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E-raamat: Introduction to Arnolds Proof of the KolmogorovArnoldMoser Theorem [Taylor & Francis e-raamat]

  • Formaat: 205 pages, 37 Line drawings, black and white; 37 Illustrations, black and white
  • Ilmumisaeg: 08-Jul-2022
  • Kirjastus: CRC Press
  • ISBN-13: 9781003287803
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  • Taylor & Francis e-raamat
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Introduction to Arnolds Proof of the KolmogorovArnoldMoser TheoremSuurem pilt
  • Formaat: 205 pages, 37 Line drawings, black and white; 37 Illustrations, black and white
  • Ilmumisaeg: 08-Jul-2022
  • Kirjastus: CRC Press
  • ISBN-13: 9781003287803
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9781003287803
INTRODUCTION TO ARNOLDS PROOF OF THE KOLMOGOROVARNOLDMOSER THEOREM

This book provides an accessible step-by-step account of Arnolds classical proof of the KolmogorovArnoldMoser (KAM) Theorem. It begins with a general background of the theorem, proves the famous LiouvilleArnold theorem for integrable systems and introduces Knesers tori in four-dimensional phase space. It then introduces and discusses the ideas and techniques used in Arnolds proof, before the second half of the book walks the reader through a detailed account of Arnolds 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 Arnolds 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).
Preface ix
Chapter 1 Hamilton Theory
1(38)
1.1 Hamilton Equations
1(1)
1.2 Generating Functions
2(2)
1.3 Infinitesimal Canonical Transformation
4(2)
1.4 Conserved Phase-Space Measure
6(2)
1.5 Coordinate Relations
8(1)
1.6 Action-Angle Variables
9(2)
1.7 Integrable Systems and Phase-Space Tori
11(20)
1.8 Kneser's Theorems for Four-Dimensional Phase Space
31(3)
1.9 Celestial Mechanics
34(5)
Chapter 2 Preliminaries
39(8)
2.1 Notation
39(4)
2.2 Domain Reduction
43(4)
Chapter 3 Outline of the KAM Proof
47(28)
3.1 Perturbation Series vs. Fast Iteration
48(7)
3.2 Perturbation Theory with New Frequencies
55(3)
3.3 Frequency Diffeomorphism
58(2)
3.4 Diophantine Condition
60(2)
3.5 Fast Iteration Plus Diophantine Condition
62(2)
3.6 Resonance Cutoff
64(3)
3.7 Domains without Interior
67(3)
3.8 Analytic Functions
70(1)
3.9 Motion on a Torus
71(1)
3.10 Small Numbers
72(3)
Chapter 4 Proof of the KAM Theorem
75(40)
4.1 Fundamental Theorem
76(13)
4.2 Inductive Theorem
89(12)
4.3 KAM Theorem
101(14)
Chapter 5 Analytic Lemmas
115(12)
5.1 Lemma A1. Exponentials vs. Powers
115(1)
5.2 Functions in Several Complex Variables
116(2)
5.3 Lemma A2. Fourier Coefficients
118(4)
5.4 Lemma A3. Cauchy Estimates
122(2)
5.5 Lemma A4. Lagrange and Taylor Estimates
124(3)
Chapter 6 Geometric Lemmas
127(34)
6.1 An Implicit Function Theorem
127(2)
6.2 Mapping Degree
129(2)
6.3 Lemma G1. Surjectivity of Maps Near Identity
131(5)
6.4 Brouwer's Lemma
136(7)
6.5 Modem Proof of Lemma G1
143(2)
6.6 Lemma G2. Frequency Diffeomorphism
145(3)
6.7 Lemma G3. Canonical Transformation
148(5)
6.8 Lemma G4. Domain Size of a Bounded Map
153(1)
6.9 Lemma G5. Frequency Variation Lemma
154(7)
Chapter 7 Convergence Lemmas
161(14)
7.1 Lemma C1. Iterated Canonical Transformation
161(2)
7.2 Lemma C2. Integral Curves and Segments
163(1)
7.3 The Push Forward
164(5)
7.4 Lemma C3. Limiting Flow
169(2)
7.5 Lemma C4. Measure of the Limit
171(4)
Chapter 8 Arithmetic Lemmas
175(10)
8.1 Lemma D1. Measure of Boundary Layer
175(1)
8.2 Lemma D2. Measure of Inner Layer
176(1)
8.3 Lemma D3. Measure of Strip
177(1)
8.4 Lemma D4. Measure of Layers and Strips
178(2)
8.5 Lemma D5. Integral Vectors with Given Norm
180(2)
8.6 Lemma D6. Integral Vectors below Given Norm
182(1)
8.7 Lemma D7. Measure of Lost Domain
182(3)
References 185(8)
Person Index 193(4)
Subject Index 197
Author

Achim Feldmeier is a professor at Universität Potsdam, Germany.
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