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.

Normal Form and Energy Conservation of High Frequency Subsystems Without Nonresonance Conditions / D. Bambusi, A. Giorgilli, S. Paleari, T. Penati. - In: RENDICONTI. CLASSE DI SCIENZE MATEMATICHE E NATURALI. - ISSN 1974-6989. - (2014 Feb 01). [Epub ahead of print]

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

D. Bambusi;A. Giorgilli;S. Paleari
Penultimo
;
T. Penati
2014

Abstract

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.
Settore MAT/07 - Fisica Matematica
http://arxiv.org/abs/1402.0076
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/230245
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