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ANALYTICAL MECHANICS

Analytical Mechanics provides a detailed introduction to the key analytical techniques ofclassical mechanics, one of the cornerstones of physics. It deals with all the importantsubjects encountered in an undergraduate course and prepares the reader thoroughly forfurther study at the graduate level.

The authors set out the fundamentals of Lagrangian and Hamiltonian mechanics earlyon in the book and go on to cover such topics as linear oscillators, planetary orbits, rigid-body motion, small vibrations, nonlinear dynamics, chaos, and special relativity. A specialfeature is the inclusion of many "e-mail questions," which are intended to facilitate dialoguebetween the student and instructor.

Many worked examples are given, and there are 250 homework exercises to helpstudents gain confidence and proficiency in problem solving. It is an ideal textbook forundergraduate courses in classical mechanics and provides a sound foundation for graduatestudy.

Louis N. Hand was educated at Swarthmore College and Stanford University. Afterserving as an assistant professor at Harvard University during the 1964 academic year, hecame to the Physics Department of Cornell University where he has remained ever since.He is presently researching in the field of accelerator physics.

Janet D. Finch, teaching associate in the Physics Department of Cornell University,earned her BS in engineering physics from the University of Illinois, and her MS intheoretical physics and her MA in teaching from Cornell. In 1994 she began working withProfessor Hand on the Classical Mechanics course from which this book developed. Shewas the e-mail tutor for the course during the first-time implementation of this innovation.

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Cambridge University Press978-0-521-57327-6 - Analytical MechanicsLouis N. Hand and Janet D. FinchFrontmatterMore information

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Cambridge University Press978-0-521-57327-6 - Analytical MechanicsLouis N. Hand and Janet D. FinchFrontmatterMore information

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ANALYTICAL MECHANICS

LOUIS N. HAND

and

JANET D. FINCH

CAMBRIDGEUNIVERSITY PRESS

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Cambridge University Press978-0-521-57327-6 - Analytical MechanicsLouis N. Hand and Janet D. FinchFrontmatterMore information

www.cambridge.org/9780521573276www.cambridge.orgwww.cambridge.org

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town,

Singapore, So Paulo, Delhi, Tokyo, Mexico City

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org Information on this title: www.cambridge.org/9780521573276

Cambridge University Press 1998

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.

First published 19987th printing 2008

A catalogue record for this publication is available from the British Library

Library of Congress cataloguing in publication dataHand, Louis N., 1933

Analytical mechanics / Louis N. Hand, Janet D. Finch.p. cm.

Includes bibliographical references and index. ISBN 0-521-57327-0 ISBN 0-521-57572-9 (pbk.) 1. Mechanics, Analytic. I. Finch, Janet D., 1969

II. Title.QA805.H26 1998

531 01 515352 dc21 97-43334 CIP

isbn 978-0-521-57327-6 Hardbackisbn 978-0-521-57572-0 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty internet websites referred to in

this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel

timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee

the accuracy of such information thereafter.

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CONTENTS

Preface xi

1 LAGRANGIAN MECHANICS 11.1 Example and Review of Newton's Mechanics: A Block Sliding on

an Inclined Plane 11.2 Using Virtual Work to Solve the Same Problem 31.3 Solving for the Motion of a Heavy Bead Sliding on a Rotating Wire 71.4 Toward a General Formula: Degrees of Freedom and Types

of Constraints 101.5 Generalized Velocities: How to "Cancel the Dots" 141.6 Virtual Displacements and Virtual Work - Generalized Forces 141.7 Kinetic Energy as a Function of the Generalized Coordinates

and Velocities 161.8 Conservative Forces: Definition of the Lagrangian L 181.9 Reference Frames 201.10 Definition of the Hamiltonian 211.11 How to Get Rid of Ignorable Coordinates 221.12 Discussion and Conclusions - What's Next after You Get the EOM? 231.13 An Example of a Solved Problem 24

Summary of Chapter 1 25Problems 26Appendix A. About Nonholonomic Constraints 36Appendix B. More about Conservative Forces 41

2 VARIATIONAL CALCULUS AND ITS APPLICATION TO MECHANICS 442.1 History 442.2 The Euler Equation 462.3 Relevance to Mechanics 512.4 Systems with Several Degrees of Freedom 532.5 Why Use the Variational Approach in Mechanics? 542.6 Lagrange Multipliers 56

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VI CONTENTS

2.7 Solving Problems with Explicit Holonomic Constraints 572.8 Nonintegrable Nonholonomic Constraints - A Method that Works 622.9 Postscript on the Euler Equation with More Than

One Independent Variable 65Summary of Chapter 2 65Problems 66Appendix. About Maupertuis and What Came to Be Called"Maupertuis' Principle" 75

3 LINEAR OSCILLATORS 813.1 Stable or Unstable Equilibrium? 823.2 Simple Harmonic Oscillator 873.3 Damped Simple Harmonic Oscillator (DSHO) 903.4 An Oscillator Driven by an External Force 943.5 Driving Force Is a Step Function 963.6 Finding the Green's Function for the SHO 993.7 Adding up the Delta Functions - Solving the Arbitary Force 1033.8 Driving an Oscillator in Resonance 1053.9 Relative Phase of the DSHO Oscillator with Sinusoidal Drive 110

Summary of Chapter 3 113Problems 114

4 ONE-DIMENSIONAL SYSTEMS: CENTRAL FORCES ANDTHE KEPLER PROBLEM 123

4.1 The Motion of a "Generic" One-Dimensional System 1234.2 The Grandfather's Clock 1254.3 The History of the Kepler Problem 1304.4 Solving the Central Force Problem 1334.5 The Special Case of Gravitational Attraction 1414.6 Interpretation of Orbits 1434.7 Repulsive ^ Forces 151

Summary of Chapter 4 156Problems 156Appendix. Tables of Astrophysical Data 167

5 NOETHER'S THEOREM AND HAMILTONIAN DYNAMICS 1705.1 Discovering Angular Momentum Conservation from

Rotational Invariance 1705.2 Noether's Theorem 1725.3 Hamiltonian Dynamics 1755.4 The Legendre Transformation 1755.5 Hamilton's Equations of Motion 1805.6 Liouville's Theorem 1845.7 Momentum Space 189

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CONTENTS VII

5.8 Hamiltonian Dynamics in Accelerated Systems 190Summary of Chapter 5 195Problems 196Appendix A. A General Proof of Liouville's TheoremUsing the Jacobian 202Appendix B. Poincare Recurrence Theorem 204

6 THEORETICAL MECHANICS: FROM CANONICALTRANSFORMATIONS TO ACTION-ANGLE VARIABLES 207

6.1 Canonical Transformations 2086.2 Discovering Three New Forms of the Generating Function 2136.3 Poisson Brackets 2176.4 Hamilton-Jacobi Equation 2186.5 Action-Angle Variables for 1-D Systems 2306.6 Integrable Systems 2356.7 Invariant Tori and Winding Numbers 237

Summary of Chapter 6 239Problems 240Appendix. What Does "Symplectic" Mean? 248

7 ROTATING COORDINATE SYSTEMS 2527.1 What Is a Vector? 2537.2 Review: Infinitesimal Rotations and Angular Velocity 2547.3 Finite Three-Dimensional Rotations 2597.4 Rotated Reference Frames 2597.5 Rotating Reference Frames 2637.6 The Instantaneous Angular Velocity co 2647.7 Fictitious Forces 2677.8 The Tower of Pisa Problem 2677.9 Why Do Hurricane Winds Rotate? 2717.10 Foucault Pendulum 272

Summary of Chapter 7 275Problems 276

8 THE DYNAMICS OF RIGID BODIES 2838.1 Kinetic Energy of a Rigid Body 2848.2 The Moment of Inertia Tensor 2868.3 Angular Momentum of a Rigid Body 2918.4 The Euler Equations for Force-Free Rigid Body Motion 2928.5 Motion of a Torque-Free Symmetric Top 2938.6 Force-Free Precession of the Earth: The "Chandler Wobble" 2998.7 Definition of Euler Angles 3008.8 Finding the Angular Velocity 3048.9 Motion of Torque-Free Asymmetric Tops: Poinsot Construction 305

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