Principles of mechanics - Synge

This edition differs in no essential way from the first. The principal revision occurs in Chap XIII, where the account of the motion of a particle in an electromagnetic field has been completely rewritten. The treatment of principal axes of inertia in Chap XI has been amplified, and some revisions have been made in the treatments of Foucault's pendulum, the spinning projectile, and the gyrocompass. The emphasis on units and dimensions has been increased by the inclusion in the earlier part of the book of a few short paragraphs, with references to the Appendix, where these matters are discussed in detail. A few additional exercises have been inserted, and numerous minor corrections have been made. We wish to thank all those readers who have contributed to the improvement of this second edition by their suggestions, arid, in particular, Professors L. Infeld, A. E. Sehild, and A. Weinstein.

In a sense this is a book for the beginner in mechanics, but in another sense it is not. From the time we make our first movements, crude ideas on force, mass, and motion take shape in our minds. This body of ideas might be reduced to some order at high school (as crude ideas of geometry are reduced to order), but that is not the educational practice in North America. There is rather an accumulation of miscellaneous facts bearing on mechanics, some mathematical and some experimental, until a state is reached where the student is in danger of being repelled by the subject, as a chaotic jumble which is neither mathematics nor physics.

This book is intended primarily for students at this stage. The authors' ambition is to reveal mechanics as an orderly self-contained subject. It may not be quite so logically clear as pure mathematics, but it stands out as a model of clarity among all the theories of deductive science.

The art of teaching consists largely in isolating difficulties and overcoming them one by one, without losing sight of the main problem while attending to the details. In mechanics, the main problem is the problem of equilibrium or motion under given forces the details are such things as the vector notation, the kinematics of a rigid body, or the theory of moments of inertia. If we rush straight at the main problem, we become entangled in the details and have to retrace our steps in order to deal with them. If, on the other hand, we decide to settle all details first, we are apt to find them uninteresting because we do not see their connection with the main problem. A compromise is necessary, and in this book the compromise consists of the division into Plane Mechanics (Part I) and Mechanics in Space (Part II). These titles must, however, be regarded only as rough indications of the contents. Part I includes some of the easier portions of three-dimensional theory, while Part II contains an introduction to the special theory of relativity, with mechanics in only one spatial dimension !

There is, of course, nothing novel in regarding plane mechanics as the preliminary field; but it is rather unusual to divide the subject in this way in a single volume, or even in a sequence of volumes. It has made the task of writing more difficult, but the authors have felt it worthwhile. Many of the most interesting results in statics and dynamics belong to the plane theory, and it is unfair to deny the reader access to them until he has mastered the more elaborate technique required for three dimensions.

Part I is complete in itself and might be used as a textbook in plane statics and dynamics, with some excursions into three dimensional theory. Vector notation is introduced, but used sparingly. The reader should have a fair knowledge of calculus, elementary differential equations, and some analytical geometry. Practical experience in physics is not essential but very desirable; mechanics is at root a physical subject and should not be treated merely as an excuse for the exercise of mathematical techniques. In Part II the language of vectors is used extensively. A knowledge of three-dimensional analytical geometry is required and greater power in the use of mathematical processes. This part is complete in itself, except for occasional references to Part I. The selection of particular applications follows conventional lines, except for one novel feature a section on electron optics. Chapters on Lagrange's equations and on the special theory of relativity are included.

The book has developed from lectures delivered by both authors to Honor Students in their second and third years at the University of Toronto. These lectures cover about 110 periods of 50 minutes, and it has been found that the work can be done fairly adequately in that time. But this does not allow sufficiently for the working of problems with the classes; it is felt that 150 periods might well be spent on the contents of the book, were it not for other demands on the students' time.

Each chapter is followed by a summary. The summaries to the chapters dealing with ijiethods are naturally the more fundamental there is little hope of being able to attack problems unless one is thoroughly familiar with the general principles outlined there. On the other hand, the summaries to the chapters dealing with applications are intended to provide only a synopsis of what has been done.

Why do we study mechanics? There are at least three reasons. First, we live in an age of machinery, which cannot be designed without a knowledge of mechanics; in fact, it is the most fundamental subject in engineering. Secondly, mechanics plays a basic part in physics and astronomy, contributing to our knowledge of the working of nature. Thirdly, the mathematician is interested in mechanics, both in the logic of its foundations and in the methods employed; a considerable portion of mathematics was developed for the express purpose of solving mechanical problems.

The subject of mechanics is not a mere collection of facts. From certain simple hypotheses an elaborate theory is built up. Anyone who has studied the subject should be able to answer questions of interest to engineers and physicists; that is to say, he should be able to apply his knowledge. But he should also have a fair idea of the logical structure. A successful textbook has to steer a middle course between undue concentration on the mere working out of problems on the one hand, and an over elaborate development of logical structure on the other.

What the student of mechanics requires more than anything else is the development of a certain point of view which is difficult to describe in a few words. Since the reader is expected to have a fair knowledge of geometry, it will be helpful to consider the ways in which we think about that subject.

Every student of geometry learns to think in two ways. First, there is the physical way, in which a point is a small dot on a sheet of paper, a straight line a mark made by drawing a sharp pencil along a straight edge, a circle a mark made by a pair of compasses, and so on. Secondly, there is the ideal or mathematical way, in which a point is no longer a dot on paper, but an ideal thing which the dot serves only to suggest. Anyone who uses geometry has both these ways of thinking at his disposal, switching from one to the other without confusion. The engineer and the physicist generally think in the physical way, but when there is a theorem to be proved they subconsciously switch to the mathematical way. On the other hand the mathematician will think primarily in the mathematical way, but he will change to the physical way when he wants to aid his thought with a diagram.

This duality in point of view is confusing to the beginner in geometry. But it is fortunate that he has to face this difficulty at an early stage in his career, because it prepares him for a similar duality in mechanics, about which he has also to learn to think in two different ways.

First, there is the physical way. We think of actual physical things, natural or man-made. We seek to understand the laws governing their behavior and to predict how they will behave under given circumstances to be able to trace the paths of comets in advance, or design machinery and bridges with confidence as to their behavior when constructed.

On the other hand, there is the mathematical way. Often without realizing it consciously, the physicist, astronomer, or engineer slips over from the physical way of thinking to the mathematical. Thus the astronomer may treat the earth as a perfect sphere an abstract mathematical concept which does not exist in nature or the engineer may discuss a wheel as if it were a perfect circle.

The transition from the physical to the mathematical and back again is a source of more confusion than may be suspected, but it is unavoidable. There is no doubt that the physical way of thought is the more natural; but as long as it is the only way, progress is slow. Physical things are very complicated and hard to think about. Slowly we come to distinguish between properties which are essential and properties which are incidental. We learn to simplify problems by forgetting the incidental properties and concentrating on those which are essential.

To illustrate, suppose we are interested in the periodic time of a bar suspended from one end, oscillating as a pendulum. Which properties of the bar is it essential for us to bear in mind, and which may we neglect as incidental? Can we predict the periodic time of oscillation without knowing the material of which the bar is constructed? Does the form of the cross section of the bar matter? Does it make any difference whether the bar is supported on a knife-edge or by bearings? The cautious well-informed physicist would say that all these things mattered and many others. One material yields more than another, the form of cross section influences the distribution of material, and a change in the mode of suspension may alter the axis about which the pendulum oscillates. But if we were as cautious as this we should have no science of mechanics. To start on the problem, at any rate, we must simplify it ruthlessly. So we think of the bar as a rigid mathematical straight line and the support as a fixed mathematical point. Now we have a problem which is reasonably simple to handle mathematically. Strictly speaking, no properties are incidental. Even the color of the bar affects the pressure of light on it; a subway train stopping five hundred miles away may cause a vibration in the support and affect the motion of the bar. Common sense, which is the accumulated experience of centuries, gives us some guide as to the factors which we may neglect.

Gradually stripping physical things of attributes which are unimportant for the question in hand, we arrive at a mathematical way of thinking about nature. The particular mathematical model to be used on a given occasion depends on that occasion. Consider the earth, for example. The simplest model of the earth is a particle, a mathematical point with mass. This model suffices to obtain the earth's orbit round the sun, but obviously will not do for the discussion of tides or lunar eclipses. For these phenomena we may think of the earth as a rigid sphere, but this model will not serve for the discussion of the precession of the equinoxes (for which we require an ellipsoidal rigid body) or for the discussion of earthquakes (for which we require an elastic sphere). Thus there are many mathematical models for the earth, and the one which we choose depends on the question we are discussing at the moment.

In fact, mechanics and indeed all theoretical science is a game of mathematical make-believe. We say: If the earth were a homogeneous rigid ellipsoid acted on by such and such forces, how would it behave? Working out the answer to this mathematical question, we compare our results with observation. If there is agreement, we say that we have chosen a good model; if disagreement, then the model or the laws assumed are bad.

Let us now sum up the general procedure in theoretical mechanics in the following five steps.

(1) A physical system is an object of curiosity; we wish to predict its behavior under various circumstances. (The system in question might be a pendulum, or a pair of stars attracting one another.)

(2) An ideal or mathematical model of the physical system is constructed mentally. (The pendulum is regarded as a rigid straight line, and the stars are regarded as two particles.)

(3) Mathematical reasoning is applied to the mathematical model. (This means that differential or finite equations are set up and solved. Formulas are developed to give answers to interesting questions, such as those concerning the periodic time of the pendulum or the orbit of one star relative to the other.)

(4) The mathematical results are interpreted physically in terms of the physical problem.

(5) The results are compared with the results of observation, if possible.

Certain remarks should be made about these five steps. First, (1) implies a physical curiosity. In spite of the fact that theoretical mechanics is a part of mathematics, we should not forget that its roots lie in physics and the actual world around us.

Secondly, as has been remarked above, the construction of a mathematical model (2) at once simple and adequate is by no means easy in all cases. However, mechanics is an old subject, and there is much accumulated experience to fall back on. The concepts of particles, rigid bodies, forces, etc. (all mathematical idealizations), have be n designed for this purpose.

Step (3) belongs largely to pure mathematics, requiring no particular knowledge of, or interest in, the physical problem. Nevertheless, it is often of the greatest assistance to the mathematician to bear the physical problem continually in mind; in this way, methods of attack may be suggested to him.

The fourth step in general presents no difficulty, provided that we are clear as to the things in nature which correspond to the things in our mathematical model.