Fundamental Principle of Classical Mechanics - A Geometrical Perspective

591 Pages·2014·4.778 MB·English

by  Kai S. Lam

FUNDAMENTAL PRINCIPLES
OF CLASSICAL MECHANICS
A Geometrical Perspective
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FUNDAMENTAL PRINCIPLES
OF CLASSICAL MECHANICS
A Geometrical Perspective
Kai S. Lam
California State Polytechnic University, Pomona, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Library of Congress Cataloging-in-Publication Data
Lam, Kai S. (Kai Shue), 1949– author.
Fundamental principles of classical mechanics : a geometrical perspective / by Kai S. Lam,
California State Polytechnic University, USA.
pages cm
Includes bibliographical references and index.
ISBN 978-981-4551-48-9 (hardcover : alk. paper)
1. Mechanics. I. Title.
QA805.L245 2014
531.01'515--dc23
2014005901
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
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Preface
This book is written with the belief that classical mechanics, as a theoretical
discipline, possessesan inherent beauty, depth, and richnessthat far transcend
its immediate applications in mechanical systems, although these are no doubt
important in their own right. These properties are manifested, by and large,
through the coherence and elegance of the mathematical structure underlying
the discipline, and, at least in the opinion of this author, areeminently worthy
of being communicated to physics students at the earliest stage possible. The
present text is therefore addressed mainly to advanced undergraduate and be-
ginning graduate physics students who are interested in an appreciation of the
relevance of modern mathematical methods in classical mechanics, in particu-
lar, those derived from the much intertwined fields of topology and differential
geometry, and also to the occasional mathematics student who is interested in
important physics applications of these areas of mathematics. Its chief pur-
poseistoofferanintroductoryandbroadglimpseofthemajesticedificeofthe
mathematicaltheoryofclassicaldynamics,notonlyinthetime-honoredanalyt-
ical tradition of Newton, Laplace, Lagrange,Hamilton, Jacobi,and Whittaker,
but alsothe moretopological/geometricaloneestablishedbyPoincar´e,and en-
riched by Birkhoff, Lyapunov, Smale, Siegel, Kolmogorov, Arnold, and Moser
(as well as many others). The latter tradition has been somewhat inexplicably
and politely ignored for many decades in the 20th century within the realm
of physics instruction, and has only relatively recently regained favor in some
physics textbooks under the guise of the more fashionable topics of chaos and
complexity in dynamical systems theory. This unfortunate circumstance may
perhaps in hindsight be attributable to related historical events: the rise of
quantummechanicsjustasPoincar´e’scontributionsincelestialmechanicswere
comingtotheforeatthedawnofthe20thcentury,andthesubsequentcompe-
titionforlimited“curricularspace”inphysicsinstructionbetweenclassicaland
quantum mechanics. The irony from a historical perspective, of course, is that
it was precisely the Hamilton-Jacobi theory within the Hamiltonian formula-
tion of classicalmechanics and its relationshipto waveoptics that precipitated
the development of non-relativistic quantum theory, and that the formalism of
action-anglevariablesinHamiltonianmechanics,throughtheBohr-Sommerfeld
quantizationrules,provedtobetheroyalroadtothe“oldquantumtheory”. In
addition,itwastheLagrangianformulationofclassicalmechanicsthatprovided
the groundworkfor the Feynman path approachin quantum field theory.
vii
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viii Fundamental Principlesof ClassicalMechanics
We hope that the present text will make a modest contribution, along with
many other excellent ones that have already appeared, towards the rehabilita-
tion of Poincar´e’s tradition in the mainstream physics curricula. Because of
the pervasive topological and geometrical character of this tradition, it is in-
evitable that a coherent, if somewhat spotty, introductory exposition (without
straying too far into the broader field of dynamical systems) of a key group
of concepts and tools of topology and differential geometry will have to be a
central feature of the presentation, as well as an earnest attempt to convince
thereaderofthefundamentalrelevanceofthesemathematicaltoolsinclassical
mechanics. Theseobjectivesgiverise tothe adoptionof a“dual-track”charac-
ter of the book, alternating between physics and mathematics. Our hope is to
strike a delicate balance between the intuitive/physically specific and the ab-
stract/mathematically general. The presentation will be mainly at a heuristic
level,largelyunburdenedbyarigorous,systematic,andlengthypresentationof
themathematicalbackground. Theinevitablelossofgenerality,rigor,andeven
accuracy entailed by such a style of “physical” presentation will hopefully be
compensatedforbyasounddegreeofcontinuityfromthe physicstothemath-
ematics (and vice versa), and by ample explicit calculations based on carefully
chosen applicationsof the abstractmathematical machinery.
To achieve the aforementioned goals, the text seeks to build upon the con-
ventional analytical treatment of the subject (based on vector calculus, mul-
tivariable calculus, and the solutions of differential equations) as presented in
almost all undergraduate texts and most graduate ones, and substantially en-
rich the exposition with a healthy dose of the modern language and tools of
topologyanddifferentialgeometry,especiallythoserelatedtothe basicnotions
of differentiable manifolds, tangent and cotangent spaces, the exterior differen-
tialcalculus,homotopy,homologyandcohomology, connectionsonfiberbundles,
Riemannian geometry, symplectic geometry, and Lie groups and algebras. The
main pedagogical route to the introduction and elucidation of most of these
mathematical topics will be the use of Cartan’s Method of Moving Frames in
theclassicalmechanicalcontextofrigid-bodydynamics,aspromulgatedbythe
geometerS.S.Chern. Historically,themathematicaldevelopmentofthediffer-
entialgeometrictheoryofmovingframeswasinfactoriginallymotivatedbythis
classicalmechanics problem. We seek to erect a pedagogicalbridge, as it were,
betweenthetime-honoredtextsof,forexample,Goldstein’s“Classical Mechan-
ics”ontheonehandandAbrahamandMarsden’s“Foundationsof Mechanics”
on the other – allbeit at a more elementary and much less rigorous level. In
this vein, the classic work “Mathematical Methods of Classical Mechanics” by
Arnold immediately comes to mind as a sort of “gold standard”; but again,
our aim is more modest: not only do we seek to build a bridge between the
physical/analytical and the mathematical/geometrical, but also one between
the undergraduate and graduate curricula. Due to the immense richness and
depth of the field, however, we cannot lay claim to any degree of completeness
or originalityof treatment.
At the risk of alienating the more mathematically oriented reader – this is,
after all, primarily a physics text – I have laid down the following guidelines
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Fundamental Principlesof ClassicalMechanics ix
for the writing of this book in regard to the sometimes tortured relationship
between the physicsand the mathematics contained therein.
Except on occasion, when clarity, conciseness, and precision trump intu-
•
itiveappeal,the usualabruptdefinition–theorem–proofsequenceencoun-
tered in the mathematical literature will be replaced by a more casual
build-up with physical motivation and justification leading to the rele-
vantmathematicalconceptsandfacts. Forexample,thediscussionofthe
mathematicalequivalenceofgaugefieldsandconnectionsonfiberbundles
is essentially built up from the elementary mechanical context of rigid
bodies movingin Euclidean space. As another example, the discussionof
the relevance of symplectic structures on manifolds and the crucial im-
portance of Darboux’s Theorem regarding symplectic manifolds will be
preceded by a good amount of justification in more familiar analytical
language. Once the informal build-up has achieved its purpose, however,
we will not shy away from exploiting the logical sharpness, elegance and
beauty of abstract concepts to arrive at results quickly, for example, in
demonstratingtheexistenceofcertainintegralinvariantsundercanonical
transformations(symplectomorphisms) in phase space.
The analytical mode of development (because of its familiarity and con-
•
creteness)willalmostalwaysbepresentedfirst,asaprecursortoandmo-
tivation for the more abstract and unfamiliar geometrical development.
For example, Poisson brackets will be introduced first in the traditional
“physics”manner(intermsofpartialderivativeswithrespecttocanonical
coordinatesandmomenta)beforebeingdefinedintermsofthevalueofthe
symplecticformontwovectorfieldsonphasespace. Oncethegeometrical
notionshavebeenfirmlygroundedinanalyticalrepresentations,however,
the strong interplay between the analytical (local) and the geometrical
(global) viewpoints will be stressed.
Within the geometricalexposition, (local) coordinates will almost always
•
be usedfirst in favorofthe coordinate-freeapproachusuallypreferredby
mathematicians. Darboux’s Theorem guarantees that this can be done
with impunity in the case of Hamiltonian mechanics, but in many other
situations, such as the use of moving frames for rigid-body dynamics as
a precursor to the introduction of frame bundles and the introduction of
connections on principal bundles (gauge fields), the description in terms
of local coordinates is usually much more intuitive. In addition, it yields
easilyinterpretableanalyticalformulasforcalculations,eventhoughthey
mayonly be locally valid.
Whenprecisemathematicaldefinitionsareunavoidable,westrivetoavoid
•
layering unfamiliar, abstract, and overly technical ones in quick succes-
sion. If possible, specific examples will be used to motivate definitions
of abstract concepts. For example, the general notion of connections on
vectorbundles is introduced throughthe much morefamiliarkinematical