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Advanced Mechanics From Euler's Determinism to Arnold's Chaos • • • s. G. RAJEEV Advanced Mechanics From Euler's Determinism to Arnold's Chaos S. G. Rajeev Department of Physics and Astronomy; Department of Mathematics University of Rochester, Rochester. NY14627 OXFORD UNrvERS]TY PRESS OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © S. G. Rajeev 2013 The moral rights of the author have been asserted First Edition published in 2013 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013938985 ISBN 978-0-19-967085-7 ISBN 978-0-19-967086-4 (pbk.) Printed and bound by CPI Group (UK) Ltd, Croydon, CRO 4YY This book is dedicated to Samda Amma and Gangadhamn Pillai, my parents. Preface Classical mechanics is the oldest and best understood part of physics. This does not mean that it is cast in marble yet, a museum piece to be admired reverently from a distance. Instead, mechanics continues to be an active area of research by physicists and mathemati cians. Every few years, we need to re-evaluate the purpose of learning mechanics and look at old material in the light of modern developments. The modern theories of chaos have changed the way we think of mechanics in the last few decades. Previously formidable problems (three body problem) have become easy to solve numerically on personal computers. Also, the success of quantum mechan ics and relativity gives new insights into the older theory. Examples that used to be just curiosities (Euler's solution of the two center problem) become starting points for physically interesting approximation methods. Previously abstract ideas (Julia sets) can be made accessible to everyone using computer graphics. So, there is a need to change the way classical mechanics is taught to advanced undergraduates and beginning graduate students. Once you have learned basic mechanics (Newton's laws, the solution of the Kepler problem) and quantum mechanics (the Schrodinger equation, hydrogen atom) it is time to go back and relearn classical mechanics in greater depth. It is the intent of this book to take you through the ancient (the original meaning of "classical") parts of the subject quickly: the ideas started by Euler and ending roughly with Poincare. Then we take up the developments of twentieth century physics that have largely to do with chaos and discrete time evolution (the basis of numerical solutions). Although some knowledge of Riemannian geometry would be helpful, what is needed is developed here. We will try to use the minimum amount of mathematics to get as deep into physics as possible. Computer software such as Mathematica, Sage or Maple are very useful to work out examples, although this book is not about computational methods. Along the way you will learn about: elliptic functions and their connection to the arithmetic-geometric-mean; Einstein's calculation of the perihelion shift of Mercury; that spin is really a classical phenomenon; how Hamilton came very close to guessing wave mechanics when he developed a unified theory of optics and mechanics; that Riemannian geometry is useful to understand the impossibility of long range weather prediction; why the maximum of the potential is a stable point of equilibrium in certain situations; the similarity of the orbits of particles in atomic traps and of the Trojan asteroids. By the end you should be ready to absorb modern research in mechanics, as well as ready to learn modern physics in depth. The more difficult sections and problems that you can skip on a first reading are marked with asterisks. The more stars, the harder the material. I have even included some problems whose answers I do not know, as research projects. Mechanics is still evolving. In the coming years we will see even more complex problems solved numerically. New ideas such as renormalization will lead to deeper theories of chaos. Symbolic computation will become more powerful and change our very definition of what viii Preface constitutes an analytic solution of a mechanical problem. Kon-commutative geometry could become as central to quantum mechanics as Riemannian geometry is to classical mechanics. The distinction between a book and a computer will disappear, allowing us to combine text with simulations. A mid-twenty-first century course on mechanics will have many of the ingredients in this book, but the emphasis will be different. Acknowledgements My teacher, A. P. Balachandran, as well as my own students, formed my view of mechanics. This work is made possible by the continued encouragement and tolerance of my wife. I also thank my departmental colleagues for allowing me to pursue various directions of research that must appear esoteric to them. For their advice and help I thank Sonke Adlung and Jessica White at Oxford University Press, Gandhimathi Ganesan at Integra, and the copyeditor Paul Beverley. It really does take a village to create a book. Contents List of Figures xv 1 The variational principle 1 1.1 Euler-Lagrange equations 3 1.2 The Variational principle of mechanics 4 1.3 Deduction from quantum mechanics* 5 2 Conservation laws 7 2.1 Generalized momenta 7 2.2 Conservation laws 7 2.3 Conservation of energy 8 2.4 Minimal surface of revolution 9 3 The simple pendulum 11 3.1 Algebraic formulation 12 3.2 Primer on Jacobi functions 13 3.3 Elliptic curves* 14 3.4 Imaginary time 15 3.5 The arithmetic-geometric mean* 16 3.6 Doubly periodic functions* 19 4 The Kepler problem 21 4.1 The orbit of a planet lies on a plane which contains the Sun 21 4.2 The line connecting the planet to the Sun sweeps equal areas in equal times 21 4.3 Planets move along elliptical orbits with the Sun at a focus 22 4.4 The ratio of the cube of the semi-major axis to the square of the period is the same for all planets 22 4.5 The shape of the orbit 23 5 The rigid body 27 5.1 The moment of inertia 27 5.2 Angular momentum 28 5.3 Euler's equations 29 5.4 Jacobi's solution 30 6 Geometric theory of ordinary differential equations 33 6.1 Phase space 33 6.2 Differential manifolds 34 6.3 Vector fields as derivations 36 6.4 Fixed points 38 xii Contents 7 Hamilton's principle 43 7.1 Generalized momenta 43 7.2 Poisson brackets 45 7.3 The star product* 46 7.4 Canonical transformation 47 7.5 Infinitesimal canonical transformations 48 7.6 Symmetries and conservation laws 51 7.7 Generating function 52 8 Geodesics 55 8.1 The metric 55 8.2 The variational principle 56 8.3 The sphere 58 8.4 Hyperbolic space 60 8.5 Hamiltonian formulation of geodesics 61 8.6 Geodesic formulation of Newtonian mechanics* 62 8.7 Geodesics in general relativity* 63 9 Hamilton-Jacobi theory 69 9.1 Conj ugate variables 69 9.2 The Hamilton-Jacobi equation 70 9.3 The Euler problem 71 9.4 The classical limit of the Schrodinger equation* 73 9.5 Hamilton-Jacobi equation in Riemannian manifolds* 74 9.6 Analogy to optics' 75 10 Integrable systems 77 10.1 The simple harmonic oscillator 78 10.2 The general one-dimensional system 79 10.3 Bohr-Sommerfeld quantization 80 10.4 The Kepler problem 81 10.5 The relativistic Kepler problem* 83 10.6 Several degrees of freedom 83 10.7 The heavy top 84 11 The three body problem 87 11.1 Preliminaries 87 11.2 Scale invariance 88 11.3 Jacobi co-ordinates 89 11.4 The ,1- potential 92 11.5 Montgomery's pair of pants 93 12 The restricted three body problem 95 12.1 The motion of the primaries 95 12.2 The Lagrangian 95 12.3 A useful identity 97 12.4 Equilibrium points 97 12.5 Hill's regions 98 Contents xiii 12.6 The second derivative of the potential 98 12.7 Stability theory 100 13 Magnetic fields 103 13.1 The equations of motion 103 13.2 Hamiltonian formalism 103 13.3 Canonical momentum 105 13.4 The Lagrangian 105 13.5 The magnetic monopole* 106 13.6 The Penning trap 108 14 Poisson and symplectic manifolds 111 14.1 Poisson brackets on the sphere 111 14.2 Equations of motion 112 14.3 Poisson manifolds 113 14.4 Liouville's theorem 114 15 Discrete time 117 15.1 First order symplectic integrators 117 15.2 Second order symplectic integrator 119 15.3 Chaos with one degree of freedom 121 16 Dynamics in one real variable 125 16.1 Maps 125 16.2 Doubling modulo one 126 16.3 Stability of fixed points 128 16.4 Unimodal maps 130 16.5 Invariant measures 133 17 Dynamics on the complex plane 137 17.1 The Riemann sphere 137 17.2 Mobius transformations 138 17.3 Dynamics of a Mobius transformation 139 17.4 The map Z f---t z2 141 17.5 i\1etric on the sphere 142 17.6 Metric spaces 143 17.7 Julia and Fatou sets 144 18 KAM theory 149 18.1 Newton's iteration 149 18.2 Diagonalization of matrices 150 18.3 Normal form of circle maps 152 18.4 Diophantine approximation 155 18.5 Hamiltonian systems 156 Further reading 159 Index 161