Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation

Autores
Grillo, Sergio Daniel
Año de publicación
2020
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Consider a symplectic manifold M, a Hamiltonian vector field X and a fibration Π:M→N. Related to these data we have a generalized version of the (time-independent) Hamilton–Jacobi equation: the Π-HJE for X, whose unknown is a section σ:N→M of Π. The standard HJE is obtained when the phase space M is a cotangent bundle T∗Q (with its canonical symplectic form), Π is the canonical projection πQ:T∗Q→Q and the unknown is a closed 1-form dW:Q→T∗Q. The function W is called Hamilton's characteristic function. Coming back to the generalized version, among the solutions of the Π-HJE, a central role is played by the so-called isotropic complete solutions. This is because, if a solution of this kind is known for a given Hamiltonian system, then such a system can be integrated up to quadratures. The purpose of the present paper is to prove that, under mild conditions, an isotropic complete solution exists around almost every point of M. Restricted to the standard case, this gives rise to an alternative proof for the local existence of a complete family of Hamilton's characteristic functions.
Fil: Grillo, Sergio Daniel. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Universidad Nacional de Cuyo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina
Materia
HAMILTON–JACOBI THEORY
INTEGRABLE SYSTEMS
SYMPLECTIC GEOMETRY
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/126678

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spelling Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equationGrillo, Sergio DanielHAMILTON–JACOBI THEORYINTEGRABLE SYSTEMSSYMPLECTIC GEOMETRYhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Consider a symplectic manifold M, a Hamiltonian vector field X and a fibration Π:M→N. Related to these data we have a generalized version of the (time-independent) Hamilton–Jacobi equation: the Π-HJE for X, whose unknown is a section σ:N→M of Π. The standard HJE is obtained when the phase space M is a cotangent bundle T∗Q (with its canonical symplectic form), Π is the canonical projection πQ:T∗Q→Q and the unknown is a closed 1-form dW:Q→T∗Q. The function W is called Hamilton's characteristic function. Coming back to the generalized version, among the solutions of the Π-HJE, a central role is played by the so-called isotropic complete solutions. This is because, if a solution of this kind is known for a given Hamiltonian system, then such a system can be integrated up to quadratures. The purpose of the present paper is to prove that, under mild conditions, an isotropic complete solution exists around almost every point of M. Restricted to the standard case, this gives rise to an alternative proof for the local existence of a complete family of Hamilton's characteristic functions.Fil: Grillo, Sergio Daniel. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Universidad Nacional de Cuyo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; ArgentinaElsevier Science2020-02info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/zipapplication/pdfhttp://hdl.handle.net/11336/126678Grillo, Sergio Daniel; Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation; Elsevier Science; Journal Of Geometry And Physics; 148; 103544; 2-2020; 1-110393-0440CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0393044019302256info:eu-repo/semantics/altIdentifier/doi/10.1016/j.geomphys.2019.103544info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1902.02280v1info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:37:03Zoai:ri.conicet.gov.ar:11336/126678instacron: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 10:37:03.851CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
title Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
spellingShingle Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
Grillo, Sergio Daniel
HAMILTON–JACOBI THEORY
INTEGRABLE SYSTEMS
SYMPLECTIC GEOMETRY
title_short Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
title_full Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
title_fullStr Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
title_full_unstemmed Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
title_sort Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation
dc.creator.none.fl_str_mv Grillo, Sergio Daniel
author Grillo, Sergio Daniel
author_facet Grillo, Sergio Daniel
author_role author
dc.subject.none.fl_str_mv HAMILTON–JACOBI THEORY
INTEGRABLE SYSTEMS
SYMPLECTIC GEOMETRY
topic HAMILTON–JACOBI THEORY
INTEGRABLE SYSTEMS
SYMPLECTIC GEOMETRY
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Consider a symplectic manifold M, a Hamiltonian vector field X and a fibration Π:M→N. Related to these data we have a generalized version of the (time-independent) Hamilton–Jacobi equation: the Π-HJE for X, whose unknown is a section σ:N→M of Π. The standard HJE is obtained when the phase space M is a cotangent bundle T∗Q (with its canonical symplectic form), Π is the canonical projection πQ:T∗Q→Q and the unknown is a closed 1-form dW:Q→T∗Q. The function W is called Hamilton's characteristic function. Coming back to the generalized version, among the solutions of the Π-HJE, a central role is played by the so-called isotropic complete solutions. This is because, if a solution of this kind is known for a given Hamiltonian system, then such a system can be integrated up to quadratures. The purpose of the present paper is to prove that, under mild conditions, an isotropic complete solution exists around almost every point of M. Restricted to the standard case, this gives rise to an alternative proof for the local existence of a complete family of Hamilton's characteristic functions.
Fil: Grillo, Sergio Daniel. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Universidad Nacional de Cuyo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina
description Consider a symplectic manifold M, a Hamiltonian vector field X and a fibration Π:M→N. Related to these data we have a generalized version of the (time-independent) Hamilton–Jacobi equation: the Π-HJE for X, whose unknown is a section σ:N→M of Π. The standard HJE is obtained when the phase space M is a cotangent bundle T∗Q (with its canonical symplectic form), Π is the canonical projection πQ:T∗Q→Q and the unknown is a closed 1-form dW:Q→T∗Q. The function W is called Hamilton's characteristic function. Coming back to the generalized version, among the solutions of the Π-HJE, a central role is played by the so-called isotropic complete solutions. This is because, if a solution of this kind is known for a given Hamiltonian system, then such a system can be integrated up to quadratures. The purpose of the present paper is to prove that, under mild conditions, an isotropic complete solution exists around almost every point of M. Restricted to the standard case, this gives rise to an alternative proof for the local existence of a complete family of Hamilton's characteristic functions.
publishDate 2020
dc.date.none.fl_str_mv 2020-02
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/11336/126678
Grillo, Sergio Daniel; Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation; Elsevier Science; Journal Of Geometry And Physics; 148; 103544; 2-2020; 1-11
0393-0440
CONICET Digital
CONICET
url http://hdl.handle.net/11336/126678
identifier_str_mv Grillo, Sergio Daniel; Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation; Elsevier Science; Journal Of Geometry And Physics; 148; 103544; 2-2020; 1-11
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/url/https://www.sciencedirect.com/science/article/pii/S0393044019302256
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.geomphys.2019.103544
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1902.02280v1
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/zip
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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