Table Of ContentTHE 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
