

We mention that, in this theorem, both the derivative



and the resultant of the

given external forces are free vectors. As well, the first relation (11.1.18) maintains its

validity if it is projected on a plane or on an axis (the second group of relations

(11.1.18)).

Starting from the relation (3.1.2) which gives the position vector of the centre of

mass

C of the discrete mechanical system S, differentiating with respect to time in the

fixed (inertial frame) and taking into consideration (11.1.1'), we obtain



(11.1.19)

and we can state

Theorem 11.1.6 The momentum of a mechanical system is equal to the momentum of

the centre of mass of this system, at which we consider that the whole mass of it is

concentrated.

We notice that the relation (11.1.19) is of the type of relation (3.1.9) for the polar

static moments; we put thus in evidence the importance of the centre of mass

C, which

can replace – in certain situations – the whole mechanical system

S. The Theorem

11.1.6 is valid for any mechanical system, either if it is free or it is subjected to

constraints.

Differentiating the relation (11.1.19) with respect to time, in the considered fixed

frame, and taking into account (11.1.18), we may write

∑



FRρ= ,

(11.1.19')

stating thus

Theorem 11.1.7 (theorem of motion of the centre of mass; Newton). The centre of mass

of a free discrete mechanical system is moving as a free particle at which would be

concentrated the whole mass of the system and which would be acted upon by the

resultant of the given external forces.

This theorem allows an independent study of the motion of the centre of mass

(even if this one does not belong to the mechanical system S ), in a first stage of study

of the system in its totality; in a second stage, one can consider the motion of the system

with respect to the centre of mass

C. As well, the Theorem 11.1.7 represents a

justification for the modelling as a particle in the study of mechanical systems.

Starting from the relation (11.1.18), we may write

11 Dynamics of Discrete Mechanical Systems