Geometrical structures of higher-order dynamical systems and field theories
Geometrical physics is a relatively young branch of applied mathematics that was initiated by the 60's and the 70's when A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, among many others, began to study various topics in physics using methods of differential geometry. This "geometri...
| Autor: | |
|---|---|
| Tipo de recurso: | tesis doctoral |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/95520 |
| Acceso en línea: | https://hdl.handle.net/2117/95520 https://dx.doi.org/10.5821/dissertation-2117-95520 |
| Access Level: | acceso abierto |
| Palabra clave: | Higher-order autonomous and non-autonomous dynamical systems Higher-order field theory Hamilton-Jacobi equations Variational principles Skinner-Rusk formalism Lagrangian and Hamiltonian formalisms Symplectic, presymplectic and multisymplectic manifolds Camps, Teoria dels (Física) Equacions de Hamilton-Jacobi Física matemàtica Lagrange, Funcions de Varietats (Matemàtica) Principis variacionals Sistemes dinàmics diferenciables Sistemes hamiltonians Àrees temàtiques de la UPC::Matemàtiques i estadística |
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Geometrical structures of higher-order dynamical systems and field theories |
| title |
Geometrical structures of higher-order dynamical systems and field theories |
| spellingShingle |
Geometrical structures of higher-order dynamical systems and field theories Prieto Martínez, Pedro Daniel|||0000-0001-8325-2462 Higher-order autonomous and non-autonomous dynamical systems Higher-order field theory Hamilton-Jacobi equations Variational principles Skinner-Rusk formalism Lagrangian and Hamiltonian formalisms Symplectic, presymplectic and multisymplectic manifolds Camps, Teoria dels (Física) Equacions de Hamilton-Jacobi Física matemàtica Lagrange, Funcions de Varietats (Matemàtica) Principis variacionals Sistemes dinàmics diferenciables Sistemes hamiltonians Àrees temàtiques de la UPC::Matemàtiques i estadística |
| title_short |
Geometrical structures of higher-order dynamical systems and field theories |
| title_full |
Geometrical structures of higher-order dynamical systems and field theories |
| title_fullStr |
Geometrical structures of higher-order dynamical systems and field theories |
| title_full_unstemmed |
Geometrical structures of higher-order dynamical systems and field theories |
| title_sort |
Geometrical structures of higher-order dynamical systems and field theories |
| dc.creator.none.fl_str_mv |
Prieto Martínez, Pedro Daniel|||0000-0001-8325-2462 |
| author |
Prieto Martínez, Pedro Daniel|||0000-0001-8325-2462 |
| author_facet |
Prieto Martínez, Pedro Daniel|||0000-0001-8325-2462 |
| author_role |
author |
| dc.contributor.none.fl_str_mv |
Román Roy, Narciso |
| dc.subject.none.fl_str_mv |
Higher-order autonomous and non-autonomous dynamical systems Higher-order field theory Hamilton-Jacobi equations Variational principles Skinner-Rusk formalism Lagrangian and Hamiltonian formalisms Symplectic, presymplectic and multisymplectic manifolds Camps, Teoria dels (Física) Equacions de Hamilton-Jacobi Física matemàtica Lagrange, Funcions de Varietats (Matemàtica) Principis variacionals Sistemes dinàmics diferenciables Sistemes hamiltonians Àrees temàtiques de la UPC::Matemàtiques i estadística |
| topic |
Higher-order autonomous and non-autonomous dynamical systems Higher-order field theory Hamilton-Jacobi equations Variational principles Skinner-Rusk formalism Lagrangian and Hamiltonian formalisms Symplectic, presymplectic and multisymplectic manifolds Camps, Teoria dels (Física) Equacions de Hamilton-Jacobi Física matemàtica Lagrange, Funcions de Varietats (Matemàtica) Principis variacionals Sistemes dinàmics diferenciables Sistemes hamiltonians Àrees temàtiques de la UPC::Matemàtiques i estadística |
| description |
Geometrical physics is a relatively young branch of applied mathematics that was initiated by the 60's and the 70's when A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, among many others, began to study various topics in physics using methods of differential geometry. This "geometrization" provides a way to analyze the features of the physical systems from a global viewpoint, thus obtaining qualitative properties that help us in the integration of the equations that describe them. Since then, there has been a strong development in the intrinsic treatment of a variety of topics in theoretical physics, applied mathematics and control theory using methods of differential geometry. Most of the work done in geometrical physics since its first days has been devoted to study first-order theories, that is, those theories whose physical information depends on (at most) first-order derivatives of the generalized coordinates of position (velocities). However, there are theories in physics in which the physical information depends explicitly on accelerations or higher-order derivatives of the generalized coordinates of position, and thus more sophisticated geometrical tools are needed to model them acurately. In this Ph.D. Thesis we pretend to give a geometrical description of some of these higher-order theories. In particular, we focus on dynamical systems and field theories whose dynamical information can be given in terms of a Lagrangian function, or a Hamiltonian that admits Lagrangian counterpart. More precisely, we will use the Lagrangian-Hamiltonian unified approach in order to develop a geometric framework for autonomous and non-autonomous higher-order dynamical system, and for second-order field theories. This geometric framework will be used to study several relevant physical examples and applications, such as the Hamilton-Jacobi theory for higher-order mechanical systems, relativistic spin particles and deformation problems in mechanics, and the Korteweg-de Vries equation and other systems in field theory. |
| publishDate |
2014 |
| dc.date.none.fl_str_mv |
2014 2014-10-02 2014 2014-11-26 |
| dc.type.none.fl_str_mv |
doctoral thesis http://purl.org/coar/resource_type/c_db06 VoR http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
| format |
doctoralThesis |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/2117/95520 https://dx.doi.org/10.5821/dissertation-2117-95520 |
| url |
https://hdl.handle.net/2117/95520 https://dx.doi.org/10.5821/dissertation-2117-95520 |
| 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 http://creativecommons.org/licenses/by-nc/3.0/es/ |
| dc.rights.openaire.fl_str_mv |
info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 http://creativecommons.org/licenses/by-nc/3.0/es/ |
| eu_rights_str_mv |
openAccess |
| dc.format.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Universitat Politècnica de Catalunya |
| publisher.none.fl_str_mv |
Universitat Politècnica de Catalunya |
| dc.source.none.fl_str_mv |
reponame:UPCommons. Portal del coneixement obert de la UPC instname:Universitat Politècnica de Catalunya (UPC) |
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Universitat Politècnica de Catalunya (UPC) |
| reponame_str |
UPCommons. Portal del coneixement obert de la UPC |
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UPCommons. Portal del coneixement obert de la UPC |
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1869413630719557632 |
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Geometrical structures of higher-order dynamical systems and field theoriesPrieto Martínez, Pedro Daniel|||0000-0001-8325-2462Higher-order autonomous and non-autonomous dynamical systemsHigher-order field theoryHamilton-Jacobi equationsVariational principlesSkinner-Rusk formalismLagrangian and Hamiltonian formalismsSymplectic, presymplectic and multisymplectic manifoldsCamps, Teoria dels (Física)Equacions de Hamilton-JacobiFísica matemàticaLagrange, Funcions deVarietats (Matemàtica)Principis variacionalsSistemes dinàmics diferenciablesSistemes hamiltoniansÀrees temàtiques de la UPC::Matemàtiques i estadísticaGeometrical physics is a relatively young branch of applied mathematics that was initiated by the 60's and the 70's when A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, among many others, began to study various topics in physics using methods of differential geometry. This "geometrization" provides a way to analyze the features of the physical systems from a global viewpoint, thus obtaining qualitative properties that help us in the integration of the equations that describe them. Since then, there has been a strong development in the intrinsic treatment of a variety of topics in theoretical physics, applied mathematics and control theory using methods of differential geometry. Most of the work done in geometrical physics since its first days has been devoted to study first-order theories, that is, those theories whose physical information depends on (at most) first-order derivatives of the generalized coordinates of position (velocities). However, there are theories in physics in which the physical information depends explicitly on accelerations or higher-order derivatives of the generalized coordinates of position, and thus more sophisticated geometrical tools are needed to model them acurately. In this Ph.D. Thesis we pretend to give a geometrical description of some of these higher-order theories. In particular, we focus on dynamical systems and field theories whose dynamical information can be given in terms of a Lagrangian function, or a Hamiltonian that admits Lagrangian counterpart. More precisely, we will use the Lagrangian-Hamiltonian unified approach in order to develop a geometric framework for autonomous and non-autonomous higher-order dynamical system, and for second-order field theories. This geometric framework will be used to study several relevant physical examples and applications, such as the Hamilton-Jacobi theory for higher-order mechanical systems, relativistic spin particles and deformation problems in mechanics, and the Korteweg-de Vries equation and other systems in field theory.La física geomètrica és una branca relativament jove de la matemàtica aplicada que es va iniciar als anys 60 i 70 qua A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, entre molts altres, van començar a estudiar diversos problemes en física usant mètodes de geometria diferencial. Aquesta "geometrització" proporciona una manera d'analitzar les característiques dels sistemes físics des d'una perspectiva global, obtenint així propietats qualitatives que faciliten la integració de les equacions que els descriuen. D'ençà s'ha produït un fort desenvolupamewnt en el tractament intrínsic d'una gran varietat de problemes en física teòrica, matemàtica aplicada i teoria de control usant mètodes de geometria diferencial. Gran part del treball realitzat en la física geomètrica des dels seus primers dies s'ha dedicat a l'estudi de teories de primer ordre, és a dir, teories tals que la informació física depèn en, com a molt, derivades de primer ordre de les coordenades de posició generalitzades (velocitats). Tanmateix, hi ha teories en física en les que la informació física depèn de manera explícita en acceleracions o derivades d'ordre superior de les coordenades de posició generalitzades, requerint, per tant, d'eines geomètriques més sofisticades per a modelar-les de manera acurada. En aquesta Tesi Doctoral ens proposem donar una descripció geomètrica d'algunes d'aquestes teories. En particular, estudiarem sistemes dinàmics i teories de camps tals que la seva informació dinàmica ve donada en termes d'una funció lagrangiana, o d'un hamiltonià que prové d'un sitema lagrangià. Per a ser més precisos emprarem la formulació unificada Lagrangiana-Hamiltoniana per tal de desenvolupar marcs geomètrics per a sistemes dinàmics d'ordre superior autònoms i no autònoms, i per a teories de camps de segon ordre. Amb aquest marc geomètric estudiarem alguns exemples físics rellevants i algunes aplicacions, com la teoria de Hamilton-Jacobi per a sistemes mecànics d'ordre superior, partícules relativístiques amb spin i problemes de deformació en mecànica, i l'equació de Korteweg-de Vries i altres sistemes en teories de camps.Universitat Politècnica de CatalunyaRomán Roy, Narciso20142014-10-0220142014-11-26doctoral thesishttp://purl.org/coar/resource_type/c_db06VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/doctoralThesisapplication/pdfhttps://hdl.handle.net/2117/95520https://dx.doi.org/10.5821/dissertation-2117-95520reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2http://creativecommons.org/licenses/by-nc/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/955202026-05-27T15:37:01Z |
| score |
15,300719 |