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 sig­ nificant 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 demonstrat­ ing 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 prin­ ciple. 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 non­ contemporaneous 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 Mechanical Engineering and Mathematical Sciences Rice University, Houston, Texas Volume 1 INTRODUCTION TO VECTORS AND TENSORS Volume 1: Linear and Multilinear Algebra Ray M Bowen and c.-c. Wang Volume 2 INTRODUCTION TO VECTORS AND TENSORS Volume 2: Vector and Tensor Analysis Ray M Bowen and C.-C. Wang Volume 3 MULTICRITERIA DECISION MAKING AND DIFFERENTIAL GAMES Edited by George Leitmann Volume 4 ANALYTICAL DYNAMICS OF DISCRETE SYSTEMS Reinhardt M. Rosenberg VolumeS TOPOLOGY AND MAPS Taqdir Husain Volume 6 REAL AND FUNCTIONAL ANALYSIS A. Mukherjea and K. Pothoven Volume 7 PRINCIPLES OF OPTIMAL CONTROL THEORY R. V. Gamkrelidze Volume 8 INTRODUCTION TO THE LAPLACE TRANSFORM Peter K. F. Kuhfittig Volume 9 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 1st 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 inter­ mediate 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 im­ part to the student the techniques needed to solve problems. Similarly, a sec­ ond 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 con­ siderable 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 Lagrangeant 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 illustra­ tion 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.