To my father, who taught me to write

To my father, who taught me to write

Stephanie Frank Singer Symmetry in Mechanics A Gentle, Modern Introduction Springer Science+Business Media, LLC

Stephanie Frank: Singer Philadelphia, PA www.symmetrysinger.com Library of Congress Cataloging.in.Publication Data Singer, Stepbanie frank, 1964 Symmetry in mecbanics : a gentle, modem introduction I Stephanie Frank Singer. p.cm. Includes bibliograpbica1 references and index. ISBN 978-0 8176-4145 0 978-8176-4145-0 ISBN 978-1-4612-0189-2 (ebook) DOI 10.1007/978-1-4612-0189-2 1. Mecbanics, Ana1ytic. 2. Geometry, Differential. I. Title. QA805.S622oo1 531-dc21 00 049398 CIP AMS Subject Classifications: 70F05, 70H33, 70H05, 22E, 53C15, 53A25, 53C80 ISBN 978-0-8176-4145-0 Printed on acid free paper. 2OO1 Stepbanie frank Singer 2004 Stepbanie frank Singer, 2nd printing OriginalIy published by Birkhiiuser Boston in 2004 AlI rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expres sion of opinion as to whether or not they are subject to property rights. 98765432 SPIN 11013433 www.birkhauser-science.com

Contents Preface o Preliminaries 0.1 Notation and Conventions. 0.2 Physics and Math Background 1 The Two-Body Problem 1.1 First Simplification 1.2 Second Simplification... 1.3 Recovering Kepler's Laws ix 1 1 2 5 7 12 14 2 Phase Spaces are Symplectic Manifolds 19 2.1 The Plane and the Area Form.... 20 2.2 Vectors, Covectors and Antisymmetric Bilinear Forms 24 2.3 Vector Fields and Differential Forms 29 2.4 The Cylinder and Jacobian Matrices 34 2.5 More Examples of Phase Spaces 39 2.6 Pullback Calculations...... 41 3 Differential Geometry 3.1 Manifolds... 47 48

vi Contents 3.2 Differentiable Functions.... 3.3 Vector Fields and Forms on Manifolds 4 Total Energy Functions are Hamiltonian Functions 4.1 Total Energy and Equations of Motion 4.2 Particles on the Line.... 4.3 Magnetism in Three-Space....... 4.4 Energy Conservation and Applications. 4.5 An Example from the 1\vo-Body Problem 5 Symmetries are Lie Group Actions 5.1 From Symmetries to Groups.. 5.2 Matrix Lie Groups.... 5.3 Abstract Lie Groups (Optional). 5.4 Group Actions.... 5.5 Orbits of Group Actions.... 5.6 Symmetries of Hamiltonian Systems. 6 Infinitesimal Symmetries are Lie Algebras 6.1 Matrix Lie Algebras................. 6.2 Vector Fields Associated to Lie Algebra Elements. 6.3 Abstract Lie Algebras (Optional).......... 7 Conserved Quantities are Momentum Maps 7.1 Definition of Momentum Map...... 7.2 Examples from the Two-Body Problem 7.3 Momentum Maps from Group Actions (Optional) 8 Reduction and The Two-Body Problem 8.1 First Reduction...... 8.2 Second Reduction..... 8.3 Recovering Kepler's Laws 8.4 Zero Angular Momentum. 8.5 Symplectic Geometry. 8.6 Concluding Remarks... 55 58 65 66 69 71 72 81 83 84 86 90 91 95 96 101 102 107 111 113 114 117 119 123 124 129 133 134 137 138

Contents vii Recommended Reading 141 Solutions 147 References 183 Index 189

Preface "And what is the use," thought Alice, "of a book without pictures or conversations in it?" -Lewis Carroll This book is written for modem undergraduate students - not the ideal students that mathematics professors wish for (and who occasionally grace our campuses), but the students like many the author has taught: talented but appreciating review and reinforcement of past course work; willing to work hard, but demanding context and motivation for the mathematics they are learning. To suit this audience, the author eschews density of topics and efficiency of presentation in favor of a gentler tone, a coherent story, digressions on mathematicians, physicists and their notations, simple examples worked out in detail, and reinforcement of the basics. Dense and efficient texts play a crucial role in the education of budding (and budded) mathematicians and physicists. This book does not presume to improve on the classics in that genre. Rather, it aims to provide those classics with a large new generation of appreciative readers. This text introduces some basic constructs of modern symplectic geometry in the context of an old celestial mechanics problem, the two-body problem. We present the derivation of Kepler's laws of planetary motion from Newton's laws of gravitation, first in the style of an undergraduate physics course, and

x Preface then again in the language of symplectic geometry. No previous exposure to symplectic geometry is required: we introduce and illustrate all necessary constructs. Sir Isaac Newton analyzed completely the two-body problem (and many other problems) in his famous and ultimate book, Philosophiae Naturalis Principia Mathematica [N]. From his own fundamental laws of motion and Kepler's laws summarizing observations of the planets, Newton deduced the inverse square law for the attractive gravitational force of the sun. Our analysis is more modest: we use the fundamental laws of motion and the inverse square law to deduce Kepler's laws. In the process Newton performs an early (the first, to the author's knowledge) symplectic reduction. He uses the concept of the center-of-mass and conservation of linear momentum to show that "the motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forward in a right line without any circular motion" [N, Corollary V of "Axioms, or Laws of Motion"]. And while Newton's (and others', as found in most current physics textbooks) exploitation of the conservation of angular momentum is not, strictly speaking, symplectic reduction, the same results can be obtained by symplectic reduction. So the two-body problem provides two natural examples of symplectic reduction. Chapter 0 covers some preliminary material. Chapter 1 presents the derivation of Kepler's laws of planetary motion from Newton's laws of gravitation in the style of a typical American undergraduate physics text. Chapter 8 presents the same argument in the language of modern symplectic geometry. The chapters in between develop the concepts and terminology necessary for the final chapter, providing a detailed translation between the quite different languages of mathematics and physics. Warning This book is not intended as a comprehensive introduction to symplectic geometry or classical mechanics. For strong undergraduate students it will not suffice as the sole text for a full semester course. Instead of formulating a general theory and treating a variety of problems, this text makes explicit the ties between mathematics and physics, as well as the ties between powerfully abstract formulations and concrete calculations. Like the trunk of a Japanese maple, it should support and unite readers who will branch off in various directions. There are many excellent treatments of the two-body problem in the physics literature, and excellent treatments of symplectic reduction in the graduate

Preface xi mathematics literature. Readers desiring broader or more sophisticated texts should consult the Recommended Reading section. The book in your hands, accessible to anyone who has studied multivariable calculus and the rudiments of linear algebra, should provide a useful complement to the existing sources. Guide to the Instructor This book can be used as a supplement to courses on differential geometry or Lie theory. In particular, Chapters 3, 5 and 6 are self-contained units, with concrete examples of sophisticated ideas and useful exercises. The book could be a major component of a course on symplectic geometry or classical mechanics. The book (in its entirety or with Chapter 3 omitted) could serve to structure the first part of a course in symplectic geometry, providing motivation for a more standard exposition of the mathematics. It would also be appropriate at the end of an example-driven semester course on classical mechanics, in which case students should be encouraged to work out the symplectic versions of examples treated earlier. For more advanced courses in symplectic geometry, Chapters I and 8 link the mathematics to its historical roots and current physics terminology. Likewise, in an advanced physics course on mechanics these chapters would make the relevant mathematics more accessible. Guide to the Reader Readers who have no particular background in symplectic geometry or classical mechanics may wish to start at the beginning and read through, doing the exercises as they go. They should feel free to ignore anything marked "optional," including all of Chapter 3. They should not be discouraged by the occasional exercise requiring a branch of mathematics they may not know (such as techniques for solving differential equations), or by the occasional paragraph aimed at more sophisticated readers (usually discussing links to other areas of mathematics ). Readers working through a standard mathematical exposition of symplectic geometry may find this book a useful auxiliary source of explicitly calculated examples. Readers familiar with mechanics or with symplectic geometry may wish to start with Chapter 8 to get the whole story quickly. The references in that chapter should make it easy to dip back into the rest of the text to fill any gaps in understanding.

xii Preface The reader will find references to many other texts in the Bibliography and Recommended Reading section. Listed there are good books for solidifying the prerequisites, interesting collateral reading, alternative sources for the material presented, major classics in the field and research articles by currently active mathematicians. Acknowledgments Thanks to several people for specific contributions to this book: Karen Uhlenbeck and Chuu-Lian Terng for the invitation to give a series of lectures on symplectic geometry for undergraduates at the IASlPark City Mentoring Program for Women in 1997 at the Institute for Advanced Study (las) in Princeton; Haverford College and the American Mathematical Society Centennial Fellowship program for financial support; Kathleen McGoldrick, Anne Humes, Catherine Jordan and the las staff for nontechnical support; Jerrold Marsden and Ann Kostant for their encouragement; Eugene Lerman for suggesting references; J. J. Duistermaat for copious historical insights; Ben Allen for writing solutions to exercises; and those who critiqued the manuscript along the way, especially Allen Knutson, Tanya Schmah, Alan Weinstein and the anonymous reviewers. Thanks also to all those who taught me, supported me and believed in me, especially the community of symplectic geometers and my students at Haverford College.