Parametrized KAM Theorem for Differentiable Hamiltonian Vector Fields without Action-Angle Variables

TL;DR

This paper extends the Kolmogorov-Arnold-Moser (KAM) theorem to differentiable Hamiltonian systems without relying on action-angle variables, broadening its applicability beyond real analytic cases.

Contribution

It generalizes the KAM theorem to differentiable Hamiltonians without action-angle coordinates, expanding the theoretical framework for Hamiltonian systems.

Findings

Proved a parametrized KAM theorem for differentiable Hamiltonians

Extended KAM results beyond real analytic Hamiltonians

Provided a new approach for Hamiltonian systems without action-angle variables

Abstract

We proved a parametrized KAM theorem in Hamiltonian system which has differentiable Hamiltonian without action-angle coordinates. It is a generalization of the result of [Llave et al. 2005] that deals with real analytic Hamiltonians.