A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds

Autores
Grillo, Sergio Daniel; Padrón, Edith
Año de publicación
2016
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper we develop, in a geometric framework, a Hamilton–Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton–Jacobi Theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton–Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is established. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.
Fil: Grillo, Sergio Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina
Fil: Padrón, Edith. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; España
Materia
HAMILTON–JACOBI EQUATIONS
INTEGRABLE SYSTEMS
POISSON MANIFOLD
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/78115

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spelling A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifoldsGrillo, Sergio DanielPadrón, EdithHAMILTON–JACOBI EQUATIONSINTEGRABLE SYSTEMSPOISSON MANIFOLDhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper we develop, in a geometric framework, a Hamilton–Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton–Jacobi Theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton–Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is established. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.Fil: Grillo, Sergio Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; ArgentinaFil: Padrón, Edith. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; EspañaElsevier Science2016-12-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/78115Grillo, Sergio Daniel; Padrón, Edith; A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds; Elsevier Science; Journal Of Geometry And Physics; 110; 1-12-2016; 101-1290393-0440CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.geomphys.2016.07.010info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0393044016301760info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1512.03121info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:14:02Zoai:ri.conicet.gov.ar:11336/78115instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:14:02.654CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
title A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
spellingShingle A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
Grillo, Sergio Daniel
HAMILTON–JACOBI EQUATIONS
INTEGRABLE SYSTEMS
POISSON MANIFOLD
title_short A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
title_full A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
title_fullStr A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
title_full_unstemmed A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
title_sort A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
dc.creator.none.fl_str_mv Grillo, Sergio Daniel
Padrón, Edith
author Grillo, Sergio Daniel
author_facet Grillo, Sergio Daniel
Padrón, Edith
author_role author
author2 Padrón, Edith
author2_role author
dc.subject.none.fl_str_mv HAMILTON–JACOBI EQUATIONS
INTEGRABLE SYSTEMS
POISSON MANIFOLD
topic HAMILTON–JACOBI EQUATIONS
INTEGRABLE SYSTEMS
POISSON MANIFOLD
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this paper we develop, in a geometric framework, a Hamilton–Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton–Jacobi Theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton–Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is established. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.
Fil: Grillo, Sergio Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina
Fil: Padrón, Edith. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; España
description In this paper we develop, in a geometric framework, a Hamilton–Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton–Jacobi Theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton–Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is established. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.
publishDate 2016
dc.date.none.fl_str_mv 2016-12-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/78115
Grillo, Sergio Daniel; Padrón, Edith; A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds; Elsevier Science; Journal Of Geometry And Physics; 110; 1-12-2016; 101-129
0393-0440
CONICET Digital
CONICET
url http://hdl.handle.net/11336/78115
identifier_str_mv Grillo, Sergio Daniel; Padrón, Edith; A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds; Elsevier Science; Journal Of Geometry And Physics; 110; 1-12-2016; 101-129
0393-0440
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.geomphys.2016.07.010
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0393044016301760
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1512.03121
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier Science
publisher.none.fl_str_mv Elsevier Science
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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