A Course in Mathematical Physics 1 and 2 Walter Thirring A Course in Mathematical Physics 1 and 2 Classical Dynamical Systems and Classical Field Theory Second Edition Translated by Evans M. Harrell With 144 lllustrations Springer-Verlag New York Wien Dr. Walter Thirring Dr. Evans M. Harrell Institute for Theoretical Physics The lohns Hopkins University University of Vienna Baltimore, Maryland Austria U.S.A. Printed on acid-free paper. © 1992 Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA) 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Volume I © 1978, 1992 by Springer-Vedag/Wien Volume 2 © 1979, 1986 by Springer-VedaglWien Typeset by Asco Trade Typesetting Ltd., Hong Kong. 9 8 7 6 5 4 3 2 1 ISBN-13: 978-0-387-97609-9 e-ISBN-13: 978-1-4684-0517-0 DOl: 10.1007/978-1-4684-0517-0 Summary of Contents A Course in Mathematical Physics 1: Classical Dynamical Systems, Second Edition 1 Introduction 2 Analysis on Manifolds 3 Hamiltonian Systems 4 Nonrelativistic Motion 5 Relativistic Motion 6 The Structure of Space and Time A Course in Mathematical Physics 2: Classical Field Theory, Second Edition 1 Introduction 2 The Electromagnetic Field of a Known Charge Distribution 3 The Field in the Presence of Conductors 4 Gravitation v Walter Thirring A Course in Mathematical Physics 1 Classical Dynamical Systems Translated by Evans M. Harrell Preface to the Second Edition The last decade has seen a considerable renaissance in the realm of classical dynamical systems, and many things that may have appeared mathematically overly sophisticated at the time of the first appearance of this textbook have since become the everyday tools of working physicists. This new edition is intended to take this development into account. I have also tried to make the book more readable and to eradicate errors. Since the first edition already contained plenty of material for a one semester course, new material was added only when some of the original could be dropped or simplified. Even so, it was necessary to expand the chap ter with the proof of the K-A-M Theorem to make allowances for the cur rent trend in physics. This involved not only the use of more refined mathe matical tools, but also a reevaluation of the word "fundamental." What was earlier dismissed as a grubby calculation is now seen as the consequence of a deep principle. Even Kepler's laws, which determine the radii of the planetary orbits, and which used to be passed over in silence as mystical nonsense, seem to point the way to a truth unattainable by superficial observation: The ratios of the radii of Platonic solids to the radii of inscribed Platonic solids are irrational, but satisfy algebraic equations of lower order. These irrational numbers are precisely the ones that are the least well approximated by rationals, and orbits with radii having these ratios are the most robust against each other's perturbations, since they are the least affected by resonance effects. Some surprising results about chaotic dynamics have been discovered recently, but unfortunately their proofs did not fit within the scope of this book and had to be left out. In this new edition I have benefited from many valuable suggestions of colleagues who have used the book in their courses. In particular, I am deeply grateful to H. Grosse, H.-R. Griimm, H. Narnhofer, H. Urbantke, and above IX x Preface to the Second Edition all M. Breitenecker. Once again the quality of the production has benefited from drawings by R. Bertlmann and J. Ecker and the outstanding word processing of F. Wagner. Unfortunately, the references to the literature have remained sporadic, since any reasonably complete list of citations would have overwhelmed the space allotted. Vienna, July, 1988 Walter Thirring Preface to the First Edition This textbook presents mathematical physics in its chronological order. It originated in a four-semester course I offered to both mathematicians and physicists, who were only required to have taken the conventional intro ductory courses. In order to be able to cover a suitable amount of advanced material for graduate students, it was necessary to make a careful selection of topics. I decided to cover only those subjects in which one can work from the basic laws to derive physically relevant results with full mathematical rigor. Models which are not based on realistic physical laws can at most serve as illustrations of mathematical theorems, and theories whose pre dictions are only related to the basic principles through some uncontrollable approximation have been omitted. The complete course comprises the following one-semester lecture series: I. Classical Dynamical Systems II. Classical Field Theory III. Quantum Mechanics of Atoms and Molecules IV. Quantum Mechanics of Large Systems Unfortunately, some important branches of physics, such as the rela tivistic quantum theory, have not yet matured from the stage of rules for calculations to mathematically well understood disciplines, and are there fore not taken up. The above selection does not imply any value judgment, but only attempts to be logically and didactically consistent. General mathematical knowledge is assumed, at the level of a beginning graduate student or advanced undergraduate majoring in physics or mathe matics. Some terminology of the relevant mathematical background is xi xii Preface to the First Edition collected in the glossary at the beginning. More specialized tools are intro duced as they are needed; I have used examples and counterexamples to try to give the motivation for each concept and to show just how far each assertion may be applied. The best and latest mathematical methods to appear on the market have been used whenever possible. In doing this many an old and trusted favorite of the older generation has been forsaken, as I deemed it best not to hand dull and worn-out tools down to the next generation. It might perhaps seem extravagant to use manifolds in a treat ment of Newtonian mechanics, but since the language of manifolds becomes unavoidable in general relativity, I felt that a course that used them right from the beginning was more unified. References are cited in the text in square brackets [ ] and collected at the end of the book. A selection of the more recent literature is also to be found there, although it was not possible to compile a complete bibliography. I am very grateful to M. Breitenecker, J. Dieudonne, H. Grosse, P. Hertel, J. Moser, H. Narnhofer, and H. Urbantke for valuable suggestions. F. Wagner and R. Bertlmann have made the production of this book very much easier by their greatly appreciated aid with the typing, production and artwork. Walter Thirring Note about the Translation In the English translation we have made several additions and corrections to try to eliminate obscurities and misleading statements in the German text. The growing popularity of the mathematical language used here has caused us to update the bibliography. We are indebted to A. Pflug and G. Siegl for a list of misprints in the original edition. The translator is grateful to the Navajo Nation and to the Institute for Theoretical Physics of the University of Vienna for hospitality while he worked on this book. Evans M. Harrell Walter Thirring Contents of Volume 1 Preface to the Second Edition ix Preface to the First Edition xi Glossary xv Symbols Defined in the Text xix 1 Introduction 1.1 Equations of Motion 1.2 The Mathematical Language 4 1.3 The Physical Interpretation 5 2 Analysis on Manifolds 8 2.1 Manifolds 8 2.2 Tangent Spaces 19 2.3 Flows 32 2.4 Tensors 42 2.5 Differentiation 56 2.6' Integrals 61 73 3 Hamiltonian Systems 84 3.1 Canonical Transformations 84 3.2 Hamilton's Equations 91 3.3 Constants of Motion 100 3.4 The Limit t -+ I ± 00 117 3.5 Perturbation Theory: Preliminaries 141 3.6 Perturbation Theory: The Iteration 153 xiii