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

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so that

cot

∞



;

the relation (13.1.71''') leads thus to Rutherford’s formula

cosec

∞

⎛⎞

⎜⎟

⎝⎠



(13.1.75)

which gives the efficacious diffusion section in the frame

R . Using this formula and

measuring experimentally, by recording on negatives in the Wilson room, the initial and

final directions of the particles, hence the angles of diffraction

, E. Rutherford has put

in evidence the existence of the atomic nucleus (charged by a positive electric charge

equal to the number order – after Mendeleev’s table – of the chemical element which is

in Wilson’s room and produces this effect). These theoretical results have been verified

by numerous experimental researches due to H. Geiger, E. Marsden, van der Boek

(1913), J. Chadwick (1920) etc.

It is interesting to see that the above results hold, with a good approximation, in the

frame of the model of quantum mechanics too, so far as one can write a conservation

theorem of momentum, the diffusion being an elastic one.

13.1.2.2 Plastic Collision of Discrete Mechanical Systems. Space of Plastic

Collisions

Let us firstly consider the case of two particles (e.g., two small spheres of masses

and

m ), which are in collision with the velocities

v and

v , respectively, remaining

then glued together and continuing their motion as a single particle of mass

Mm m=+

, with the common velocity

11 22

()mm

=+vvv

Mm m=+,

(13.1.76)

obtained from the conservation theorem of momentum (assuming the absence of

external percussions). The loss of kinetic energy is

()

0222

11 22 1 2

() ( )

TTT mvmv mmv

′′′

Δ=−= + − + ,

and the kinetic energy of lost velocities is given by

[]

01122

()()

Tm m=−+−vv vv ;

in this case, we can write the generalized theorem of Carnot in the form (13.1.23'') (the

restitution coefficient

k is equal to zero), so that

012

() ( )

TT mΔ== −vv,

111

mm m

(13.1.77)

13 Other Considerations on Dynamics of Mechanical Systems

163

MECHANICAL SYSTEMS, CLASSICAL MODELS

having

′′′

≥ .

The phenomenon of plastic collision of two particles leads thus, from a mathematical

point of view, to a law of composition of masses and velocities. Following this order of

ideas, let be

P a particle of mass m and velocity v; we denote the element mass-velocity

which characterizes the particle in motion by

(,)Sm≡ v . We assume that the mass

and the velocity are functions of time of class

C , the mass m (t ) being a positive

function (

() 0mt > ). In the set

{}

S=M of elements S we introduce a relation of

equivalence denoted by

K; we say thus that two elements

,SS∈ M are equivalent

(we denote

SS∼ ) if they have the same momentum (

11 22

mm=vv). This is written

in the form

() ()

KS KS= .

(13.1.78)

One can easily prove that the relation

K defined on M is an equivalence relation,

because it is reflexive, symmetrical and transitive. The equivalence relation

introduced in the set

M realizes a partition of M into classes of equivalence. Let be

/KM this set of classes of equivalence; it represents the quotient set of M by K.

Thus, corresponding to the relation

K, to each element S ∈ M will correspond the

class of equivalence



, hence



SS K→∈M .

(13.1.79)

In the set

/KM of classes of equivalence, we define a law of internal composition,

denoted additively in the form (



(,)Sm≡ v ,



(,)Sm≡ v ,



,/SS K∈ M )



()



11 22

12 1 2

,,(,)

SS S S m m M

⎛⎞

→+= + ≡

⎜⎟

⎝⎠

(13.1.80)

One can verify the property of commutativity (



12 21

SS SS+=+) and the property of

associativity (



()



()

12 3 1 23

SS S S SS++=++). The neutral element



/OK∈ M

is given by the relation



(,)Om= 0

, hence by



()

KO = 0 . From the definition of that

element, it results



11 1

(,)(,) ,

SOm m mm S

⎛⎞

+= + = + =

⎜⎟

⎝⎠

because



()



()

KS O m KS+= =v , and the property of null effect of the neutral

element is put in evidence. The law of internal composition defines thus an Abelian

group on

/KM .

164

We define a law of external composition in a multiplicative notation (

(,)Sm≡ v ,

α ∈  )



()



()

,SSmαα α→= v

(13.1.80')

The element

′



, opposite to the element



S is specified by



(, ) (,)Sm m S

′

=−=− =−



; obviously, we have



(,) (, )SS m m

′

+= + −





(2 , )mO==0 , so that



()

KS S

′



and we can write



′

=−



. We notice the

property



1(,)Sm S==v for a real number 1; as well, one has the property of

associativity (



() ()() (, )SS Smα β β α αβ αβ===v , ,αβ∈  ). Analogously, we

can put in evidence a property of distributivity with respect to addition of real numbers

(





()SSSαβ α β+=+,

,αβ∈ 

) and a property of distributivity with respect to the

law of internal composition (



()



12 1 2

SS S Sααα+= +,

α ∈ 

). We have shown

thus that the quotient set

/KM , in which we have defined internal and external laws

of composition, constitutes a vector space on the field R of real numbers.

It is important to observe that the internal composition law introduced above

corresponds to the plastic collision, since the mass of the sum element is equal to the

sum of the masses of the component elements, while the velocity after collision

corresponds to that given by (13.1.76). Also, the quotient set

/KM obtained with the

aid of the equivalence law

()KS m= v corresponds to the mechanical meaning of the

collision phenomenon; indeed, if the elements

S and

S are equivalent, then they

have the same momentum too. This justifies the denomination of space of plastic

collisions given to the vector space thus introduced by W. Kecs and P.P. Teodorescu.

Applying the second principle of mechanics to two equivalent elements

S and

S ,

we may write

11 22 1 2

()()

===vvFF;

hence, the differential equation of motion of two elements of the same class of

equivalence is the same, and this illustrates the mechanical sense of the equivalence

We can write



111 222 333

123

123 123

123

mmm

SSSmmm

mmm

αα α

ααα

⎛⎞

++=++

⎜⎟

⎝⎠

vvv

;

if we take into account the definition of the neutral element, it follows that the relation



()



()

123

KS S S KOααα++ = =0

holds if and only if

13 Other Considerations on Dynamics of Mechanical Systems

165

MECHANICAL SYSTEMS, CLASSICAL MODELS

11 1 22 2 33 3

()()( )mmmαα α++=vvv0.

In this way, the relation



123

SSSOααα++=,

123

,,ααα∈  ,

cannot occur for arbitrary



123

,,SSS, unless

123

0ααα===.

On the other hand, we may have a relation of the form



SOα

∑

α ∈ 

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

3n >

even if not all

α are zero; it results that the space of plastic collisions is three-

dimensional.

Let be now a mapping of

/KM onto the real positive half-straight line

 ,

defined in the form



SS mv→= ;

(13.1.81)

we remark that we have thus introduced the kinetic energy of the element. It may be

shown that the mapping defined by the relation (13.1.81) satisfies the following

properties:



0S ≥ ,

(13.1.82)





12 1 2

SS S S+≤ +



,/SS K∈ M ,

(13.1.82')



SSαα= , α ∈  ,

(13.1.82'')

the equality in relation (13.1.82) involving



/SO K=∈M . Indeed, the first and the

third property follow immediately from the relation of definition (13.1.81); observing

that

 

SST

′

+= and



SS T

′′

+= and taking into account Carnot’s

generalized theorem, one obtains the second property too.

We remark that the space of plastic collisions may be normed by introducing a norm

defined by the relation



Sm= v ; in this case, the distance in the respective space is

given by



()

11 22

d,SS m m=−vv. Although the mapping (13.1.81) does not

constitute a norm in the space of plastic collisions, it allows the introduction of the

notion of distance in the set

/KM , and thus the space of plastic collisions becomes a

metric space. Therefore, we shall call distance in the set

/KM the number



()



11 22

12 1 2

()

SS S S

−

=−=

;

(13.1.83)

166

Taking into account the mapping (13.1.81), it may be easily seen that the distance, as

defined above, satisfies the following conditions:



()

d, 0SS =



()



()

12 21

d, d,SS SS= ,



,/SS K∀∈M ,



()



()



()

13 12 23

d, d, d,SS SS SS≤+,



123

,, /SSS K∀∈M ,

that is the conditions of reflexivity, symmetry and triangle relation, respectively.

Let



((),())Smt t= v

be an element of /KM ; we assume that ()mt and ()tv are

functions of class

C with respect to the time t. If t

′

is a value belonging to the domain

of definition, then there follows



( )( ) ()()

() () () (),

() ()

mt t mt t

St St mt mt

mt mt

′′

−

⎛⎞

′′

−= +

⎜⎟

′

⎝⎠

Multiplying by

1/( )tt

′

− , we obtain



() () ( )

() (),

() ()

St St m

mt mt

tt t

mt mt

′

−Δ

⎛⎞

′

⎜⎟

′

−Δ

⎝⎠

where

() ()() ()()mmttmtt

′′

Δ= −vv v and tt t

′

Δ= −; passing to the limit, we

have



()

d1d

2, ( ) 2,

d2d 2

mmm

tmt m

(13.1.84)

where we took into account the second principle of Newton. The corresponding

equivalence relation will be



⎛⎞

⎜⎟

⎝⎠

(13.1.84')

being thus led to the force acting upon the particle.

Introducing the mapping (13.1.81) for the derived element (13.1.84), it results



() ()

d1 1

22(2)

d22 2

tmm

(13.1.85)

where the acceleration energy (corresponding to the case

m = const) has been marked

out.

We notice that the derivative is a linear operator in the space of plastic collisions and

verifies the relations

13 Other Considerations on Dynamics of Mechanical Systems

167

MECHANICAL SYSTEMS, CLASSICAL MODELS



()



αα= ,

constα =



()



ddd

ttt

+= +

If two elements



S and



S are acted upon by the forces

F and

F , respectively,

then the acceleration energy before collision will be

⎛⎞ ⎛⎞

⎜⎟ ⎜⎟

⎝⎠ ⎝⎠

′

;

(13.1.86)

the acceleration energy after the plastic collision is given by

12 12

() ()

mm mm

⎛⎞

⎜⎟

⎛⎞

=+ =+

⎜⎟

⎝⎠

⎜⎟

⎝⎠

′′

(13.1.86')

The loss of acceleration energy is, in this case,

ححح

()Δ=−

′′′

. The acceleration

energy of the lost accelerations will be, analogously, given by

12 1 12 2

01 2

121 12 2

mm m mm m

⎡⎤

⎛⎞⎛⎞

=−+−

⎜⎟⎜⎟

⎢⎥

⎝⎠⎝⎠

⎣⎦

FF F FF F

(13.1.86'')

It results

حح

() m

⎛⎞

Δ= = −

⎜⎟

⎝⎠

111

mm m

(13.1.87)

hence

حح≥

′′′

, so that the acceleration energy after the plastic collision of two

particles is smaller or at most equal to the acceleration energy before collision (an

analogue of the generalized theorem of Carnot). This relation can be written also in the

form

12 1 2

()

111

222mm m m

≤+

(13.1.87')

We have introduced the notion of derivative in the quotient set

/KM , but we

cannot introduce, analogously, the notion of Riemann integral; instead, we can

introduce the concept of primitive. Thus,



/SK

∗

∈ M is a primitive for



(,) /Sm K=∈v M if the relation



∗



dSSt

∗

∫

(13.1.88)

168

takes place. In the same order of ideas, we say that the element



is constant if its

momentum is constant (



()

constKS m===





vc ); hence, the element



(,/ )Smm= c

is constant in /KM . Corresponding to the relation (13.1.84), we

deduce that



()

d/d d/dKS t t==c0, so that for a constant element



S we may

write



d/dStO=

. We notice that



()



ddd

SS SOS

ttt

∗

+= + =+=;

hence, if



∗

is a primitive of



S , then



∗

+ is a primitive for



S too. Let be



((),())Smt t= v ,



((),())Smtt

∗

= v , where ()mt and ()tv are functions of class

C ; from the first relation (13.1.88) we obtain (2 ,(1/2 )d( )/d ) ( , )mmmtm=vv, so

that, by the equality of momenta, we can write

d( )/dmtm=vv. Integrating, we get

dmmt=

∫

vv, so that the general form of the primitive



∗

is expressed by



d (), ()()d

()

SStmt mttt

∗

⎛⎞

⎜⎟

⎝⎠

∫∫

v ,

(13.1.89)

where

()mt is an arbitrary positive function.

13.2 Dynamics of Mechanical Systems of Variable Mass

The results obtained in Chap. 10, Sect. 3 for the dynamics of the particle of variable

mass will be extended, in what follows, to the case of a discrete mechanical system of

variable mass; thus, the general theorems of dynamics and some interesting particular

problems will be presented. Some cases of continuous mechanical systems will be dealt

with too.

13.2.1 Discrete Mechanical Systems

The problem of a particle of variable mass (studied in Chap. 10, Sect. 3.1.1) is

considered again in the space of plastic collisions, introduced by W. Kecs and P.P.

Teodorescu, modelling the discontinuity of mass in the frame of the theory of

distributions; the general problem of a discrete mechanical system is then presented,

deducing the corresponding universal theorems.

13.2.1.1 Particle of Variable Mass

Let be an element



() ( (), ()) /St mt t K=∈v M , where the momentum corresponds

to a particle

P; we assume that the functions ()mt and ()tv are of class

. If the

mass of the particle

P increases, due to a phenomenon of capture at the moment t of an

13 Other Considerations on Dynamics of Mechanical Systems

169

MECHANICAL SYSTEMS, CLASSICAL MODELS

element

() ( , )St m

′

≡Δ



u , 0mΔ>, of absolute velocity ()tu and momentum

()

KS m

′

=Δ



u , then the external composition law at that moment leads to



()

() () ,

St S t m m

+Δ

′

+=+Δ

+Δ



of momentum



()

KS S m m

′

+=+Δ



; at the moment tt+Δ will exist only one

element



(,)Smm

′

≡+Δ +Δvv, where

′

Δ v

represents the variation of the velocity

of the particle

P in the time interval Δt, its momentum, after neglecting the terms of

higher order, being



()

()()KS m m m m m

′′

=+Δ +Δ= +Δ+Δvv v v v

The equivalence



SS S

′



∼

leads to



()



()

KS S KS

′



, so that mm+Δvu

mmm

′

=+Δ+Δvv v; introducing also the influence of the forces F, given in the

form

′′

Δ=ΔvF, where

′′

v is the corresponding variation of the velocity of the

particle

P in the same time interval Δt, and using the principle of the parallelogram, we

get the resultant variation of the velocity

v in the form

′′′

Δ=Δ +Δ

vvv. Passing to

limit for

0tΔ→ , we find again Meshcherskiĭ’s equation (10.3.3'), where

=−wuv

is the relative velocity of the element with respect to a non-inertial frame of reference

attached to the particle

If an element

() ( , )St m

′′

≡−Δ



u , 0mΔ>, of momentum

()

KS m

′′

=−Δ



detached from the particle

P at the moment t, then we can write



()

SS m m

−−Δ

⎛⎞

′′

−= −Δ

⎜⎟

−Δ

⎝⎠



the momentum being



()

KS S m m

′′

−=+Δ



; at the moment tt+Δ , after

emission, one obtains the element



(( ), )Sm m

′

≡−−Δ +Δvv of momentum



()

KS m m m

′

=+Δ+Δvv v, neglecting the terms of higher order. Proceeding as

before and passing to limit for

0tΔ→ , we find again Meshcherskiĭ’s equation

(10.3.1).

Analogously, one can obtain the generalized equation (10.3.8) of Meshcherskiĭ;

taking into account (10.3.11), one can write this equation also in the form

(

mm m

+−

 

)

()

mmm

−+

−

== +



vF u u, 0m

−



, 0m



(13.2.1)

170

We assume, in what follows, that the equation (13.2.1) maintains its form in

distributions; for distributions of function type, the derivative in the sense of the theory

of distributions will be given by (see the formula (1.1.51)

() () ()( )

mm tt

=+Δ−

∑



vvH

(13.2.2)

where the sign “tilde” corresponds to the derivative in the usual sense; the momentum

H = mv has only discontinuities of the first kind, with the jumps

() ( 0)( 0) ( 0)( 0)

ii i i i

mt t mt tΔ= + +− − −Hv v,

(13.2.2')

the moments of discontinuity being

t , i = 1,2,…,n. If the first n

′

moments of

discontinuity correspond to the mass discontinuities

()mt

−

, while the other nn

′

−

moments of discontinuities correspond to the mass discontinuities

()mt

, we can write

d()d()

()( )

mt mt

mtt

−−

′

−

=+Δ −

∑



d()d()

()( )

mt mt

mtt

′

=+Δ −

∑



(13.2.2'')

where we have introduced the jumps

() (0) (0)

jj j

mmt mt

−− −

Δ= +− −

, 1,2,...,jn

′

() (0) (0)

kk k

mmt mt

++ +

Δ= +− −, 1, 2,...,kn n n

′′

=+ + .

(13.2.2''')

The equation (13.2.1) becomes

ddd

() ()( )

ddd

mtt

ttt

−+

−

+Δ −=+ +

∑



vH Fu u

()( ) ()( )

mtt mttδδ

′

−− + +

′

+Δ −+ Δ −

∑∑

uu ;

finally, this equation may be replaced by the equation

ddd

()

ddd

ttt

−+

−

=+ +



vF u u,

(13.2.3)

corresponding to the moments of continuity, the derivatives being calculated in the

usual sense and by the jump relations

() ( )

−

Δ=ΔHu,

1,2,...,jn

′

() ( )

Δ=ΔHu, 1, 2,...,kn n n

′′

=+ +

(13.2.3')

13 Other Considerations on Dynamics of Mechanical Systems

171

MECHANICAL SYSTEMS, CLASSICAL MODELS

corresponding to the discontinuities of the mass. One obtains thus the generalized

equation of Meshcherskiĭ for a particle of discontinuous variable mass.

13.2.1.2 Theorems of Momentum and Moment of Momentum

In the case of a discrete mechanical system

S of variable mass it is convenient to use

the universal theorems of mechanics. We assume, in this case, that the constraint

percussive forces appear only in case of detachment or capture of some elements, by

their contact with the system

S (in fact, by the contact with a subsystem of that one); as

well, the elements which have not a relative velocity with respect to this system are

attached to it. We suppose that, at a given moment, the particles of the system

S have

not a relative velocity with respect to a given frame

R, so that the latter one is rigidly

linked to the system at the respective moment; thus, we take not in consideration the

state previous to that at the moment

t. The origin of the movable frame R will not be

taken, in general, at the centre of mass

C, because the masses of the particles are

variable, so that the position of this centre varies with respect to their positions. Using

the notations in Sect. 11.2.2.1, we can express the velocity

′

of a particle of position

vector

′′

=+rr r

, i = 1,2,…,n, with respect to the inertial (fixed) frame

′

in the

form (see the formula (11.2.10'))

′′

=+×vv rω

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

(13.2.4)

the relative velocity vanishing (

=v0 at the moment t); the corresponding

acceleration is given by

()

iii

′′

=+×+××



aa r rωωω, 1,2,...,in= .

(13.2.4')

The absolute velocity of the point which coincides with the centre of mass

C at the

moment

t is given by

′′

=+×vvω

(13.2.5)

corresponding to the formula (11.2.14) in which we make

=v0. The momentum of

the mechanical system

S will be given, in this case, by (

m is the mass of the particle

P at the moment t, considered not to have a relative velocity with respect to the

system

S )

′′′

∑

Hvv

(13.2.6)

where we took into account the formula (3.1.2), which gives the mass centre; hence, the

momentum of the discrete mechanical system

S of variable mass, at a given moment, is

equal to the momentum of the point which coincides with the mass centre at the

respective moment and at which the whole mass of the system would be concentrated.

172