Analytical Dynamics of Discrete Systems

Reinhardt M. Rosenberg University of California. Berkeley

Volume 4 in Mathematical Concepts and Methods

in Science and Engineering

Analytical Dynamics of Discrete Systems. providing seniors and beginning graduate students in engineering and the natural sciences with an advanced textbook in dynamics, examines the development of classical particle mechanics from Newton to Lagrange. Every concept is clearly defmed before it is used, every sig­nificant result is stated mathematically as well as verbally, and the domain of applicability of each is explicitly stated. Most of the chapters contain a large number of carefully worked out examples, as well as a set of suggested exercises.

The author adopts a geometrical approach to the study of dynamics. He provides an orderly transition from Newtonian to Lagrangean mechanics by demonstrat­ing the need for a baSically different classification of forces in these two theories and the necessity of replacing Newton's third law by d'Alembert's prin­ciple. In the first seven chapters, he includes detailed reviews of

• Newtonian mechanics, with attention paid to the historical setting in which it developed

• the representation of motion as a trajectory in configuration space, event space, and other spaces

• constraints • rigid body kinematics and kinetics

The major portion of the book deals with the theory and application of Lagrangean mechanics, beginning with precise defmitions of "virtual displacements," "virtual velocity," and "virtual work." The author then discusses the principles of Hamilton and of least action, the theory of contemporaneous and non­contemporaneous variations, the theory of generalized coordinates and forces, and derivations of Lagrange's equations. Special chapters on celestial problems, gyrodynamics, and impulsive motion are included as well.

Analytical Dynamics of Discrete Systems

MATHEMA TICAL CONCEPTS AND METHODS IN SCIENCE AND ENGINEERING

Series Editor: Angelo Miele

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Mechanical Engineering and Mathematical Sciences Rice University, Houston, Texas

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

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

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.

Analytical Dynamics of Discrete Systems

Reinhardt M. Rosenberg University of California, Berkeley

PLENUM PRESS . NEW YORK AND LONDON

Library of Congress Cataloging in Publication Data

Rosenberg, Reinhardt Mathias. Analytical dynamics of discrete systems.

(Mathematical concepts and methods in science and engineering) Bibliography: p. Includes index. 1. Dynamics. I. Title.

QA845.R63 531 '.11 '01515 77-21894

ISBN 978-1-4684-8320-8 ISBN 978-1-4684-8318-5 (eBook) DOl 10.1007/978-1-4684-8318-5

© 1977 Plenum Press, New York Softcover reprint of the hardcover 1 st edition 1977

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

All rights reserved

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

If p be the distance to 0

Mr. Newton said he could show,

That the force of attraction

Behaves like the fraction Of one over the square of rho.

R.M.R.

Preface

This book is to serve as a text for engineering students at the senior or beginning graduate level in a second course in dynamics. It grew out of many years experience in teaching such a course to senior students in mechanical engineering at the University of California, Berkeley. While temperamentally disinclined to engage in textbook writing, I nevertheless wrote the present volume for the usual reason-I was unable to find a satisfactory English-language text with the content covered in my inter­mediate course in dynamics.

Originally, I had intended to fit this text very closely to the content of my dynamics course for seniors. However, it soon became apparent that that course reflects too many of my personal idiosyncracies, and perhaps it also covers too little material to form a suitable basis for a general text. Moreover, as the manuscript grew, so did my interest in certain phases of the subject. As a result, this book contains more material than can be studied in one semester or quarter. My own course covers Chapters 1 to 5 (Chapters 1,2, and 3 lightly) and Chapters 8 to 20 (Chapter 17 lightly). Insofar as the preparation of the student is concerned, the demands are satisfied by present-day methods of teaching mathematics, physics, and mechanics during the first three undergraduate years of an engineering curriculum. Students are expected to have studied kinematics and kinetics in a first course at the sophomore or junior level by the methods now current, and to be familiar with the fundamental principles of Newtonian mechanics and their applications in two and three dimensions. Their preparation in mathematics should include the elements of determinant and matrix theory, the calculus, and a first course in ordinary differential equations, and they must know how to manipulate, multiply, and differentiate vectors. It may

vii

viii Preface

be of some slight help to them to be familiar with set-theoretical symbols, but the demands in this respect are so modest that they can easily acquire this familiarity while studying its application.

In my opinion, a first course in dynamics should do more than only im­part to the student the techniques needed to solve problems. Similarly, a sec­ond course in dynamics should do more than help the student learn new techniques more sophisticated than those he or she knows already; it should also deepen his or her understanding of the fundamentals. And so, a con­siderable portion of this text is devoted to a new, a longer, and a more penetrating look at a familiar subject-Newtonian mechanics. Not only does this seem to me to be one of the proper functions of a second course in dynamics, but it becomes altogether unavoidable when the transition is made from the Newtonian to the Lagrangean t point of view.

In the review of Newtonian mechanics some attention is paid to the foundations of that discipline, the problem of classical mechanics is defined with some precision, and much care is devoted to the theory of constraints. In all this I have stressed geometric interpretations not only because they appeal to me, but because I have found that they appeal to the student as well.

Rigid body mechanics has been touched lightly, as has motion relative to moving frames, because these subjects are usually discussed in a first course in dynamics. The theory of rotations has been treated as an illustra­tion of orthogonal matrix transformations because, to my knowledge, that theory is almost never included in a first course in mechanics; Poinsot's representation is included for the same reason.

This book is intended for the student unfamiliar with Lagrangean mechanics; the theory and application of that theory forms the major portion new to him.

I regard Lagrangean mechanics not primarily as a mechanical process for producing equations of motion, but as a bold departure from Newtonian viewpoints, as the crowning touch to a development begun by Bernoulli and d' Alembert. Its formulation of the general theory of a constrained dynamical system is a subtle and aesthetically satisfying product. I have attempted to describe it that way.

Almost every chapter contains solved problems illustrating the theory in it. For one thing, I regard the application of theories as an important

t This spelling is phonetically equivalent to the more common "Lagrangian." It reflects my reluctance to mutilate Lagrange's name and was agreed to by the publisher to please me.

Preface ix

learning aid; for another, it is essential that knowledge of a way to solve a problem (or merely one's faith in the possession of this knowledge) not be

confused with actually producing the solution.

Every author setting out to write a textbook must make certain decisions with respect to notation. Whatever they are, he is sure not to please everyone. In this respect, his position is perhaps not unlike that of the elected official, judged in a public-opinion poll; some readers will approve, some will disapprove, and some will have no opinion.

In general, I have followed conventional, and perhaps old-fashioned, notation. I have not used the double index summation convention even though it would have resulted in more compact formulas. It seems to me that the added burden placed on the student by its use should be reserved for fields in which most of the quantities dealt with are tensors, and in which tensor transformations form an essential part of the theory. Also, I have not used special symbols to differentiate between a function and the value of a function at a point. Thus, having defined a function I on some domain X, I say that the value of I at x is I(x). On the rare occasions where the distinction is important I write: I(x;) is the value of I at Xi E X.

Perhaps the only departure in my notation from that commonly used in elementary texts on dynamics is that I do not use bold print to denote a vector, and I use unit vectors sparingly. Thus, an n vector is usually written as x = (Xl' X 2 , ... ,xn ). When unit vectors are useful, I use the symbol er for the unit vector in the r direction; thus,

n

X = L Xiei' i~l

When the space is ?':f3 and Cartesian coordinates are used I write xl+ y j+zk without explanatory phrases.

On some occasions the same symbol is used in different places in the text to denote different quantities. Where this has been done intentionally, some explanatory text has been added pointing to this change. If it has also occurred unintentionally, I apologize in advance, and I would be grateful to readers who call my attention to it.

It is evident that a new text on an old and well-established subject can contain little that is new. This book is no exception; many of the problems treated here are classical, and much of the contents can also be found elsewhere. I hope, nevertheless, that some of the material will be novel to many readers.

In writing this book, lowe a great deal to others. Some of the excellent

x Preface

books which I found of great help are t : An Introduction to the Use of Generalized Coordinates in Mechanics and Physics by W. E. Byerly, Classical Mechanics by H. Goldstein, Mechanics by L. D. Landau and E. M. Lifshitz, Analytical Mechanics (in Russian) by A. I. Lur'e, Classical Dynamics by J. L. Synge, The Dynamics of Particles by A. G. Webster, and A Treatise on the Analytical Mechanics of Particles and Rigid Bodies by E. T. Whittaker. There are, however, two books without which the present volume could not have been written. One of these is a truly great and an altogether ad­mirable book by L. A. Pars: A Treatise on Analytical Dynamics; the other is a very deep book (in German) by G. Hamel entitled Theoretische Mechanik.

Many of the examples in this book have originated with one of the sources listed above. Where this is the case, the source has been acknowl­edged. However, the treatment here is never a direct quotation and it is usually done differently as well as more extensively than in the quoted source.

Undoubtedly, this book owes more to Pars' treatise than to any other source. I have learned much from his careful and clear definitions and from his unambiguous treatment of concepts which emerge from most books as somewhat nebulous and indistinct shapes.

My love of dynamics came initially from a study of Hamel's inspiring book. It is from him that I first learned that the comparison arcs in Hamil­ton's principle are not, in general, possible paths, and that therefore Hamilton's is not, in general, a variational principle. Hamel had a very thorough understanding of dynamics and he had the rare ability of com­bining analytical skill with physical insight. Moreover, his book contains one of the most interesting collections of problems to be found anywhere. Many of the problems in this text are due to Hamel. His book is not easily read, his notation is often cumbersome and rarely conventional, and the organization of its content could be improved. Nevertheless, it makes rewarding reading for those who make the effort.

Unfortunately, neither Pars' nor Hamel's book is suitable as an under­graduate text. Pars' treatise is too comprehensive for this purpose (as he says, it gives "a reasonably complete account of the subject [the entire subject of dynamics] as it now stands") and it lacks the problem collection expected in such texts. Hamel's book not only has the disadvantage (to English-speaking students) of being written in German, but it is perhaps

t Books which are frequently referred to are listed in the bibliography. Where only rare references are made to a source, it is given in a footnote.

Preface xi

more suitable for the student who already has an acquaintance with the subject matter.

It gives me great pleasure to acknowledge my debt to many. First and foremost, I want to thank the students who have attended my second course of dynamics; I have learned much from them.

My special thanks go to my colleagues Professors C. S. Hsu and G. Leitmann, and to my teaching assistants Messrs. Wen-Fan Lin and James Casey; they have read the manuscript critically and have made many suggestions which have materially improved it. In particular, Messrs. Lin and Casey have checked all formulas, equations, and examples. Without their devoted effort, this book would contain a myriad of errors which are not now in it.

Finally, I wish to acknowledge with gratitude aid from the National Science Foundation, which has supported my work in the geometry of dynamics, some of which is published here for the first time.

Reinhardt M. Rosenberg

Contents

1. Introduction. . . .

2. Dynamical Systems

2.1. Particles . . .

3.

4.

2.2. Systems of Particles 2.3. Forces and Laws of Motion 2.4. Galilean Transformations 2.S. Arguments of the Forces 2.6. The Problems of Particle Mechanics

Representations of the Motion

3.t. The Configuration Space 3.2. The Event Space 3.3. The State Space 3.4. The State-Time Space 3.S. Notions on the Concept of Stability 3.6. Problems.

Constraints

4.t. General Observations 4.2. Holonomic Constraints 4.3. Nonholonomic Constraints . 4.4. The Pfaffian Forms 4.S. When is a System of Constraints Holonomic? . 4.6. Accessibility (of the Configuration Space) 4.7. Problems.

xiii

1

7

7 9

10 12 IS 17

19

19 21 24-26 26 28

29

29 31 39 43 4S 48 SI

xiv Contents

5. The Strictly Newtonian Mechanics Problem 55

5.1. General Remarks 55 5.2. The Given Quantities and Relations 55 5.3. The First Problem 57 5.4. The Second Problem 59 5.5. Other Problems . 59 5.6. Concluding Remarks 60

6. Some Rigid Body Kinematics 61

6.1. The Rigid Body 61 6.2. Finite Rotation . 63 6.3. The Direction Cosines . 66 6.4. Orthogonal Transformations 69 6.5. The Matrix Notation 71 6.6. Properties of the Rotation Matrix 73 6.7. The Composition of Rotations 75 6.8. Applications 76

(a) The Euler Angles 77 (b) The Rodrigues Formulas 82

6.9. Problems. 84

7. Some Rigid Body Kinetics 87

7.1. Introductory Remarks 87 7.2. The Inertial Parameters in Rotated Axes 92 7.3. Angular Momentum and Principal Axes 94 7.4. The Ellipsoids of Cauchy and Poinsot 96 7.5. The General Motion of Rigid Bodies . 107 7.6. Problems. 111

8. The Nature of Lagrangean Mechanics . 115

8.1. General Remarks 115 8.2. The Generalizations by Lagrange . 115

9. Virtual Displacement and Virtual Work 119

9.1. General Observations 119 9.2. Classification of Displacements 119 9.3. D' Alembert's Principle 121

9.4. The Nature of the Forces of Constraint 128

9.5. The Virtual Velocity 139 9.6. The Variation 145

Contents xv

9.7. Possible Velocities and Accelerations 147 9.8. The Fundamental Equation . . . . 149 9.9. The Nature of the Given Forces . . 150 9.10. Given Forces Which Are Functions of Constraint Forces. 152 9.11. Problems. . . . . . . . . . . . . . . . . . . . . .. 157

10. Hamilton's Principle 161

10.1. The Kinetic Energy 161 10.2. Kinetic Energy in Catastatic Systems 162 10.3. The Energy Relations in Catastatic Systems 163 lOA. The Central Principle . . . . . 167 10.5. The Principle of Hamilton. . . . . 169 10.6. Noncontemporaneous Variations . . 174 10.7. Lagrange's Principle of Least Action 176 10.8. Jacobi's Principle of Least Action 179 10.9. Problems. . . . . . . . . . . . 181

11. Generalized Coordinates 185

11.1. Introductory Remarks 185 11.2. The Theory of Generalized Coordinates . 186 11.3. The Nature of Generalized Coordinates. ]90 11.4. The J Operator for Generalized Coordinates . 194 11.5. Exceptional Cases 195 11.6. Problems. . . . . . . . . . . . . . . . . 199

12. The Fundamental Equation in Generalized Coordinates 201

12.1. The Kinetic Energy . . . . 201 12.2. Two Equalities . . . . . . 204 12.3. The Fundamental Equation 204 12.4. Generalized Potential Forces 206 12.5. Velocity-Dependent Potentials 207 12.6. Problems. . . . . . . . . . 209

13. Lagrange's Equations 211

13.1. The Dynamical Problem. 211 13.2. The Multiplier Rule 212 13.3. Derivation from the Fundamental Equation 214 13.4. Derivation from the Central Principle 216 13.5. Derivation from Hamilton's Principle 217 13.6. Dynamic Coupling and Decoupling 221

xvi

13.7. Special Forms of Lagrange's Equations (a) Existence of a Potential . . . . (b) Holonomic Systems . . . . . . . (c) Rayleigh's Dissipation Function. . (d) The Dissipation Function of Lur'e

13.8. The Principle of Least Action Reconsidered 13.9. Problems . . . . . . . . . . . . . . . .

14. Embedding Constraints

14.1. Introductory Remarks 14.2. A Fallacy . . . .. 14.3. Embedding of Nonholonomic Constraints 14.4. Problems. . . . . . . . . . . . . . .

15. Formulating Problems by Lagrange's Equations

15.1. General Remarks . . . . . . . . . . . 15.2. The Unconstrained Particle 15.3. The Holonomically Constrained Particle 15.4. The Nonholonomically Constrained Particle 15.5. Systems of Particles and Rigid Bodies 15.6. Problems. . . . . . . . . . . . . .

16. The Integration

16.1. The Meaning of an Integral 16.2. Jacobi's Integral ..... 16.3. The Routhian Function and the Momentum Integrals

(a) The Legendre Transformation ..... (b) The Routhian Function . . . . . . . .

16.4. Partial and Complete Separation of Variables 16.5. Solution in Quadratures 16.6. Qualitative Integration. 16.7. Problems. . . . . . .

17. Stability

17.1. Introductory Remarks 17.2. Definition of Stability 17.3. The Variational Equations 17.4. Some Remarks on Indirect Methods 17.5. Some Remarks on Liapunov's Direct Method

(a) The Autonomous Case . . (b) The Nonautonomous Case

17.6. Problems. . . . . . . . ..

Contents

226 226 226 227 229 232 235

239

239 239 243 246

247

247 249 253 256 259 270

273

273 276 279 280 283 286 290 293 298

301

301 302 304 305 311 312 319 320

Contents xvii

18. Applications. . . . . . . 323

18.1. Introductory Remarks 323 18.2. The Single Particle . 324 18.3. Systems of Particles . 328 18.4. Nonholonomic Systems 334 18.5. Problems. . . . . . 346

19. About Celestial Problems. 349

19.1. Historical Notes 349 19.2. The Central Force Problem 351 19.3. The Central Force Problem Continued-The Apsides . 354 19.4. The Central Force Problem Continued-On Bertrand's Theorem 356 19.5. The n-Body Problem. . . . . . . . . . . . . 364 19.6. The Two-Body Problem . . . . . . . . . . . 368 19.7. Some Information about the Three-Body Problem 369 19.8. Problems. . . . . . . . . . . . . . . . . . 370

20. Topics in Gyrodynamics 373

20.1. Introduction . . . 373 20.2. The Heavy Symmetrical Top 374 20.3. The Gyroscope . . 383 20.4. The Gyrocompass . 385 20.5. Problems . . 388

21. Impulsive Motion 391

21.l. General Remarks . . . . . 391 2l.2. The Fundamental Equation 393 2l.3. Impulsive Constraints . . . 395 21.4. The Fundamental Equation with Impulsive Constraints 400 21.5. Impulsive Motion Theorems . . . . . . . 401 2l.6. Lagrange's Equations for Impulsive Motion 411 21.7. Problems . . . . . . . . . . . . . . . . 412

BIBLIOGRAPHY. 415

INDEX . . . . 417