Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures

Autores
Grillo, Sergio Daniel; Marrero, Juan Carlos; Padrón, Edith
Año de publicación
2021
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that π(U) has a manifold structure and π|U:U→π(U), the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If X|U is not vertical with respect to π|U, we show that such complete solutions solve the reconstruction equations related to X|U and G, i.e., the equations that enable us to write the integral curves of X|U in terms of those of its projection on π(U). On the other hand, if X|U is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of X|U up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve exp(ξt), different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of exp(ξt) is valid for all ξ inside an open dense subset of g.
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. Universidad Nacional de Cuyo; Argentina
Fil: Marrero, Juan Carlos. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; España
Fil: Padrón, Edith. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; España
Materia
Hamilton–Jacobi Theory
Lie group
Symplectic geometry
Integrability by quadratures
First integrals
Quadratures
Reconstruction
Lie group exponential map
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/157388

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spelling Extended Hamilton-Jacobi Theory, Symmetries and Integrability by QuadraturesGrillo, Sergio DanielMarrero, Juan CarlosPadrón, EdithHamilton–Jacobi TheoryLie groupSymplectic geometryIntegrability by quadraturesFirst integralsQuadraturesReconstructionLie group exponential maphttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that π(U) has a manifold structure and π|U:U→π(U), the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If X|U is not vertical with respect to π|U, we show that such complete solutions solve the reconstruction equations related to X|U and G, i.e., the equations that enable us to write the integral curves of X|U in terms of those of its projection on π(U). On the other hand, if X|U is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of X|U up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve exp(ξt), different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of exp(ξt) is valid for all ξ inside an open dense subset of g.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. Universidad Nacional de Cuyo; ArgentinaFil: Marrero, Juan Carlos. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; EspañaFil: Padrón, Edith. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; EspañaMultidisciplinary Digital Publishing Institute2021-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/157388Grillo, Sergio Daniel; Marrero, Juan Carlos; Padrón, Edith; Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures; Multidisciplinary Digital Publishing Institute; Mathematics; 9; 12; 6-2021; 1-342227-7390CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.mdpi.com/2227-7390/9/12/1357info:eu-repo/semantics/altIdentifier/doi/10.3390/math9121357info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/2105.02130info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:09:18Zoai:ri.conicet.gov.ar:11336/157388instacron: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 10:09:18.596CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
title Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
spellingShingle Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
Grillo, Sergio Daniel
Hamilton–Jacobi Theory
Lie group
Symplectic geometry
Integrability by quadratures
First integrals
Quadratures
Reconstruction
Lie group exponential map
title_short Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
title_full Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
title_fullStr Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
title_full_unstemmed Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
title_sort Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures
dc.creator.none.fl_str_mv Grillo, Sergio Daniel
Marrero, Juan Carlos
Padrón, Edith
author Grillo, Sergio Daniel
author_facet Grillo, Sergio Daniel
Marrero, Juan Carlos
Padrón, Edith
author_role author
author2 Marrero, Juan Carlos
Padrón, Edith
author2_role author
author
dc.subject.none.fl_str_mv Hamilton–Jacobi Theory
Lie group
Symplectic geometry
Integrability by quadratures
First integrals
Quadratures
Reconstruction
Lie group exponential map
topic Hamilton–Jacobi Theory
Lie group
Symplectic geometry
Integrability by quadratures
First integrals
Quadratures
Reconstruction
Lie group exponential map
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 study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that π(U) has a manifold structure and π|U:U→π(U), the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If X|U is not vertical with respect to π|U, we show that such complete solutions solve the reconstruction equations related to X|U and G, i.e., the equations that enable us to write the integral curves of X|U in terms of those of its projection on π(U). On the other hand, if X|U is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of X|U up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve exp(ξt), different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of exp(ξt) is valid for all ξ inside an open dense subset of g.
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. Universidad Nacional de Cuyo; Argentina
Fil: Marrero, Juan Carlos. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; España
Fil: Padrón, Edith. Universidad de La Laguna; España. Consejo Superior de Investigaciones Científicas; España
description In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that π(U) has a manifold structure and π|U:U→π(U), the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If X|U is not vertical with respect to π|U, we show that such complete solutions solve the reconstruction equations related to X|U and G, i.e., the equations that enable us to write the integral curves of X|U in terms of those of its projection on π(U). On the other hand, if X|U is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of X|U up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve exp(ξt), different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of exp(ξt) is valid for all ξ inside an open dense subset of g.
publishDate 2021
dc.date.none.fl_str_mv 2021-06
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/11336/157388
Grillo, Sergio Daniel; Marrero, Juan Carlos; Padrón, Edith; Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures; Multidisciplinary Digital Publishing Institute; Mathematics; 9; 12; 6-2021; 1-34
2227-7390
CONICET Digital
CONICET
url http://hdl.handle.net/11336/157388
identifier_str_mv Grillo, Sergio Daniel; Marrero, Juan Carlos; Padrón, Edith; Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures; Multidisciplinary Digital Publishing Institute; Mathematics; 9; 12; 6-2021; 1-34
2227-7390
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.mdpi.com/2227-7390/9/12/1357
info:eu-repo/semantics/altIdentifier/doi/10.3390/math9121357
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/2105.02130
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Multidisciplinary Digital Publishing Institute
publisher.none.fl_str_mv Multidisciplinary Digital Publishing Institute
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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