On Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes

TL;DR

This paper extends the averaging method for Hamiltonian systems with one fast phase and small amplitudes, making it applicable in non-analytic neighborhoods with uniform convergence.

Contribution

It introduces a transformation that allows Neishtadt's averaging procedure to be used in non-analytic regions with uniform limits.

Findings

A new transformation enabling averaging in non-analytic neighborhoods

Uniform convergence of the averaging process in these neighborhoods

Extension of existing averaging methods to broader classes of Hamiltonian systems

Abstract

The problem of averaging for systems with one fast phase was considered from various points of view in many papers. The averaging method of Krylov and Bogolyubov and methods of KAM theory originated this line of research, the most complete results were obtained by Neishtadt, where the coefficients are assumed real analytic. However, in many problems which are interesting from the point of view of applications, analytic dependence ceases to hold in the neighborhoods of some points. We show that one can choose a transformation such that the averaging procedure of Neishtadt is applicable in these neighborhoods and such passage to limit is uniform.