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

Achim Feldmeier

• Publisher: CRC Press
• Year: 2022
• Print ISBN: 9781032260655
• eBook ISBN: 9781000610000
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Applied Mathematics

INTRODUCTION TO ARNOLD'S PROOF OF THE KOLMOGOROV–ARNOLD–MOSER THEOREM

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.

Features

• 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

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• 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
• References
• Person Index
• Subject Index