What books should I read to fully understand KAM theory?

[Andrea85]

Hi all, I've read the last capters of Arnold's "mathematical methods of classic mechanics" (so I know the basic facts about differential and symplectic manifolds and the Liouville-Arnold theorem) but this didn't give me enough background to go through KAM theorem: I don't know which is a "resonant" torus, I have not clear in which sense periodic orbits are dense on the torus, so I really can't properly understand this theorem. What should I read further? Which are the best books providing the necessary background to master KAM theory?

 

[Valerie Park]

I recommend the book "KAM and Foliation Theory" by Jürgen Moser. It provides a comprehensive overview of KAM theory and is written in an accessible style. Additionally, I would suggest looking at the lecture notes from the course "Topics in Hamiltonian Dynamics" by Serge Tabachnikov, which covers many of the same topics in a more tutorial style.

 

[charng]

To fully understand KAM theory, I recommend reading the following books:

1. "Introduction to KAM Theory" by Antonio Delshams, Rafael de la Llave, and Tere M. Seara. This book provides a comprehensive introduction to KAM theory, starting from the basics of classical mechanics and differential equations. It covers the main concepts and results of KAM theory, including resonant tori and the density of periodic orbits.

2. "Dynamics in One Non-Autonomous Dimension" by Vassili Gelfreich and Carles Simó. This book focuses on the one-dimensional case of KAM theory, which is a good starting point for understanding the more general case. It also includes many examples and exercises to help deepen your understanding.

3. "Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions" by Darryl D. Holm, Jerrold E. Marsden, and Tudor S. Ratiu. This book provides a more advanced treatment of KAM theory, using the language of symplectic and Poisson geometry. It also covers topics such as integrable systems and Hamiltonian dynamics, which are closely related to KAM theory.

In addition, I recommend studying the original papers by Kolmogorov, Arnold, and Moser, who first developed KAM theory. These can be found in the book "Selected Works of A. N. Kolmogorov: Volume III: Dynamical Systems and Mechanics" and "Selected Works of V. I. Arnold: Volume III: Topology and Hamiltonian Mechanics."

Overall, mastering KAM theory requires a strong background in classical mechanics, differential equations, and symplectic geometry. So, I suggest also reading books on these topics to supplement your understanding. Good luck with your studies!