Analytical Mechanics Analytical mechanics is the foundation of many areas of theoretical physics, including quantum theory and statistical mechanics, and has wide-ranging applications in engineer- ing and celestial mechanics. This introduction to the basic principles and methods of analytical mechanics covers Lagrangian and Hamiltonian dynamics, rigid bodies, small oscillations, canonical transformations and Hamilton-Jacobi theory. This fully up-to-date textbook includes detailed mathematical appendices and addresses a number of advanced topics, some of them of a geometric or topological character. These include Bertrand’s theorem, proof that action is least, spontaneous symmetry breakdown, constrained Hamil- tonian systems, non-integrability criteria, KAM theory, classical field theory, Lyapunov functions, geometric phases and Poisson manifolds. Providing worked examples, end-of- chapter problems and discussion of ongoing research in the field, it is suitable for advanced undergraduate students and graduate students studying analytical mechanics. Nivaldo A. Lemos is Associate Professor of Physics at Federal Fluminense University, Brazil. He was previously a visiting scholar at the Massachusetts Institute of Technology. His main research areas are quantum cosmology, quantum field theory and the teaching of classical mechanics. Analytical Mechanics NIVALDO A. LEMOS Fluminense Federal University University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108416580 DOI: 10.1017/9781108241489 © 2004, Editora Livraria da F´ısica, Sao˜ Paulo English translation © Cambridge University Press 2018 Revised and enlarged translation of Mecanicaˆ Anal´ıtica by Nivaldo A. Lemos originally published in Portuguese by Editora Livraria da F´ısica, Sao˜ Paulo 2004 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. English edition first published 2018 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Lemos, Nivaldo A., 1952– author. Title: Analytical mechanics / Nivaldo A. Lemos, Fluminense Federal University. Other titles: Mecanicaˆ anal´ıtica. English Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018011108| ISBN 9781108416580 (hb) | ISBN 1108416586 (hb) Subjects: LCSH: Mechanics, Analytic. Classification: LCC QA805 .L4313 2018 | DDC 531.01/515–dc23 LC record available at https://lccn.loc.gov/2018011108 ISBN 978-1-108-41658-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To my wife Marcia and my daughters Cintia, Luiza and Beatriz Contents Preface page xi 1 Lagrangian Dynamics 1 1.1 Principles of Newtonian Mechanics 1 1.2 Constraints 7 1.3 Virtual Displacements and d’Alembert’s Principle 11 1.4 Generalised Coordinates and Lagrange’s Equations 17 1.5 Applications of Lagrange’s Equations 24 1.6 Generalised Potentials and Dissipation Function 28 1.7 Central Forces and Bertrand’s Theorem 31 Problems 37 2 Hamilton’s Variational Principle 42 2.1 Rudiments of the Calculus of Variations 42 2.2 Variational Notation 49 2.3 Hamilton’s Principle and Lagrange’s Equations 50 2.4 Hamilton’s Principle in the Non-Holonomic Case 56 2.5 Symmetry Properties and Conservation Laws 65 2.6 Conservation of Energy 71 2.7 Noether’s Theorem 73 Problems 78 3 Kinematics of Rotational Motion 86 3.1 Orthogonal Transformations 86 3.2 Possible Displacements of a Rigid Body 92 3.3 Euler Angles 95 3.4 Infinitesimal Rotations and Angular Velocity 96 3.5 Rotation Group and Infinitesimal Generators 102 3.6 Dynamics in Non-Inertial Reference Frames 103 Problems 109 4 Dynamics of Rigid Bodies 112 4.1 Angular Momentum and Inertia Tensor 112 4.2 Mathematical Interlude: Tensors and Dyadics 114 4.3 Moments and Products of Inertia 119 vii viii Contents 4.4 Kinetic Energy and Parallel Axis Theorem 120 4.5 Diagonalisation of the Inertia Tensor 122 4.6 Symmetries and Principal Axes of Inertia 126 4.7 Rolling Coin 129 4.8 Euler’s Equations and Free Rotation 131 4.9 Symmetric Top with One Point Fixed 139 Problems 145 5 Small Oscillations 149 5.1 One-Dimensional Case 149 5.2 Anomalous Case: Quartic Oscillator 154 5.3 Stationary Motion and Small Oscillations 161 5.4 Small Oscillations: General Case 163 5.5 Normal Modes of Vibration 165 5.6 Normal Coordinates 170 5.7 Mathematical Supplement 176 Problems 178 6 Relativistic Mechanics 183 6.1 Lorentz Transformations 183 6.2 Light Cone and Causality 188 6.3 Vectors and Tensors 191 6.4 Tensor Fields 194 6.5 Physical Laws in Covariant Form 196 6.6 Relativistic Dynamics 199 6.7 Relativistic Collisions 204 6.8 Relativistic Dynamics in Lagrangian Form 207 6.9 Action at a Distance in Special Relativity 209 Problems 211 7 Hamiltonian Dynamics 215 7.1 Hamilton’s Canonical Equations 215 7.2 Symmetries and Conservation Laws 220 7.3 The Virial Theorem 221 7.4 Relativistic Hamiltonian Formulation 224 7.5 Hamilton’s Equations in Variational Form 226 7.6 Time as a Canonical Variable 228 7.7 The Principle of Maupertuis 234 Problems 237 8 Canonical Transformations 242 8.1 Canonical Transformations and Generating Functions 242 8.2 Canonicity and Lagrange Brackets 248 ix Contents 8.3 Symplectic Notation 250 8.4 Poisson Brackets 254 8.5 Infinitesimal Canonical Transformations 258 8.6 Angular Momentum Poisson Brackets 262 8.7 Lie Series and Finite Canonical Transformations 264 8.8 Theorems of Liouville and Poincare´ 268 8.9 Constrained Hamiltonian Systems 272 Problems 281 9 The Hamilton-Jacobi Theory 288 9.1 The Hamilton-Jacobi Equation 288 9.2 One-Dimensional Examples 291 9.3 Separation of Variables 294 9.4 Incompleteness of the Theory: Point Charge in Dipole Field 299 9.5 Action as a Function of the Coordinates 303 9.6 Action-Angle Variables 306 9.7 Integrable Systems: The Liouville-Arnold Theorem 312 9.8 Non-integrability Criteria 316 9.9 Integrability, Chaos, Determinism and Ergodicity 318 9.10 Prelude to the KAM Theorem 320 9.11 Action Variables in the Kepler Problem 323 9.12 Adiabatic Invariants 325 9.13 Hamilton-Jacobi Theory and Quantum Mechanics 329 Problems 332 10 Hamiltonian Perturbation Theory 338 10.1 Statement of the Problem 338 10.2 Generating Function Method 341 10.3 One Degree of Freedom 342 10.4 Several Degrees of Freedom 345 10.5 The Kolmogorov-Arnold-Moser Theory 348 10.6 Stability: Eternal or Long Term 353 10.7 Lie Series Method 354 Problems 358 11 Classical Field Theory 360 11.1 Lagrangian Field Theory 360 11.2 Relativistic Field Theories 365 11.3 Functional Derivatives 367 11.4 Hamiltonian Field Theory 371 11.5 Symmetries of the Action and Noether’s Theorem 374 11.6 Solitary Waves and Solitons 378 11.7 Constrained Fields 381 Problems 385 x Contents Appendix A Indicial Notation 392 Appendix B Frobenius Integrability Condition 398 Appendix C Homogeneous Functions and Euler’s Theorem 406 Appendix D Vector Spaces and Linear Operators 408 Appendix E Stability of Dynamical Systems 421 Appendix F Exact Differentials 426 Appendix G Geometric Phases 428 Appendix H Poisson Manifolds 433 Appendix I Decay Rate of Fourier Coefficients 440 References 442 Index 452 Preface It is no exaggeration to say that analytical mechanics is the foundation of theoretical physics. The imposing edifice of quantum theory was erected on the basis of analytical mechanics, particularly in the form created by Hamilton. Statistical mechanics and the quantum field theory of elementary particles are strongly marked by structural elements extracted from classical mechanics. Moreover, the modern development of the theories of chaos and general dynamical systems has given rise to a renaissance of classical mechanics that began in the middle of the twentieth century and is ongoing today. Thus, the study of virtually any branch of contemporary physics requires a solid background in analytical mechanics, which remains important in itself on account of its applications in engineering and celestial mechanics. For more than two decades, intermittently, the author has taught analytical mechanics to undergraduate physics students at Universidade Federal Fluminense in Niteroi,´ Brazil. The present book, fruit of the author’s teaching experience and ceaseless fascination with the subject, aims at advanced undergraduate or graduate students who have previously taken a course in intermediate mechanics at the level of Marion & Thornton (1995) or Taylor (2005). As to mathematical prerequisites, the usual courses on calculus of one and several variables, ordinary differential equations and linear algebra are sufficient. Analytical mechanics is much more than a mere reformulation of Newtonian mechanics, and its methods have not been primarily conceived to help solve the equations of motion of specific mechanical systems. To make this clear, symmetry properties, invariance and general structural features of mechanics are highlighted in the course of the presentation. As remarked by Vladimir I. Arnold, numerous modern mathematical theories, with which some of the greatest names in the history of mathematics are associated, owe their existence to problems in mechanics.