Table Of ContentCELESTIAL MECHANICS
Volume I
Dynamical Principles
and Transformation Theory
CELESTIAL
MECHANICS
Yusuke Hagihara
VOLUME I
Dynamical Principles
and Transformation Theory
The MIT Press
CAMBRIDGE, MASSACHUSETTS, AND LONDON, ENGLAND
Copyright © 1970 by The Massachusetts Institute of Technology
Designed b;• Dwight E. Agner. Set in Monotype Baskerville. Printed and bound in the United
States of America by The Maple Press Company.
All rights reserved. No part of this book may be reproduced in any form or by any means, electronic
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system, without permission in writing from the publisher.
ISBN 0 262 08037 0 (hardcover)
Library of Congress catalog card number: 74-95280
Preface
The rather ambitious aim of the present series is to recapitulate the results
of the whole field of celestial mechanics and the associated branches of
science during the last hundred years. During this period the classical
method of formal expansion into series, due to Laplace and Le Verrier,
has been severely criticized; a new class of orbits called periodic solutions
has been introduced, with applications to Hill's lunar theory. The trig
onometric series employed in the ordinary perturbation theory are
expanded in powers of the small quantities characteristic ofactual natural
bodies, and with arguments proportional to time. Bruns and Poincare
have proved that such series are not uniformly convergent, although the
existence of the solution has been established on function-theoretic
grounds by Sundman. The newly discovered class of functions called
"quasi-periodic" and "almost periodic" can be expanded in similar but
uniformly convergent form. The question naturally arises: What condi
tions should be satisfied by the equations of motion of the n-body
problem? There has been some endeavor to solve these equations by
resorting to integral equations and infinite nonlinear analysis. The
question of stability of motion, especially of the solar system, has been
discussed by means of the linearized system of equations according to the
ideas of Poincare and Liapounov. The problem is to study stability
behavior by means of nonlinear equations, even by dealing with non
convergent expansions in the vicinity of the equilibrium point. Birkhoff
has extended the idea of periodic solutions of Hill and Poincare to
recurrent motions, central motions, and wandering motions, on the
basis of measure theory. Another important aspect of the modern trend
in celestial mechanics is the topological theory initiated by Poincare and
VI PREFACE
Birkhoff, for example the invariant-point theorem and the ergodic
theorem. The present series ofv olumes undertakes the program oft reating
all such questions, extending over the whole of celestial mechanics and
associated fields, and of giving suggestions for future development of the
subject, stimulated by the launching of artificial celestial bodies. This first
volume presents introductory dynamical principles as Part I, and trans
formation theory as Part II.
Part I begins with the principles of analytical dynamics. From the
variational principle of Hamilton, and using the calculus of variations,
we deduce Hamilton's canonical form of the differential equations. As an
introduction to the topological study ofd ynamical trajectories, Riemann
ian geometry is described as it pertains to the quadratic form of the
kinetic energy. The notions ofinvariance, covariance, and contravariance
are introduced for use in dealing with the canonical transformations.
The Hamilton-Jacobi partial differential equation is derived both from
the canonical equations and from a hydrodynamical analogy. Liouville's
and Stackel's theorems are proved, and the condition for separability of
the variables in the Hamilton-Jacobi equation is obtained by using
Riemannian geometry; the theorems obtained are also applied to the
Schrodinger equation in wave mechanics.
Chapter 2 deals with quasi-periodic motions. We define a libration, a
revolution, and an asymptotic motion for a simple integrable dynamical
system; these definitions serve as keys to much more complicated types of
motion in celestial mechanics. The multiply periodic motions and
conditionally periodic motions are discussed on the basis of Weierstrass's
preparatory theorem in the theory of functions of many variables and
Kronecker's theorem in number theory. The motion of molecules in a
cubic vessel is solved as a concrete example of ergodicity. Bohl's quasi
periodic functions and Bohr's almost periodic functions are defined,
and certain of their properties which will be useful later are stated.
Schwarzschild's relativistic one-body problem in the gravitational field
and the problem of the motion of an earth satellite under the action of
the distorted earth are solved by using the Hamilton-Jacobi equation and
quasi-periodic motions. Various other methods of treating such problems
are subjects of the subsequent volumes. The theory of algebraic integrals
is briefly sketched as an introduction to the ideas used in the text.
Chapter 3 is dedicated to particular solutions of the three-body and
many-body problems. Euler's and Lagrange's types ofp articular solutions
are obtained for the n-body problem in a general manner, and the
nature of the motion is fully analyzed. The isosceles-triangular solution is
discussed on the basis of the theory of analytic functions.
PREFACE Vll
Part II of this volume begins in Chapter 4 with Lie's theory of con
tinuous groups of transformations, with application to the n-body prob
lem. The theorems due to Poisson, Liouville, and Lie are proved on the
basis of Lie's ideas of complete systems, involution, Lie's function groups
and distinguished functions; then these theorems are applied to the
integrals of the n-body problem. Integral invariants and adiabatic
invariants are introduced for the theoretical reduction of the order of the
differential equations. The existence of adiabatic invariants is proved in
a general abstract space for use in later arguments. A summary of the
modern theory of Lie groups is given with a view to suggesting a new
trend of development.
In Chapter 5, the differential equations of the n-body problem are
reduced by using the known integrals. By Lagrange's, Levi-Civita's, and
Poincare's transformations, the differential equations are reduced to the
lowest order. The two-body problem, especially, is discussed in full; and
various expansions are obtained, using the theory offunctions as applied
to Keplerian motion to establish the convergence of the expansions. Then
the equations for the variation of elements, Delaunay's, Poincare's, and
Kepler's, are derived as the fundamental principle in perturbation
theory. The role of the angular-momentum integral .is stressed in view
of the elimination of the node by Jacobi. Various kinds of canonical
variables are defined, along with their dynamical meanings.
Chapter 6 is devoted to discussions ofBruns's and Poincare's theorems.
The generality ofBruns's proof of the theorem of the nonexistence of new
algebraic integrals is extended from three bodies to n bodies. The proof
of Painleve's extension of Bruns's theorem is given. The integrals of the
restricted three-body problem are discussed with special reference to the
ergodic theorem. Poincare's theorem concerning the nonexistence of new
uniform integrals is proved and discussed in full detail. Exceptions to
Poincare's theorem are also dealt with. The top motions of Euler,
Lagrange, and Kowalewski are discussed from the point of view of
Bruns's and Poincare's theorems, and the solution, given with integrals,
is based on the theory of algebraic functions, especially on the Riemann
0-functions and the hyperelliptic functions.
Volume II, which will be published shortly after Volume I, will
contain perturbation theories in classical celestial mechanics for the
motions of planets, asteroids, satellites, comets, and the moon, that is,
theories of Laplace, Lagrange, Delaunay, von Zeipel, Hansen, Gylden,
Hill, Brown, and others, and their applications to the motions of artificial
celestial bodies. Questions will be raised on the gaps in the asteroidal
distribution and on the capture of comets.
Vlll PREFACE
In Volume III, the form of the solutions of these perturbation theories
and the convergence of the series expansions for the solutions will be
discussed on mathematical grounds. The question of the third integral of
motion will be touched upon. Differential equations with periodic co
efficients and with quasi-periodic coefficients will be used in connec
tion with perturbation theory. Krylov-Bogoliubov's averaging method
and Diliberto's periodic surface theory will be applied to the motion of
celestial bodies.
In Volume IV, Poincare's theory of periodic solutions will be discussed
in full with various modern versions of the theory and with practical
applications to the three- and many-body problems. Periodic solutions
will be dealt with on the basis ofWintner's nonlinear infinite analysis and
Lichtenstein's nonlinear integral equations. Motions in the neighborhood
of an equilibrium point will be discussed according to the theories of
Birkhoff and Siegel. The conditional existence of quasi-periodic solutions
in the manner of Kolmogorov, Arnold, and Moser will be shown. The
stability theory of Liapounov and its modern development will be
described.
Volume V will contain proofs of the existence of the solution of the
three-body problem by Sundman, Levi-Civita, Chazy, and Merman, as
well as the proofofthe rigorous capture theory of Merman; finally, there
will be discussed the topological theory of Poincare and Birkhoff,
including the theory of characteristics, manifolds of motion, surface
transformations, the invariant-point theorem, the ergodic theorem, and
almost periodic motions.
The author wishes to express his appreciation for a unique editorial
collaboration: Dr. Peter Musen of the National Aeronautics and Space
Administration's Goddard Space Flight Center (GSFC) kindly read the
manuscript and offered aid and encouragement as the work progressed.
Subsequently, Dr. Musen gave generously of his time and wisdom to
Mr. R. L. Tanner, chief editor of GSFC, and his associate Mr. P. Barr,
who with NASA support jointly prepared the work for the publishers.
Yusuke Hagihara
Tokyo, July 1968
