Multiphase Averaging for Classical Systems: With Applications To Adiabatic Theorems: 72 - Brossura

Lochak, P.

Brossura

ISBN 10: 0387967788 ISBN 13: 9780387967783

Casa editrice: Springer, 2013

Riassunto

In the past several decades many significant results in averaging for systems of ODE's have been obtained.

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Contenuti

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.- 8.6 Perpetual 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.

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2013
• Lingua: Inglese
• ISBN 10: 0387967788
• ISBN 13: 9780387967783
• Rilegatura: Copertina flessibile
• Numero di pagine: 376
• Contatto del produttore: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3; Heidelberg; 69115; Germania