Singular KAM theory for convex Hamiltonian systems

TL;DR

This paper extends singular KAM theory to convex real analytic nearly integrable Hamiltonian systems, broadening its applicability beyond the mechanical case to more general convex Hamiltonians.

Contribution

It introduces an extension of singular KAM theory to convex Hamiltonian systems in action-angle variables, generalizing previous work focused on mechanical systems.

Findings

Extended singular KAM theory to convex Hamiltonians.

Applicable to a broader class of nearly integrable systems.

Provides theoretical foundation for future research in Hamiltonian dynamics.

Abstract

In this note, we briefly discuss how singular KAM Theory - which was worked out in a previous work by L.B. and L.C. for the mechanical case $\frac12 |y|^2+\varepsilon f(x)$ - can be extended to convex real analytic nearly integrable Hamiltonian systems with Hamiltonian in action-angle variables given by $h(y)+\varepsilon f(x)$ with $h$ convex and generic $f$.