Understanding KAM Theorem: What to Read?

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In summary, for those looking to understand KAM theory, it is important to have a strong understanding of differential and symplectic manifolds, as well as the Liouville-Arnold theorem. It is also important to have a good grasp on the concepts of resonant tori and periodic orbits. Some recommended books for mastering KAM theory include "Dynamical Systems and KAM Theory" by Yuri Kifer, "Classical Mechanics: An Introduction" by David Morin, and "Hamiltonian Dynamical Systems" by Morris W. Hirsch.
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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?
 
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For a good introduction to KAM theory, I would recommend looking at the book "Dynamical Systems and KAM Theory" by Yuri Kifer. This book provides a comprehensive introduction to both the theory of dynamical systems and KAM theory, as well as providing examples and applications. Additionally, there are many other books that deal with this subject, such as "Classical Mechanics: An Introduction" by David Morin and "Hamiltonian Dynamical Systems" by Morris W. Hirsch. These books provide a more in-depth look at the topics of KAM theory and Hamiltonian dynamics.
 

1. What is the KAM theorem?

The KAM theorem, or Kolmogorov-Arnold-Moser theorem, is a mathematical theorem that explains the behavior of certain dynamical systems, particularly those with multiple degrees of freedom. It states that in these systems, near a certain type of invariant torus, there exist quasi-periodic motions that are stable under small perturbations.

2. How does the KAM theorem relate to Hamiltonian systems?

The KAM theorem specifically applies to Hamiltonian systems, which are systems that describe the motion of particles under the influence of conservative forces. The theorem explains the existence of stable quasi-periodic orbits in these systems, even when they are perturbed.

3. What is the significance of the KAM theorem?

The KAM theorem has significant implications in both mathematics and physics. It provides a deeper understanding of the behavior of dynamical systems, and has been used to prove the stability of the solar system. It also has applications in fields such as fluid mechanics and statistical mechanics.

4. What are some recommended readings for understanding the KAM theorem?

There are many resources available for understanding the KAM theorem, including books, articles, and lectures. Some recommended readings include "The KAM Story" by Hans Koch, "Dynamics of Small Solar System Bodies and Exoplanets" by Jürgen Blum and John A. Burns, and "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem" by Kenneth R. Meyer and Glen R. Hall.

5. Are there any current developments or controversies surrounding the KAM theorem?

While the KAM theorem has been widely accepted and used in various fields, there are some ongoing debates and developments surrounding it. Some researchers are exploring extensions and generalizations of the theorem, while others are studying its limitations and potential failures. There is also ongoing research on the numerical implementation and applications of the KAM theorem.

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