The Sheer Joy of Celestial Mechanics

193 Pages·1996·13.049 MB·English

by  Nathaniel Grossman (auth.)

Cover design by Joseph Sherman, Hamden, Ct.
Nathaniel Grossman
The Sheer Joy
of Celestial Mechanics
Birkhauser
Boston • Basel • Berlin
Nathaniel Grossman
Department of Mathemalics
University of California at Los Angeles
Los Angeles, CA 90024-1555
Library of Congress Cataloging-in-Publication Data
Grossman, Nathaniel, 1937-
TIle sheer joy of celestial mechanics I Nathaniel Grossman.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-14612-8647-9 e-ISBN-13: 978-146124090-7
DOl: 10.107/978-146124090-7
acid free paper)
I. Celestial mechanics. I. Title.
QB35l.G69 1995 95-34467
521--dc20 CIP
Printed on acid-free paper Qaalv) ®
Birkhiiuser
e
1996 Birkhauser Boston
Softco\'er reprint of the hardcover 1st edition 1996
Copyright is not claimed for works of U.S. Government employees.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
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or otherwise, without prior pennission of the copyright owner.
Pennission to photocopy for internal or personal use of specific clients is granled by
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addressed directly to B irkhauser Boslon, 675 Massachusetts Av enue, Cambridge, MA 02139,
U.S.A.
Refonnalted from author's disk by TeXniques, IIIC., Boslon, MA
9 8 7 654 321
Contents
List of Figures viii
Preface ix
Prepublication Review xiii
Chapter I. Rotating Coordinates . 1
1. Some kinematics 1
2. Dynamics 8
3. Newton's Laws of Motion 11
4. The Laws of Motion and conservation laws 12
5. Simple harmonic motion 15
6. Linear motion in an inverse square field 16
7. Pendulum in a uniform gravitational field 17
8. Foucault's pendulum 22
Chapter II. Central Forces . 25
1. Motion in a central field 25
2. Force and orbit 28
3. The integrable cases of central forces 32
4. Bonnet's Theorem .. 35
5. Miscellaneous exercises 36
6. Motion on a surface of revolution 38
Chapter III. Orbits under the Inverse Square Law 41
1. Kepler's three laws and Newton's Law 41
2. The orbit from Newton's Law 44
3. The true, eccentric, and mean anomalies 46
4. Kepler's equation .... 50
5. Solution of Kepler's equation 52
6. The velocity of a planet in its orbit 56
7. Drifting of the gravitational constant 57
vi CONTENTS
Chapter IV. Expansions for an Elliptic Orbit 63
1. The general problem 63
2. Lagrange's expansion theorem 64
3. Bessel coefficients 69
4. Fourier series . . 72
5. Preliminaries for expansions 75
6. Some algebraically-derived expansions 77
7. Expansions in terms of the mean anomaly 83
Chapter V. Gravitation and Closed Orbits 89
1. Bertrand's characterization of a universal gravitation 89
2. Circular motions . . . . . . 91
3. Neighbors of circular motions 92
4. Higher perturbations; completion of the proof 94
5. From differential geometry in the large . 97
6. Ovals described under a central attraction 102
Chapter VI. Dynamical Properties of Rigid Bodies 105
1. From discrete to continuous distributions of mass 105
2. Moments of inertia . . . . 106
3. Particular moments of inertia 108
4. Euler's equations of motion 110
5. Euler free motion of the Earth 110
6. Feynman's wobbling plate 113
7. The gyrocompass 115
8. Euler angles 117
Chapter VII. Gravitational Properties of Solids . 123
1. The gravitational potential of a sphere 123
2. Potential of a distant body; MacCullagh's formula 127
3. Precession of the equinoxes .... ... 128
4. Internal potential of a homogeneous ellipsoid 135
5. External potential of a homogeneous ellipsoid 143
CONTENTS vii
Chapter VIll. Shape of a Self-Gravitating Fluid 149
1. Hydrostatic equilibrium . . . . . . . . 149
2. Distortion of a liquid sphere by a distant mass 150
3. Tide-raising on a ringed planet 154
4. Clairaut and the variation of gravity 157
5. Poincare's inequality for rotating fluids 160
6. Lichtenstein's symmetry theorem . 164
7. Rotundity of a rotating fluid 169
8. Ellipsoidal figures of rotating fluids 172
Index . . . . . . . . . . . . . . . 179
List of Figures
1.1 Infinitesimal rotation 3
1.2 Fixed and moving frames 4
1.3 Plane frame 6
1.4 Areal velocity 8
1.5 Simple pendulum 18
1.6 Foucault's pendulum 23
11.1 Tangent line and radius of curvature 29
11.2 Acceleration components 30
111.1 Elliptical orbit 42
111.2 True and eccentric anomalies 47
V.1 Tangent and normal vectors 98
V.2 Minkowski support function 100
V.3 Parallel tangents 101
VI.1 The gyrocompass (in idealized suspension) 116
VI. 2 Euler angles 120
VII. 1 Concentric spheres 124
VII.2 Ring element of a spherical shell 125
VII. 3 Point outside a body 128
VII.4 Graph of c,o(A) 144
VII. 5 Elementary strips 145
VIII. 1 Sphere and distant mass 151
VIII.2 Planetary ring(s) 155
VIII.3 Deflection of a plumb line 159
VIII.4 Midpoint locus 165
VIII.5 Qo inside T 166
VIII.6 Converging chords 167
VIII.7 Singular points 169
VIII.8 Ball and bulge 170
VIII. 9 Maclaurin and Jacobi series 174
Preface
Dear Reader,
Here is your book. Take it, run with it, pass it, punt it, enjoy all the
many things that you can do with it, but-above all-read it. Like all
textbooks, it was written to help you increase your knowledge; unlike all
too many textbooks that you have bought, it will be fun to read.
A preface usually tells of the author's reasons for writing the book and
the author's goals for the reader, followed by a swarm of other important
matters that must be attended to yet fit nowhere else in the book. I am
fortunate in being able to include an insightful prepublication review that
goes directly to my motivations and goals. (Look for it following this
preface.) That leaves only those other important matters.
In preparing the text, I consulted a number of books, chief of which
included these:
• S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale Uni
versity Press, 1969.
• J .M.A. Danby, Fundamentals of Celestial Mechanics, Macmil
lan, 1962. Now available in a 2nd edition, 3rd printing, revised,
corrected and enlarged, Willmann-Bell, 1992.
• Y. Hagihara, Theories of Equilibrium Figures of a Rotating Ho
mogeneous Fluid Mass, NASA, 1970.
• R.A. Lyttleton, The Stability of Rotating Liquid Masses, Cam-
ix
x PREFACE
bridge University Press, 1953.
• C.B. Officer, Introduction to Theoretical Geophysics, Springer
Verlag, 1974.
• A.S. Ramsey, Newtonian Attraction, Cambridge University Press,
1949.
• W.M. Smart, Celestial Mechanics, Longmans, Green, and Co,
1953.
• E. T. Whittaker, Analytical Dynamics, Cambridge University Press,
1927.
Readers familiar with these books will recognize the great debt that lowe
to them. Other books have offered both comfort and enjoyment; they are
credited in footnotes.
I suggest to instructors that this course be taught without either midterm
or final examinations. Those medieval rituals may still be necessary to
goad recalcitrant calculus students into studying, but they are no longer
needed for upper-division students who, after all, are volunteers. In lieu
of the examinations, require students to complete a pre specified number
of the problems, exactly which ones being each student's choice. These
problems may be handed in singly or in batches at any time during the
term, and they must be essentially correct in order to receive credit. Prob
lems that are not worked correctly are to be quickly recycled back to the
student marked with helpful suggestions for his or her emendation and
resubmission. Both the mathematical content and the presentation must
be scrutinized. It may be useful to the students to distribute copies of
J.J. Price's 'Learning Mathematics Through Writing: Some Guidelines,'
The College Mathematics Journal, 20 (1989) 393-402, which will help
them to write well in the journalese that is the received style nowadays.
The effort that they put in to improve their mathematical writing will pay
off in all of their writing.
One of the useful byproducts of the xerographic reproduction age is
a plentiful supply of scratch paper. Students are often reluctant to make
sketches as they study, perhaps because they feel that their sketches are
too crude. But even the crudest sketching can be helpful. After all,