Multiphase Averaging for Classical Systems: With Applications to Adiabatic Theorems

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9780387967783: Multiphase Averaging for Classical Systems: With Applications to Adiabatic Theorems

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.

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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.

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P. Lochak; C. Meunier; Translator-H.S. Dumas
Editore: Springer (1988)
ISBN 10: 0387967788 ISBN 13: 9780387967783
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Descrizione libro Springer, 1988. Paperback. Condizione libro: New. This item is printed on demand. Codice libro della libreria SONG0387967788

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Pierre Lochak, Claude Meunier
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Descrizione libro Springer-Verlag New York Inc., United States, 1988. Paperback. Condizione libro: New. 232 x 232 mm. Language: English . Brand New Book ***** Print on Demand *****.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. Softcover reprint of the original 1st ed. 1988. Codice libro della libreria AAV9780387967783

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Pierre Lochak, Claude Meunier
Editore: Springer-Verlag New York Inc., United States (1988)
ISBN 10: 0387967788 ISBN 13: 9780387967783
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Descrizione libro Springer-Verlag New York Inc., United States, 1988. Paperback. Condizione libro: New. 232 x 232 mm. Language: English . Brand New Book ***** Print on Demand *****. 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. Softcover reprint of the original 1st ed. 1988. Codice libro della libreria AAV9780387967783

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Pierre Lochak
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Descrizione libro Springer-Verlag New York Inc., 1988. PAP. Condizione libro: New. New Book. Delivered from our UK warehouse in 3 to 5 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Codice libro della libreria LQ-9780387967783

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P. LOCHAK
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Descrizione libro Springer, 1988. Paperback. Condizione libro: NEW. 9780387967783 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Codice libro della libreria HTANDREE0411316

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Lochak, P.; Meunier, C.
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Descrizione libro Springer, 1988. Paperback. Condizione libro: New. book. Codice libro della libreria 0387967788

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Descrizione libro Springer-Verlag New York Inc., 1988. PAP. Condizione libro: New. New Book. Shipped from US within 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Codice libro della libreria IQ-9780387967783

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Lochak, P., Meunier, C.
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Descrizione libro Springer, 2017. Paperback. Condizione libro: New. This item is printed on demand. Codice libro della libreria P110387967788

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P. Lochak, C. Meunier, H.S. Dumas (Translator)
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Descrizione libro Springer, 1988. Paperback. Condizione libro: New. Softcover reprint of the origina. This item is printed on demand. Codice libro della libreria DADAX0387967788

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