Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

TL;DR

This paper demonstrates that small divisor effects are negligible in normal form theory for nearly-integrable Hamiltonians with decaying aperiodic time-dependent perturbations, simplifying the normalization process.

Contribution

It extends normalization techniques to a broader class of Hamiltonians without requiring geometric conditions, using adapted perturbative methods.

Findings

Normal form exists regardless of frequency vector properties.

Decaying aperiodic perturbations allow for exact normalization.

No geometric assumptions needed beyond analyticity.

Abstract

The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called "geometric part" is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by A.Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems.