Table Of ContentCELESTIAL MECHANICS
Volume III, Part 1
Differential Equations 1n
Celestial Mechanics
CELESTIAL
MECHANICS
Yusuke Hagihara
VOLUME III, Part 1
Diflerential Equations in
Celestial Mechanics
JAPAN SOCIETY FOR THE PROMOTION OF SCIENCE
KOJIMACHI, CHIYODA-KU, TOKYO
1974
Copyright© 1974 by Yusuke Hagihara
Published by Japan Society for the Promotion of Science
PRINTED BY TOKYO PRESS
Preface
Volume III, Part l, and its companion, Volume III, Part 2, discuss
formal solutions of the differential equations in celestial mechanics
and show that none of them is uniformly convergent. A chapter on
stability theory is adjointed; this theory is also related to the nature
of the solutions of differential equations.
Part l contains the discussion of differential equations to which
the problems on celestial mechanics are reduced.
We begin in Chapter 12 with simpler differential equations in
perturbation theory, such as Hill's, Lindstedt's, and Mathieu's, with
various methods for discussing the form of solution. The conver
gence of Hill's infinite determinant is discussed. Various methods of
solution of Lindstedt's equation, which is the reduced form of the
equations of the three-body problem, are described. The form of
solution of the three-body problem is closely related to the solution
of such simplified differential equations with periodic coefficients.
The solution of the Mathieu equation is discussed in detail, includ
ing the zeropoints of the solution and the stability domains. After
a description of Maxwell's theory and its modern criticisms and
refinements, the method is applied to the motion of the particles
constituting Saturn's rings. Then the matrix method of integrating
a system of linear differential equations with periodic coefficients is
described, along with various successful applications, such as to the
problems of Saturn's rings, libration, higher-degree secular pertur
bation, the motion of Saturn's satellites, and electron motion inside
a magnetron.
vi PREFACE
In Chapter 13 the averaging method of Krylov-Bogoliubov for
solving nonlinear differential equations is treated. The characteristic
feature of the method is discussed in comparison with other methods
described in Volume II. The advantage of the method, as well as
Ritz-Galerkin's, is to enable us to estimate the error of approxima
tion. The method, as well as the Hale-Cesari modified method, is
applied to the motion of earth satellites. For proving the existence
of periodic solutions we sometimes refer to Poincare's invariant
point theorem and Birkhoff's rotation numbers, which will be
described in Chapter 25 (Volume V). In particular, systems of dif
ferential equations with slowly varying variables and systems with
resonance are discussed, based on the averaging method with a view
to applying the results to the three-body problem. Diliberto's
periodic-surface method is described with application to the motion
of earth satellites. The existence of periodic solutions of the motion
of earth satellites is proved. The formal stability of the three-body
problem is discussed according to Diliberto. The Ritz-Galerkin
method and Kruskal's theory are sketched.
Part 2 contains the discussion on the form of solution in the per
turbation theory and the three-body problem, and the stability
theory of Liapounov with its modern developments.
In Chapter 14 particularly the forms of solution in the planetary
theory are discussed in the light of various proposed theories, such
as those of Newcomb, Lindstedt, von Zeipel, Poincare, Bohlin,
Whittaker, and Cherry. These theories are based on the idea of
irregular integrals, abnormal integrals, and asymptotic representa
tion of the solution of differential equations according to Poincare,
Horn, Kneser and Birkhoff. The forms of the formal solution of the
three-body problem are trigonometric series with linear functions of
time as arguments. If they were uniformly convergent then they
would be the quasi-periodic functions of Bohl. It will be shown in
Volume IV that they are actually quasi-periodic if certain conditions
are satisfied. The adelphic integral of Whittaker is described. The
recent discovery of the existence of the third integral for a galactic
system is discussed in detail, including the numerical computation
by Contopoulos and his colleagues and by Henon. Remarkable fea
tures of this nonuniform third integral are revealed. The solution
of a linear differential equation with quasi-periodic or almost
periodic coefficients is treated on the basis of Bochner's normal func
tions in view of applications to perturbation theory.
In Chapter 15 we show the general divergence of the series thus
PREFACE Vil
obtained. Bruns's ingenious proof of nonuniform convergence of the
series for the solution of perturbation theory is discussed, together
with its modern development by Petersson and Wintner. Gylden's
question on probability in connection with this series is described.
The nonuniform convergence is shown to be related to Borel series,
Liouville's numbers and Mahler's U-numbers. The character of such
trigonometric series is shown in detail with several examples.
Methods of promoting the more rapid formal convergence and of
grouping the terms of the series are discussed. The divergence of
Lindstedt's and Bohlin's series is proved according to Poincare. The
asymptotic integral of Littlewood for the three-body problem is de
rived. This integral, as well as the so-called third integral and
Diliberto's periodic surface, may throw some light on the asymptotic
behavior of the solution of the problem.
Chapter.16 is devoted to the discussion of stability of the states of
motion. Liapounov's theory and its modern developments are dis
cussed. Refined notions of stability are presented and are used in
deriving the converse of Liapounov's theorems. Lefschetz's necessary
and sufficient condition is stated with his generalized Liapounov
function. Concepts such as uniform and global stability, semi-definite
and decrescent functions, intensive behavior and stability probability
are defined. Several examples of Liapounov's function for the cri
terion of stability are given. Bohl's works on the motion in the
neighborhood of an equilibrium point are described in connection
with quasi-periodic solutions. Perron's study is explained with his
notion of order numbers and kinematic similarity. The stability
theory is extended to abstract spaces with the idea of linear operators
and to the topology of the stability domains. Application to controls
is briefly sketched. The recent development on the stability of rigid
body rotation is described.
Having treated the unfortunate circumstance of the divergence
or rather the nonuniform convergence-of the derived series, we turn
to the discussion of rigorous periodic and conditioned quasi-periodic
solutions, together with asymptotic solutions, of the three-body prob
lem in the next Volume. The topological theories of Poincare and
Birkhoff will be treated in Volume V.
