Celestial Mechanics

AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 18 18

Celestial Mechanics Harry Pollard Celestial This monograph presents the basic mathematics underlying the subject of celestial mechanics. Chapter 1 formulates the central force problem and then deals with Kepler's rst and second laws, orbits Mechanics under non-Newtonian attraction, elements of an orbit, the two-body system, the solar system, and disturbed motion. Chapter 2 introduces the n-body problem. Included in Chapter 2 are the Lagrange—Jacobi formula, Sundman's theorem on total collapse, the three-body Harry Pollard problem, and Lagrange's and Euler's solutions to the three-body problem. Chapter 3 is an introduction to Hamilton-Jacobi Theory. M A PRESS / MAA AMS

4-color Process 145 pages spine: 5/16" finish size: 5.5" X 8.5" 50 lb stock 10.1090/car/018

CELESTIAL MECHANICS

By HARRY POLLARD THE

CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA

Committee on Publications E. F. BECKENBACH, Chairman

Subcommittee on Carus Monographs D. T. F1NKBEINER II, Chairman R. P. BOAS ARGELIA V. ESQUIREL GEORGE PIRANIAN HE CARUS MATHEMATICAL MONOGRAPHS are an expres- sion of the desire of Mrs. Mary Hegeler Cams, and of her son, Dr. TEdward H. Cams, to contribute to the dissemination of mathematical knowledge by making accessible at nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics. The publication of the first four of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Cams as sole trustee of the Edward C. Hegeler Trust Fund. The sales from these have resulted in the Cams Monograph Fund, and the Mathematical Association has used this as a revolving book fund to publish the succeeding monographs. The expositions of mathematical subjects which the monographs contain are set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises. The scope of this series includes also historical and biographical monographs.

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No. 18. Celestial Mechanics, by HARRY POLLARD

Library of Congress Catalog Card Number 76-51507

eISBN 978-1-61444-018-5 Paperback ISBN 978-0-88385-140-1

Hardcover (out of print) 978-0-88385-019-0

Printed in the United States of America The Carus Mathematical Monographs

NUMBER EIGHTEEN

CELESTIAL MECHANICS

By HARRY POLLARD Purdue University

Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA NOTE ON THE USE OF THIS BOOK

1. Vectors are printed in bold-face. Where possible the length of a vector is indicated by the same letter in italic. Thus the length of y is o. When this cannot be done, the length is indicated by the absolute-value symbol. Thus the length of a x b is |a x b|.

2. Starred exercises are not necessarily difficult. The star indicates an important final result, or a result to be used later. Therefore starred exercises should not be omitted.

3. Unless otherwise stated, all references to formulas and exercises are made to the same chapter where they occur. PREFACE

This is a corrected version of Chapters I—III of my Mathematical Introduction to Celestial Mechanics (Prentice-Hall, Inc., 1966). The acknowledgements made in the preface to that book apply equally well to this one. In addition, I am especially indebted to Professor D. G. Saari of Northwestern University for his thorough criti- cism of the original version.

HARRY POLLARD To Helen and my family CONTENTS

THChapteE CENTRArL OnFORCeE PROBLEM 1

/. Formulation of the problem 1 2. The conservation of angular momentum: Kepler's second law 2 3. The conservation of energy 5 4. The inverse square law: Kepler's first law 6 5. Relations among the constants 10 6. Orbits under non-Newtonian attraction 14 7. Position on the orbit: the case h = 0 16 8. Position on the orbit: the case h Φ0 20 9. Position on the orbit: the case h >0 23 10. Position on the orbit: the case h <0 24 11. Determination of the path of a particle 28 12. Expansions in elliptic motion 33 13. Elements of an orbit 36 14. The two-body problem 38 15. The solar system 43 16. Disturbed motion 46 17. Disturbed motion: variation of the elements 49 18. Disturbed motion: geometric effects 53

ix χ CONTENTS Chapter Two

INTRODUCTION TO THE fl-BODY PROBLEM

/. The basic equations: conservation of linear momentum 57 2. The conservation of energy: the Lagrange-Jacobi formula 60 3. The conservation of angular momentum 63 4. Sundman's theorem of total collapse 65 5. The virial theorem 67 6. Growth of the system 69 7. The three-body problem: Jacobi coordinates 71 8. The Lagrange solutions 74 9. Euler's solution 76 10. The restricted three-body problem 78 11. The circular restricted problem: the Jacobi constant 82 12. Equilibrium solutions 85 13. The curves of zero velocity 90

Chapter Three

INTRODUCTION TO HAMILTON-JACOBI THEORY

/. Canonical transformations 93 2. An application of canonical transformations 98 3. Canonical transformations generated by a function 102 4. Generating functions 106 5. Application to the central force and restricted problems 111 6. Equilibrium points and their stability 117 7. Infinitesimal stability 121 8. The characteristic roots 124 9. Conditions for stability 126 10. The stability of the libration points 129

Index

INDEX

Angular momentum, 2, 63 Eccentricity, 8 Anomaly: Ecliptic, plane of, 43 eccentric, 16 Elements of orbit, 36 mean, 23 table, 45 true, 9 Elliptic orbit, 9 Area, Kepler's law on, 4 position on, 20 Argument of pericenter, 9 Energy, 5, 60 Asteroids, Trojan, 82 kinetic, 6, 61 Attraction, law of, 1 potential, 6, 60 Equations of motion, 1, 57 Barycentric coordinates, 40 Euler, L., 76, 89 Existence theorems, 57, 81 Canonical transformations, 93 Expansions in elliptic motion, generating function of, 10S 33-4 Center of mass, uniform motion of, 59 Force, central, 1 Central force, 1 Fourier series, 28 (Ex. 10.2), 33-4 Characteristic roots, 124 Collapse, total, 65 G, the gravitational constant, 39 Collision, 19 Generating function, 105 Coordinates: Gravitation, Newton's law of, 1 barycentric, 40 Gravitational constant, 39 Jacobi, 71 relative, 39 Hamilton-Jacobi theory, (Chap. Curves of zero velocity, 90 3), 93 Herget, P., 24 Disturbed motion, 46 Hill curves (— curves of zero velocity), 90 Eccentric anomaly, 16 Hyperbolic orbit, 9 Eccentric axis, 7 position on, 23

133 134 INDEX

Identity of Lagrange-Jacobi, 61 under non-Newtonian attrac- Inertia, moment of, 61 tion, 14 osculating, 46 Jacobi constant, 82 parabolic, 9, 16 Jacobi, Hamilton-, theory of, 93 types of, 9 Jacobi, Lagrange-, identity of, 61 Osculating orbit, 46 Jupiter, 82 Parabolic orbit, 9, 16 Kepler's equations, 22 (Ex. 8.4) Pericenter, 9 Kepler's laws: Perihelion, 9 first, 6 Planets, 43 second, 4 table, 45 third, 13, 43 Plummer, H. C, 32 (footnote) Kilroy, W. H., 91 Points of libration, 89 Kinetic energy, 6, 61 Pollard, H„ 67 Kolmogoroff, Α., 131 Potential energy, 6, 60 Problem: Lagrange-Jacobi identity, 61 of η bodies, 57 Lagrange, J. L., 74, 89 of three bodies, 71 Liapounov stability, 118 of two bodies, 38 Libration points, 87 Line of nodes, 38 Restricted three-body problem, 78 Longitude: Self-potential, 60 of ascending node, 37 Solar system, 43 of pericenter, 38 Sun, 43 Sundman, K., 65 Matrix, symplectic, 94 Symplectic matrix, 93 Mean anomaly, 23 Mean motion, 21 Theorem: Momentum, angular, conservation existence, 57, 81 of, 2, 63 Sundman's, 65 Moon, 82 Virial, 67, 71 (Ex. 6.1) Motion relative to center of mass, Three-body problem, 71 59 Transformation, canonical, 95 Trojan asteroids, 82 η-body problem, Chapter 2, p. 57 True anomaly, 9 Newtonian gravitation, 1 Two-body problem, 38 Nodes, line of, 36 Virial theorem, 67, 71 (Ex. 6.1) Orbit: definition, 9 Weierstrass, K., 66 elliptic, 9, 24 Widder, D. V., 68 hyperbolic, 9, 23 Wintner, Α., 35 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 18 18

Celestial Mechanics

4-color Process 145 pages spine: 5/16" finish size: 5.5" X 8.5" 50 lb stock