Normal Form and Energy Conservation of High Frequency Subsystems Without Nonresonance Conditions

We consider a system in which some high frequency harmonic oscillators are coupled with a slow system. We prove that up to very long times the energy of the high frequency system changes only by a small amount. The result we obtain is completely independent of the resonance relations among the frequencies of the fast system. More in detail, denote by $\epsilon^{-1}$ the smallest high frequency. In the first part of the paper we apply the main result of [BG93] to prove almost conservation of the energy of the high frequency system over times exponentially long with ${\epsilon^{-1/n}} $ ($n$ being the number of fast oscillators). In the second part of the paper we give e new self-contained proof of a similar result which however is valid only over times of order $\epsilon^{-N}$ with an arbitrary $N$. Such a second result is very similar to the main result of the paper [GHL13], which actually was the paper which stimulated our work.