On the Geometry of the Hamilton-Jacobi Equation and Generating Functions

Autores
Ferraro, Sebastián José; de León, Manuel; Marrero, Juan Carlos; Martin de Diego, David; Vaquero, Miguel
Año de publicación
2017
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80-88, 1991), Ge (Indiana Univ. Math. J. 39(3):859-876, 1990), Ge and Marsden (Phys Lett A 133(3):134-139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.
Fil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina
Fil: de León, Manuel. Consejo Superior de Investigaciones Científicas; España. Instituto de Ciencias Matemáticas; España
Fil: Marrero, Juan Carlos. Universidad de La Laguna; España
Fil: Martin de Diego, David. Instituto de Ciencias Matemáticas; España. Consejo Superior de Investigaciones Científicas; España
Fil: Vaquero, Miguel. Consejo Superior de Investigaciones Científicas; España. Instituto de Ciencias Matemáticas; España
Materia
HAMILTON-JACOBI THEORY
SYMPLECTIC GROUPOIDS
LAGRANGIAN SUBMANIFOLDS
SYMMETRIES
POISSON MANIFOLDS
POISSON INTEGRATORS
GENERATING FUNCTIONS
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:11336/43043
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/43043
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling On the Geometry of the Hamilton-Jacobi Equation and Generating FunctionsFerraro, Sebastián Joséde León, ManuelMarrero, Juan CarlosMartin de Diego, DavidVaquero, MiguelHAMILTON-JACOBI THEORYSYMPLECTIC GROUPOIDSLAGRANGIAN SUBMANIFOLDSSYMMETRIESPOISSON MANIFOLDSPOISSON INTEGRATORSGENERATING FUNCTIONShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80-88, 1991), Ge (Indiana Univ. Math. J. 39(3):859-876, 1990), Ge and Marsden (Phys Lett A 133(3):134-139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.Fil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: de León, Manuel. Consejo Superior de Investigaciones Científicas; España. Instituto de Ciencias Matemáticas; EspañaFil: Marrero, Juan Carlos. Universidad de La Laguna; EspañaFil: Martin de Diego, David. Instituto de Ciencias Matemáticas; España. Consejo Superior de Investigaciones Científicas; EspañaFil: Vaquero, Miguel. Consejo Superior de Investigaciones Científicas; España. Instituto de Ciencias Matemáticas; EspañaSpringer2017-10-09info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/43043Ferraro, Sebastián José; de León, Manuel; Marrero, Juan Carlos; Martin de Diego, David; Vaquero, Miguel; On the Geometry of the Hamilton-Jacobi Equation and Generating Functions; Springer; Archive For Rational Mechanics And Analysis; 226; 1; 9-10-2017; 243-3020003-9527CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s00205-017-1133-0info:eu-repo/semantics/altIdentifier/doi/10.1007/s00205-017-1133-0info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1606.00847info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:47:45Zoai:ri.conicet.gov.ar:11336/43043instacron: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 10:47:45.367CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
title On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
spellingShingle On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
Ferraro, Sebastián José
HAMILTON-JACOBI THEORY
SYMPLECTIC GROUPOIDS
LAGRANGIAN SUBMANIFOLDS
SYMMETRIES
POISSON MANIFOLDS
POISSON INTEGRATORS
GENERATING FUNCTIONS
title_short On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
title_full On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
title_fullStr On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
title_full_unstemmed On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
title_sort On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
dc.creator.none.fl_str_mv Ferraro, Sebastián José
de León, Manuel
Marrero, Juan Carlos
Martin de Diego, David
Vaquero, Miguel
author Ferraro, Sebastián José
author_facet Ferraro, Sebastián José
de León, Manuel
Marrero, Juan Carlos
Martin de Diego, David
Vaquero, Miguel
author_role author
author2 de León, Manuel
Marrero, Juan Carlos
Martin de Diego, David
Vaquero, Miguel
author2_role author
author
author
author
dc.subject.none.fl_str_mv HAMILTON-JACOBI THEORY
SYMPLECTIC GROUPOIDS
LAGRANGIAN SUBMANIFOLDS
SYMMETRIES
POISSON MANIFOLDS
POISSON INTEGRATORS
GENERATING FUNCTIONS
topic HAMILTON-JACOBI THEORY
SYMPLECTIC GROUPOIDS
LAGRANGIAN SUBMANIFOLDS
SYMMETRIES
POISSON MANIFOLDS
POISSON INTEGRATORS
GENERATING FUNCTIONS
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 a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80-88, 1991), Ge (Indiana Univ. Math. J. 39(3):859-876, 1990), Ge and Marsden (Phys Lett A 133(3):134-139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.
Fil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina
Fil: de León, Manuel. Consejo Superior de Investigaciones Científicas; España. Instituto de Ciencias Matemáticas; España
Fil: Marrero, Juan Carlos. Universidad de La Laguna; España
Fil: Martin de Diego, David. Instituto de Ciencias Matemáticas; España. Consejo Superior de Investigaciones Científicas; España
Fil: Vaquero, Miguel. Consejo Superior de Investigaciones Científicas; España. Instituto de Ciencias Matemáticas; España
description In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80-88, 1991), Ge (Indiana Univ. Math. J. 39(3):859-876, 1990), Ge and Marsden (Phys Lett A 133(3):134-139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.
publishDate 2017
dc.date.none.fl_str_mv 2017-10-09
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/11336/43043
Ferraro, Sebastián José; de León, Manuel; Marrero, Juan Carlos; Martin de Diego, David; Vaquero, Miguel; On the Geometry of the Hamilton-Jacobi Equation and Generating Functions; Springer; Archive For Rational Mechanics And Analysis; 226; 1; 9-10-2017; 243-302
0003-9527
CONICET Digital
CONICET
url http://hdl.handle.net/11336/43043
identifier_str_mv Ferraro, Sebastián José; de León, Manuel; Marrero, Juan Carlos; Martin de Diego, David; Vaquero, Miguel; On the Geometry of the Hamilton-Jacobi Equation and Generating Functions; Springer; Archive For Rational Mechanics And Analysis; 226; 1; 9-10-2017; 243-302
0003-9527
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://link.springer.com/article/10.1007/s00205-017-1133-0
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00205-017-1133-0
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1606.00847
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
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
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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score 13.070432

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