An elementary treatise on theoretical mechanics

The primary aim of the present book is to supply for students beginning the study of Theoretical Mechanics a course of such a nature as shall emphasize the fundamental physical principles of the subject. Different students will of course approach the study of mechanics with different interests, different aims, and different amounts of mathematical equipment, so that it may not be possible to produce a single book which shall exactly fit the requirements of every class of student. But I believe that all students of mechanics, no matter what their aims and intentions may be, will be in the same position in one respect, namely that they will best begin the study of the subject by trying to acquire a firm grasp of the physical principles, leaving aside at first all mathematical developments and all practical applications, except in so far as these contribute to the elucidation of the fundamental physical principles.

I am aware that this belief is not held by all teachers of mechanics, some of whom regard the laws of mechanics simply as working rules to be acquired as rapidly as possible for their utilitarian value, while others appear to regard them in the same light as the rules of a game, the game consisting in the solution of mathematical puzzles, most of which have no conceivable reference to the facts of nature. I find it hard to believe that there can be any considerable class of students for whom either of these points of view is the best. As regards the former, I feel that a student who cannot get, or does not wish to get, a clear understanding of mechanical principles would be well advised not to enter a profession in which his work will consist in the handling of mechanical problems ; and as regards the latter, that a student who wishes merely to obtain material for puzzle solving would do better to turn his attention to chess or double acrostics. If I have taken some space to express my private convictions, it is because the method I have embodied in the present book arises directly out of these convictions. Mathematical analysis is, of course, not excluded from the book, because without mathematics there can be no serious study of mechanics, but I have tried to reduce the amount of mathematics to a minimum, and I have regarded it (in the present book) as the servant and not as the master. Again, practical applications of mechanics have not been excluded, - on the contrary, these have been introduced wherever possible as illustrations of principles or results, - but I have tried to place principles first and applications second. And problems have not been excluded: I have inserted a great number, because the solution of problems seems to me to be the one and indispensable way of emphasizing a group of abstract principles and of fixing them in the mind of the student. But I have regarded the problems as an adjunct to the study of the principles, and not the principles as a framework round which to build problems.

Besides explaining the method and objects of a book, a preface may be expected to explain where the book starts and where it ends. The present book is intended to start from the very beginning of its subject, assuming no previous knowledge of mechanics on the part of the student. The question of how much knowledge of mathematics ought to be assumed has been a more difficult one to settle. I finally decided to rely as little as possible on the student's knowledge of trigonometry, and to employ the calculus as little as possible in the earlier chapters, but felt that the subjects of the later chapters could not be advantageously treated without a very considerable use of the calculus. Until the later chapters the use of the calculus is confined almost exclusively to unimportant branches and extensions of the subject, and to the working of illustrative examples. Thus a student who has no knowledge at all of the calculus will, I hope, be able to omit the sections of the book in which it is used, while at the same time acquiring a considerable and continuous knowledge of the essentials of theoretical mechanics.

The point at which the book ought to close seemed in the present instance to be determined by the method, of the book itself. If, as I believe, a study of physical principles ought to be the common preliminary to the study of every branch and every application of mechanical science, then the book might clearly try to cover all this common ground, and ought to stop at the point at which detailed specialization becomes feasible and profitable. It ought, in fact, to cover the range which will be covered by all students, and stop short of subjects which wiU be of interest or importance only to a few. Judged by this criterion the book will perhaps be thought by some to be open to the criticism of covering too much ground; it may be thought that the final chapter on generalized coordinates can hardly be regarded as essential to the student whose study of mechanics is a preliminary to his entering the profession of, say, engineering. I am nevertheless convinced that, even if the study of generalized coordinates is not absolutely indispensable to such students, it is of extreme value and ought not to be neglected by a student, possessed of the requisite ability, who can possibly find time for it. The student who omits it shuts himself off from a point of view which sums up and illuminates the whole of dynamical theory; at the same time he denies himself the opportunity of studying, or at least of fully understanding, the theory of electricity and magnetism. And as regards the student who intends to continue his studies in the direction of theoretical physics, the theory of generalized coordinates forms so essential a preliminary to the study of most branches of physics that the advantages of including a short treatment of this subject in the preliminary mechanics course will hardly be disputed.

1. Uniformity of nature. If we place a stone in water, it will sink to the bottom; if we place a cork in water, it will rise to the top. These two statements will be admitted to be true not only of stones and corks which have been seen to sink or rise in water but of all stones and corks. Given a piece of stone which has never been placed in water, we feel confident that if we place it in water it will sink. What justification have we for supposing that this' new and untried piece of stone will sink in water? We know that millions of pieces of stone have at different times been placed in water; we know that not a single one of these has ever been known to do anything but sink. From this we infer that nature treats all pieces of stone alike when they are placed in water, and so feel confident that a new and untried piece of stone will be treated by the forces of nature in the same way as the innumerable pieces of stone of which the behavior has been tested, and hence that it will sink in water. This principle is known as that of the uniformity of nature; what the forces of nature have been found to do once, they will, under similar conditions, do again.

2. Laws of nature. The principle just stated amounts to saying that the action of the forces of nature is governed by certain laws; these we speak of as laws of nature. For instance, if it has been found that every stone which has ever been placed in water has sunk to the bottom, then, as has already been said, the principle of uniformity of nature leads us to suppose that every stone which at any future time is placed m water will sink to the bottom; and we can then announce, as a law of nature, that any stone, placed in water, will sink to the bottom.

That part of science which deals with the laws of nature is called natural science. Natural science is divided into two parts, experimental and theoretical. Experimental science tries to discover laws of nature by observing the action of the forces of nature time after time. Theoretical science takes as its material the laws of nature discovered by experimental science, and aims at reducing them, if possible, to simpler forms, and then discovering how to predict from these laws what the action of the forces of nature will be in cases which have not actually been subjected to the test of experiment. For example, experimental science discovers that a stone sinks, that a cork floats, and a number of similar laws. From these theoretical physics arrives at the simple laws of nature which govern all phenomena of sinking or floating, and, going further, shows how these laws enable us to predict, before the experiment has been actually tried, whether a given body will sink or float. For instance, experimental science cannot discover whether a 50,000-ton ship will float or sink, because no 50,000-ton ship exists with which to experiment. The naval architect, relying on the uniformity of nature, on the laws of nature deter- mined by experimental science, and on the method of handling these laws taught by theoretical science, may build a 50,000-to ship with every confidence that it will behave in the way predicted by theoretical science.

3. The science of mechanics. The branch of science known as mechanics deals with the motion of bodies in space, and with the forces of nature which cause or tend to cause this motion. The laws of nature which govern the action of these forces and the motion of bodies have long been known, and' were reduced to their simplest form by Newton. Thus we may say that experimental mechanics is a completed branch of science.

The present book deals with theoretical mechanics. We start from the laws supplied by experimental mechanics, and have to discuss how these laws can be used to predict the motion of bodies, - for instance, the falling of bodies to the ground, the firing of projectiles, the motion of the earth and the planets round the sun. An important class of problems which we shall have to discuss will be those in which no motion takes place, the forces of nature which tend to cause motion being so evenly balanced that no motion occurs. Such problems are known as statical.


DOWNLOAD FREE BOOK:  An elementary treatise on theoretical mechanics