低维环面的KAM理论

Chapter 1 Hamiltonian Mechanics

1．1 The Basic Concepts Of Dynamical System

1．2 Differential Form

Chapter 2 Hamiltonian System

2．1 Classical Hamiltonian and Generalized Hamiltonian

2．2 Completely Integrable Systems and Nearly Integrable Systems

Chapter 3 KAM Theorem

3．1 The Classical KAM Theorem

3．2 Lower-Dimensional KAM Theorem

3．3 The Generalized KAM Theorem

Chapter 4 Lower-Dimensional Invariant Tori in Hamiltonian Svstem

4．1 Introduction

4．2 KAM Step

4．3 Iteration Lemma

4．4 Proof of the Main Result

4．5 Example

Chapter 5 KAM-Type Theorem with a Quasi-Periodic Perturbation

5．1 Introduction

5．2 The KAM Step

5．3 Iteration Lemma

5．4 Proof of the Main Result

5．5 Some Examples

Chapter 6 The Persistence of Lower-Dimensional Tori for Generalized Hamiltonian

6．1 Introduction and Main Result

6．2 KAM Step

6．3 Iteration Lemma

6．4 Proof of the Main Result

Chapter 7 Hyperbolic Invariant Tori for Generalized Hamiltonian Systems

7．1 Introduction and Main Result

7．2 KAM Step

7．3 Iteration Lemma

7．4 Proof of the Main Result

7．5 Examples

Chapter 8 The Hyperbolic Invariant Tori of the Symplectic Mappings

8．1 Introduction

8．2 Outline of the Proof of Theorem 8．1．1

8．3 Estimates in One Step

8．4 Iteration Lemma

8．5 Proof of the Main Result

Chapter 9 Appendix

9．1 Some Facts About Analytic Functions

9．2 Technical Lemma

Bibliography

《KAM Theory for Lower-Dimensional Tori》讨论了在经典的KAM理论中几个没有讨论的问题。主要介绍KAM理论中的低维不变环面的保持性。介绍了广义Hamilton系统双曲型低维不变环面和具有一般形式的广义Hamilton系统的低维不变环面的保持性。介绍了具有拟周期摄动的近可积Hamilton系统的KAM理论。介绍了具有相交性小扭转的辛映射的不变环面的保持性。介绍了辛映射的双曲型低维不变环面的保持性定理。