Classical mechanics for physics graduate students

301 Pages·1999·12.715 MB·English

by  Ernesto Corinaldesi

CLASSICAL
MECHANlCS
FOR PHYSICS
GRADUATE
STUDENTS
Ernesto Corinaldesi
World Scientific Publishing
ClASSl~AMl ECHANICS
FOR PHYSI~S~ RADUATES TUDENTS
http://avaxhome.ws/blogs/ChrisRedfield
ClASSl~AMl ECHANICS
FOR PHYSI~S~ RADUATES TUDENTS
CLASSICAL MECHANlCS
FOR PHYSICS CRADUATE ~TUDENTS
Ernest0 Carinaldesi
Published by
World Scientific Publishing Co. Pte. Ltd.
P 0 Box 128, Fmr Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress ~a~Ioging-~-PublicaD~aitoan
Corinaldesi, E.
Classical mechanics for physics graduate students I Emesto
Corinaldesi.
p. cm.
lnciudes index.
ISBN 9810236255
I. Mechanics. 1. Title.
Q125.C655 1998
531-dc21 98-48670
CIP
British Library Catalo~ing-in-Publ~~tDioatna
A catalogue record for this book is available from the British Library.
Copyright Q 1998 by World Scientific Publishing Co. Pte. Ltd.
A11 rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means,
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This book is printed on acid-free paper.
Printed in Singapore by Uto-Print
to Abner Shimony, gratefully

Preface
This book evolved from my occasional teaching of graduate Classical Me-
chanics at Boston University.
In the late 1960’s I recommended the well-known texts by H. Goldstein,
L.D. Landau and E.M. Lifshitz (?), and A. Sommerfeld. In the late 1980’s
I learned some modern differential geometry while catching up with gauge
field theory. Thus I became able to appreciate the elegance of V.I. Arnold’s
Mathematical Methods of Classical Mechanics.
I took the educational risk of presenting Hamiltonian mechanics ex-
pressed in Cartan’s notation as a kind of appendix following the traditional
treatment. Instructors in other universities also seem to have recognized
that the graduate teaching of classical mechanics in physics departments
should be updated. In this book I have tried to satisfy both tradition-
alists and modernists by approaching each subject at successive levels of
abstraction.
My work is designed for a first-year physics graduate student at Boston
University, who has taken “Intermediate Mechanics”. Knowledge of curvi-
linear coordinates, vector analysis, advanced calculus etc. is assumed. The
wealth of detail offered should not lull the reader into thinking that the
material can be learned by leafing through the book. As much time as
possible should be devoted to going through the calculations and solving
problems.
Chapter 1 is an introduction meant to establish a rapport between the
reader and my way of presenting the material. Chapter 2 builds up a stock
of formulae to be drawn from in later chapters. Chapter 3 is mostly devoted
to oscillations. It ends with a sketchy account of chaos for discrete maps
to introduce the reader to a field founded a century ago by Poincare and
cultivated intensively in recent years.
Chapters 4 and 5 cover coordinate systems, inertial forces, and rigid
bodies.
Analytical mechanics begins with chapter 6, Lagrangians. Problems al-
ready worked out in previous chapters are again solved by using Lagrangian
methods. The reader is invited to compare the amount of labor involved in
the two versions.
Chapter 7 presents Hamiltonian mechanics. Sections 7.1 to 7.4 give a
simple account of the traditional treatment. Section 7.5 introduces Cartan’s
notation, which is used throughout the following sections as well as in
chapter 8, action-angle variables and adiabatic invariants.
Chapter 9 is on classical perturbation theory with emphasis on the sim-
ilarity with the perturbation theory of quantum mechanics and field theory.
viii PREFACE
Relativistic dynamics is discussed in chapter 10, with attention to the
(‘spinor connection”, the spin, and the Thomas precession.
Chapter 11 illustrates Lagrangian and Hamiltonian methods for contin-
uous systems by discussing two case studies, the vibrating string and the
ideal incompressible non-viscous fluid. The Lagrangian description of the
latter is not widely known. I worked unsuccessfully trying to formulate it,
until I was lucky enough to find D.E. Soper’s Classical Field Theory and
reference to the original 1911 paper by G. Herglotz.
Acknowledgements
I wish to express my gratitude to my students, especially to Nathaniel R.
Greene and Dinesh Loomba, for asking challenging questions and bringing
to my attention interesting material. After taking my course Gregg Jaeger
and John B. Ross became my graders and helped me in the preparation of
the monthly exams from which some of the problems in this text originated.
Adriana Ruth Corinaldesi painstakingly checked parts of my work point-
ing out errors and lack of clarity. Abner Shimony kindly read the final
version of the manuscript and suggested improvements. Neither Abner nor
Adriana are responsible for any residual errors and imperfections, which
are indeed possible because of the unusual number of formulae presented.
Anthony P. French, John Stachel, and Edwin F. Taylor kindly helped
me to dispel doubts about one of the relativity problems.
In addition to the books mentioned in the Preface I learned from many
others, but I wish to mention J.D. Jackson Classical Electrodynamics, C.W.
Kilmister and J.E. Reeves Rational Mechanics, R.A. Mann The Classical
Dynamics of Particles - Galilean and Lorentz relativity, B.F. Schutz Geo-
metrical methods of mathematical physics, W.Thirring Classical Dynamical
Systems.
I was able to prepare the printed manuscript only because of the gener-
ous coaching in Latex given me by Jok Leao. I am also indebted to Jinara
Reyes and Guoan Hu for frequent help.
Wei Chen and Lakshmi Narayanan of World Scientific have been an
example of amicable and helpful editorship.
Contents
1 INTRODUCTION 1
1.1 Motion in phase space . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motion of a particle in one dimension . . . . . . . . . . . . 2
1.3 Flow in phase space . . . . . . . . . . . . . . . . . . . . . . 5
1.4 The action integral . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 The Maupertuis principle . . . . . . . . . . . . . . . . . . . 12
1.6 Thetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Fermat’s principle . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 Chapter 1 problems . . . . . . . . . . . . . . . . . . . . . . 18
2 EXAMPLES OF PARTICLE MOTION 23
2.1 Central forces . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Circular and quasi-circular orbits . . . . . . . . . . . . . . . 25
2.3 Isotropic harmonic oscillator . . . . . . . . . . . . . . . . . . 26
2.4 The Kepler problem . . . . . . . . . . . . . . . . . . . . . . 27
2.5 The L-R-L vector . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Open Kepler-Rutherford orbits . . . . . . . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Integrability 34
2.8 Chapter 2 problems . . . . . . . . . . . . . . . . . . . . . . 36
3 FIXED POINTS. OSCILLATIONS. CHAOS 41
3.1 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Small oscillations . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Parametric resonance . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Periodically jerked oscillator . . . . . . . . . . . . . . . . . . 50
3.5 Discrete maps. bifurcation. chaos . . . . . . . . . . . . . . . 52
3.6 Chapter 3 problems . . . . . . . . . . . . . . . . . . . . . . 56
ix