Discrete Mechanics for Forced and Constrained Systems

Geometric mechanics is a branch of mathematics that studies classical mechanics of particles and fields from the point of view of geometry and its relation to symmetry. One of its most interesting developments was bringing together numerical analysis and geometry by relating what is known as discret...

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Detalles Bibliográficos
Autor: Sato Martín de Almagro, Rodrigo T.
Tipo de recurso: tesis doctoral
Fecha de publicación:2020
País:España
Institución:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/11039
Acceso en línea:https://hdl.handle.net/20.500.14352/11039
Access Level:acceso abierto
Palabra clave:531(043.2)
Discrete Mechanics
Diferential geometry
Numerical analysis
Mecánica Discreta
Geometría diferencial
Análisis numérico
1206 Análisis Numérico
1204.04 Geometría Diferencial
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spelling Discrete Mechanics for Forced and Constrained SystemsMecánica Discreta para Sistemas Forzados y LigadosSato Martín de Almagro, Rodrigo T.531(043.2)Discrete MechanicsDiferential geometryNumerical analysisMecánica DiscretaGeometría diferencialAnálisis numéricoAnálisis numéricoGeometría diferencial1206 Análisis Numérico1204.04 Geometría DiferencialGeometric mechanics is a branch of mathematics that studies classical mechanics of particles and fields from the point of view of geometry and its relation to symmetry. One of its most interesting developments was bringing together numerical analysis and geometry by relating what is known as discrete mechanics with numerical integration. This is called geometric integration. In the last 30 years this latter field has exploded with researchfrom the purely theoretical to the strictly applied. Variational integrators are a type of geometric integrators arising naturally from the discretization process of variational principles in mechanics. They display some of the most salient features of the theory, such as symplecticity, preservation of momenta and quasi-preservation of energy. These methods also apply very naturally to optimal control problems, also based on variational principles. Unfortunately, not all mechanical systems of interest admit a variational formulation. Such is the case of forced and nonholonomic mechanical systems. In this thesis we study both of these types of systems and obtain several new results. By geometrizing a new technique of duplication of variables and applying it, we were able to definitely prove the order of integrators for forced systems by using only variational techniques. Furthermore, we could also extend these results to the reduced setting in Lie groups, leading us to a very interesting geometric structure, Poisson groupoids. In addition, we developed new methods to geometrically integrate nonholonomic systems to arbitrary order preserving their constraints exactly. These methods can be seen as nonholonomic extensions of variational methods, and we were able to prove their order, although not through variational means. These methods have a nice geometric interpretation and thanks to their closeness to variational methods, they can be easily generalized to other geometric settings, such as Lie group integration. Finally, we were able to apply these new methods to optimal control problems...Universidad Complutense de MadridMartín de Diego, DavidUniversidad Complutense de Madrid20202020-03-0920202020-03-09doctoral thesishttp://purl.org/coar/resource_type/c_db06info:eu-repo/semantics/doctoralThesisapplication/pdfhttps://hdl.handle.net/20.500.14352/11039reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/110392026-06-02T12:44:21Z
dc.title.none.fl_str_mv Discrete Mechanics for Forced and Constrained Systems
Mecánica Discreta para Sistemas Forzados y Ligados
title Discrete Mechanics for Forced and Constrained Systems
spellingShingle Discrete Mechanics for Forced and Constrained Systems
Sato Martín de Almagro, Rodrigo T.
531(043.2)
Discrete Mechanics
Diferential geometry
Numerical analysis
Mecánica Discreta
Geometría diferencial
Análisis numérico
Análisis numérico
Geometría diferencial
1206 Análisis Numérico
1204.04 Geometría Diferencial
title_short Discrete Mechanics for Forced and Constrained Systems
title_full Discrete Mechanics for Forced and Constrained Systems
title_fullStr Discrete Mechanics for Forced and Constrained Systems
title_full_unstemmed Discrete Mechanics for Forced and Constrained Systems
title_sort Discrete Mechanics for Forced and Constrained Systems
dc.creator.none.fl_str_mv Sato Martín de Almagro, Rodrigo T.
author Sato Martín de Almagro, Rodrigo T.
author_facet Sato Martín de Almagro, Rodrigo T.
author_role author
dc.contributor.none.fl_str_mv Martín de Diego, David
Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 531(043.2)
Discrete Mechanics
Diferential geometry
Numerical analysis
Mecánica Discreta
Geometría diferencial
Análisis numérico
Análisis numérico
Geometría diferencial
1206 Análisis Numérico
1204.04 Geometría Diferencial
topic 531(043.2)
Discrete Mechanics
Diferential geometry
Numerical analysis
Mecánica Discreta
Geometría diferencial
Análisis numérico
Análisis numérico
Geometría diferencial
1206 Análisis Numérico
1204.04 Geometría Diferencial
description Geometric mechanics is a branch of mathematics that studies classical mechanics of particles and fields from the point of view of geometry and its relation to symmetry. One of its most interesting developments was bringing together numerical analysis and geometry by relating what is known as discrete mechanics with numerical integration. This is called geometric integration. In the last 30 years this latter field has exploded with researchfrom the purely theoretical to the strictly applied. Variational integrators are a type of geometric integrators arising naturally from the discretization process of variational principles in mechanics. They display some of the most salient features of the theory, such as symplecticity, preservation of momenta and quasi-preservation of energy. These methods also apply very naturally to optimal control problems, also based on variational principles. Unfortunately, not all mechanical systems of interest admit a variational formulation. Such is the case of forced and nonholonomic mechanical systems. In this thesis we study both of these types of systems and obtain several new results. By geometrizing a new technique of duplication of variables and applying it, we were able to definitely prove the order of integrators for forced systems by using only variational techniques. Furthermore, we could also extend these results to the reduced setting in Lie groups, leading us to a very interesting geometric structure, Poisson groupoids. In addition, we developed new methods to geometrically integrate nonholonomic systems to arbitrary order preserving their constraints exactly. These methods can be seen as nonholonomic extensions of variational methods, and we were able to prove their order, although not through variational means. These methods have a nice geometric interpretation and thanks to their closeness to variational methods, they can be easily generalized to other geometric settings, such as Lie group integration. Finally, we were able to apply these new methods to optimal control problems...
publishDate 2020
dc.date.none.fl_str_mv 2020
2020-03-09
2020
2020-03-09
dc.type.none.fl_str_mv doctoral thesis
http://purl.org/coar/resource_type/c_db06
dc.type.openaire.fl_str_mv info:eu-repo/semantics/doctoralThesis
format doctoralThesis
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/11039
url https://hdl.handle.net/20.500.14352/11039
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
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Universidad Complutense de Madrid
publisher.none.fl_str_mv Universidad Complutense de Madrid
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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