General KAM theorems and their applications to invariant tori with prescribed frequencies

TL;DR

This paper introduces new KAM theorems applicable to nearly integrable Hamiltonian systems without non-degeneracy assumptions, enabling the construction of invariant tori with prescribed frequencies.

Contribution

The authors develop general KAM theorems that do not require non-degeneracy, extending classical results and allowing for prescribed frequency invariant tori.

Findings

New KAM theorems applicable without non-degeneracy conditions

Classical KAM results are special cases of these theorems

Results on invariant tori with prescribed frequencies

Abstract

In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some non-degeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.