A-posteriori KAM theory with optimal estimates for partially integrable systems

TL;DR

This paper develops a covariant a-posteriori KAM theory for partially integrable Hamiltonian systems, providing optimal estimates and avoiding action-angle variables, facilitating numerical and computer-assisted proofs.

Contribution

It introduces a covariant formulation of KAM theory for partially integrable systems that bypasses the need for action-angle variables and symplectic reduction.

Findings

Proves existence of invariant tori in partially integrable systems.

Provides a covariant approach suitable for numerical computations.

Includes both ordinary and iso-energetic KAM theorems.

Abstract

In this paper we present a-posteriori KAM results for existence of $d$-dimensional isotropic invariant tori for $n$-DOF Hamiltonian systems with additional $n-d$ independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.