Energy separation in oscillatory Hamiltonian systems without any non-resonance condition

TL;DR

This paper proves that in multiscale Hamiltonian systems with multiple high frequencies, the oscillatory energy remains nearly conserved over very long times without needing non-resonance conditions, regardless of frequency choices.

Contribution

It demonstrates long-term near conservation of oscillatory energy in multiscale Hamiltonian systems without non-resonance assumptions, uniform across frequencies.

Findings

Oscillatory energy nearly preserved over long timescales

Results are uniform in high frequencies

No non-resonance conditions required

Abstract

We consider multiscale Hamiltonian systems in which harmonic oscillators with several high frequencies are coupled to a slow system. It is shown that the oscillatory energy is nearly preserved over long times eps^{-N} for arbitrary N>1, where eps^{-1} is the size of the smallest high frequency. The result is uniform in the frequencies and does not require non-resonance conditions.