Table Of ContentAnalytical 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
c.-c.
Ray M Bowen and 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 1s t 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. Thus, an n vector is usually written
as x = (Xl' X2, ... ,xn). When unit vectors are useful, I use the symbol
e
r for the unit vector in the r direction; thus,
n
L
X = 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 aret: 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 ofP articles 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 ad
mirable 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 acknowl
edged. 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 Hamil
ton'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 com
bining 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 under
graduate 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.
