The analytical foundations of celestial mechanics PDF
Preview The analytical foundations of celestial mechanics
THE ANALYTICAL FOUNDATIONS OF CELESTIAL MECHANICS PRINCETON MATHEMATICAL SERIES Editors: MARSTON MORSE, H.P. ROBERTSON, A. w. TUCKER 1. The Classical Groups Their Invariants and Representations BY HERMANN WEYL 2. Topological Groups BY L. PONTR.J AGIN 8. An Introduction to Differential Geometry with Use of the Tensor Calculus BY LUTHER PFAHLER EISENHART 4. Dimension Theory BY WITOLD HUREWICZ AND HENRY WALLMAN 5. The Analytical Foundations of Celestial Mechanics BY AUREL WINTNER 6. The Laplace Transform BY DAVID VERNON WIDDER 7. Integration BY EDWARD .J. MC SHANE 8. Theory of Lie Groups BY CLAUDE CHEVALLEY 9. Mathematical Methods of Statistics BY HARALD CRAMER THE ANALYTICAL FOUNDATIONS OF CELESTIAL MECHANICS BY AUREL WINTNER 1947 PRINCETON; NEW JERSEY PRINCETON UNIVERSITY PRESS LONDON: GEOFFREY CUMBERLEGE OXFORD UNIVERSITY PRESS COPYRIGHT 1941 BY PRINCETON UNIVERSITY PRESS SECOND PRINTING, 1947 Printed in the United States of America LEON LICHTENSTEIN IN MEMORIAM PREFACE It was more than twelve years ago that, at the suggestion of the late Professor Lichtenstein, I began working on a book on the prob lem of three bodies. The original plan was to present a systematic ac.count of the methods and results of the theory of the periodic and related particular solutions of the restricted problem of three bodies and its extensions, and to arrange everything else around these fun damental solutions. However, during the progress of the work it became more and more clear that a systematic presentation of the mathematical the ory of periodic solutions and their applications to the problem of the solar system, on the one hand, and to Stromgren's numerical investi gations, on the other hand, must be preceded by a modernized treat ment of those analytiCal aspects of the general theory of canonical systems which were originated by, and are still fundamental for, Celestial Mechanics as a whole. Through repeated dis·~ussions of the plan of the book with Professor G. D. Birkhoff, I became still more convinced of the necessity of such an approach. I am greatly indebted to him for the friendly and helpful interest which he has always taken in this book. The title is intended to imply that the general topological methods in proofs of existence, as initiated by Poincare, are not dis cussed in this volume .. Nevertheless, this book could not have been written without the investigations of Levi-Civita and Birkhoff. Ac tually, the theory of periodic solutions will be illustrated only by the case of Hill's lunar theory; a case historically and methodically so fundamental as to necessitate an exception. Approximately the first third of the book is based on a course of lectures on analytical mechanics, given for graduate students in physics and mathematics. It is therefore hoped that these chapters can serve as an introduction into the pure analysis of theoretical dynamics and of the theory of perturbations. Throughout the book (and especially in Chapter VI), I have tried not to repel that re grettable majority of younger mathematicians who have had no con tact with theoretical astronomy. Chapter I is perhaps unusual, in that it develops only the dynami cal operators of canonical systems of differential equations, without disguising the actual content of the formalism by an introduction of vii viii PREFACE these equations themselves. In fact, the differential equations and their solutions are introduced only in Chapter II. Correspondingly, the theory of the canonical variation of constants in the theory of perturbations is not subordinated to the characteristic partial differ ential equation which, in fact, appears only as a by-product of the general theory of the transformations of phase space. In Ch~pter II, emphasis is laid on a careful distinction between formal questions, which are always local in nature, and questions in the large, which are the actual problems of mathematical dynamics. While it is true that in most cases more is known about the possible nature of the non-local problems in Celestial Mechanics than about a workable approach to them, the sections dealing with the nature of non-local problems appeared to be rather necessary. In fact, with out these sections it would have been hardly possible even to indicate in later chapters what, in case of n bodies, are actual problems and what must be considered at present pseudo-problems. While Chapter I and Chapter II concern an arbitrary canonical system, Chapter III takes into account the peculiar quadratic struc ture of a dynamical Hamiltonian function. The only non-trivial case for which an explicit analytical formalism is available at pres ent, namely, the case of two degrees of freedom, is conaj.dered in some detail in order that it may become available for application to the restricted problem of three bodies. Chapter IV presents the problem of two bodies, as far as it is of theoretical interest and does not involve the practice of the deter mination of preliminary orbits. The treatment of this elementary case is focused on the fact that the Newtonian choice of the law of attraction is exceptional in every respect. While historical rema.rks are deferred to the end of the book in most cases, in this chapter it seemed to be advisable to put a few remarks of this nature into the text; for it is almost forgotten how much the theory of analytic func tions, for instance, owes to the "elementary" problem of two bodies. Chapter V is the longest chapter of the book. It is somewhat heterogeneous, since it attempts to give an account of our present knowledge of the problem of three or more bodies (with the excluf)ion of the theory of certain periodic and related motions). However, in a few cases I did not succeed in finding short-cuts to certain results for which lengthy proofs are available in original memoirs. In these few cases, I was content to mention (sometimes among the historical notes at the end of the book) the result without proof, but with an explanation of the role of the result or of the apparent reasons for
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