## Classical Mechanics

John R. Taylor University of Colorado

### Preface

The first two questions about any text book are: “What is it about?” and “For what audience is it intended.” The brief answer to the first question is that this book is about classical mechanics — a subject I’ll describe in a moment. The answer to the second is that this book is intended for students of the physical sciences, especially physics, who have already met mechanics as part of an introductory physics course (“freshman physics” at a typical American university) and are now ready for a deeper look at the subject. The book grew out of the junior-level mechanics course which is offered by the Physics Department at Colorado and is taken mainly by physics majors, but also by some mathematicians, chemists, and engineers. Almost all of these students have taken a year of freshman physics, and so have at least a nodding acquaintance with Newton’s laws, energy and momentum, simple harmonic motion, and so on. In this book I build on this nodding acquaintance to give a deeper understanding of these basic ideas, and then go on to develop more advanced topics, such as the Lagrangian and Hamiltonian formulations, the mechanics of non-inertial frames, motion of rigid bodies, coupled oscillators, chaos theory, and a few more.

Mechanics is, of course, the study of how things move — how an electron moves down your TV tube, how a baseball flies through the air, how a comet moves round the sun. Classical mechanics is the form of mechanics developed by Galileo and Newton in the seventeenth century and reformulated by Lagrange and Hamilton in the eighteenth and nineteenth centuries. For almost three hundred years, it seemed that classical mechanics was the form of mechanics, that it could explain the motion of all conceivable systems. Then, in two great revolutions of the early twentieth century it was shown that classical mechanics cannot account for the motion of objects traveling close to the speed of light, nor of subatomic particles moving inside atoms. The years from about 1900 to 1930 saw the development of relativistic mechanics primarily to describe fast moving bodies and of quantum mechanics primarily to describe subatomic systems. Faced with this competition, one might expect classical mechanics to have lost much of its interest and importance. In fact, however, classical mechanics is now, at the start of the twenty-first century, just as important and glamorous as ever. This resilience is due to two facts: First, there are just as many interesting physical systems as ever that are best described in classical terms. To understand the orbits of space vehicles and of charged particles in modern accelerators, you have to understand classical mechanics. Second, recent developments in classical mechanics, mainly associated with the growth of chaos theory, have spawned whole new branches of physics and mathematics and have changed our understanding of the notion of causality. It is these new ideas that have attracted some of the best minds in physics back to the study of classical mechanics.

Physicists tend to use the term “classical mechanics” rather loosely. Many use it for the mechanics of Newton, Lagrange, and Hamilton, and for these people, “classical mechanics” excludes relativity and quantum mechanics. On the other hand, in some areas of physics, there is a tendency to include relativity as a part of “classical mechanics”; for people of this persuasion, “classical mechanics” means “non-quantum mechanics.” Perhaps as a reflection of this second usage, some courses called “classical mechanics” include an introduction to relativity, and for the same reason, I have included one chapter on relativistic mechanics, which you can use or not, as you please.

An attractive feature of a course in classical mechanics is that it is a wonderful opportunity to learn to use many of the mathematical techniques needed in so many other branches of physics — vectors, vector calculus, differential equations, complex numbers, Taylor series, Fourier series, calculus of variations, and matrices. I have tried to give at least a minimal review or introduction for each of these topics (with references to further reading) and to teach their use in the usually quite simple context of classical mechanics. I hope you will come away from this book with an increased confidence that you can really use these important tools.

Inevitably, there is more material in the book than could possibly be covered in a one-semester course. I have tried to ease the pain of choosing what to omit. A number of sections are marked with an asterisk to indicate that they can be omitted without loss of continuity. (This is not to say that this material is unimportant. I certainly hope you’ll come back and read it later!) And the last seven chapters are designed to be mutually independent, so that you can choose to read any one of them without reference to any of the others.

As always in a physics text, it is crucial that you do lots of the exercises at the end of each chapter. I have included a large number of these to give both teacher and student plenty of choice. Some of them are simple applications of the ideas of the chapter and some are extensions of those ideas. I have listed the problems by section, so that as soon as you have read any given section you could (and probably should) try a few problems listed for that section. (Naturally, problems listed for a given section may require knowledge of earlier sections. I promise only that you shouldn’t need material from later sections.) I have tried to grade the problems to indicate their level of difficulty, ranging from one star,\st, (meaning a straightforward exercise usually involving just one main concept) to three stars,\ssst, (meaning a challenging problem that involves several concepts and will probably take considerable time and effort). This kind of classification is quite subjective and is necessarily only very approximate. Several of the problems require the use of computers to plot graphs, solve differential equations, and so on. None of these requires any specific software; some can be done with a relatively simple system such as MathCad; some require more sophisticated systems, such as Mathematica, Maple, or Matlab. (Incidentally it is my experience that the course for which this book was written is a wonderful opportunity for the students to learn to use one of these fabulously useful systems.) Problems requiring the use of a computer are indicated thus: [Computer]. I have tended to grade them as \ssst, or at least \sst, on the grounds that it takes time to set up the necessary code. Naturally, these problems will be easier for students who are experienced with the necessary software.

There are many people I wish to thank for their help and suggestions. At the University of Colorado, these include Professors John Cox, Scott Parker, Steve Pollock, Mike Dubson, Mike Ritzwoller, and Larry Baggett. From other institutions, I want to thank the following: Professors E. Stern at the University of Washington, John Markert and Tom Griffy at the University of Texas, R. Pompi at SUNY at Binghampton, Meagan Aronson at the University of Michigan, and Peter Blunden at the University of Manitoba. I particularly want to thank my two friends and colleagues, Dave Goodmanson at the Boeing Aircraft Company and Mark Semon at Bates College, who both reviewed the manuscript with the finest of combs and gave me literally hundreds of suggestions; for their help and advice I am especially grateful. Bruce Armbruster and Jane Ellis of University Science Books are an author’s dream come true. Finally and most of all, I want to thank my wife Debby. Being married to an author can be quite trying, and she puts up with it very graciously. And, as an English teacher with the highest possible standards, she has taught me most of what I know about writing and editing. I am eternally grateful.

This preliminary edition is missing the last three chapters, and there are some other additions and changes that I would have made if there had been more time. For example, I fully intend to add some more end-of-chapter problems to some of the later chapters. The purpose of this preliminary edition is to find ways to improve the book before its official launch. All suggestions, both large and small, will be most welcome.

John R. Taylor

Department of Physics

University of Colorado

Boulder, Colorado 80309

John.Taylor@Colorado.edu