Classical Mechanics

Tom W B Kibble, Frank H Berkshire

• Publisher: ICP
• Year: 2004
• ISBN:
  9781911298281
• Edition: 5
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This is the fifth edition of a well-established textbook. It is intended to provide a thorough coverage of the fundamental principles and techniques of classical mechanics, an old subject that is at the base of all of physics, but in which there has also in recent years been rapid development. The book is aimed at undergraduate students of physics and applied mathematics. It emphasizes the basic principles, and aims to progress rapidly to the point of being able to handle physically and mathematically interesting problems, without getting bogged down in excessive formalism. Lagrangian methods are introduced at a relatively early stage, to get students to appreciate their use in simple contexts. Later chapters use Lagrangian and Hamiltonian methods extensively, but in a way that aims to be accessible to undergraduates, while including modern developments at the appropriate level of detail. The subject has been developed considerably recently while retaining a truly central role for all students of physics and applied mathematics.

This edition retains all the main features of the fourth edition, including the two chapters on geometry of dynamical systems and on order and chaos, and the new appendices on conics and on dynamical systems near a critical point. The material has been somewhat expanded, in particular to contrast continuous and discrete behaviours. A further appendix has been added on routes to chaos (period-doubling) and related discrete maps. The new edition has also been revised to give more emphasis to specific examples worked out in detail.

Classical Mechanics is written for undergraduate students of physics or applied mathematics. It assumes some basic prior knowledge of the fundamental concepts and reasonable familiarity with elementary differential and integral calculus.

Contents:

• Linear Motion
• Energy and Angular Momentum
• Central Conservative Forces
• Rotating Frames
• Potential Theory
• The Two-Body Problem
• Many-Body Systems
• Rigid Bodies
• Lagrangian Mechanics
• Small Oscillations and Normal Modes
• Hamiltonian Mechanics
• Dynamical Systems and Their Geometry
• Order and Chaos in Hamiltonian Systems
• Appendices:
  • Vectors
  • Conics
  • Phase Plane Analysis Near Critical Points
  • Discrete Dynamical Systems — Maps

Readership: Undergraduates in physics and applied mathematics.
Classical Mechanics;Dynamical Systems;Order and Chaos;Lagrangian Methods;Hamiltonian Methods0 Key Features:

• Further development of the chapters on dynamical systems and their geometry, and on order and chaos, introduced in the fourth edition.
• Examples with solutions in the text, supplementing the wide range of problems with answers.
• Appendices on vectors, on conic sections, on dynamical systems near a critical point and, new in this edition, on routes to chaos and related discrete maps.
• Emphasis on basic principles of wide applicability.
• End-of-chapter summaries.
• A comprehensive index and list of symbols.

Chapter 1

Introduction

Classical mechanics is one of the most familiar of scientific theories. Its basic concepts — mass, acceleration, force, and so on — have become very much a part of our everyday modes of thought. So we may easily regard their physical meaning as more obvious than it really is. For this reason, a large part of this introductory chapter will be devoted to a critical examination of the fundamental concepts and principles of mechanics.

Every scientific theory starts from a set of hypotheses, which are suggested by our observations, but represent an idealization of them. The theory is then tested by checking the predictions deduced from these hypotheses against experiment. When persistent discrepancies are found, we try to modify the hypotheses to restore the agreement with observation. If many such tests are made and no serious disagreement emerges, then the hypotheses gradually acquire the status of ‘laws of nature’. When results that apparently contradict well-established laws appear, as they often do, we tend to look for other possible explanations — for simplifying assumptions we have made that may be wrong, or neglected effects that may be significant.

It must be remembered however that, no matter how impressive the evidence may be, we can never claim for these laws a universal validity. We may only be confident that they provide a good description of that class of phenomena for which their predictions have been adequately tested. One of the earliest examples is provided by Euclid’s axioms. On any ordinary scale, they are unquestionably valid, but we are not entitled to assume that they should necessarily apply on either a cosmological or a sub-microscopic scale. Indeed, they have been modified in Einstein’s theory of gravitation (‘general relativity’).

The laws of classical mechanics are no exception. Since they were first formulated by Galileo and by Newton in his Principia, their range of known validity has been enormously extended, but in two directions they have been found to be inadequate. For the description of the small-scale phenomena of atomic and nuclear physics, classical mechanics has been superseded by quantum mechanics, and for phenomena involving speeds approaching that of light, by relativity.

This is not to say that classical mechanics has lost its value. Indeed both quantum mechanics and the special and general theories of relativity are extensions of classical mechanics in the sense that they reproduce its results in appropriate limiting cases. Thus the fact that these theories have been confirmed actually reinforces our belief in the correctness of classical mechanics within its own vast range of validity. Indeed, it is a remarkably successful theory, which provides a coherent and satisfying account of phenomena as diverse as the planetary orbits, the tides and the motion of a gyroscope. Moreover, even outside this range, many of the results of classical mechanics still apply. In particular, the conservation laws of energy, momentum and angular momentum are, so far as we yet know, of universal validity.

1.1Space and Time

The most fundamental assumptions of physics are probably those concerned with the concepts of space and time. We assume that space and time are continuous, that it is meaningful to say that an event occurred at a specific point in space and a specific instant of time, and that there are universal standards of length and time (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements). These assumptions are common to the whole of physics, and, though all are being challenged, there is as yet no compelling evidence that we have reached the limits of their range of validity.

In ‘classical’ physics, we assume further that there is a universal time scale (in the sense that two observers who have synchronized their clocks will always agree about the time of any event), that the geometry of space is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all positions and velocities. These assumptions have been somewhat modified in quantum mechanics and relativity. Here, however, we shall take them for granted, and concentrate our attention on the more specific assumptions of classical mechanics.

The relativity principle

In Aristotle’s conception of the universe, the fact that heavy bodies fall downwards was explained by supposing that each element (earth, air, fire, water) has its own appointed sphere, to which it tends to return unless forcibly prevented from so doing. The element earth, in particular, tends to get as close as it can to the centre of the Universe, and therefore forms a sphere about this point. In this kind of explanation, the central point plays a special, distinguished role, and position in space has an absolute meaning.

In Newtonian mechanics, on the other hand, bodies fall downward because they are attracted towards the Earth, rather than towards some fixed point in space. Thus position has a meaning only relative to the Earth, or to some other body. In just the same way, velocity has only a relative significance. Given two bodies moving with constant relative velocity, it is impossible in principle to decide which of them is at rest, and which moving. This statement, which is of fundamental importance, is the principle of relativity.

Acceleration, however, still retains an absolute meaning, since it is experimentally possible to distinguish between motion with uniform velocity (i.e., constant in magnitude and direction) and accelerated motion. If we are sitting inside an aircraft, we can easily detect its acceleration, but we cannot measure its velocity — though by looking out we can estimate its velocity relative to objects outside. (In Einstein’s theory of general relativity, even acceleration becomes a relative concept, at least on a small scale. This is made possible by the fact that, to an observer in a confined region of space, the effects of being accelerated and of being in a gravitational field are indistinguishable.)

If two unaccelerated observers perform the same experiment, they must arrive at the same results. It makes no difference whether it is performed on the ground or in a smoothly travelling vehicle. However, an accelerated observer who performs the experiment may well get a different answer. The relativity principle asserts that all unaccelerated observers are equivalent; it says nothing about accelerated observers.

Inertial frames

It is useful at this point to introduce the concept of a frame of reference. To specify positions and time, each observer may choose a zero of the time scale, an origin in space, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a frame of reference. The position and time of any event may then be specified with respect to this frame by the three Cartesian co-ordinates x, y, z and the time t. If we are located on a solid body, such as the Earth, we may, for example, choose some point of the body as the origin, and take the axes to be rigidly fixed to it (though, as we discuss later, this frame is not quite unaccelerated).

In view of the relativity principle, the frames of reference used by different unaccelerated observers are completely equivalent. The laws of physics expressed in terms of our x, y, z, t must be identical with those of someone else’s x′, y′, z′, t′. They are not, however, identical with the laws expressed in terms of the co-ordinates used by an accelerated observer. The frames used by unaccelerated observers are called inertial frames.

We have not yet said how we can tell whether a given observer is unaccelerated. We need a criterion to distinguish inertial frames from the others. Formally, an inertial frame may be defined to be one with respect to which any isolated body, far removed from all other matter, would move with uniform velocity. This is of course an idealized definition, since in practice we can never get infinitely far away from other matter. For all practical purposes, however, an inertial frame is one whose orientation is fixed relative to the ‘fixed’ stars, and in which the Sun (or more precisely the centre of mass of the solar system) moves with uniform velocity. It is an essential assumption of classical mechanics that such frames exist. Indeed, this assumption (together with a definition of inertial frames) is the real physical content of Newton’s first law (a body acted on by no forces moves with uniform velocity in a straight line).

It is generally convenient to use only inertial frames, but there is no necessity to do so. Sometimes it proves convenient to use a non-inertial (in particular, rotating) frame, in which the laws of mechanics take on a more complicated form. For example, we sha...