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Space Mathematics Part 1 Volume 5 Lectures in Applied Mathematics SPACE MATHEMATICS PART 1 J. Barkley Rosser, EDITOR MATHEMATICS RESEARCH CENTER THE UNIVERSITY OF WISCONSIN 1966 AMERICAN MATHEMATICAL SOCIETY, PROVIDENCE, RHODE ISLAND Supported by the National Aeronautics and Space Administration under Research Grant NsG 358 Air Force Office of Scientific Research under Grant AF-AFOSR 258-63 Army Research Office (Durham) under Contract DA-3 I- 12 4-ARO(D)-82 Atomic Energy Commission under Contract AT(30-I)-3164 Office of Naval Research under Contract Nonr(G)O0025-63 National Science Foundation under NSF Grant GE-2234 All rights reserved except those granted to the United States Government, otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number 66-20435 Copyright © 1966 by the American Mathematical Society Printed in the United States of America Contents Part 1 FOREWORD ix ELLIPTIC MOTION J. M. A. Danby MATRIX METHODS 32 _-'/ J. M. A. Danby THE LAGRANGE-HAMILTON-JACOBI MECHANICS 40 Boris Garfinkel STABILITY AND SMALL OSCILLATIONS ABOUT EQUILIBRIUM .,,J AND PERIODIC MOTIONS 77 P. J. Message LECTURES ON REGULARIZATION 100 j,- Paul B. Richards THE SPHEROIDAL METHOD IN SATELLITE ASTRONOMY 119 J John P. Vinti PRECESSION AND NUTATION 130 u/" Alan Fletcher ON AN IRREVERSIBLE DYNAMICAL SYSTEM WITH TWO DEGREES J OF FREEDOM: THE RESTRICTED PROBLEM OF THREE BODIES 150 Victor Szebehely PROBLEMS OF STELLAR DYNAMICS 169 j George Contopoulos //,- QUALITATIVE METHODS IN THE n-BODY PROBLEM 259 Harry Pollard INDEX 293 vi CONTENTS Part2 MOTION IN THE VICINITY OF THE TRIANGULAR LIBRATION CENTERS Andr_ Deprit MOTION OF A PARTICLE IN THE VICINITY OF'A TRIANGULAR LIBRATION POINT IN THE EARTH-MOON SYSTEM 31 J. Pieter deVries THE DOMINANT FEATURES OF THE LONG-PERIOD LIBRATIONS OF THE TROJAN MINOR PLANETS 7O P. J. Message OUTLINE OF A THEORY OF NONPERIODIC MOTIONS IN THE NEIGHBORHOOD OF THE LONG-PERIOD LIBRATIONS ABOUT THE EQUILATERAL POINTS OF THE RESTRICTED PROBLEM OF THREE BODIES 79 Eugene Rabe ELEMENTS OF A THEORY OF LIBRATIONAL MOTIONS IN THE ELLIPTICAL RESTRICTED PROBLEM 102 Eugene Rabe THE EQUILIBRIUM SHAPE OF THE EARTH IN THE LIGHT OF RECENT DISCOVERIES IN SPACE SCIENCE 119 John A. O'Keefe THE STABILITY OF A ROTATING LIQUID MASS 155 John A. O'Keefe GEODETIC PROBLEMS AND SATELLITE ORBITS 170 W. H.Guier ELEMENTS OF CALCULUS OF VARIATIONS AND OPTIMUM CONTROL THEORY 212 Magnus R. Hestenes INDEX 255 CONTENTS vii Part3 BASIC FLUID DYNAMICS S. F. Shen SHOCK WAVES IN RAREFIED GASES 54 S. F. Shen MODELS OF GAS FLOWS WITH CHEMICAL AND RADIATIVE EFFECTS 94 F. K. Moore DECAY OF ORBITS 153 P. J. Message THE EFFECT OF RADIATION PRESSURE ON THE MOTION OF AN ARTIFICIAL SATELLITE 167 Gen-ichiro Hori SPECIAL COMPUTATION PROCEDURES FOR DIFFERENTIAL EQUATIONS 179 S. V. Parter THE TWO VARIABLE EXPANSION PROCEDURE FOR THE APPROXIMATE SOLUTION OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS 206 J. Kevorkian RENDEZVOUS PROBLEMS 276 J. C. Houbolt INDEX 301 AUTHOR INDEX FOR THE THREE VOLUMES 305 Foreword In 1962 a committee of the Division of Mathematics of the National Research Council presented to the Space Science Board of the National Academy of Sciences a report on the current and anticipated uses of mathematics in space activities. A key finding was the need for much more intensive work in various mathematical areas, both to develop more powerful results in these areas and to discover ways of using the existing knowledge more effectively. The Summer Seminar of 1963 was planned to fill this need in part. It was the third Summer Seminar in Applied Mathematics sponsored by the American Mathematical Society. In order to keep the Seminar within bounds and to achieve reasonable coherence, many mathematical areas of great importance and urgency for space activity were not considered at the Seminar. The most notable omissions are the area of nonlinear differential equations, which is of use in the study of guidance systems for propulsive space vehicles, and various areas of statistics, such as those involved in the design of experiments to be performed in space, analysis of large amounts of data from experiments aboard space vehicles, etc. Primarily, the topics considered at the Seminar were those having to do with the behavior of nonpropulsive space vehicles. Inevit- ix x FOREWORD ably, this led to heavy emphasis on the theory of orbits. For this reason, the Seminar was combined with the fifth annual Summer Institute in Dynamical Astronomy. Four previous Summer Institutes in Dynamical Astronomy have been sponsored by Yale University, in the summers of 1959 through 1962. They were devoted primarily to topics in the theory of orbits, and were more restricted in scope than the present Seminar. The orbits of most present and planned space vehicles differ so markedly from the orbits of the classical celestial bodies that only parts of the classical theories of celestial mechanics are applicable. Much effort is being expended to determine which of the classical methods are applicable, to find suitable modifications of some of the classical methods to make them more widely applicable, and to find new methods to handle novel situations. Such matters were a main concern of the four previous Summer Institutes in Dynamical Astronomy, and received a considerable amount of attention at the present Seminar; it was to assure this that the Seminar was combined with the fifth Institute. Notes were compiled for the Institute of 1959, but their supply is exhausted. Notes for the Institute of 1960 were edited by William E. Felling and published in 1961 by The McDonnell Aircraft Corporation, St. Louis, Missouri, under the title, "Notes of the Summer Institute in Dynamical Astronomy at Yale University, July 1960". Copies of these notes can be obtained from The McDonnell Aircraft Corporation. The Institute of 1961 was held at Tucson, Arizona, and the Institute of 1962 was held at Yale University. Only scattered notes, mostly in mimeographed form, survive from these meetings. Some of these were collected and have been included in the present Proceedings. Specifically, from the Institute of 1961 come the chapters: Precession and nutation by Alan Fletcher; Lectures on regularization by Paul B. Richards. From the Institute of 1962 come the chapters: Problems of stellar dynamics by G. Contopoulos; Matrix methods by J.M.A. Danby; The effect of radiation pressure on the motion of an artificial satellite by Gen-ichiro Hori; The two variable expansion procedure for the approximate solution of certain nonlinear differential equations by J. Kevorkian; Stability and small oscillations about equilibrium and periodic motions by P. J. Message; The spheroidal method in satellite astronomy by John P. Vinti. FOREWORD xi Some of the lecturesof the earlierInstitutes,and of the present Seminar, have been published elsewhere, either separately or as parts of larger works. Indeed, some lectures excerpted particularly relevant material already in print. Besides the Notes of the 1960 Institute already cited, a listof publications containing material not covered in the present Proceedings which was presented in lecturesat the Institutesand the Seminar follows: R. F. Arenstorf, Periodic solutions of the restricted three body prob- lem presenting analytic continuations of Keplerian motions, Amer. J. Math 85(1963), 27-35. V. I. Arnol'd, The stability of the equilibrium position of a Hamil- tonian system of ordinary differential equations in the general elliptic case, Dokl. Akad. Nauk SSSR 137(1961), 255-257 = Soviet Math. Dokl. 2(1961), 247. __, Generation of quasi-periodic motion from a family of periodic motions, Dokl. Akad. Nauk SSSR 138(1961), 13-15 = Soviet Math. Dokl. 2(1961), 501. __, The classical theory of perturbations and the problem of stability of planetary systems, Dokl. Akad. Nauk SSSR 145(1962), 487-490 = Soviet Math. Dokl. 3(1962), 1008. __, Proof of A. N. Kolrnogorov's theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian, Uspehi Mat. Nauk SSSR 18, Ser. 5 (113), 1963, pp. 13-40. __, Small divisor and stability problems in classical and celestial mechanics, Uspehi Mat. Nauk SSSR 18, Ser. 6(119), 1963, pp. 81-192. R. E. Bellman, Adaptive control processes; a guided tour, Princeton Univ. Press, Princeton, N. J., 1961. R. E. Bellman and S. Dreyfus, Applied dynamic programming, Princeton Univ. Press, Princeton, N. J., 1962. D. Brouwer, Solution of the problem of artificial satellite motion without drag, Astronom. J. 64(1959), 378-397. D. Brouwer and G. M. Clemence, Methods of celestial mechanics, Academic Press, New York, 1961. D. Brouwer and G. Hori, Theoretical evaluation of atmospheric drag effects in the motion of an artificial satellite, Astronom. J. 66(1961), 193-225; 264-265. C. J. Cohen and E. C. Hubbard, A non-singular set of orbit elements, Astronom. J. 67(1962), 10-15. C. C. Conley, A disk mapping associated with the satellite problem, xii FOREWORD Comm. Pure Appl. Math. 17(1964), 237-243. S. Dreyfus, Dynamic programming and the calculus of variations, J. Math. Anal. Appl. 1(1960), 228-239. W. J. Eckert and D. Brouwer, The use of rectangular coordinates in the differential correction of orbits, Astronom. J. 46(1937), 125-132. B. Garfinkel, Motion of a satellite in the vicinity of the critical in- clination, Astronom. J. 65(1960), 624-627. Y. Hagihara, Theories of equilibrium figures of a homogeneous rotat- ing fluid mass, Blaisdell, New York, forthcoming. __, Celestial mechanics, Blaisdell, New York, forthcoming in four volumes. Paul Herget, The computation of orbits, Observatory of the Univer- sity of Cincinnati, 1948. __, Computation of preliminary orbits, Astronom. J. 70(1965), 1-2. G. Hori, Motion of an artificial satellite in the vicinity of the critical inclination, Astronom. J. 65(1960), 291-300.
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