A Kolmogorov theorem for nearly-integrable Poisson systems with asymptotically decaying time-dependent perturbation

TL;DR

This paper extends Kolmogorov's theorem to nearly-integrable Poisson systems with aperiodic, asymptotically vanishing time-dependent perturbations, demonstrating the persistence of Diophantine flows under these conditions.

Contribution

It generalizes previous results for canonical Hamiltonian systems to Poisson systems using Lie series methods, accommodating asymptotically decaying time-dependent perturbations.

Findings

Proves persistence of Diophantine flows in Poisson systems with time-dependent perturbations.

Extends Kolmogorov theorem to a broader class of dynamical systems.

Utilizes Lie series method for the generalization.

Abstract

The aim of this paper is to prove the Kolmogorov theorem of persistence of Diophantine flows for nearly-integrable Poisson systems associated to a real analytic Hamiltonian with aperiodic time dependence, provided that the perturbation is asymptotically vanishing. The paper is an extension of an analogous result by the same authors for canonical Hamiltonian systems; the flexibility of the Lie series method developed by A. Giorgilli et al., is profitably used in the present generalisation.