Multiphase Averaging for Classical Systems

1. Front Matter

   Pages i-xi

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2. Introduction and Notation

   • Pierre Lochak, Claude Meunier

   Pages 1-9

3. Ergodicity

   • Pierre Lochak, Claude Meunier

   Pages 11-24

4. One Frequency Systems and First Result for Two Frequency Systems

   • Pierre Lochak, Claude Meunier

   Pages 25-66

5. Two Frequency Systems; Neistadt’s Results

   • Pierre Lochak, Claude Meunier

   Pages 67-116

6. N Frequency Systems; Neistadt’s Result Based on Anosov’s Method

   • Pierre Lochak, Claude Meunier

   Pages 117-132

7. N Frequency Systems; Neistadt’s Results Based on Kasuga’s Method

   • Pierre Lochak, Claude Meunier

   Pages 133-152

8. Hamiltonian Systems

   • Pierre Lochak, Claude Meunier

   Pages 153-182

9. Adiabatic Theorems in One Dimension

   • Pierre Lochak, Claude Meunier

   Pages 183-227

10. The Classical Adiabatic Theorems in Many Dimensions

    • Pierre Lochak, Claude Meunier

    Pages 229-247

11. The Quantum Adiabatic Theorem

    • Pierre Lochak, Claude Meunier

    Pages 249-268

12. Back Matter

    Pages 269-361

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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.

• Fourier series
• KAM theory
• Kolmogorov–Arnold–Moser theorem
• hamiltonian system
• integrable system
• maximum
• stability

• Centre de Mathematiques, Ecole Normale Superieure, Paris Cedex 05, France

  Pierre Lochak

• Centre de Physique Theorique, Ecole Polytechnique, Palaiseau Cedex, France

  Claude Meunier

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