CLASSICAL MECHANICS A & Slllllllll f <! I>! John Michael Finn

Transcription

1 -^ CLASSICAL MECHANICS John Michael Finn Slllllllll f <! I>! A &

2 CLASSICAL MECHANICS

3 LICENSE, DISCLAIMER OF LIABILITY, AND LIMITED WARRANTY By purchasing or using this book (the "Work''), you agree that this license grants permission to use the contents contained herein, but does not give you the right of ownership to any of the textual content in the book or ownership to any of the information or products contained in it. This license does not permit use of the Work on the Internet or on a network (of any kind) without the written consent of the Publisher. Use of any third party code contained herein is limited to and subject to licensing terms for the respective products, and permission must be obtained from the Publisher or the owner of the source code in order to reproduce or network any portion of the textual material (in any media) that is contained in the Work. Jones and Bartlett Publishers, LLC ("the Publisher ") and anyone involved in the creation, writing, or production of the accompanying algorithms, code, or computer programs ("the software"), and any accompanying Web site or software of the Work, cannot and do not warrant the performance or results that might be obtained by using the contents of the Work. The authors, developers, and the Publisher have used their best efforts to insure the accuracy and functionality of the textual material and/or programs contained in this package; we, however, make no warranty of any kind, express or implied, regarding the performance of these contents or programs. The Work is sold "as is" without warranty (except for defective materials used in manufacturing the book or due to faulty workmanship). The authors, developers, and the publisher of any accompanying content, and anyone involved in the composition, production, and manufacturing of this work will not be liable for damages of any kind arising out of the use of (or the inability to use) the algorithms, source code, computer programs, or textual material contained in this publication. This includes, but is not limited to, loss of revenue or profit, or other incidental, physical, or consequential damages arising out of the use of this Work. The sole remedy in the event of a claim of any kind is expressly limited to replacement of the book, and only at the discretion of the Publisher. The use of "implied warranty" and certain "exclusions" vary from state to state, and might not apply to the purchaser of this product.

4 CLASSICAL MECHANICS John Michael Finn JONES AND BARTLETT PUBLISHERS Sudbury. Massachusetts BOSTON TORONTO LONDON SINGAPORE

5 World Headquarters Jones and Bartleu Publishers Jones and Bartlett Publistiers Jones and Bartlett Publishers 40 Tall Pine Drive Canada International Sudbury, MA Ormindale Way Barb House. Barb Mews Mississauga, ON L5V 1J2 London W6 7PA inf0(a)jbpub.com Canada United Kingdom Jones and Bartlett's books and products are available through most bookstores and online booksellers. To contact Jones and Bartlett Publishers directly, call , fax , or visit our website at Substantial discounts on bulk quantities of Jones and Bartlett's publications are available to corporations, professional associations, and other qualified organizations. For details and specific discount information, contact the special sales department at Jones and Bartlett via the above contact information or send an to Copyright 2010 by Jones and Bartlett Publishers, LLC Copyright 2008 by Infinity Science Press, LLC ISBN All rights reserved. No part of the material protected by this copyright may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarks or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc. is not an attempt to infringe on the property of others. Production Credits Publisher: David Pallai Cover Design: Tyler Creative Printing and Binding: Malloy, Inc. Cover Printing: Malloy, Inc. Library of Congress Cataloging-in-Publication Data Finn, J. Michael. Classical mechanics / J. Michael Finn. p. cm. ISBN (hardcover) 1. Mechanics. I. Title. QCI25.2.F dc Printed in the United States of America

6 In Memoriam: Galileo Galilei ( ) Once again, the Inquisition called me in for questioning, "Are you quite ceitain Of what you have claimed to see?" I tell you, I sweated, despite the chill in the air. Misunderstanding their intent, I persisted, "Through my lenses, I saw what I saw: spots on the sun; Valleys, mountains, and plains on the moon; More moons circhng Jupiter; the phases of Venus. "The Heavens are a sight to behold. God's creation is grander than we dared imagine. Would anyone here care to see for himself? Perhaps make use of the telescope lying before you on the table?" Their response, when it finally came, was solemn and sure, "This Council will not be lectured On that, of which We have already decided. "Let us speak plainly: You are disturbing the people. The truth will be what we sa\' it is to be. Neither one word more, nor one word less." Defeated at last, Bhnded by age, Broken in spirit, I consented to their demands. "Your will be done, but, nevertheless. Earth turns round the sun still. No matter what )'ou choose to beheve, Or what I am forced to confess." John Michael Finn, adapted from The Butterfi)- Girl 2002

7 CONTENTS Preface Chapter 1. Review of Newtonian Particle Mechanics 1 LI Introduction to Classical Mechanics 1 L2 Historical Context Newton s Principia Motion of a System of Particles Solution of the Initial Value Problem One-Dimensional Motion Summar)^' Exercises 51 Chapter 2. Vector Spaces and Coordinate Systems Points and Vectors Inner Product Spaces Vector Operators Curvilinear Coordinate Systems Manifolds *Covariant Differentiation Summary Exercises 100 Chapter 3. Lagrangian and Hamiltonian Dynamics Generalized Coordinates and Constraints Principle of Virtual Work Lagrange's Equations of Motion Electromagnetic Interactions Hamilton's Equations of Motion Routh's Procedure for Eliminating Cyclic Variables Summary Exercises 140 xi VII

8 VIII CONTENTS Chapter 4. Hamilton's Principle Hamilton's Action Principle Role of tlic Action in Qi-ii"!*"'^^ Fiekl Theor>' Calculus of Wiriations Lagrange's Method of Undetermined Multipliers Noether's Theorem *Svnmieti-v and Scale in Experimental Design Summaiv 1'5 4.8 Exercises 176 Chapter 5. Central Force Motion Central Force Interactions Lagrangian Fonmilation of the Two Body Problem Iinerse-Square Force Law Restricted Three-Body Problem Gra\ itation Field Sources Precession of the Planetan Orbits Sunnnary Exercises 227 Chapter 6. Small Oscillations Small Oscillations about an Equilibrium Configuration Re\icw of the One-Dimensional Problem Normal Mode Analysis of Coupled Oscillators Finite Lattice Structures Forced and Damped Oscillations SunnnaiA Exercises 265 Chapter 7. Rotational Geometry and Kinematics Rotational Geometnof Euclidean Space Pauli Spin Algebra 2S7 7.3 Wctor Algebra Notation Rotating Frames of Motion Kinematics of Rotating Bodies Snnunai'v Exercises 317 Chapter 8. Rigid Rody Motion General Motion of a Rigid Bod> Rotational Kinetic Energy of a Rigid Body 326

9 CONTENTS ix 8.3 Moment of Inertia Tensor Euler's Equation of Motion Lagrangian Formulation *Precession of Magnetic Dipoles Summary Exercises 360 Chapter 9. Canonical Transformation Theory Generalizing the Action Poisson Brackets Generating Functions Infinitesimal Canonical Transformations Lie Algebra of the Rotation Group S}Tnplectic Structure of the Theory * Elevating Time to a Coordinate Summary Exercises 412 Chapter 10. Hamilton-Jacobi Theory Hamilton-Jacobi Equation Bohm Interpretation of Quantum Mechanics Action-Angle Variables Adiabatic Invariants Summary Exercises 446 Chapter 11. Special Relativity Unifying Space and Time Minkowski Space Geometric Structure of Spacetime Proper Lorentz Transformations Relati\dstic Particle Dynamics Impulse Approximation Relativistic Lagrangians Summary Exercises 505 Chapter 12. Waves, Particles, and Fields Introduction to the Classical Theory of Fields Non Relativistic Wave Equation Non-Linear Sine-Gordon Equation 527

10 X CONTENTS 12.4 Electromagnetic Waves *Gra\itational Field Theor\' Summary Exercises 542 Appendix A. International System of Units and Conversion Factors 545 A.l CGS Units 547 A.2 Conversion Factors between Various Electromagnetic 548 Systems of Units A.3 Dimensionless Units 550 Appendix R. Physical Constants and Solar System Data 553 Appendix C. Geometric Algebras in N Dimensions 557 References 561 Index 567