Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows

We show that certain mechanical systems, including a geodesic °ow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy. The assumptions we make in the case of geodesic °ows are: a) The metric and the external perturbation are smooth enough. b) The geode...

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Detalles Bibliográficos
Autores: Delshams Valdés, Amadeu|||0000-0003-4134-8882, Llave Canosa, Rafael de la, Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
Tipo de recurso: artículo
Fecha de publicación:2003
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:2117/1204
Acceso en línea:https://hdl.handle.net/2117/1204
Access Level:acceso abierto
Palabra clave:Hamiltonian dynamical systems
Lagrangian functions
Differential geometry
Hamiltonian systems
Differentiable dynamical systems
orbits
quasi-periodic perturbations
Hamilton, Sistemes de
Lagrange, Funcions de
Geometria diferencial
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior
Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
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repository_id_str
spelling Orbits of unbounded energy in quasi-periodic perturbations of geodesic flowsDelshams Valdés, Amadeu|||0000-0003-4134-8882Llave Canosa, Rafael de laMartínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717Hamiltonian dynamical systemsLagrangian functionsDifferential geometryHamiltonian systemsDifferentiable dynamical systemsorbitsquasi-periodic perturbationsHamilton, Sistemes deLagrange, Funcions deGeometria diferencialSistemes dinàmics diferenciablesClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systemsClassificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behaviorClassificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometryClassificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanicsWe show that certain mechanical systems, including a geodesic °ow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy. The assumptions we make in the case of geodesic °ows are: a) The metric and the external perturbation are smooth enough. b) The geodesic °ow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection. c) The frequency of the external perturbation is Diophantine. d) The external potential satisØes a generic condition depending on the periodic orbit considered in b). The assumptions on the metric are C2 open and are known to be dense on many manifolds. The assumptions on the potential fail only in inØnite codimension spaces of potentials. The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques). We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension.20032003-01-0120072007-10-01journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/1204reponame: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:2117/12042026-05-27T15:37:01Z
dc.title.none.fl_str_mv Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
title Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
spellingShingle Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
Delshams Valdés, Amadeu|||0000-0003-4134-8882
Hamiltonian dynamical systems
Lagrangian functions
Differential geometry
Hamiltonian systems
Differentiable dynamical systems
orbits
quasi-periodic perturbations
Hamilton, Sistemes de
Lagrange, Funcions de
Geometria diferencial
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior
Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
title_short Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
title_full Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
title_fullStr Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
title_full_unstemmed Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
title_sort Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
dc.creator.none.fl_str_mv Delshams Valdés, Amadeu|||0000-0003-4134-8882
Llave Canosa, Rafael de la
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author Delshams Valdés, Amadeu|||0000-0003-4134-8882
author_facet Delshams Valdés, Amadeu|||0000-0003-4134-8882
Llave Canosa, Rafael de la
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author_role author
author2 Llave Canosa, Rafael de la
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author2_role author
author
dc.subject.none.fl_str_mv Hamiltonian dynamical systems
Lagrangian functions
Differential geometry
Hamiltonian systems
Differentiable dynamical systems
orbits
quasi-periodic perturbations
Hamilton, Sistemes de
Lagrange, Funcions de
Geometria diferencial
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior
Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
topic Hamiltonian dynamical systems
Lagrangian functions
Differential geometry
Hamiltonian systems
Differentiable dynamical systems
orbits
quasi-periodic perturbations
Hamilton, Sistemes de
Lagrange, Funcions de
Geometria diferencial
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior
Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
description We show that certain mechanical systems, including a geodesic °ow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy. The assumptions we make in the case of geodesic °ows are: a) The metric and the external perturbation are smooth enough. b) The geodesic °ow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection. c) The frequency of the external perturbation is Diophantine. d) The external potential satisØes a generic condition depending on the periodic orbit considered in b). The assumptions on the metric are C2 open and are known to be dense on many manifolds. The assumptions on the potential fail only in inØnite codimension spaces of potentials. The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques). We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension.
publishDate 2003
dc.date.none.fl_str_mv 2003
2003-01-01
2007
2007-10-01
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/2117/1204
url https://hdl.handle.net/2117/1204
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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score 15,301603

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