General Physics. Mechanics and Molecular Physics

376 Pages·1967·7.535 MB·English

by  L D Landau (Auth.)

L. D. L A N D A U, A. L A K H I E Z ER
E. M. L I F S H I TZ
GENERAL PHYSICS
Mechanics and Molecular Physics
Translated by
J. B. SYKES
A. D. PETFORD, C. L. PETFORD
P E R G A M ON P R E SS
OXFORD · NEW YORK · TORONTO
SYDNEY · BRAUNSCHWEIG
Pergamon Press Ltd., Headington Hill Hall, Oxford
Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523
Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1
Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay,
N.S.W. 2011, Australia
Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig
Copyright ® 1967 Pergamon Press Ltd.
All Rights Reserved. No part of this publication may be reproduced, stored in
a retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the prior permission
of Pergamon Press Ltd.
First English edition 1967
Reprinted 1970
Translated from Kurs obshchéífiziki: mekhanika i molekulyarnaya fizika,
Izdatel'stvo "Nauka", Moscow, 1965
Library of Congress Catalog Card No. 67-30260
Filmset by Graphic Film Limited, Dublin
and printed in Great Britain by
J. W. Arrowsmith Limited, Bristol
0 08 009106 7
P R E F A CE
THE purpose of this book is to acquaint the reader with the
principal phenomena and most important laws of physics. The
authors have tried to make the book as compact as possible,
including only what is essential and omitting what is of secondary
significance. For this reason the discussion nowhere aims at
anything approaching completeness.
The derivations of the formulae are given only in so far as they
may help the reader in understanding the relations between
phenomena. Formulae are therefore derived for simple cases
wherever possible, on the principle that the systematic derivation
of quantitative formulae and equations should rather appear in a
textbook of theoretical physics.
The reader is assumed to be familiar with algebra and trig­
onometry and also to understand the fundamentals of the
differential calculus and of vector algebra. He is further expected
to have an initial knowledge of the main ideas of physics and
chemistry. The authors hope that the book will be useful to physics
students at universities and technical colleges, and also to physics
teachers in schools.
This book was originally written in 1937, but has not been
published until now, for a variety of reasons. It has now been
augmented and rewritten, but the plan and essential content
remain unchanged.
To our profound regret, L. D. Landau, our teacher and friend,
has been prevented by injuries received in a road accident from
personally contributing to the preparation of this edition. We have
everywhere striven to follow the manner of exposition that is
characteristic of him.
We have also attempted to retain as far as possible the original
choice of material, being guided here both by the book in its
original form and by the notes (published in 1948) taken from
Landau's lectures on general physics in the Applied Physics
Department of Moscow State University.
÷ PREFACE
In the original plan, in order not to interrupt the continuity of
the discussion, the methods of experimental study of thermal
phenomena were to have been placed in a separate chapter at the
end of the book. Unfortunately, we have not yet had an oppor­
tunity to carry out this intention, and we have decided, in order
to avoid further delay, to publish the book without that chapter.
A. I. AKHIEZER
June 1965 E. M. LIFSHITZ
CHAPTER I
P A R T I C LE M E C H A N I CS
§ 1. The principle of the relativity of motion
The fundamental concept of mechanics is that of motion of a
body with respect to other bodies. In the absence of such other
bodies it is clearly impossible to speak of motion, which is always
relative. Absolute motion of a body irrespective of other bodies
has no meaning.
The relativity of motion arises from the relativity of the concept
of space itself. We cannot speak of position in absolute space
independently of bodies therein, but only of position relative to
certain bodies.
A group of bodies which are arbitrarily considered to be at
rest, the motion of other bodies being taken as relative to that
group, is called in physics a frame of reference. A frame of
reference may be arbitrarily chosen in an infinite number of
ways, and the motion of a given body in different frames will in
general be different. If the frame of reference is the body itself,
then the body will be at rest in that frame, while in other frames
it will be in motion, and in different frames it will move differently,
i.e. along different paths.
Different frames of reference are equally valid and equally
admissible for investigating the motion of any given body.
Physical phenomena, however, in general occur differently in
different frames, and in this way different frames of reference may
be distinguished.
It is reasonable to choose the frame of reference such that
natural phenomena take their simplest form. Let us consider
a body so far from other bodies that it does not interact with
them. Such a body is said to be moving freely.
In reality, the condition of free motion can, of course, be
fulfilled only to a certain approximation, but we can imagine in
2 PARTICLE MECHANICS [l
principle that a body is free from interaction with other bodies to
any desired degree of accuracy.
Free motion, like other forms of motion, appears differently in
different frames of reference. If, however, the frame of reference
is one in which any one freely moving body is fixed, then free
motion of other bodies is especially simple: it is uniform motion
in a straight line or, as it is sometimes called, motion with a
velocity constant in magnitude and direction. This statement
forms the content of the law of inertia, first stated by Galileo. A
frame of reference in which a freely moving body is fixed is called
an inertial frame. The law of inertia is also known as Newton's
first law.
It might appear at first sight that the definition of an inertial
frame as one with exceptional properties would permit a defini­
tion of absolute space and absolute rest relative to that frame.
This is not so, in fact, since there exists an infinity of inertial
frames: if a frame of reference moves with a velocity constant
in magnitude and direction relative to an inertial frame, then it is
itself an inertial frame.
It must be emphasised that the existence of inertial frames of
reference is not purely a logical necessity. The assertion that
there exist, in principle, frames of reference with respect to which
the free motion of bodies takes place uniformly and in a straight
line is one of the fundamental laws of Nature.
By considering free motion we evidently cannot distinguish
between different inertial frames. It may be asked whether the
examination of other physical phenomena might in some way
distinguish one inertial frame from another and hence select
one frame as having special properties. If this were possible, we
could say that there is absolute space and absolute rest relative
to this special frame of reference. There is, however, no such
distinctive frame, since all physical phenomena occur in the same
way in different inertial frames.
All the laws of Nature have the same form in every inertial
frame, which is therefore physically indistinguishable from, and
equivalent to, every other inertial frame. This result, one of the
most important in physics, is called the principle of relativity of
motion, and deprives of all significance the concepts of absolute
space, absolute rest and absolute motion.
Since all physical laws are formulated in the same way in
§2] VELOCITY 3
every inertial frame, but in different ways in different non-inertial
frames, it is reasonable to study any physical phenomenon in
inertial frames, and we shall do so henceforward except where
otherwise stated.
The frames of reference actually used in physical experiments
are inertial only to a certain approximation. For example, the
most usual frame of reference is that in which the Earth, on which
we live, is fixed. This frame is not inertial, owing to the daily
rotation of the Earth on its axis and its revolution round the
Sun. These motions occur with different and varying velocities
at different points on the Earth, and the frame in which the Earth
is fixed is therefore not inertial. However, because of the relative
slowness of variation of the direction of the velocities in the
Earth's daily rotation on its axis and revolution round the Sun,
we in fact commit a very small error, of no importance in many
physical experiments, by assuming that the "terrestrial" frame of
reference is an inertial frame. Although the difference between
the motion in the terrestrial frame of reference and that in an
inertial frame is very slight, it can nevertheless be observed, for
example, by means of a Foucault pendulum, whose plane of
oscillation slowly moves relative to the Earth's surface (§31).
§2. Velocity
It is reasonable to begin the study of the laws of motion by
considering the motion of a body of small dimensions. The
motion of such a body is especially simple because there is no
need to take into account the rotation of the body or the relative
movement of different parts of the body.
A body whose size may be neglected in considering its motion
is called a particle, and is a fundamental object of study in
mechanics. The possibility of treating the motion of a given body
as that of a particle depends not only on its absolute size but also
on the conditions of the physical problem concerned. For
example, the Earth may be regarded as a particle in relation to
its motion round the Sun, but not in relation to its daily rotation
on its axis.
The position of a particle in space is entirely defined by
specifying three coordinates, for instance the three Cartesian
coordinates x, y, z. For this reason a particle is said to have three
degrees of freedom. The quantities x,y, ζ form the radius vector
4 PARTICLE MECHANICS [l
r from the origin to the position of the particle.
The motion of a particle is described by its velocity. In uniform
motion, the velocity is defined simply as the distance traversed
by the particle in unit time. Generally, w^hen the motion is not
uniform and varies in direction, the particle velocity must be
defined as a vector equal to the vector of an infinitesimal displace­
ment ds of the particle divided by the corresponding infinitesimal
time interval dt. Denoting the velocity vector by v, we therefore
have
V = dsldt.
The direction of the vector ν is the same as that of ds; that is,
the velocity at any instant is along the tangent to the path of the
particle in the direction of motion.
Figure 1 shows the path of a particle and the radius vectors
r and r + i/r at times t and t-l·dt. By the vector addition rule it is
easily seen that the infinitesimal displacement ds of the particle
is equal to the difference between the radius vectors at the initial
and final instants: ds = dr. The velocity ν may therefore be
written
V = dr/dt,
and is thus the time derivative of the radius vector of the moving
particle. Since the components of the radius vector r are the
coordinates x, y, z, the components of the velocity along these
axes are the derivatives
Vj. = dx/dt, Vy = dyldt, v^ = dzldt.
The velocity, like the position, is a fundamental quantity
§3] MOMENTUM 5
describing the state of motion of a particle. The state of the par­
ticle is therefore defined by six quantities: three coordinates and
three velocity components.
The relation between the velocities ν and v' of the same particle
in two different frames of reference Κ and K' may be determined
as follows. If in a time dt the particle moves an amount ds relative
to the frame K, and the frame Κ moves an amount dS relative to
the frame Κ', the vector addition rule shows that the displace­
ment of the particle relative to the frame K' is ds' = ds-^dS.
Dividing both sides by the time interval dt and denoting the
velocity of the frame A'' relative to Κ by V, we find
v' = v-hV.
This formula relating the velocities of a given particle in diflFerent
frames of reference is called the velocity addition rule.
At first sight the velocity addition rule appears obvious, but
in fact it depends on the tacitly made assumption that the passage
of time is absolute. We have assumed that the time interval during
which the particle moves by an amount ds in the frame Κ is equal
to the time interval during which it moves by ds' inK'. In reality,
this assumption proves to be not strictly correct, but the conse­
quences of the non-absoluteness of time begin to appear only at
very high velocities, comparable with that of light. In particular,
the velocity addition rule is not obeyed at such velocities. In
what follows we shall consider only velocities so small that the
assumption of absolute time is quite justified.
The mechanics based on the assumption that time is absolute
is called Newtonian or classical mechanics, and we shall here
discuss only this mechanics. Its fundamental laws were stated
by Newton in his Principia (1687).
§3. Momentum
In free motion of a particle, i.e. when it does not interact with
other bodies, its velocity remains constant in any inertial frame
of reference. If particles interact with one another, however, their
velocities will vary with time; but the changes in the velocities
of interacting particles are not completely independent of one
another. In order to ascertain the nature of the relation between
them, we define a closed system —a. group of particles which
6 PARTICLE MECHANICS [l
interact with one another but not with surrounding bodies. For a
closed system there exist a number of quantities related to the
velocities which do not vary with time. These quantities naturally
play a particularly important part in mechanics.
One of these invariant or conserved quantities is called the total
momentum of the system. It is the vector sum of the momenta
of each of the particles forming a closed system. The momentum
of a single particle is simply proportional to its velocity. The
proportionality coefficient is a constant for any given particle and
is called its mass. Denoting the particle momentum vector by
ρ and the mass by m, we can write
p= mv,
where ν is the velocity of the particle. The sum of the vectors
ρ over all particles in the closed system is the total momentum
of the system:
Ρ = Pi + Ρ2 + * · * = ^iVi-\-m2\2~^ · · s
where the suffixes label the individual particles and the sum
contains as many terms as there are particles in the system. This
quantity is constant in time:
Ρ = constant.
Thus the total momentum of a closed system is conserved.
This is the law of conservation of momentum. Since the momen­
tum is a vector, the law of conservation of momentum separates
into three laws expressing the constancy in time of the three
components of the total momentum.
The law of conservation of momentum involves a new quantity,
the mass of a particle. By means of this law, we can determine
the ratios of particle masses. For let us imagine a collision
between two particles of masses m^ and m2, and let Vi and V2
denote their velocities before the collision, v/ and Vg' their
velocities after the collision. Then the law of conservation of
momentum shows that
miVi -hm2V2 = mi\i -\-m2y2!-