Analytical Mechanics : A Comprehensive Treatise on the Dynamics of Constrained Systems (Reprint Edition)

1417 Pages·2014·10.09 MB·English

by  John G Papastavridis

(cid:2) R e p r i n t E d i ti o n (cid:3)
A
NALYTICAL
M
ECHANICS
A Comprehensive Treatise
on the
Dynamics of Constrained Systems
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
(cid:2) (cid:3)
Reprint Edition
A
NALYTICAL
M
ECHANICS
A Comprehensive Treatise
on the
Dynamics of Constrained Systems
J G. P ,
OHN APASTAVRIDIS Ph.D.
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Papastavridis, J. G. (John G.)
Analytical mechanics : a comprehensive treatise on the dynamics of constrained systems / by John G Papastavridis.
-- Reprint edition.
pages cm
Reprint of 2002 edition.
Includes bibliographical references and index.
ISBN 978-981-4338-71-4 (hardcover : alk. paper)
1. Mechanics, Analytic. I. Title.
QA805.P355 2014
531.01'515--dc22
2013017316
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
This edition is a corrected reprint of the work first published by Oxford University Press in 2002.
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to be
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For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222
Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
Printed in Singapore by Mainland Press Pte Ltd.
To the living and loving memory of my father,
GEORGE S. PAPASTAVRIDIS
(!E:P!IOY D. &A&ADTAYPI"H)
A lawyer and fearless maverick, who, throughout
his life, fought with fortitude, conviction, and class
to better his world;
and to whom I owe a critical part of my Weltanschauung.
[Theauthor’sfatherisshownhereinthetraditionalGreekfoustane´lla,
inAthens,Greece(Themistokle´ousstreet,nearLo´fosStre´fi),ca.1924]
PREFACE TO THE CORRECTED REPRINT
Thisisacorrectedreprintofaworkfirstpublishedinearly2002,byOxfordUniversity
Press,andwhichwentoutofprintshortlythereafter.
Afewsignmisprintsandsimilarerrorshavebeencorrected;somenotationshavebeen,
hopefully,improved(especiallyinChs.1,2);ausefuladditionhasbeenmadeonp.336,
andacoupleofsectionshavebeenthoroughlyrevised(e.g.§3.12,§8.13).
I am gratefulto(a) the manyreviewers,in someof themostprestigiousprofessional
journalsandelsewhere(e.g.BulletinoftheAmericanMathematicalSociety,IEEEControl
Systems Magazine, Zentralblatt fu¨r Mathematik;amazon.comand amazon.co.uk,private
communications,and referencesin advancedworks of mechanics), for their enthusiastic
comments;(b)theAmericanAssociationofPublishersforselecting,inJanuary2003,the
bookfortheir“AnnualAwardforOutstandingProfessionalandScholarlyTitlesof2002,
inEngineering”;and(c)lastbutnotleast,myWSPCeditor,Dr.S.W. Lim,andhismost
capableandcourteousstaff,fortheircontinuousandeffectivesupport.Allthesehavebeen
ofessentialmoralandpracticalhelptomeinthemakingofthis“new”edition!
Here,Itaketheopportunitytorestatethat,inthisbook,Ihave:
(a) Soughttocombinethebestoftheoldandnew,i.e.noagediscrimination;noknee-jerk
disdainfor“dustyoldstuff”norautomaticfollowingof“progress/modernity”—even
intheexactsciences,andespeciallyinmechanics,newisnotnecessarilyoruniformly
better;and
(b) Avoideddevelopmentsofconsiderablebutneverthelesspurelymathematicalinterest,
especiallythoseofthea-historicalandintuition-deadening(“epsilonic”)type.
May this treatise, as well as my other two works on mechanics [“Tensor Calculus
and Analytical Dynamics” (CRC, 1999) and “Elementary Mechanics” (WSPC, under
production)],keepmakingmanyandloyalreaders!
JohnG.Papastavridis
Atlanta,Georgia,Spring2014
PREFACE
Many ofthescientific treatises todayare formulated in a half-
mystical language, as thoughto impress thereader withthe
uncomfortable feeling that heis in thepermanent presence ofa
superman. Thepresent bookis conceivedin a humble spirit and
is written for humble people.
(Lanczos, 1970, pp.vii–viii)
GENERALDESCRIPTION
This book is a classical and detailed introduction to advanced analytical mechanics
(AM),withspecialemphasisonitsbasicprinciplesandequationsofmotion,asthey
apply to the most general constrained mechanical systems with a finite number of
degrees of freedom (this term is explained in Chapter 2). For the reasons detailed
below,andinspiteoftheageofthesubject,Ithinkthatnoothersinglevolumeexists,
inEnglishandinprint,thatiscomparabletotheoneathandinbreadthanddepthofthe
materialcovered—and,inthisnontrivialsense,thisca.1400-pageand174-figurelong
work is unique.
The book is addressed to graduate students, professors, and researchers, in the
areas of applied mechanics, engineering science, and mechanical, aerospace, struc-
tural, (even) electrical engineering, as well as physics and applied mathematics.
Advanced undergraduates are also very welcome to browse, and thus get initiated
intohigherdynamics.Thesoletechnicalprerequisitehere,arelativelymodestone,is
a solid working knowledge of ‘‘elementary/intermediate’’ (i.e., undergraduate)
dynamics; roughly, equivalent to the (bulk of the) material covered in, say,
Spiegel’s Theoretical Mechanics, part of the well-known Schaum’s outline series.
Also, familiarity with the simplest aspects of Lagrange’s equations, that is, how to
takethepartial andtotalderivativesofscalarenergetic functions,wouldbehelpful;
although, strictly, it is not necessary. [See also ‘‘Suggestions to the Reader’’
(Introduction, }3).]
CONTENTS
Specifically,thebookcoversinwhatIconsidertobeamostlogicalandpedagogical
sequence, the following topics:
Introduction: Introduction to analytical mechanics, brief summary of the history of
theoretical mechanics; suggestions to the reader; and list of symbols/notations, abbre-
viations,and basicformulae.
viii PREFACE
Chapter 1: Background: Algebra of vectors and Cartesian tensors, and basic concepts
and equations of Newton–Euler (or momentum) mechanics of particles and rigid
bodies; that is, a highly selective compendium of undergraduate dynamics, and (some
of)itsmathematics, from a mature viewpoint.
Chapter 2: Kinematics of constrained systems (i.e., Lagrangean kinematics); including
thegeneraltheoryofuptolinearvelocity(i.e.,Pfaffian)constraints,inbothholonomic
(ortrue)andnonholonomic(orquasi)coordinates;andauniquelyreadableaccountof
the fundamental theorem of Frobenius, for testing the nonholonomicity of such con-
straints.
Chapter 3: Kinetics of constrained systems (i.e., Lagrangean kinetics); including the
fundamentalprinciplesofAM;thatis,thoseofd’Alembert–Lagrangeandofrelaxation
of the constraints, the central equation of Heun–Hamel; equations of motion with or
without reactions, with or without multipliers, in true or quasi system variables; an
introduction to servoconstraints (theories of Appell–Beghin, et al.); and rigid-body
applications. This is the key chapter of the entire book, as far as engineering readers
areconcerned.
Chapter4:Impulsivemotion,underidealconstraints;includingtheassociatedextremum
theoremsof Carnot,Kelvin, Bertrand,Robin, etal.
Chapter 5: Nonlinear nonholonomic constraints; that is, kinematics and kinetics under
nonlinear,and generally nonholonomic, velocityconstraints.
Chapter6:Differentialvariationalprinciples,ofJourdain,Gauss,Hertz,etal.,andtheir
derivativehigher-order equations of motionof Nielsen,Tsenov, et al.
Chapter 7: Time-integral theorems and variational principles, of Lagrange, Hamilton,
Jacobi, O. Ho¨lder, Voss, Suslov, Voronets, Hamel, et al.; for linear and nonlinear
velocityconstraintsintrueandquasivariables,withorwithoutmultipliers;plusenergy
andvirialtheorems.
Chapter 8: Introduction to Hamiltonian/Canonical methods; that is, equations of
HamiltonandRouth–Helmholtz,cyclicsystems,steadymotionanditsstability,varia-
tion of constants, canonical transformations and Poisson’s brackets, Hamilton–Jacobi
integration theory, integral invariants, Noether’s theorem, and action–angle variables
andtheir applicationsto adiabatic invariants andperturbationtheory.
Chapters 2–8 each contain a large number of completely solved examples, and
problems with their answers (and, occasionally, hints), to illustrate and extend the
previous theories; short onesareintegrated within eachchapter section, andlonger,
moresynthetic,onesarecollectedateachchapter’send;andalso,criticalcomments/
references for further study. The exposition ends with a relatively extensive, cumu-
lative, and alphabetical list of References and Suggested Reading, including every-
thing from standard textbooks all the way to epoch-making memoirs of the last
(more than) two hundred years. This list complements those found in such well-
known references as Neimark and Fufaev (1967/1972) and Roberson and
Schwertassek (1988).
Parts of the text have, unavoidably, state-of-the-art flavor. However, as far as
fundamental ideas go, very little, if anything, of the topics covered is truly new—
today,noonecanclaimmuchoriginalityinclassicalmechanics!Thenewnesshere,a
nontrivial one, I think, consists in restoring, clarifying, putting together, and pre-
senting,inwhatIhopeisareadableform,materialmostofwhichhasappearedover
the past one hundred fifty, or so, years; frequently in little known, and/or hard to
findanddecipher,sources.(Inviewofthethousandsofbooks,lecturenotes,articles,
and so on, used in the writing of this work, failure to acknowledge an author’s
PREFACE ix
particularcontributionisnotintentional,merelyanoversight.)But,giventheaston-
ishing unfamiliarity, confusion, and intellectual provincialism so prevalent in many
theoretical and applied mechanics circles today, even in the fundamental concepts
and principles ofanalytical dynamics(like virtual displacements/work and principle
of d’Alembert–Lagrange, which is, by far, the most misunderstood ‘‘principle’’ of
physics!),Ifeltverystronglythatthisnoble,beautiful,andpowerfulbodyofknowl-
edge, that diamond of our cultural heritage, should be accurately preserved and
represented, so as to benefit present and future workers in dynamics.
No single volume can even pretend to cover satisfactorily all aspects of this vast
and fascinating subject; in particular, both its theoretical and applied aspects, let
alonethecurrentlypopularcomputationalones.Sincethisisnotanencyclopediaof
theoretical and applied dynamics, an inescapable and necessary selection has oper-
ated, and so, the following important topics are not covered: applications of differ-
ential forms/exterior calculus (of Cartan, Gallissot, et al.) and symplectic geometry
toLagrangeanandHamiltonianmechanics;grouptheoreticapplications;nonlineardynam-
ics (incl.regularandstochastic/chaoticmotion)and stability of motion;theoryoforbits;
and computational/numericaltechniques. For all these, there already exists an enormous
andcompetentliterature(see“SuggestionstotheReader”).However,withthehelpofthis
treatise,theconscientiousreaderwillbeabletomovequicklyandconfidentlyintoanypar-
ticulartheoreticaland/orcomputationalareaofmoderndynamics.Inthissense,thework
athandconstitutesanoptimalinvestmentofthereader’spreciousenergies.
RAISOND’EˆTRE,ANDSOMEPHILOSOPHY
The customary words of explanation, or apology, for writing ‘‘another’’ book on
advanceddynamicsarenowinorder.Themainthemeofthiswork,likeaWagnerian
leitmotiv,isdeductiveorder,formalstructure,andphysicalideas,astheypertainto
that particular energetic form of mechanics of constrained systems founded by
Lagrange and known as analytical (¼deductive) mechanics; to be differentiated
fromthealsoanalyticalbutmomentum,or‘‘elementary,’’formofsystemmechanics,
foundedbyEuler.Itisabookforpeoplewhoplacetheory(theories),ideas,knowledge,
and understanding above all else—and do not apologize for it. Here, AM is studied
not as the “maid” of some other (allegedly) more important discipline, but as a sub-
ject worth knowing in its own right; that is, as a “king or queen.” As such, it will
attract those with a qualitative and theoretical bent of mind; while it may not be as
agreeabletothosewithpurelycomputationaland/orintellectuallylocalpredilections.[In
the words of the late Professor R.M. Rosenberg (University of California, Berkeley):
‘‘The field of dynamics is plowed by two classes of people: those who enjoy the
inherent beauty, symmetry and consistency of this discipline, and those who are
satisfied with having a machine that manufactures equations of motion of complex
mechanisms’’ (private communication, 1986).] Generally, science is more than a
collection of particular problems and special techniques, even involved ones—it is
much more than mere information. However, practical people should be reminded
that theory and application are mutually complementary rather than adversarial; in
fact, contemporary important practical problems and the availability of powerful
computationalcapabilitieshavemadethethoroughunderstandingofthefundamen-
tal principles of mechanics more necessary today than before. Applications and
computers have, among other things, helped resurrect, restore, and sharpen
old academic curiosities (for engineers anyway), such as the differential variational