Introduction to Arnold’s Proof of the Kolmogorov–Arnold–Moser Theorem  book cover
1st Edition

Introduction to Arnold’s Proof of the Kolmogorov–Arnold–Moser Theorem

ISBN 9781032260655
Published July 8, 2022 by CRC Press
217 Pages 37 B/W Illustrations

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Book Description


This book provides an accessible step-by-step account of Arnold’s classical proof of the Kolmogorov–Arnold–Moser (KAM) Theorem. It begins with a general background of the theorem, proves the famous Liouville–Arnold theorem for integrable systems and introduces Kneser’s tori in four-dimensional phase space. It then introduces and discusses the ideas and techniques used in Arnold’s proof, before the second half of the book walks the reader through a detailed account of Arnold’s proof with all the required steps. It will be a useful guide for advanced students of mathematical physics, in addition to researchers and professionals.


• Applies concepts and theorems from real and complex analysis (e.g., Fourier series and implicit function theorem) and topology in the framework of this key theorem from mathematical physics.

• Covers all aspects of Arnold’s proof, including those often left out in more general or simplifi ed presentations.

• Discusses in detail the ideas used in the proof of the KAM theorem and puts them in historical context (e.g., mapping degree from algebraic topology). 

Table of Contents

Chapter 1. Hamilton Theory

Chapter 2. Preliminaries

Chapter 3. Outline of the KAM Proof

Chapter 4. Proof of the KAM Theorem

Chapter 5. Analytic Lemmas

Chapter 6. Geometric Lemmas

Chapter 7. Convergence Lemmas

Chapter 8. Arithmetic Lemmas

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Achim Feldmeier is a professor at Universität Potsdam, Germany.