A Course of Theoretical Physics, Volume 1: Fundamental Laws - Mechanics, Electrodynamics, Quantum Mechanics [1]

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A. S. KOMPANEYETS

A COURSE OF THEORETICAL PHYSICS VOLUME l FUNDAMENTAL LAWS MECHANICS ELECTRODYNAMICS QUANTUM MECI-IANIC~ MIR PUBLISHERS • MOSCOW

Alexander S. Kompaneyets (1914-1974) Professor Alexander Solomonovich Kompaneyets was a leading Soviet theoretical physicist from 1946 Until his untimely death he worked at the Institute of Chemical Physics of the USSR Academy of Sciences, contributing, among other things, to the development of nuclear energy in the Soviet Union in all its aspects.

A S. KOMPANEYETS A COURSE OF THEORETICAL PHYSICS

VOLUME

MIR PUBLISHERS MOSCOW

A. S. KOMPANEYETS FUNDAMENTAL LAWS Translated from the Russian

by

v.

MIR PUBLISHERS MOSCOW

TALMY

A. C. KOMIIAHEEU

K ypc

TEOPETOTECKOfl OH3HKH TOM 1 3JIEMEHTAPHHE 3AKOHH

English translation first published 1978 Revised from the 1972 Russian edition

Ha amAiiucKOM H3UKe

#01> • • • » #0n)

(2.22)

Elimination of all the initial values of the coordinates and veloc­ ities, that is, solution of Eqs. (2.21) and (2.22) with respect to the initial coordinates and velocities, yields 2n equations of the form •

?0a (t;

•

• • •. qn; q\, • • •, ?n) = q0a = constant

(2.23)

Functions of the coordinates and velocities of a system that re­ main constant- throughout the motion are known as the integrals of the motion. In the right-hand side they can have any constant coordinates and velocities, which need not necessarily be the initial ones. Determination of the integrals of the motion is one of the problems of mechanics. The Determinancy of the Lagrange Function. As is apparent from its definition, the Lagrange function (or simply Lagrangian), con­ tains two terms:

i The first term, which is quadratically dependent upon the veloci­ ties, is called the kinetic energy of the system; the second, which describes the interaction between particles, is called the potential energy. The meaning of both will be made clearer in Section 4. The Lagrange equation (2.20) involves not the function L itself but its derivatives. This gives rise to the question of the determinancy

Mechanics

23

of L, that is, of possible supplementary terms not affecting the equations of motion. It is obvious, for example, that in any case an additional constant term does not affect the equations of motion; also, a total time derivative of any function of all qa's and tfs can be added to the Lagrangian without in any way affecting the system of equations (2.20). This is easily verified by direct substitution as well as in the fol­ lowing simple way. The term df (qa, t)/dt, which has the form of a total derivative, can be integrated; as a result it is added to the action in the form of a difference between the function values at the limits: (2.25) to

to

But since the variations of the coordinates vanish at the limits, these values remain constant in the variation of qa. Hence the derivative of a function of coordinates and time is not involved in the variation of the action and does not affect the equations of motion. This property can be used to determine the form of L, if it is not given in advance as (2.24a), on the basis of Hamilton’s principle and certain other general propositions of mechanics. The Principle of Relativity. The concept of an inertial frame of reference was defined above as a frame in which all the accelerations of particles are due solely to interactions between them. Suppose we have such a frame. Then all other inertial reference frames must be moving uniformly in a straight line relative to it. Otherwise bodies moving relative to the initial frame with a velocity constant in magnitude and direction would be found to be moving with an acceleration relative to another reference frame. But in that case the latter would not by definition be inertial. Thus, all inertial reference frames are in rectilinear uniform motion relative to one another. Any one of them can be legitimately assumed at rest, and all the others moving. The equations of motion of a mechanical system have the same form in any inertial reference frame. A common example is that of a passenger in a train travelling at a uniform velocity: he sees all physical phenomena in the coach exactly the same as if the train were at rest. It would be better to say that this is not an example demonstrating the equivalence of two inertial frames of reference but experimental proof of the fundamental mechanical principle known as the principle of relativity. As applied to Newtonian mechanics, which reflects simple facts known to us from everyday life, the principle seems self-evident. But when it was applied to the theory of electromagnetism, it led to a fundamental revision of physical concepts (see Part II).

24

Fundamental laws

The Symmetry of the Laws of Motion. The property whereby an equation expressing a known physical relationship retains its form in a transformation is known as symmetry with respect to that trans­ formation. The relativity principle declares that the equations of motion are symmetrical with respect to the substitution of one inertial frame for another. Experience shows that the laws of me­ chanics possess other forms of symmetry as well. In a mechanical system that is sufficiently distant from other bodies motion is always the same wherever the system is located. This means the following. Let there be two identical mechanical systems with identical initial conditions of motion, both of which are very far away from any other bodies capable of affecting them. In that case, if they are taken in the same reference frame, motion in them occurs in strictly the same way. In other words, the motion is not affected by the transfer of all the moving bodies over the same distance, along parallel lines, at the same time. This assertion is, of course, based on the vast experience accumulated by mechanics in the whole course of its development. More briefly the property is known as the homogeneity of space. Two equivalent mechanical systems like the ones described here can be taken not only displaced relative to one another but also turned through any angle. Again, if the two systems are sufficiently far away from all bodies capable of affecting them, motion in them takes place in the same way. In other words all directions in space are equivalent. This property of space is known as the isotropy of space. Like homogeneity, the isotropy of space also follows from the sum-total of experience. Homogeneity and isotropy are an expres­ sion of specific properties of the laws of motion: their symmetry with respect to displacements and rotations. Mathematically, displace­ ments and rotations in space are represented by corresponding transformations of the coordinate system. There is one more type of symmetry of the laws of motion. They are homogeneous with respect to time transfer: the laws of motion do not change with time. If this property of the laws of motion of mechanical systems did not hold, it would be impossible to design any machine. Determination of the Form of the Lagrangian. The laws of sym­ metry of motion listed above, that is, space and time homogeneity, space isotropy, the relativity principle, and Hamilton’s principle, can be used to determine the form of the Lagrangian without pre­ liminary reference to Eqs. (1.1). Let us start with a free particle sufficiently far away from all other bodies (which is the definition of a free particle). By virtue of space homogeneity, its Lagrangian cannot be explicitly dependent on the coordinates, since otherwise at different spatial points the

Mechanics

25

particle would move according to different laws. For the same reason the Lagrangian does not explicitly involve time, and this refers not only to an individual free particle but to any assembly of particles not subject to external forces. Thus the Lagrangian of a free particle can depend only upon its velocity. But L is a scalar quantity. A sca­ lar can be obtained from a vector in one of two ways: by taking the absolute value of the vector or by multiplying it scalarly by another vector. But there is no such preferred vector in isotropic space since all directions in it are equivalent. Thus the only possible form of the Lagrangian of a free particle is L = L (| r |). It remains to determine what the function is. According to the relativity principle the character of motion should not change in passing to another inertial reference frame. As was pointed out, the latter must be travelling rectilinearly and uniformly relative to the initial one. If its velocity is V, the particle under consideration moves relative to it with the velocity r + V. We have made use of the simple law of velocity composition which, as will be shown in Part II, holds only when both | r | and | V | are substantially below the speed of light. Thus in the new inertial frame the Lagrangian is L = L (| r + V |). For the law of motion to remain the same the difference between the two expressions must be equal to the total derivative of a certain function of the coordinates and time. It is immediately apparent that for a free particle this leaves only one possibility: L = — \r ^

-

(2.26)

where m is a constant quantity. Indeed, we then obtain m| ( r + V)|* 2

m \i\* _

, m | V |* 2

2 d (

= ‘d r

m | V I2 t \

(mrV+—2

)

What is the sign of m? Let us determine it. But first we must somewhat refine Hamilton’s principle by requiring that along short paths the action be not simply extremal but minimal. Then the sign of m is positive. At negative m the action could decrease limitlessly with the increase of | r |. We have thus finally determined the first term in the Lagrangian for a free particle. If we now take a system of interacting particles, to describe their interaction we must introduce an additional term into the Lagran­ gian.

26

Fundamental laws

We assumed that the action of the particles on one another de­ pends only on their position at a given time. It is, however, sig­ nificant that it is determined only by their relative position, that is, it depends only on the separation r £ — rh and not on each vector separately. Only the differences between vectors remain constant in transfers of a coordinate system. In addition, only the differences r i — rk satisfy the relativity principle: the products V£, which are added to each radius vector r £, Tk in going over to another inertial frame, cancel out. Since the Lagrangian is a scalar quantity, it can depend either on the absolute values of the differences | r* — rh | or on scalar products of the type (r* — rk) (rt — rm). But the latter case is not encountered in practice and need not be considered. Hence the Lagrangian of a system of material points not interacting with other bodies is | r i - r k| ...)

(2.246)

i

We have not restricted ourselves to developing the Lagrange equa­ tions from Eqs. (1.1) and we have performed all of the complex reasoning needed to deduce formula (2.24b) because in this way it is easier to arrive at the necessary generalizations required by Einstein’s relativity principle and electromagnetic field theory. The special significance of Hamilton’s principle in mechanics consists in that it makes it possible to express all symmetry prop­ erties of mechanical systems in the most clear and concise form. Although they can be derived from the differential equations of motion as well, the integral principle expresses them much more distinctly. Since the symmetry of the conditions of motion is a generalization of certain experimentally established laws, Hamil­ ton’s principle provides the most convenient means of formulating all the general laws of mechanics. It should, of course, be borne in mind that this formulation is a reflection of a tendency to seek the most concise and convenient notation, not of any natural “striving” for minimum action. Symmetry properties substantially restrict the possible behaviour of mechanical systems. As will be shown later in Section 4, different types of symmetry are associated with certain quantities (dependent upon dynamic variables) whose values, determined at the initial instant of time, are conserved. This substantially restricts the domain of variables in the problems considered. In a number of important cases these quantities are best found with the help of the variation principle, according to the symmetry properties inherent in it.

Mechanics

27

Hamilton’s principle, with due account of symmetry require­ ments, can be used to determine the form of the Lagrangian, and thereby the form of the equations of motion. In this sense it posses­ ses great heuristic force, that is, makes it possible to find unknown quantities from general considerations. Finally, the variation principle is extremely convenient in solving specific problems of mechanics with the help of the Lagrange equa­ tions obtained by variation.

3

EXAMPLES OF CONSTRUCTING THE LAGRANGE EQUATIONS The Rules for Constructing the Lagrange Equations. Let us sum up the sequence of operations for developing the Lagrange equations for a specific mechanical system: (i) Express the Cartesian coordinates in terms of the generalized coordinates: (Qh • • •»

Xi

• • •* Qn)

(ii) By differentiating these equalities obtain the Cartesian veloci­ ty components expressed in terms of the generalized coordinates and generalized velocities (bearing in mind the summation rule for the subscript a):

(iii) Substitute generalized coordinates for the Cartesian coordi­ nates involved in the potential energy formula: £ / ( • • • | Xi Xk | • • •) = U ( * ^his is true, for example, of Newtonian gravitational forces or for electrostatic Coulomb forces between charged bodies.

Mechanics

53

Let us summarize the assumptions we have made regarding UM (r): (i) at r -> 0 the “centrifugal” term isjpredominant, hence V M (r) isj'infinite and positive; (ii) at r -^oo, where | U(r) | > M 2/(2mr2), UM (r) tends to zero from the side of negative values. Consequently, the curve UM (r) has the form shown in Figure 5. On the side of small values of r it decreases away from zero as 1/r2;

as it approaches large values of r it increases as —air, approaching the r axis from below. The curve must have a minimum in the domain of median values of r. In the interaction of a charged body with a neutral one (for instance, with an atom which has not lost a single electron) U(r) decreases faster than the “centrifugal” energy. Therefore at large distances UM (r) approaches the r axis from the positive side. Then the curve UM (r) first passes through a minimum, then increases, and after passing a maximum decreases again, tending to zero as r oo. This holds if there is a domain where U(r) predominates over the “centrifugal” energy. Otherwise the decrease is monotonic. The total energy of a system of converging particles can also be plotted in Figure 5. Since E is conserved in motion, the curve has the form of a horizontal straight line lying above or below the z axis, depending on the sign of E. For positive values of energy, the line E = constant lies above the curve UM (r) everywhere to the right of point A. In this case the difference E — UM (r) is positive. The particles can approach each other from infinity and recede from each other to infinity. Such motion is termed infinite. As we shall see

54

Fundamental laws

later in this section, in the case of Newtonian attraction we obtain hyperbolic orbits. At E < 0 , but higher than the minimum of the curve UM (r), the difference E — UM (r) = mr2/2 remains positive only between points B and B '. Consequently, between the corresponding two values of the radius lies a physically possible domain of motion with the given negative total energy. The motion is in this case termed finite. In the case of Newtonian attraction elliptical orbits correspond to it. In the motion of a planet about the sun point B corresponds to the perihelion, point B ' to the aphelion. At E = 0 the motion is infinite. As r increases the velocity tends to zero, remaining positive. In the case of Newtonian attraction parabolic orbits correspond to this value of E . Falling Onto the Centre. From the preceding reasoning it is appa­ rent that r cannot decrease to zero owing to “centrifugal” energy. Only if the particles are “targeted” on each other does the arm vanish, so that M = 0, and the curve UM (r) is replaced by the curve U (r). Then nothing prevents the particles from colliding. Let us now investigate an imaginary rather than a real case, when —U (r) tends to infinity as r -^ 0 faster than 1/r2. In this case UM (r) is negative for all r ’s close to zero. From (5.7), r2 is positive at in­ finitesimal values of r and tends to infinity as r -^0 . But the particle cannot collide head-on because such a collision would violate the law of conservation of angular momentum. The angular momentum is equal to mpz;^, where p is the arm. For M to retain its given finite value when the arm decreases to infinitesimal values the velocity component v^ perpendicular to the radius must tend to infinity as 1/p. Then the product mpz;^, which defines the angular momentum, remains finite. Thus, if UM = —oo at r = 0, the radial component of the velocity tends to zero, and the azimuthal component tends to infinity. The path of the particle has the form of a helix winding around the attracting centre, but never reaching it. The coils of the helix de­ crease, but the rotation speed increases. The force of “centrifugal” repulsion cannot prevent the particles from drawing gradually closer, which takes place the slower the smaller r is. In the motion of three bodies gravitating towards one another according to Newton’s law, two of them may collide even if at the initial time their motion was not purely radial. Indeed, only the total angular momentum of relative motion is conserved, and this does not preclude the collision of the two bodies. Finally, for repulsive forces tending to infinity as r 0, falling onto the centre is impossible. Obviously in this case the motion is only infinite.

Mechanics

55

Reducing to Quadrature. Let us now find the equation of thejpath in general form. To do this we must in (5.7) change from differen­ tiation with respect to time to differentiation with respect to cp. Using (5.4) we have

Separating the variables and passing to q> in (5.7) yields r

(5.10) r0 As can be seen from the equation, angle

r0, which correr sponds to an impermeable sphere. But as can be seen from Exercise j at n = oo the scattering is completely isotropic. If n is not infinity but sufficiently large, the particles are distributed almost isotro pically over all angles, and a sharp maximum appears only at small deflection angles, receding into infinity when the deflection angle tends to zero. Consequently, a scattering law approximating the isotropic law is indicative of a rapid diminishing of the forces with distance. This circumstance played an important part in the inves­ tigation of nuclear forces. EXERCISES 1. Find the differential cross section for particles scattering on an impermeable sphere of radius r0. S o l u t i o n . An impermeable sphere can be described in terms of mechanics by stating the potential energy in the form U(r) = 0 at r > r0 (outside the sphere) and U(r) = oo at r < • r0 (inside the sphere). Then whatever a par­ tic le ’s kinetic energy, its penetration into the domain r < r0 is impossible. Reflection from the sphere takes place in the following way. The radial momentum component reverses its sign, while the tangential component is conserved, since, given radial symmetry of the potential, no forces can be perpendicular to the radius. The absolute value of the linear momentum is conserved, insofar as the impact is elastic and the kinetic energy does not change. A simple construction reveals that the impact parameter is related to the deflection angle by the dependence p = r0 cos (x/2) if p < r#. Hence the general formula yields do = (ri/A )d Q

so that the scattering is uniform over all angles, that is, it is isotropic. The total scattering cross section a is equal to Jtrj in this case, as could be expect­ ed. Significant here is the fact that the interaction forces identically vanish at a finite distance. 2. A collision of two particles is observed, jr?x and m2 being their mas­ ses (mx is the mass of the incident particle). As a result of the collision par­ ticles are foimed whose angular momenta lie at angles

) = mgl (1 — cos