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By the same author: VON D E N G R E N Z E N D E S W I S S E N S ( B a s e l 1953) ZUM W E L T B I L D D E R ASTRONOMIE ( W i t h M . S c h ü r e r . B e r n 1954 a n d 1957) E N T W U R F E I N E R METAPHYSIK ( B e r n 1955) CHRISTENTUM U N D STAAT ( B e r n 1957) INTRODUCTION TO T H E MECHANICS OF STELLAR SYSTEMS ( L o n d o n 1957) INTRODUCTION TO THE MECHANICS OF THE SOLAR SYSTEM RUDOLF KURTH Department of Astronomy University of Manchester PERGAMON PRESS N E W Y O R K · LONDON · PARIS · LOS ANGELES 1959 PERGAMON P R E S S INC. 122 East 55th Street, New York 22, N. Y. P.O. Box 47715, Los Angeles, California PERGAMON P R E S S LTD. 4 ώ 5 Fitzroy Square, London W.l. PERGAMON P R E S S S.A.R.L. 24 Rue des Écoles, Paris Ve Copyright © 1959 Rudolf Kurth Library of Congress Card No. 59-12064 Printed in Northern Ireland at The Universities Press, Belfast PREFACE THIS book is perhaps somewhat unconventional. One of the principal reasons is that, like Socrates, the author dislikes long speeches—and long formulae; and the formulae of celestial mechanics can indeed be very long. No doubt these formulae are very useful or necessary for predicting the motions of the planets; I am not, however, interested in that here, but in understanding the fundamental principles and methods. Anyone, therefore, wishing to copy out ready-made recipes from this book will probably consult it in vain; but I hope that it will be possible to learn a methodical approach from it. My first concern has been with the needs of young students. I have had the following aims: (i) Nearness to natural reality—as is seen, for example, in the occasional use of such names as Jupiter and Saturn. (ii) Insight through simplicity: direct elementary methods and simply formulated results, expressed by short approximate formulae, as is both usually and fruitfully done in theoretical physics. The following requirements were made: a. The results must be methodically derived. b. They must be able to be interpreted easily. c. They shall correctly describe the phenomena both qualitatively, and to the right order of magnitude quantitatively. d. They must be capable of being improved to arbitrary accuracy, for example by iterative procedures or the inclusion of higher terms, by the same method by which they were derived. (Insight is obtained from the approximate formulae which are then improved for the needs of astronomical practice.) (iii) Rigour: although proofs of convergence have been at most hinted at, I hope that at least a breath of the esprit de géométrie may be felt in this book. (iv) Development of the basic concepts: for example the laws of motion and of gravitation were not to be presented as having fallen down ready-made from Heaven, but were to be derived from the observations. VI PREFACE (v) Approach by an axiomatic treatment: each experience or hypothesis should be exhausted before any new experiences or ideas are introduced. For didactical reasons, however, this maxim is not applied rigorously. I should add to (ii) that I have not discussed Hamilton-Jacobi Theory. It seems to me, after detailed critical examination, that it is avast detour from the way to the perturbation equations. The beauty of its general formulation (so simple in principle) is rather useless in face of the concrete problem of integrating a given system of ordinary differential equations. It is only in statistical problems—which are outside traditional celestial mechanics, and so beyond the scope of this book—that canonic variables offer a decided advantage, in conjunction with Liouville's Theorem. So much for my intentions. Little previous knowledge is expected of the reader beyond the elements of analysis, analytical geometry and linear algebra. My references to literature are of a