Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system
We give a precise statement for KAM theorem in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system. If the frequencies of the elliptic point are nonresonant up to a certain order $K\ge4$, and a nondegeneracy condition is fulfilled, we get an estimate for the measure of the compl...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 1996 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/832 |
| Acceso en línea: | https://hdl.handle.net/2117/832 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations Global analysis (Mathematics) Hamiltonian dynamical systems Lagrangian functions invariant tor elliptic equilibrium point Hamiltonian system Equacions diferencials ordinàries Varietats (Matemàtica) Hamilton, Sistemes de Lagrange, Funcions de Classificació AMS::58 Global analysis, analysis on manifolds Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics |
| Sumario: | We give a precise statement for KAM theorem in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system. If the frequencies of the elliptic point are nonresonant up to a certain order $K\ge4$, and a nondegeneracy condition is fulfilled, we get an estimate for the measure of the complement of the KAM tori in a neighbourhood of given radius. Moreover, if the frequencies satisfy a Diophantine condition, with exponent $\tau$, we show that in a neighbourhood of radius $r$ the measure of the complement is exponentially small in $(1/r)^{1/(\tau+1)}$. We also give a related result for quasi-Diophantine frequencies, which is more useful for practical purposes. The results are obtained by putting the system in Birkhoff normal form up to an appropiate order, and the key point relies on giving accurate bounds for its terms. |
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