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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/157388
Ver los metadatos del registro completo
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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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1844613970100486144 |
score |
13.070432 |