Exponentially small estimates for KAM theorem near an elliptic equilibrium point
We give a precise statement of KAM theorem for a Hamiltonian system in a neighborhood of an elliptic equilibrium point. If the frequencies of the elliptic point satisfy a Diophantine condition, with exponent $\tau$, and a nondegeneracy condition is fulfilled, we show that in a neighborhood of radius...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 1997 |
| 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:2117/963 |
| Acceso en línea: | https://hdl.handle.net/2117/963 |
| Access Level: | acceso abierto |
| Palabra clave: | Hamiltonian systems Hamiltonian dynamical systems Lagrangian functions KAM theorem elliptic equilibrium point Hamilton, Sistemes de Lagrange, Funcions de Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics |
| Sumario: | We give a precise statement of KAM theorem for a Hamiltonian system in a neighborhood of an elliptic equilibrium point. If the frequencies of the elliptic point satisfy a Diophantine condition, with exponent $\tau$, and a nondegeneracy condition is fulfilled, we show that in a neighborhood of radius $r$ the measure of the complement of the KAM tori is exponentially small in $(1/r)^{1/(\tau+1)}$. This result is obtained by putting the system in Birkhoff normal form up to an appropriate order, and the key point relies on giving accurate estimates for its terms. |
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