Geometry of KAM tori for nearly integrable Hamiltonian systems

TL;DR

This paper develops a global Hamiltonian KAM theorem for nearly integrable systems, enabling the preservation of geometric invariants and providing a smooth conjugacy between near and exact integrable systems.

Contribution

It introduces a method to extend local KAM conjugacies into a global Whitney smooth conjugacy, preserving geometric invariants in nearly integrable Hamiltonian systems.

Findings

Established a global Whitney smooth conjugacy for nearly integrable systems.

Preserved geometric invariants such as monodromy and Chern classes.

Unified local and global KAM results for Hamiltonian systems.

Abstract

We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly-integrable system and an integrable one. This leads to preservation of geometry, which allows us to define all the nontrivial geometric invariants like monodromy or Chern classes of an integrable system also for near integrable systems.