Multiphase Averaging for Classical Systems
With Applications to Adiabatic Theorems
Published on 7. September 1988
Book
Paperback/Softback
XI, 360 pages
978-0-387-96778-3 (ISBN)
Description
In the past several decades many significant results in averaging for systems of ODE's have been obtained. These results have not attracted a tention in proportion to their importance, partly because they have been overshadowed by KAM theory, and partly because they remain widely scattered - and often untranslated - throughout the Russian literature. The present book seeks to remedy that situation by providing a summary, including proofs, of averaging and related techniques for single and multiphase systems of ODE's. The first part of the book surveys most of what is known in the general case and examines the role of ergodicity in averaging. Stronger stability results are then obtained for the special case of Hamiltonian systems, and the relation of these results to KAM Theory is discussed. Finally, in view of their close relation to averaging methods, both classical and quantum adiabatic theorems are considered at some length. With the inclusion of nine concise appendices, the book is very nearly self-contained, and should serve the needs of both physicists desiring an accessible summary of known results, and of mathematicians seeing an introduction to current areas of research in averaging.
More details
Series
Edition
Softcover reprint of the original 1st ed. 1988
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
10 s/w Abbildungen
XI, 360 p. 10 illus.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 21 mm
Weight
569 gr
ISBN-13
978-0-387-96778-3 (9780387967783)
DOI
10.1007/978-1-4612-1044-3
Schweitzer Classification
Persons
Content
1 Introduction and Notation.- 1.1 Introduction.- 1.2 Notation.- 2 Ergodicity.- 2.1 Anosov's result.- 2.2 Method of proof.- 2.3 Proof of Lemma 1.- 2.4 Proof of Lemma 2.- 3 On Frequency Systems and First Result for Two Frequency Systems.- 3.1 One frequency; introduction and first order estimates.- 3.2 Increasing the precision; higher order results.- 3.3 Extending the time-scale; geometry enters.- 3.4 Resonance; a first encounter.- 3.5 Two frequency systems; Arnold's result.- 3.6 Preliminary lemmas.- 3.7 Proof of Arnold's theorem.- 4 Two Frequency Systems; Neistadt's Results.- 4.1 Outline of the problem and results.- 4.2 Decomposition of the domain and resonant normal forms.- 4.3 Passage through resonance: the pendulum model.- 4.4 Excluded initial conditions, maximal separation, average separation.- 4.5 Optimality of the results.- 4.6 The case of a one-dimensional base.- 5 N Frequency Systems; Neistadt's Result Based on Anosov's Method.- 5.1 Introduction and results.- 5.2 Proof of the theorem.- 5.3 Proof for the differentiable case.- 6 N Frequency Systems; Neistadt's Results Based on Kasuga's Method.- 6.1 Statement of the theorems.- 6.2 Proof of Theorem 1.- 6.3 Optimality of the results of Theorem 1.- 6.4 Optimality of the results of Theorem 2.- 7 Hamiltonian Systems.- 7.1 General introduction.- 7.2 The KAM theorem.- 7.3 Nekhoroshev's theorem; introduction and statement of the theorem.- 7.4 Analytic part of the proof.- 7.5 Geometric part and end of the proof.- 8 Adiabatic Theorems in One Dimension.- 8.1 Adiabatic invariance; definition and examples.- 8.2 Adiabatic series.- 8.3 The harmonic oscillator; adiabatic invariance and parametric resonance.- 8.4 The harmonic oscillator; drift of the action.- 8.5 Drift of the action for general systems.- 8.6Perpetual stability of nonlinear periodic systems.- 9 The Classical Adiabatic Theorems in Many Dimensions.- 9.1 Invariance of action, invariance of volume.- 9.2 An adiabatic theorem for integrable systems.- 9.3 The behavior of the angle variables.- 9.4 The ergodic adiabatic theorem.- 10 The Quantum Adiabatic Theorem.- 10.1 Statement and proof of the theorem.- 10.2 The analogy between classical and quantum theorems.- 10.3 Adiabatic behavior of the quantum phase.- 10.4 Classical angles and quantum phase.- 10.5 Non-communtativity of adiabatic and semiclassical limits.- Appendix 1 Fourier Series.- Appendix 2 Ergodicity.- Appendix 3 Resonance.- Appendix 4 Diophantine Approximations.- Appendix 5 Normal Forms.- Appendix 6 Generating Functions.- Appendix 7 Lie Series.- Appendix 8 Hamiltonian Normal Forms.- Appendix 9 Steepness.- Bibliographical Notes.