Regular and Stochastic Motion

517 Pages·1983·10.64 MB·English

by  M. A. Lieberman (auth.)

Applied Mathematical Sciences
Volume 38
Editors
F. John J. E. Marsden L. Sirovich
Advisors
H. Cabannes M. Ghil J. K. Hale
J. Keller J. P. LaSalle G. B. Whitham
Applied Mathematical Sciences
1. John: Partial Differential Equations, 4th ed. (cloth)
2. Sirovich: Techniques of Asymptotic Analysis.
3. Hale: Theory of Functional Differential Equations, 2nd ed. (cloth)
4. Percus: Combinatorial Methods.
5. von Mises/Friedrichs: Fluid Dynamics.
6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics.
7. Pipkin: Lectures on Viscoelasticity Theory.
8. Giacaglia: Perturbation Methods in Non-Linear Systems.
9. Friedrichs: Spectral Theory of Operators in Hilbert Space.
10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations.
11. Wolovich: Linear Multivariable Systems.
12. Berkovitz: Optimal Control Theory.
13. Bluman/Cole: Similarity Methods for Differential Equations.
14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost
Periodic Solutions.
15. Braun: Differential Equations and Their Applications, 2nd ed. (cloth)
16. Lefschetz: Applications of Algebraic Topology.
17. CollatzlWetterling: Optimization Problems.
18. Grenander: Pattern SyntheSis: Lectures in Pattern Theory, Vol. I.
19. Marsden/McCracken: The Hopf Bifurcation and Its Applications.
20. Driver: Ordinary and Delay Differential Equations.
21. Courant/Friedrichs: Supersonic Flow and Shock Waves. (cloth)
22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method.
23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory.
24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II.
25. Davies: Integral Transforms and Their Applications.
26. Kushner/Clark: Stochastic Approximation Methods for Constrained and
Unconstrained Systems.
27. de Boor: A Practical Guide to Splines.
28. Keilson: Markov Chain Models-Rarity and Exponentiality.
29. de Veubeke: A Course in ElastiCity.
30. Sniatycki: Geometric Quantization and Quantum Mechanics.
31. Reid: Sturm ian Theory for Ordinary Differential Equations.
32. Meis/Marcowitz: Numerical Solution of Partial Differential Equations.
33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III.
34. Kevorkian/Cole: Pertubation Methods in Applied Mathematics. (cloth)
35. Carr: Applications of Centre Manifold Theory.
36. Bengtsson/Ghil/Kallen: Dynamic Meteorology: Data Assimilation Methods.
37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces.
38. Lichtenberg/Lieberman: Regular and Stochastic Motion. (cloth)
A. J. Lichtenberg
M. A. Lieberman
Regular and
Stochastic Motion
With 140 Figures
Springer Science+Business Media, LLC
A. J. Lichtenberg
M. A. Lieberman
Department of Electrical Engineering
and Computer Sciences
University of California
Berkeley, CA 94720
USA
Editors
F. John J. E. Marsden L. Sirovich
Courant Institute of Department of Division of
Mathematical Sciences Mathematics Applied Mathematics
New York University University of California Brown University
New York, NY 10012 Berkeley, CA 94720 Providence, RI 02912
USA USA USA
AMS Subject Classifications: 70Kxx, 60Hxx, 34Cxx, 35Bxx
Library of Congress Cataloging in Publication Data
Lichtenberg, Allan J.
Regular and stochastic motion.
(Applied mathematical sciences; v. 38)
Bibliography: p.
Includes index.
1. Nonlinear oscillations. 2. Stochastic processes.
3. Hamiltonian systems. I. Lieberman, M. A (Michael A)
II. Title. III. Series: Applied
mathematical sciences (Springer-Verlag New York Inc.); v. 38.
QA1.A647 vol. 38 [QA867.5] 510s [531'.322] 82-19471
©1983 by Springer Science+Business Media New York
Originally published by Springer-Verlag N ew York Heidelberg Berlin in 1983
Softcover reprint of the hardcover 1st edition 1983
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission from Springer Science+Business Media. LLC.
Typeset by Computype, Inc., St. Paul, MN.
9 8 7 6 5 432 1
ISBN 978-1-4757-4259-6 ISBN 978-1-4757-4257-2 (eBook)
DOI 10.1007/978-1-4757-4257-2
To Elizabeth and Marlene
Preface
This book treats stochastic motion in nonlinear oscillator systems. It
describes a rapidly growing field of nonlinear mechanics with applications
to a number of areas in science and engineering, including astronomy,
plasma physics, statistical mechanics and hydrodynamics. The main em-
phasis is on intrinsic stochasticity in Hamiltonian systems, where the
stochastic motion is generated by the dynamics itself and not by external
noise. However, the effects of noise in modifying the intrinsic motion are
also considered. A thorough introduction to chaotic motion in dissipative
systems is given in the final chapter.
Although the roots of the field are old, dating back to the last century
when Poincare and others attempted to formulate a theory for nonlinear
perturbations of planetary orbits, it was new mathematical results obtained
in the 1960's, together with computational results obtained using high speed
computers, that facilitated our new treatment of the subject. Since the new
methods partly originated in mathematical advances, there have been two
or three mathematical monographs exposing these developments. However,
these monographs employ methods and language that are not readily
accessible to scientists and engineers, and also do not give explicit tech-
niques for making practical calculations.
In our treatment of the material, we emphasize physical insight rather
than mathematical rigor. We present practical methods for describing the
motion, for determining the transition from regular to stochastic behavior,
and for characterizing the stochasticity. We rely heavily on numerical
computations to illustrate the methods and to validate them.
The book is intended to be a self contained text for physical scientists
and engineers who wish to enter the field, and a reference for those
V111 Preface
researchers already familiar with the methods. It may also be used as an
advanced graduate textbook in mechanics. We assume that the reader has
the usual undergraduate mathematics and physics background, including a
mechanics course at the junior or senior level in which the basic elements of
Hamiltonian theory have been covered. Some familiarity with Hamiltonian
mechanics at the graduate level is desirable, but not necessary. An exten-
sive review of the required background material is given in Sections 1.2 and
1.3.
The core ideas of the book, concerning intrinsic stochasticity in Hamilto-
nian systems, are introduced in Section 1.4. Our subsequent exposition in
Chapters 2-6 proceeds from the regular to the stochastic. To guide the
reader here, we have "starred" (*) the sections in which the basic material
appears. These "starred" sections form the core of our treatment. Section
2.4a on secular perturbation theory is of central importance. The core
material has been successfully presented as a 30 lecture-hour graduate
course at Berkeley.
In addition to the core material, other major topics are treated. The
effects of external noise in modifying the intrinsic motion are presented in
Section 5.5, (using the results of Section 5.4d) for two degrees of freedom,
in Section 6.3 for more than two degrees of freedom, and an application is
given in Section 6.4. Our description of dissipative systems in Chapter 7 can
be read more-or-less independently of our treatment of Hamiltonian sys-
tems. For studying the material of Chapter 7, the introduction in Section
1.5 and the material on surfaces of section in Section 1.2b and on Liapunov
exponents in Sections 5.2b and 5.3 should be consulted. The topic of period
doubling bifurcations is presented in Section 7.2b, 7.3a and Appendix B
(see also Section 3.4d). Other specialized topics such as Lie perturbation
methods (Section 2.5), superconvergent perturbation methods (Section 2.6),
aspects of renormalization theory (Sections 4.3, 4.5), non-canonical meth-
ods (Section 2.3d), global removal of resonances (Section 2.4d and part of
2.5c), variational methods (Sections 2.6b and 4.6), and modulational diffu-
sion (Section 6.2d) can generally be deferred until after the reader has
obtained some familiarity with the core material.
This book has been three and a half years in the writing. We have
received encouragement from many friends and colleagues. We wish to
acknowledge here those who reviewed major sections of the manuscript.
The final draft has been greatly improved by their comments. Thanks go to
H. D. I. Abarbanel, J. R. Cary, B. V. Chirikov, R. H. Cohen, D. F.
Escande, J. Ford, J. Greene, R. H. G. Helleman, P. J. Holmes, J. E.
Howard, O. E. Lanford, D. B. Lichtenberg, R. Littlejohn, B. McNamara,
H. Motz, C. Sparrow, J. L. Tennyson and A. Weinstein. Useful comments
have also been received from G. Casati, A. N. Kaufman, I. C. Percival and
G. R. Smith. We are also pleased to acknowledge the considerable influ-
ence of the many published works in the field by B. V. Chirikov. Many of
Preface ix
the ideas expressed herein were developed by the authors while working on
grants and contracts supported by the National Science Foundation, the
Department of Energy, and the Office of Naval Research. One of the
authors (A.J.L.) acknowledges the hospitality of St. Catherine's College,
Oxford, and one of the authors (M.A.L.) acknowledges the hospitality of
Imperial College, London, where much of the manuscript was developed.
Contents
List of Symbols xvii
Chapter 1
Overview and Basic Concepts
1.1. An Introductory Note 1
* 1.2. Transformation Theory of Mechanics 7
* 1.2a. Canonical Transformations 7
* 1.2b. Motion in Phase Space 12
*1.2c. Action-Angle Variables 20
1.3. Integrable Systems 23
*1.3a. One Degree of Freedom 23
1.3b. Linear Differential Equations 28
1.3c. More than One Degree of Freedom 31
*1.4. Near-Integrable Systems 41
*1.4a. Two Degrees of Freedom 42
*l.4b. More than Two Degrees of Freedom 54
1.5. Dissipative Systems 56
l.5a. Strange Attractors 57
1.5b. The Lorenz System 59
Chapter 2
Canonical Perturbation Theory 63
2.1. Introduction 63
2.1 a. Power Series 66
* Starred sections indicate core material.
xii Contents
2.1 b. Asymptotic Series and Small Denominators 68
2.1 c. The Effect of Resonances 70
*2.2. Classical Perturbation Theory 71
*2.2a. One Degree of Freedom 71
*2.2b. Two or More Degrees of Freedom 76
2.3. Adiabatic Invariance 85
*2.3a. Introduction and Basic Concepts 85
*2.3b. Canonical Adiabatic Theory 88
*2.3c. Slowly Varying Harmonic Oscillator 92
2.3d. Noncanonical Methods 94
2.4. Secular Perturbation Theory 100
*2.4a. Removal of Resonances 101
*2.4b. Higher-Order Resonances 107
*2.4c. Resonant Wave-Particle Interaction 112
2.4d. Global Removal of Resonances 119
2.5. Lie Transformation Methods 123
2.5a. General Theory 125
2.5b. Deprit Perturbation Series 126
2.5c. Adiabatic Invariants 130
2.6. Superconvergent Methods 138
2.6a. Kolmogorov's Technique 141
2.6b. Singly Periodic Orbits 143
Chapter 3
Mappings and Linear Stability 150
*3.1. Hamiltonian Systems as Canonical Mappings 151
*3.la. Integrable Systems 151
*3.lb. Near-Integrable Systems 155
*3.lc. Hamiltonian Forms and Mappings 157
*3.2. Generic Behavior of Canonical Mappings 158
*3.2a. Irrational Winding Numbers and KAM Stability 159
*3.2b. Rational Winding Numbers and Their Structure 168
*3.2c. Complete Description of a Nonlinear Mapping 172
*3.2d. A Numerical Example 176
3.3. Linearized Motion 178
3.3a. Eigenvalues and Eigenvectors of a Matrix 179
*3.3b. Two-Dimensional Mappings 183
*3.3c. Linear Stability and Invariant Curves 186
*3.4. Fermi Acceleration 190
*3.4a. Physical Problems and Models 191
*3.4b. Numerical Results 194
*3.4c. Fixed Points and Linear Stability 198
*3.4d. Bifurcation Phenomena 201
*3.4e. Hamiltonian Formulation 205
*3.5. The Separatrix Motion 206
*3.5a. Driven One-Dimensional Pendulum 208
*3.5b. The Separatrix Mapping 211