Mathematical Principles of Mechanics and Electromagnetism

Part A: Analytical and Continuum Mechanics

MATHEMATICAL CONCEPTS AND METHODS IN SCIENCE AND ENGINEERING

Series Editor: Angelo Miele Mechanical Engineering and Mathematical Sciences Rice University

Volume 1 INTRODUCTION TO VECTORS AND TENSORS Volume 1: Linear and Multilinear Algebra

Volume 2

Volume 3

Volume 4

Volume 5

Volume 6

Volume 7

Volume 8

Volume 9

Volume 11

Volume 12

Volume 14

Volume 16

Volume 17

Ray M. Bowen and c..c. Wang

INTRODUCTION TO VECTORS AND TENSORS Volume 2: Vector and Tensor Analysis Ray M. Bowen and c..c. Wang

MULTICRITERIA DECISION MAKING AND DIFFERENTIAL GAMES Edited by George Leitmann

ANALYTICAL DYNAMICS OF.DISCRETE SYSTEMS Reinhardt M. Rosenberg

TOPOLOGY AND MAPS Taqdir Husain

REAL AND FUNCTIONAL ANALYSIS A. Mukherjea and K. Pothoven

PRINCIPLES OF OPTIMAL CONTROL THEORY R. V. Gamkrelidze

INTRODUCTION TO THE LAPLACE TRANSFORM Peter K. F. Kuhjittig

MATHEMATICAL LOGIC: An Introduction to Model Theory A. H. Lightstone

INTEGRAL TRANSFORMS IN SCIENCE AND ENGINEERING Kurt Bernardo Wolf

APPLIED MATHEMATICS: An Intellectual Orientation Francis J. Mu"ay

PRINCIPLES AND PROCEDURES OF NUMERICAL ANALYSIS Ferenc Szidarovszky and Sidney Yakowitz

MATHEMATICAL PRINCIPLES OF MECHANICS AND ELECTROMAGNETISM, Part A: Analytical and Continuum Mechanics c..c. Wang

MATHEMATICAL PRINCIPLES OF MECHANICS AND ELECTROMAGNETISM, Part B: Electromagnetism and Gravitation c..c. Wang

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

Mathematical Principles of Mechanics and Electromagnetism

Part A: Analytical and Continuum Mechanics

c.-c. Wang Rice University Houston, Texas

PLENUM PRESS • NEW YORK AND LONDON

Library of Congress Cataloging in Publication Data

VVang,Cha~heng, 1938-Mathematical principles of mechanics and electromagnetism.

(Mathematical concepts and methods in science and engineering; v. 16-17) Bibliography: pt. A, p. ; pt, B, p. Includes Indexes. CONTENTS: pt. A. Analytical and continuum mechanics.-pt. B. Electromag­

netism and gravitation. 1. Mechanics, Analytic. 2. Continuum mechanics. 3. Electromagnetism. 4.

Gravitation. I. Title. QA805.VV26 ISBN-13: 978-1-4684-3538-2 DOl: 10.1007/978-1-4684-3536-8

531'.0151 e-ISBN-13: 978-1-4684-3536-8

© 1979 Plenum Press, New York Sof'tcover reprint of the hardcover 1st edition 1979

A Division of Plenum Publishing Corporation 227 VVest 17th Street, New York, N.Y. 10011

All rights reserved

79-11862

No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming,

recording, or otherwise, without written permission from the Publisher

To my teacher

CLIFFORD TRUESDELL

Preface to Part A

The purpose of this work is to give an introduction to the mathematical principles of mechanics and of electromagnetism. Part A is concerned with two main subjects in classical mechanics: analytical mechanics and con­tinuum mechanics.

I start in Chapter I from Newtonian space-time, which is the basic mathematical model for the event world in all classical theories of physics. On the basis of this model I present the equations of motion for mass points and rigid bodies. Then I derive Lagrange's equations for holonomic systems of mass points and rigid bodies.

My derivation differs in one important aspect from the traditional approach followed by many authors. In the traditional approach a rigid body is regarded as a limiting case of a rigid system of mass points; the number of mass points becomes infinite and the mass of each point becomes infinitesimal in such a way that a finite mass density may be assigned to each part of the body. Lagrange's equations are then derived on the basis of the equations of motion for mass points only, but the results are applied to rigid bodies by using the aforementioned limiting process. Since this limiting process amounts to only a motivation of the equations of motion for a rigid body, I feel that the traditional derivation is not rigorous. In my opinion rigid bodies are primitive concepts like mass points, so that their equations of motion are independent of those for mass points. Hence I derive Lagrange's equations for systems of mass points and rigid bodies from the equations of motion for mass points and for rigid bodies separately. In this way I do not use any argument based on the limiting process in my derivation.

vii

viii Preface to Part A

For conservative systems, Lagrange's equations may be transformed into a set of autonomous first-order differential equations, known as the Hamiltonian equations, in phase space. In Chapter 2, I consider the solution operator of the Hamiltonian equations from the standpoint of differential geometry and in the context of a first-order partial differential equation known as the Hamilton-Jacobi equation. The contents of the first two chapters constitute the bulk of general principles in analytical mechanics.

Chapter 3 is devoted to the derivation of the governing equations of motion for deformable bodies. Unlike mass points and rigid bodies, the dynamical responses of which are determined by inertia, deformable bodies have a variety of dynamical responses, which may be described by various constitutive equations. I present just one general class of constitutive equa­tions; bodies characterized by this class are known as simple material bodies.

Constitutive equations for simple material bodies are local models in continuum mechanics, since they are formed by sets of equations of me­chanical response for individual body points of the body manifolds. Each distinguished equation of mechanical response characterizes a particular simple material; a simple material body is just a body manifold made up of simple material points. If the points of a body manifold belong to the same simple material, then the body is called materially uniform. The struc­ture of a materially uniform simple material body may be characterized by a single equation of mechanical response and by a distribution of that equation on the body manifold. That distribution may be homogeneous or inhomogeneous.

In Chapter 4, I treat three topics of interest in the theory of simple material bodies in order to illustrate the general principles in continuum mechanics. The first two topics are concerned with homogeneous bodies made up of fluids and isotropic elastic solids, respectively. The third topic is concerned with the geometric structure of inhomogeneous elastic bodies in general.

Throughout this work I have followed a simple, direct, and somewhat old-fashioned approach in order that the text may be followed by advanced undergraduate students with limited background in the elements of the subjects. I set out to present in a clear and rigorous way the basic principles in the two subjects; mathematical generality and elegance are not my primary concern. I hope that this work is helpful to students in grasping the central concepts and results in the subjects. However, I make no claim that the subjects are covered completely in this work. Most of the math­ematical preliminaries needed for the formulation of the principles con-

Preface to Part A ix

sidered in this work may be found in the two-volume work Introduction to Vectors and Tensors, * published in this Series (Mathematical Concepts and Methods in Science and Engineering) in 1976.

I am grateful to the Series editor, Angelo Miele, a long-time colleague and a good friend, for permitting me to publish a second work in his series. To Ray Bowen, who has collaborated with me on many other works, I wish to express my thanks for his comments and critical remarks on the preliminary draft.

This work is dedicated to my teacher, Clifford Truesdell, who has directed and guided me on many works, this one included, and who has been a source of inspiration to me for many years. Without his encourage­ment it would not have been possible for me to undertake and to finish this work.

I take this opportunity to acknowledge also my gratitude to the U.S. National Science Foundation for its support during the preparation of this work.

As always, it is a pleasure to express my appreciation to my wife, Sophia, and to my boys, Ferdie and Ted, for their patience and under­standing during the years this work was in progress.

C.-C. Wang Houston, Texas

• R. M. Bowen and C.-C. Wang, Plenum Press, New York.

Contents of Part A Analytical and Continuum Mechanics

Contents of Part B . . . . . . . . . . . . . . . . . . . . . . .. xiii

Chapter 1. Lagrangian Mechanics of Particles and Rigid Bodies

Section 1. Kinematics of Systems of Particles. . . . . .

Section 2. Kinematics of a Rigid Body

Section 3. Kinematics of Holonomic Systems of Particles and Rigid 7

Bodies. . . . . . . . . . . . . . . . . . . . . 15

Section 4. Dynamical Principles for Particles and Rigid Bodies . 23

Section 5. Lagrange's Equations for Constrained Systems 27

Section 6. Explicit Forms of Lagrange's Equations 34

Chapter 2. Hamiltonian Systems in Phase Space. . . . 41

Section 7. Hamilton's Principle . . . . . . . . . 41

Section 8. Phase Space and Its Canonical Differential Forms. 46

Section 9. The Legendre Transformation and the Hamiltonian System I: The Time-Independent Case ........... 51

Section 10. The Legendre Transformation and the Hamiltonian System II: The Time-Dependent Case. . . . . . . . . . 58

Section 11. Contact Transformations and the Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . 65

Section 12. The Hamilton-Jacobi Theory . . . . . . . . . . 68

Section 13. Huygens' Principle for the Hamilton-Jacobi Equation 72

Appendix. Characteristics of a First-Order Partial Differential Equation 78

xi

xii Contents of Part A

Chapter 3. Basic Principles of Continuum Mechanics 87

Section 14. Deformations and Motions . . . . 87 Section 15. Balance Principles . . . . . . . . 93 Section 16. Cauchy's Postulate and the Stress Principle 98 Section 17. Field Equations . . . . . . . 101 Section 18. Constitutive Equations . . . . . . . . . 107 Section 19. Some Representation Theorems . . . . . 114 Section 20. The Energy Principle for Hyperelastic Materials. 122 Section 21. Internal Constraints 128

Chapter 4. Some Topics in the Statics and Dynamics of Material Bodies 135

Section 22. Homogeneous Simple Material Bodies . . . . . . .. 135 Section 23. Viscometric Flows of Incompressible Simple Fluids .. 143 Section 24. Universal Solutions for Isotropic Elastic Solids I: The

Compressible Case . . . . . . . . . . 158 Section 25. Universal Solutions for Isotropic Elastic Solids II: The

Incompressible Case . . . . . . . . . . 165 Section 26. Materially Uniform Smooth Elastic Bodies 175 Section 27. Material Connections . . . . . . . . . . Section 28. Noll's Equations of Motion . . . . . . . Section 29. Inhomogeneous Isotropic Elastic Solid Bodies .

SELECTED READING FOR PART A INDEX .......•....

178 185 189

197 xv

Contents of Part B Electromagnetism and Gravitation

Contents of Part A . . . . . . . . . . . . . . . . . . . . . . .. xi

Chapter 5. Classical Theory of Electromagnetism. . . . 199

Section 30. Classical Laws of Electrostatic Fields. . 199 Section 31. Steady Currents and Magnetic Induction 207 Section 32. Time-Dependent Electromagnetic Fields, Maxwell's

Equations . . . . . . 214 Section 33. Balance Principles . . . . . 222 Section 34. Electromagnetic Waves . . . 226 Section 35. Electromechanical Interactions 236

Chapter 6. Special Relativistic Theory of Electromagnetism 245

Section 36. Newtonian, Galilean, and Ether Space-Times 245 Section 37. Minkowskian Space-Time. . . . . . . . . 252 Section 38. Lorentz Transformations 259 Section 39. Vectors and Tensors in the Minkowskian Space-Time 264 Section 40. Maxwell's Equations in Special Relativistic Form .. 269 Section 41. Lorentz's Formula and the Balance Principles in Special

Relativistic Form. . . . . . . . . . . . 278 Section 42. Doppler Effect of Electromagnetic Waves. 282

Chapter 7. General Relativistic Theory of Gravitation ...... 285

Section 43. Newton's Law of Gravitation and the Principle of Equivalence . . . . . 285

Section 44. Minkowskian Manifold . 290

xiii

xiv Contents of Part B

Section 45. The Stress-Energy-Momentum Tensor in a Material Medium. . . . . . . . . . . . . . . . . . . . 295

Section 46. Einstein's Field Equations. . . . . . . . . . . . 303

Section 47. The Schwarzschild Solution and the Problems of Planetary Orbits and the Deflection of Light . 309

Section 48. The Action Principle . . . . . . . . . . . . 316 Section 49. Action and Coaction . . . . . . . . . . . . 325

Section 50. The Nordstrom-Toupin Ether Relation and the Minkowskian Metric . . . . . . . . . . 333

Chapter 8. General Relativistic Theory of Electromagnetism 343

Section 51. Maxwell's Equations in General Relativistic Form . 343 Section 52. The Maxwell-Lorentz Ether Relation and the

Minkowskian Metric I: Toupin's Uniqueness Theorem. 350

Section 53. The Maxwell-Lorentz Ether Relation and the Minkowskian Metric II: Basic Properties and Preliminary Lemmas. . . . . . . . . . . . . . . . . . . .. 355

Section 54. The Maxwell-Lorentz Ether Relation and the Minkowskian Metric III: Toupin's Existence Theorem 362

Section 55. The Electromagnetic Action and the Electromagnetic Stress-Energy-Momentum Tensor. . . . . . . 370

Section 56. Electrogravitational Fields of an Electrically Charged Mass Point . . . . . . . . . . . . . 378

SELECTED READING FOR PART B 385

INDEX ...•........ xv