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- EM NEWTONIAN DYNAMICS Ralph Baierlein Charlotte Ayres Professor of Physics Wesleyan Uni,•ersity McGraw-Hill Book Company New York St. Louis San Francisco Auckland Bogota Hamburg Johannesburg London Madrid Mexico Montreal New Delhi Panama Paris Siio Paulo Singapore Sydney Tokyo Toronto This book was set in Times Roman by A Graphic Method Inc. The editors were John J. Corrigan and James S. Amar; the production supervisor was Leroy A. Young. The drawings were done by Allyn-Ma son. Inc. The cover was designed by Robin Hessel. Halliday Lithograph Corporation was printer and binder. NEWTONIAN DYNAMICS Copyright © 1983 by McGraw-Hill, Inc. All rights reserved . Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976. no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system. without the prior written permission of the publisher. 234567890 HALHAL 89876543 ISBN 0-07-003016-2 Library of Congress Cataloging in Publication Data Baierlein, Ralph. Newtonian dynamics. Includes index. I. Dynamics. I. Title. QA845 .B33 1983 531'.I I 82-17944 ISBN 0-07-003016-2 To MY FAMILY Jean, Eric, and Jeffrey CONTENTS Preface VII A Note to the Problem Solver IX Chapter 1 A Review of Some Basic 1.1 Qualitative Rea oning I 1.2 Vectors and the calar Product 7 1.3 To ing Rock 8 1.4 A on equence of ewton' Second Law: Energy 15 1.5 Gravitational Potential Energy 20 1.6 Potential Energy and tability 23 I. 7 ome Implication of Invariance 27 1.8 ewton' Law : Structure and Meaning 30 1.9 Reviewing the Review 34 Chapter 2 The Harmonic Oscillator 45 2.1 Damped Harmonic O cillator 45 2.2 Pha e Space: An Introduction 51 2.3 Harmonic Oscillation in Two Dimensions 54 2.4 inu oidally Driven O cillator 56 2.5 Quality Factor Q 61 2.6 A Guide to the Major Ideas 63 Chapter 3 Non linear O cillator 71 3. I A onlinear O cillator 71 3.2 Amplitude Jump and Hy teresi 81 3.3 van der Pol' Equation: The Limit Cycle 88 3.4 More U e for the Averaging Method 93 3.5 eries Expan ion 97 104 3.6 About the Method s Chapter 4 Lagrangian Formulation 116 4. I Fermat' Principle 116 4.2 alculu of Variation I 18 4.3 ewton I I a an Extremal Principle 123 4.4 Lagrangian and on traints 128 4.5 Another In tance of Con trained Motion I 31 4.6 onver ion to First-Order Equations: Hamilton's Equations 136 140 4. 7 Liouville' Theorem 4.8 The Lagrangian and Quantum Mechanic 144 4.9 A en e of Per pective 146 V vi CONTENTS Chapter 5 Two-Body Problem 156 5.1 Reduction to Motion in a Plane 156 5.2 Effective Potential Energy 161 5.3 Orbit Shape 164 5.4 Orbits around the Spherical Sun 168 5.5 The Oblate Sun 175 5.6 Stability of Circular Orbits 181 5.7 The Orbit in Time 187 5.8 Compendium on Central-Force Motion 189 Chapter 6 Rotating Frame of Reference 202 6.1 Vector in a Rotating Frame of Reference 202 6.2 Physics on a Rotating Table 209 6.3 The Rotating Earth 210 6.4 Foucault Pendulum 217 6.5 The Figure of the Earth 223 6.6 A Per pective on Rotating Frames 228 Chapter 7 Extended Bodie in Rotation 235 7.1 Equations for Location and Orientation 23 7.2 Simple Precession 240 7.3 How L Is Related tow 244 7.4 A Novel Pendulum 247 7.5 L ls Not Neces arily Parallel tow 249 7.6 Diagonal Form for the Inertia Ten or 251 7.7 Euler's Equations for a Rigid Body 254 7.8 Axisymmetric and Torque-Free 25 7.9 Chandler Wobble 260 7.10 An Interlude on Kinetic Energy 264 7 .11 The Symmetric, Supported Top 265 7 .12 Prece ion of the Equinoxe 272 7 .13 urvey of Critical Notion 275 Chapter 8 Cross Sections 286 8.1 Scattering Effectivenes : The Idea Behind the Cro s Section 286 8.2 A Capture ros Section 289 8.3 A Differential Cro s Section 291 8.4 Rutherford Scattering 295 8.5 Major ldeas 300 Appendixes 30 A Expansion , ldentitie , and Mi cellany 308 B Vector Product 311 The Averaging Method 317 D The Craft of the Physici t 321 Index 324