Teodorescu P.P. Mechanical Systems, Classical Models Volume I: Particle Mechanics

Подождите немного. Документ загружается.

MECHANICAL SYSTEMS, CLASSICAL MODELS

A field of conservative forces is

stationary, while a field of quasi-conservative forces is

non-stationary. We observe that

F0curl ,

(1.1.82')

so that a field of conservative or quasi-conservative forces is

irrotational. As an

example of conservative field of forces we may consider the

terrestrial gravitational

field

of the form

(

)

== − +Gg

grad constmmgx,

(1.1.83)

where

m and G are the mass and the weight of a particle, respectively, while g is the

gravity acceleration; the axis

Ox is along the gravity acceleration, but in the opposite

direction of it. Galileo proved experimentally that, in the same place on the Earth, all

the bodies have the same acceleration

g by a free falling in vacuum; but this

acceleration depends on the place in which this measurement is made (the latitude),

taking into account the fact that this place can be closer (at the poles) or further away (at

the equator) from the centre of the Earth (the vector

g is considered approximately

directed towards this centre, but – in reality – it is directed along the vertical of the

place; the greatest deviation is given by an angle of approximate six minutes with

respect to this vertical at the 45

° parallel). The magnitude (modulus) of this acceleration

≅

9, 781 /gms at the equator and ≅

9, 831 /gms at the poles, at the sea table;

experimentally, at the University of Bucharest it was obtained

≅

9, 806 /gms

corresponding approximately to the

45° parallel.

The formula (1.1.72

) allows us to introduce the mean unit weight and the unit

weight

, respectively, in the form

mean mean

gγμ,

Δ→

mean

lim

gγγμ,

(1.1.83')

and we may use all the considerations of Subsec. 1.1.8.

Another important example is given by the field of

Newtonian attraction forces;

Newton admitted that two particles

P and

P of masses

m and

m , respectively, are

acted upon by an attraction force

==−=− =−FF F r r

vers

ij ij

iij ij ij

ij ij

mm mm

⎛⎞

⎜⎟

⎝⎠

grad const

(1.1.84)

where

ij i j

JJJG

;

F and

F are internal forces, which verify the relation (1.1.81).

Observing that

=−rrr

ij j i

, it is obvious that

(

)

=FFr

ijij

and

()

=FFr

ij ij i

(the

second position vector

r or

r , respectively, playing the rôle of a parameter) in the

Newtonian model of mechanics

calculus of the gradient in (1.1.84); if

≡

(the corresponding particle having the

mass

M ), and ==rrr

ij j

(the particle

≡

PP being of mass m ), we may write

(

)

=− =− = +Frr

vers grad const

mM mM mM

fff

(1.1.84')

and the force

F is applied at the point P . The coefficient

> 0

is a universal constant

(sometimes denoted by

G ), which can be obtained experimentally in a particular case;

such a determination has been made for the first time by Henry Cavendish in 1798,

using the balance of torsion of Coulomb. In the CGS system,

6.673 10 cm/g s

−

≅⋅ ⋅. This is a very small value; the Newtonian attraction force

is not practically perceptible for bodies on the Earth surface (their masses are relative

small); this force has an intensity, which must be taken into consideration in case of

bodies with great masses (the case of cosmic bodies). As well, this force has an intensity

which cannot be neglected in case of a gravitational field, if one of the masses is great;

thus, the terrestrial gravitational field can be considered as been generated by the

Newtonian attraction forces, between bodies on the Earth surface and the Earth itself.

Formulae (1.1.83) and (1.1.84') lead to

()

=− =−

Grr

vers vers

fmg

where

G is the weight of a material body of mass m , at the distance r from the Earth

surface (along the vertical of the place, neglecting the deviation mentioned above),

while

R is the distance from the Earth surface to the centre (we admit that the Earth is

approximately spherical, its radius being

R ) of the Earth, the mass of which is M ; we

may thus write

(

)

(

)

⎡

⎤

=+=++

⎢

⎥

⎣

⎦

112

rrr

fM gR gR

RRR

But

rR , so that we may neglect the ratio /rR with respect to unity; finally, we

obtain

≅

MgR.

(1.1.85)

The forces which arise between two

electric charges

q and

q (attraction if they are

of contrary sign and repulsion in case of the same sign) have the same structure as the

forces (1.1.84) (

Coulomb’s law)

==−= =FF F r r

vers

ij ij

iij ij ij

ij ij

qq qq

⎛⎞

=−+

⎜⎟

⎝⎠

grad const

(1.1.84'')

MECHANICAL SYSTEMS, CLASSICAL MODELS

where

/4k πε ,

ε being the electric permittivity of vacuum and  the

rationalization factor

( = 1 in rationalized units, in the SI system (see Subsec. 2.1.3),

while

= 4π

in non-rationalized units).

In general, any force (we use formula (A.2.16))

()vers grad ()d

Fr Fρρ==

∫

(1.1.86)

exerted by a pole

upon a particle

and the intensity of which depends only on the

distance

r is conservative; such forces are encountered, for instance, in the kinetic

theory of gases. The forces of the form (1.1.86) are called

central forces, because their

supports pass through the pole

(in fact, it is a particular case of central forces; an

arbitrary central force, for which

(, )FFrθ , in polar co-ordinates, because in such a

case the trajectory of the particle is a plane curve, is – in general – no more a

conservative force). The force is attractive or repulsive as we have

0F ≶

; the

Newtonian attraction force (1.1.84') represents a particular case. Another particular case

is obtained for

=−Fr r() k , > 0k . One obtains thus an attractive elastic force of the

form

(

)

=− = − +Fr

grad const

kkr

> 0k

;

(1.1.87)

if we take the sign plus (

Frk , > 0k ), then we have a repulsive elastic force. Such

forces appear in case of a linear elastic spring and are used for the modelling of a linear

elastic continuous medium; as well, they can lead to linear oscillations, which allow the

construction of the simplest models of atoms.

The constants in the formulae (1.1.83), (1.1.84) – (1.1.84''), (1.1.87) and the constant

r in the formula (1.1.86) are determined by a given value of the potential U (which

may – eventually – be equal to zero) at a given point.

Besides the forces given by (1.1.82), we may consider also quasi-conservative or

conservative forces, the components of which are expressed in the form

∂

⎛⎞

=−

⎜⎟

∂

⎝⎠

tx

1, 2, 3j

;

(1.1.88)

in this case, the function

(

)

123123

,,,,,;UUxxxxxxt is called a generalized quasi-

potential

or a generalized potential, as it depends or not explicitly on time. Such forces

appear in the motion of the electron in an electromagnetic field (

Lorentz’s forces).

Supposing that the force

F does not depend explicitly on the acceleration



r , as it will

be seen in Subsec. 1.2.1, it follows that the generalized quasi-potential

U must have a

linear dependence with respect to the components of the velocity

0jj

UUx U ,

(

)

123

,,;

UUxxxt,

1, 2, 3j ,

(

)

123

,,;UUxxxt.

(1.1.88')

Newtonian model of mechanics

Unlike the generalized potential (quasi-potential), a potential (quasi-potential) is

called also a

simple potential or a simple quasi-potential. A system of forces

(corresponding – for instance – to a field of forces) which admits a simple potential

(quasi-potential) or a generalized potential (quasi-potential) is called also a

natural

system

. The forces which are not conservative are called non-conservative forces.

1.1.13 Elementary particle interactions

If a mechanical system S is subjected only to the action of internal forces (e.g., the

solar system, considered as independent of the other systems of the Galaxy, or a system

of elementary particles), then the respective system is closed (neither inputs, nor outputs

take place). The physical phenomena which occur in nature can be, in general, described

interactions between particles (we consider, in what follows, the elementary

particles

, which are basic constituents of the matter).

Corresponding to the experimental data, till now only four basic (elementary)

interactions (which, in fact, are forces) have been encountered, i.e.: the gravitational

interaction, the weak interaction, the electromagnetic interaction and the strong

(nuclear) interaction, which determine all the forces known in the Universe. An

interaction is characterized by its

intensity (expressed by a number) with respect to one

of them taken as unit, and determined by the properties of the particles which they

transport.

The gravitational interaction maintains the Earth as a whole, as well as the Sun and

the planets in the solar system, and connects the stars in the frame of the Galaxy; the

intensity of this interaction is

239

/10GM c

−

≅= , where

Gf is the universal

gravitational constant,

M is the mass of the nuclear particle, c is the velocity of light

in vacuum, and

/2h π== , h being Planck’s constant. This interaction is the weakest

of all interactions; it acts between all particles and is exerted at any distances (small,

great or very great, theoretically infinite), being susceptible to be propagated by a

quantum of vanishing mass, called

graviton. The corresponding forces are the only

attraction forces between particles at great distances.

The weak interaction is an interaction of contact (at a small distance of approximate

−15

10 cm, smaller than the nuclear dimensions which are of approximate

−13

10 cm), its

intensity being

−9

10 (its intensity in an elementary disintegration volume is of

approximate

10 times smaller that the intensity of the strong interaction). The vector

bosons

are carrying agents of those interactions, which are exerted between leptons and

quarks, the corresponding forces between identical particles being repulsive; thus, the

weak interaction cannot form stable states. In the absence of a conservation law of

barions (heavy particles), the weak interaction could lead to the decay as electrons and

neutrons of matter in the Universe, in an interval of time less than

−

10 s.

The electromagnetic interaction is an interaction at distance between all particles

with electric charge or/and magnetic moment. It ensures the existence of atoms,

molecules and crystalline systems as stable systems. The intensity of this interaction is

given by

the constant of fine structure

/4ecαπε= =

, where

e is the elementary

charge

and

ε is the dielectric constant of vacuum; we have

−

≅⋅

7.297351 10α

≅≅1/137.030602 1/137 , its intensity being

−

10 . The photons, which can be

coupled with any charged particle, are carrying agents of those interactions, the

MECHANICAL SYSTEMS, CLASSICAL MODELS

corresponding forces between identical particles being

repulsive; as the gravity forces,

these ones have a large radius of action and may be studied in the frame of quantum

electrodynamics.

The strong (nuclear) interaction leads to the building of all elements, by linking the

nuclear particles, and is acting at a small distance upon quarks, the carrying agent being

the gluons. It corresponds a repulsive force, the intensity of which is – conventionally –

taken equal to unity, with respect to the intensity of the other forces (the interactions

have been presented in order of growth of their intensities).

The idea of

unifying the four basic interactions, so that to use a single set of

equations to predict all their characteristics, is old and, at present, it is not known

whether such a theory can be developed. However, the most successful attempt in this

direction is

the electroweak theory (the Weinberg-Salam model) proposed during the

late 1960s by Steven Weinberg, Abdus Salam, Samuel Glashow (Nobel prize, 1979).

The carrying agents are

the vector bosons, with electric charge

W , and the neutral

vector boson

Z . In the frame of this model, the masses

±(82 2.4)

m GeV and

= 75

m GeV (Gigaelectronvolts) have been theoretically calculated. In December

1982, the mass of the vector boson has been determined experimentally, its value being

=±(85 5)

m GeV, in excellent agreement with the theoretical prediction of the

Weinberg-Salam model. The measures have been effected at CERN (Switzerland) and

represent the greatest discovery of this important European centre of research in the area

of physics. In June 1983, the existence of

the third vector boson

Z has been confirmed

experimentally. Recent experimental values of the masses of vector bosons are

=±(80.423 0.0039)

m GeV and

±(91.1876 0.0021)

m GeV.

The electroweak theory is

a unified theory of electromagnetic and weak interactions,

based on the SU(2)

×U(1) symmetry. It regards the weak force and the electromagnetic

force as different manifestations of a new fundamental force (electroweak), similarly to

electricity and magnetism that appear as different aspects of the electromagnetic force.

We mention that the electroweak theory is – in essence –

a gauge theory.

We may infer that there are only

three fundamental interactions in Nature:

gravitational, electroweak and strong (nuclear). Further, one tries to unify these three

interactions in the frame of a

unitary theory, reducing – eventually in a first stage – their

number to two; in this order of ideas, theories which try to link the electroweak

interaction to the strong one (e.g.,

Giorgi’s theory) have been developed. As well, the

supergravitation

describes gravitational phenomena in the frame of a quantum theory of

field; the problem is very difficult also because the gravitational interaction, which

plays the most important rôle in the study of mechanical systems, is the only interaction

which leads only to forces of attraction (described, e.g., by the Newtonian model).

Certain researchers assume the existence of

a fifth type of basic interaction too.

Now, the theory of electroweak interactions is a component of

the Standard Model,

that includes also the theory of strong interactions; but it seems that the current models

of strong interactions need to be revised.

However, the recent discovery of

exotic quark systems suggests that the Nature is

much more complicated; thus, in July 2003, nuclear physicists in Japan, Russia and the

USA have discovered

the pentaquark, and in November 2003, a new subatomic particle

has been discovered, while Daniel Gross, David Politzer and Frank Wilczek have been

awarded Nobel prize, 2004.

Newtonian model of mechanics

In conclusion, stress is put to set up new models for unifying the three remaining

basic interactions.

1.2 Mathematical model of mechanics

By means of the fundamental notions introduced above, one can formulate – in a new

form – Newton’s principles, which constitute the basis of the mathematical model of

mechanics. The object of study of this science of nature may be thus emphasized. A

short history is also given.

1.2.1 Newton’s principles

Continuing the results obtained by his predecessors, especially those of Galileo

Galilei (who intuits the principle of inertia and the principle of the initial conditions),

Sir Isaac Newton enounced, in his famous work “Philosophiae Naturalis Principia

Mathematica” (the first fascicle appeared in London at 5th of July 1686), the three laws

which form the basis of classical models of mechanics.

Lex I. Corpus omne perseverare in statu suo quiscendi vel movendi uniformiter in

directum, nisi quantenus illud a viribus impressis cogitur statum suum mutare

(Any

body preserves its state of rest or of uniform rectilinear motion if it is not constrained by

induced forces to change its state) (after the last enunciation of Newton, in the third

edition (1726) of his treatise).

Lex II. Mutationem motus proportionalem esse vi motrici impresse et fieri secundum

lineam rectam, qua vis illa imprimitur

(The variation of motion (of the quantity of

motion) is proportional to the induced moving force and is directed along the straight

line, which is the support of this induced force).

Lex III. Actioni contrariam semper et aequalem esse reactionem, sive corporum

duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi

(The

reaction is always opposite and equal to the action or the reciprocal actions of two

bodies are always equal and directed in contrary directions).

Newton’s laws (principles) (called sometimes the Galileo-Newton principles,

because of the important contribution of Galileo to the building of the classical model of

mechanics) by body we understand a particle (material point). As well, we admit the

existence of an inertial frame of reference with respect to which these principles are

verified and the motion is described. In these conditions, we may enounce the principles

of classical mechanics in the following new form:

Principle of inertia. A particle, which is not subjected to the action of any

force, is in a uniform and rectilinear motion or is in rest

The motion or the rest are considered with respect to the inertial frame of reference,

whose existence, independent of the particle, was postulated. Practically, we cannot take

out a body of the influence of other bodies in the Universe. But we can reduce this

influence, observing that it is diminished if one moves away the other bodies from that

on which we focus our attention or that one can annul an action by another one; other

actions (for instance, friction, resistance of the medium etc.) may be sensibly reduced

by certain technical means. We may thus imagine some experiments (reducing friction,

experiments in vacuum etc.) leading closer to the ideal conditions of this principle; this

one is thus verified with a good approximation. It follows that there can exist a motion

(an inertial motion) even in the absence of a given force; we observe that the

MECHANICAL SYSTEMS, CLASSICAL MODELS

acceleration of a particle, which is not in interaction with other particles, vanishes.

Hence, not the motion (not the velocity), but the variation of the motion (variation of the

velocity, that is of the momentum) must be put in connection with the force, as we have

seen in Subsec. 1.1.11.

ii)

Principle of action of forces. Between a force acting upon a particle and

the acceleration induced by it takes place a relation of the form

=arFmm



(1.1.89)

where m is the mass of the particle.

This constitutes

the basic law of mechanics, discovered by Newton, observing

various phenomena of mechanical nature: the falling of bodies (phenomenon of

gravitational nature), the oscillation of the pendulum, simple devices (studied

previously by Galileo and Huygens), motion of the Moon around the Earth etc. Tullio

Levi-Civita admits that

dv is along the force F , dv being proportional to F and to

dt and in inverse proportion to the weight

of the particle; hence

=vFdd

We suppose that

k is a constant, which does not depend on the force F , but only on

the considered particle (affirmation corresponding also to

the principle of equivalence

enounced by Albert Einstein). In case of a free falling in vacuum, we see that

=gG

where

G is the gravity force; it follows that

kg (g is the gravity acceleration),

hence

Observing that the ratio

> 0

is constant anywhere on the Earth, no matter where

G and g are measured, m being –

in fact – the mass of the particle, we obtain the law (1.1.89). In case of gravity forces

(taking into account the equality between the gravitational and the inertial mass), we

may write this law in the form

gGm

(1.1.89')

The mass of the bodies on the Earth is determined with the aid of the balance; for

cosmic bodies we use the law (1.1.89) and

the principle of universal attraction, which

leads to the Newtonian attraction forces (1.1.84). It is interesting to notice that, in fact,

Newtonian model of mechanics

Newton enounced the second law in the form (he intended by

mv the quantity of

motion

)

()

vF,

(1.1.89'')

relation equivalent to (1.1.89) if

constm

; this conception is the more valuable as a

relation of the form (1.1.89'') is maintained also in relativistic mechanics. If

=F0,

then the relation (1.1.89) leads to

a0, hence to a uniform and rectilinear motion with

respect to an inertial frame of reference; that does not mean that the principle i) is a

particular case of the principle ii). Indeed, the principle of inertia can lead to the right

solution if the solution of the differential equation (1.1.89) is not unique.

We will consider an example given by Jean Chazy in 1941, that is a particle of mass

m , acted upon by a force =

1/3

() 6Fx mx along the Ox -axis; the differential equation

of motion is

1/3

6xx

(1.1.90)

with three possible solutions

= 0x ,

−

xt,

(1.1.90')

which verify the initial conditions

= 0x

0x

for

. (1.1.90'')

Hence, the problem has an infinity of continuous solutions of class

C for ∈ [0, ]tT,

> 0T

, i.e.

()

0 for 0 ,

()

for ,

tt t t

≤

⎧

⎪

⎨

±− >

⎪

⎩

(1.1.91)

where

≤≤

0 tT.

Using the principle of inertia, Victor Vâlcovici solves, in 1951, the problem. Indeed,

this principle leads to the solution

0x , which verifies the conditions (1.1.90''); for

any

t , the particle remains in the initial position of rest. For

≤

, any other solution

(1.1.91) is eliminated by the first principle of mechanics, so that

0x for ∈ [0, ]tT;

hence, this principle must be considered as an independent one.

iii)

Principle of action and reaction. To any action of a particle towards

another particle, there corresponds a reaction of the second particle

towards the first one, having the same support and magnitude, but opposite

direction.

If we denote by

F the action of a particle

P upon a particle

P and by

F the

reaction

of the particle

P upon the particle

P (an action can be a reaction or

MECHANICAL SYSTEMS, CLASSICAL MODELS

inversely, the denominations being relative), then the relation (these actions are, in fact,

forces)

=FF 0

12 21

, =±F

21 21 1 2

vers FPP

JJJG

12 21

0FF ,

(1.1.92)

where we take the sign + or – as the forces are repulsive or attractive, holds; we notice

that these forces are of the nature of internal forces and enjoy the properties put in

evidence in Subsec. 1.1.11. The action and the reaction are applied to two different

particles, so that we cannot say that they are in equilibrium. Let us consider, for

instance, an agent exerting a force

F on a particle P , inducing a certain motion of it;

writing the equation (1.1.89) in the form

−

=Fa0m and taking into account the

principle of action and reaction (formula 1.1.92)), we can introduce a reaction

− am

applied on the considered agent. The force

−

am is called force of inertia and is due to

the inertia of the particle. This principle may be extended passing from a particle acting

upon another particle to a mechanical system acting upon another mechanical system;

obviously, in this case, the action as well as the reaction are systems of forces.

iv)

Principle of the parallelogram of forces. If a particle is acted upon

separately by the forces

F and

F , which induce the accelerations

a and

a , respectively, and then it is acted upon simultaneously by the two forces,

then the latter ones can be summed vectorially

(using the parallelogram

rule,

=+FF F

), the acceleration induced being obtained analogously

(

=+aa a

Indeed, starting from

m ,

(1.1.93)

we may write

(

)

=+= + =FF F a a a

12 12

mm,

(1.1.93')

the independence of the action of forces being thus put into evidence; this principle may

be easily verified experimentally and, together with the principle iii), allows – for

instance – to consider simultaneously the discrete mechanical systems in the Universe.

The principle iv) was presented by Newton as a corollary to the second law. We must

notice also that this principle is applied admitting that the particle is in the same initial

state (position, velocity, physical or chemical state etc.) if it is acted upon either by the

force

or by the force

. This principle allows us to affirm that the force and the

acceleration are vector quantities.

Principle of initial conditions. The initial state (initial position and

velocity)

of a particle determines entirely its motion at an arbitrary given

moment

(if one uses the first four principles).

In fact, it is not necessary to know the initial state (at time

); it is sufficient to

know the state of a particle at a given moment. This observation is due to Galileo, who

showed that the initial position is not sufficient to determine the motion of the particle,

but it is necessary to know also its initial velocity; indeed, we can write the

development

Newtonian model of mechanics

=+− +−

000 0

() () ( )() ( ) ()

ii i i

xt xt t txt t t xt,

()tt ttε

+−,

(0,1)ε ∈ ,

1, 2, 3i ,

(1.1.94)

for the components of the position vector

r . Hence, we see that – the initial state being

given – it is necessary to calculate also the acceleration of the particle for the

determination of its motion. This principle may be enounced also in another form,

equivalent to that given above.

v')

If two particles are isolated from the rest of the Universe, then the forces in

interaction at a given moment are determined in direction

(including

support)

and modulus if at this very moment the positions and the relative

velocities are known

In this form, the principle of initial conditions shows that the given forces

F depend

only on time, position vector and velocity, hence

(

)

FFrr,;t



(1.1.95)

In this context, we may admit that the basic law (1.1.89) of mechanics constitutes

the

representation in a normal form

(in which we express the acceleration



r , that is the

derivative of highest order as a function of the position

r , the velocity



r and time t ) of

a law of the form

(

)

rrr 0f,,;t



;

(1.1.95')

hence, in the Newtonian model of mechanics do not intervene

accelerations of higher

order

, as it is experimentally confirmed by all mechanical phenomena thus modelled.

In conclusion, from a philosophical point of view, one can state that the classical

model of mechanics is a

deterministic one.

1.2.2 Mathematical models of mechanics

Once the basic notions of mechanics (space, time, mass, motion, force) introduced,

the classical (Newtonian) model of this science of nature is born, adopting the principles

discussed in the previous subsection. This model was verified in practice in case of

bodies moving with

velocities v relatively small (negligible with respect to the velocity

c of light in vacuum).

But the Newtonian model of classical mechanics was only a step in the process of

knowledge; indeed, it was proved (by the famous experiments of Albert Abraham

Michelson in 1881, as well as together with Edward William Morley in 1887 and of E.

W. Morley and Dayton Clarence Miller in 1904, who showed the inexistence of the

ether) that, at greater velocities (comparable with the velocity of light in vacuum), the

principles of Newton must be changed or completed. To describe more exactly the

properties of the real space,

non-Euclidean geometries are introduced. As well, the

Newtonian concept of the universal time must be replaced by the relativistic

representation, which takes into account

the individual physical time, the fact that time

depends on the motion itself. The mass is also no more constant, depending on the

velocity and, implicitly, on time. Thus, following the papers of Albert Einstein of 1905