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...

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Autores: Delshams Valdés, Amadeu|||0000-0003-4134-8882, Gutiérrez Serrés, Pere|||0000-0001-8027-1166
Formato: artículo
Fecha de publicación:1997
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/963
Acesso em linha:https://hdl.handle.net/2117/963
Access Level:acceso abierto
Palavra-chave: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
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spelling Exponentially small estimates for KAM theorem near an elliptic equilibrium pointDelshams Valdés, Amadeu|||0000-0003-4134-8882Gutiérrez Serrés, Pere|||0000-0001-8027-1166Hamiltonian systemsHamiltonian dynamical systemsLagrangian functionsKAM theoremelliptic equilibrium pointHamilton, Sistemes deLagrange, Funcions deClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systemsClassificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanicsWe 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.19971997-01-0120072007-05-09journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/963reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/9632026-05-27T15:37:01Z
dc.title.none.fl_str_mv Exponentially small estimates for KAM theorem near an elliptic equilibrium point
title Exponentially small estimates for KAM theorem near an elliptic equilibrium point
spellingShingle Exponentially small estimates for KAM theorem near an elliptic equilibrium point
Delshams Valdés, Amadeu|||0000-0003-4134-8882
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
title_short Exponentially small estimates for KAM theorem near an elliptic equilibrium point
title_full Exponentially small estimates for KAM theorem near an elliptic equilibrium point
title_fullStr Exponentially small estimates for KAM theorem near an elliptic equilibrium point
title_full_unstemmed Exponentially small estimates for KAM theorem near an elliptic equilibrium point
title_sort Exponentially small estimates for KAM theorem near an elliptic equilibrium point
dc.creator.none.fl_str_mv Delshams Valdés, Amadeu|||0000-0003-4134-8882
Gutiérrez Serrés, Pere|||0000-0001-8027-1166
author Delshams Valdés, Amadeu|||0000-0003-4134-8882
author_facet Delshams Valdés, Amadeu|||0000-0003-4134-8882
Gutiérrez Serrés, Pere|||0000-0001-8027-1166
author_role author
author2 Gutiérrez Serrés, Pere|||0000-0001-8027-1166
author2_role author
dc.subject.none.fl_str_mv 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
topic 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
description 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.
publishDate 1997
dc.date.none.fl_str_mv 1997
1997-01-01
2007
2007-05-09
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/963
url https://hdl.handle.net/2117/963
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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