Computer-assisted estimates for birkhoff normal forms
| Autor: | Chiara Caracciolo, Ugo Locatelli |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Equilibrium point
Celestial Mechanics Computational Mechanics FOS: Physical sciences Motion (geometry) normal form methods Mathematical Physics (math-ph) effective stability Stability (probability) Celestial mechanics Primary: 68V05 Secondary: 37J40 37N05 70F07 70H08 Hamiltonian system Computational Mathematics Simple (abstract algebra) Settore MAT/07 Bounded function Applied mathematics Computer-assisted proofs normal form methods effective stability Hamiltonian systems Celestial Mechanics Hamiltonian systems Focus (optics) Computer-assisted proofs Mathematical Physics Mathematics |
| Popis: | Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the H\'enon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones. |
| Databáze: | OpenAIRE |
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