Author | Lochak, Pierre. author |
---|---|
Title | Multiphase Averaging for Classical Systems [electronic resource] : With Applications to Adiabatic Theorems / by Pierre Lochak, Claude Meunier |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1044-3 |
Descript | XI, 360 p. 10 illus. online resource |
1 Introduction and Notation -- 1.1 Introduction -- 1.2 Notation -- 2 Ergodicity -- 2.1 Anosovโs 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; Arnoldโs result -- 3.6 Preliminary lemmas -- 3.7 Proof of Arnoldโs theorem -- 4 Two Frequency Systems; Neistadtโs 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; Neistadtโs Result Based on Anosovโs Method -- 5.1 Introduction and results -- 5.2 Proof of the theorem -- 5.3 Proof for the differentiable case -- 6 N Frequency Systems; Neistadtโs Results Based on Kasugaโs 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 Nekhoroshevโs 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