The variational discretization of the constrained higher-order Lagrange-Poincaré equations

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete co...

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Detalles Bibliográficos
Autores: Bloch, Anthony, Colombo, Leonardo, Jiménez Alburquerque, Fernando
Tipo de recurso: artículo
Fecha de publicación:2019
País:España
Institución:Universidad Nacional de Educación a Distancia
Repositorio:e-spacio. Repositorio Institucional de la UNED
Idioma:inglés
OAI Identifier:oai:e-spacio.uned.es:20.500.14468/31475
Acceso en línea:https://hdl.handle.net/20.500.14468/31475
Access Level:acceso abierto
Palabra clave:12 Matemáticas
Variational integrators
discrete mechanical systems
LagrangePoincar´e equations
geometric integration
discrete variational calculus
ordinary differential equations
control of mechanical systems
reduction by symmetries
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spelling The variational discretization of the constrained higher-order Lagrange-Poincaré equationsBloch, AnthonyColombo, LeonardoJiménez Alburquerque, Fernando12 MatemáticasVariational integratorsdiscrete mechanical systemsdiscrete mechanical systemsLagrangePoincar´e equationsgeometric integrationdiscrete variational calculusordinary differential equationscontrol of mechanical systemsreduction by symmetriesIn this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.American Institute of Mathematical Sciencese-Spacio UNED20262026-01-1920192019-01-0120192019-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14468/31475reponame:e-spacio. Repositorio Institucional de la UNEDinstname:Universidad Nacional de Educación a DistanciaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-nd/4.0/deed.esoai:e-spacio.uned.es:20.500.14468/314752026-06-06T12:38:31Z
dc.title.none.fl_str_mv The variational discretization of the constrained higher-order Lagrange-Poincaré equations
title The variational discretization of the constrained higher-order Lagrange-Poincaré equations
spellingShingle The variational discretization of the constrained higher-order Lagrange-Poincaré equations
Bloch, Anthony
12 Matemáticas
Variational integrators
discrete mechanical systems
discrete mechanical systems
LagrangePoincar´e equations
geometric integration
discrete variational calculus
ordinary differential equations
control of mechanical systems
reduction by symmetries
title_short The variational discretization of the constrained higher-order Lagrange-Poincaré equations
title_full The variational discretization of the constrained higher-order Lagrange-Poincaré equations
title_fullStr The variational discretization of the constrained higher-order Lagrange-Poincaré equations
title_full_unstemmed The variational discretization of the constrained higher-order Lagrange-Poincaré equations
title_sort The variational discretization of the constrained higher-order Lagrange-Poincaré equations
dc.creator.none.fl_str_mv Bloch, Anthony
Colombo, Leonardo
Jiménez Alburquerque, Fernando
author Bloch, Anthony
author_facet Bloch, Anthony
Colombo, Leonardo
Jiménez Alburquerque, Fernando
author_role author
author2 Colombo, Leonardo
Jiménez Alburquerque, Fernando
author2_role author
author
dc.contributor.none.fl_str_mv e-Spacio UNED
dc.subject.none.fl_str_mv 12 Matemáticas
Variational integrators
discrete mechanical systems
discrete mechanical systems
LagrangePoincar´e equations
geometric integration
discrete variational calculus
ordinary differential equations
control of mechanical systems
reduction by symmetries
topic 12 Matemáticas
Variational integrators
discrete mechanical systems
discrete mechanical systems
LagrangePoincar´e equations
geometric integration
discrete variational calculus
ordinary differential equations
control of mechanical systems
reduction by symmetries
description In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.
publishDate 2019
dc.date.none.fl_str_mv 2019
2019-01-01
2019
2019-01-01
2026
2026-01-19
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14468/31475
url https://hdl.handle.net/20.500.14468/31475
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
http://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv American Institute of Mathematical Sciences
publisher.none.fl_str_mv American Institute of Mathematical Sciences
dc.source.none.fl_str_mv reponame:e-spacio. Repositorio Institucional de la UNED
instname:Universidad Nacional de Educación a Distancia
instname_str Universidad Nacional de Educación a Distancia
reponame_str e-spacio. Repositorio Institucional de la UNED
collection e-spacio. Repositorio Institucional de la UNED
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.812429

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