Table Of ContentNumerical Solutions of the N-body Problem
Mathema tics and Its Applications (East European Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIAL YNICKI-BIRULA, Institute of Mathematics PKIN, Warsaw, Poland
J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia
L. LEINDLER, Bolyai Institute, Szeged, Hungary
L. LOVA SZ, Bolyai Institute, Szeged, Hungary
D. S. MITRINOVIC, University of Belgrade, Yugoslavia
S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland
BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria
I. T. TODOROV, Academy of Sciences, Sofia, Bulgaria
H. TRIEBEL, Universitiit lena, D.D.R.
Andrzej Marciniak
Institute of Mathematics,
A. Mickiewicz University, Poznan, Poland
Numerical Solutions
of the
N-body Problem
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster
library of Congress cataloging in Publication Data
Marciniak, Andrzej, 1953-
Numerical solutions of the N-body problem.
(Mathematics and its applications. East European series)
Includes index.
1. Many-body problem-Numerical solutions. 2. Initial value
problems-Numerical solutions. I. Title. II. Series.
QA378.M37 1985 530.1' 44 85-10821
ISBN-13: 978-94-010-8889-3 e-ISBN-13: 978-94-009-5412-0
DOl: 10.1007/978-94-009-5412-0
Published by D. Reidel Publishing Company
P.O. Box 17,3300 AA Dordrecht, Holland
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All Rights Reserved
© 1985 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover I sl edition 1985
No part of the material protected by this copyright notice may be reproduced or utilized
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without written permission from the copyright owner
To My Wife
Contents
Series Editor's Preface ix
Preface xi
INTRODUCTION 1
Chapter 1. CONVENTIONAL NUMERICAL METHODS FOR SOLVING
THE INITIAL VALUE PROBLEM ..............•... 3
1.1. The Initial Value Problem .•..........•.•.•. 3
1.2. Discretization Methods .............•....... 5
1.3. Survey of Numerical Methods for the Initial
Value Problem ......••....•..•............•. 13
1.3.1. One-step Methods •..•....•••••......• 14
1.3.2. Linear Multistep Methods ...••....... 24
1.3.3. Extrapolation Methods •...•......•.•. 34
1.3.4. Some Other Methods ..........•...•.•• 46
Chapter 2. THE GENERAL N-BODY PROBLEM ................. 49
2.1. Equations of Motion •..••..•...•....•.....•. 49
2.2. The Application of Conventional Numerical
Methods ..•.•.••..•.....•••............•.... 56
2.3. Discrete Mechanics of Greenspan •....•...... 78
2.3.1. Basic Relations and their Properties 79
2.3.2. Stability and Convergence ........•.. 85
2.3.3. Practical Realization of Greenspan's
Method.............................. 94
2.4. Discrete Mechanics of Arbitrary Order ...•.. 100
2.5. Energy Conserving Methods .•.•....•......... 121
Chapter 3. THE RELATIVE MOTION OF N BODIES •....•...... 141
3.1. Equations of Motion ....•...........•....... 141
3.2. The Application of Conventional NumeriCal
Methods ...•........•.••.•................•• 145
3.3. The Application of Discrete Mechanics
Formulas ...•.•.....••....................•• 157
Chapter 4. EQUATIONS OF MOTION IN A ROTATING FRAME •.•. 167
4.1. Equations of Motion •.....•...............•• 167
4.2. The Application of Conventional Numerical
Methods .•..••....•....•...........•......•. 178
4.3. Discrete Mechanics in a Rotating Frame ...•. 189
vii
viii Contents
4.4. A Motion nearby the Equilibrium Points .•.... 195
4.5. Discrete Hill's Equations ••..••••...•.....•. 204
Appendix. PL/I PROCEDURES .•...•.•.••.•.....•........... 215
REFERENCES •.••.•..••.••..•.........•.•.••.•....••...... 233
INDEX .....................••..•••••....•...••.....••... 239
SERIES EDITOR'S PREFACE
Approach your problem from the right It isn't that they can't see
end and begin with the answers. the solution.
Then one day, perhaps you will find It is that they can't see the
the final question. problem.
The Hermit Clad in Crane Feathers in G.K. Chesterton. The Scandal
R. van Gulik's The Chinese Maze of Father Brown The Point of
Murders. a Pin.
Growing specialization and diversification have brought a host of
monographs and textbooks on increasingly specialized topics. However,
the "tree" of knowledge of mathematics and related fields does not
grow only by putting forth new brancheq. It also happens, quite often
in fact, that branches which were thought to be completely disparate
are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied
in various sciences has changed drastically in recent years: measure
theory is used (non-trivially) in regional and theoretical economics,
algebraic geometry interacts with physics; the Minkowsky lemma, coding
theory and the structure of water meet one another in packing and
covering theory; quantum fields, crystal defects and mathematical
programming profit from homotopy theory; Lie algebras are relevant to
filtering; and prediction and electrical engineering can use Stein
spaces. And in addition to this there are such new emerging subdisci
plines as "experimental mathematics", "CFDfI, "completely integrable
systems", "chaos, synergetics and large-scale order", which are
almost impossible to fit into the existing classification schemes.
They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging
(sub)disciplines and to such (new) interrelations as exempla gratia:
- a central concept which plays an important role in several differ
ent mathematical and/or scientific specialized areas;
- new applications of the results and ideas from one area of scien
tific endeavour into another;
- influences which the results, problems and concepts of one field
of enquiry have and have had on the development of another.
The Mathematics and Its Applications programme tries to make available
a careful selection of books which fit the philosophy outlined above.
With such books, which are stimulating rather than definitive,
intriguing rather than encyclopaedic, we hope to contribute something
towards better communication among the practitioners in diversified
fields.
ix
x Series Editor's Preface
The N-body problem is an old one and, in spite of the efforts of the
best mathematicians of all time, unsolved for N>3, and only really
solved for N=2 (where of course the solution is easy and almost 400
years old). The case N=3 is intermediate and would require at least
a full length book to describe all that is known. We know, thanks to
the work of Siegel, Kolmogorov, Arnol'd, Moser, and others, that the
dynamics are exceedingly complicated in terms of their dependence on
initial conditions.
Numerical simulation has wrought revolutions in several parts of
mathematics. Whether it can also perform similar services for this
old, exceedingly intractable, problem is something I would not dare
try to predict. But certainly insight can be generated this way
both for veterans and newcomers to the field. This requires the
appropriate numerics which takes full accourit of the special proper
ties of this particular problem. This is exactly what this book
provides.
The unreasonable effectiveness of As long as algebra and geo
mathematics in science ... metry proceeded along
separate paths, their advance
Eugene Wigner was' slow and their
applications limited.
Well, if you know of a better 'ole, But when these sciences joined
go to it. company they drew from each
other fresh vitality and
Bruce Bairnsfather thenceforward marched on at a
rapid pace towards perfection.
What is now proved was once only
imagined. Joseph Louis Lagrange.
William Blake.
Bussum, March 1985 Michiel Hazewinkel
Preface
This book is basically designed for university students
majoring in numerical methods and their applications.
It contains a review of conventional numerical methods used
for solving the initial value problem, an analysis of their
applications to different variants of the N-body problem
and analysis of some special methods of solving this problem.
In addition, it presents more than twenty computer algorithms
which can immediately be programmed in any programming
language. For this reason, this book may also be of interest
to professionals involved in research in the N-body problem.
I wrote this book during my stay at the University of Florida
in Gainesville on an exchange program with A.Mickiewicz
University in Pozna~, Poland. I express my sincere gratitude
to Prof.A.R.Bednarek, chairman of the Department of Mathema
tics, for creating conditions in which I was able to write
the book. I also address special thanks to Prof.J.Albrycht
of Pozna~'s Technical University without whose inspiration,
encouragement and advice this book would not have been
written.
Gainesville, May, 1984 Andrzej Marciniak
xi
Description:Approach your problem from the right It isn't that they can't see end and begin with the answers. the solution. Then one day, perhaps you will find It is that they can't see the the final question. problem. G.K. Chesterton. The Scandal The Hermit Clad in Crane Feathers in of Father Brown The Point o
