Geometrical aspects of contact mechanical systems and field theories

Many important theories in modern physics can be stated using the tools of differential geometry. It is well known that symplectic geometry is the natural framework to deal with autonomous Hamiltonian mechanics. This admits several generalizations for nonautonomous systems and classical field theories, both regular and singular. Some of these generalizations are the subject of the present dissertation. In recent years there has been a growing interest in studying dissipative mechanical systems from a geometric perspective by using contact structures. In the present thesis we review what has been done in this topic and go deeper, studying symmetries and dissipated quantities of contact systems, and developing the Lagrangian-Hamiltonian mixed formalism (Skinner-Rusk formalism) for these systems.With regard to classical field theory, we introduce the notion of k-precosymplectic manifold and use it to give a geometric description of singular nonautonomous field theories. We also devise a constraint algorithm for k-precosymplectic systems. Furthermore, field theories with damping are described through a modification of the De Donder-Weyl Hamiltonian field theory. This is achieved by combining both contact geometry and k-symplectic structures, resulting in what we call the k-contact formalism. We also introduce two notions of dissipation laws, generalizing the concept of dissipated quantity. The preceding developments are also applied to Lagrangian field theory. The Skinner-Rusk formulation for k-contact systems is described in full detail and we show how to recover both the Lagrangian and Hamiltonian formalisms from it. Throughout the thesis we have worked out several examples both in mechanics and field theory. The most remarkable mechanical examples are the damped harmonic oscillator, the motion in a constant gravitational field with friction, the parachute equation and the damped simple pendulum. On the other hand, in field theory, we have studied the damped vibrating string, the Burgers' equation, the Klein-Gordon equation and its relation with the telegrapher's equation, and the Maxwell's equations of electromagnetism with dissipation.Moltes teories de la física moderna es poden formular mitjançant les eines de la geometria diferencial. Com és ben sabut, la geometria simplèctica és el marc natural per treballar amb sistemes mecànics Hamiltonians autònoms. La geometría simplèctica admet diverses generalitzacions que permeten treballar amb sistemes no autònoms i amb teories de camps, tant en el cas regular com en el singular. En aquesta tesi treballarem amb algunes d'aquestes generalitzacions. En els darrers anys l'interés per l'estudi geomètric dels sistemes mecànics dissipatius mitjançant estructures de contacte ha crescut notablement. En aquesta tesi revisem el que s'ha fet sobre aquest tema i anem més enllà, estudiant les seves simetries i quantitats dissipades, i desenvolupant el formalisme mixt lagrangià-hamiltonià (formalisme de Skinner-Rusk) per aquests sistemes. En referència a la teoria de camps, introduïm la noció de varietat k-precosimplèctica i la usem per a descriure geomètricament les teories de camps no autònomes singulars. També desenvolupem un algorisme de lligams per a sistemes k-precosimplèctics. A més, descrivim les teories de camps amb dissipació mitjançant una modificació de la teoria hamiltoniana de camps de De Donder-Weyl. Aquesta descripció combina la geometria de contacte i les estructures k-simplèctiques, obtenint el que anomenem formalisme de k-contacte. També generalitzem el concepte de quantitat dissipada introduint dues nocions de llei de dissipació. Aquests desenvolupaments s'apliquen també a les teories de camps lagrangianes. Donem una descripció completa de la formulació de Skinner-Rusk dels sistemes de k-contacte i veiem com recuperar els formalismes lagrangià i hamiltonià a partir del formalisme mixt. Al llarg de la tesi hem estudiat diversos exemples tant en mecànica com en teoria de camps. Els sistemes mecànics més rellevants són l'oscil·lador harmònic esmorteït, el moviment en un camps gravitatori constant amb fregament, l'equació del paracaigudista i el pèndul simple esmorteït. Per altra banda, en teoria de camps estudiem la corda vibrant esmorteïda, l'equació de Burgers, l'equació de Klein-Gordon i la seva relació amb l'equació del telegrafista, i les equacions de Maxwell de l'electromagnetisme amb dissipació.Postprint (published version

Constraint algorithm for singular k-cosymplectic field theories

The main objective of this thesis is to develop a constraint algorithm for singular k-cosymplectic field theories

Formulación geométrica de las teorias Gauge y de Yang-Mills

Este trabajo presenta todas las herramientas propias de la geometría diferencial necesarias para poder describir adecuadamente las teorías gauge y en particular las ecuaciones de Yang-Mills. Concretamente, el trabajo empieza presentando las pseudométricas en espacios vectoriales y generalizándolas a las variedades pseudoriemannianas, incluyendo una descripción del operador estrella de Hodge y la coderivada. Como ejemplo canónico de teoría de Yang-Mills tomamos el caso del electromagnetismo, estudiándolo en detalle. También se incluye una introducción a los grupos y las álgebras de Lie. En los capítulos centrales del trabajo se introducen los dos conceptos clave para poder escribir las ecuaciones de Yang-Mills: los fibrados y las conexiones principales. En estos capítulos se estudian en detalle estos conceptos ya que son la piedra angular de la teoría de Yang-Mills. Terminamos el trabajo escribiendo las ecuaciones de Yang-Mills en lenguaje geométrico

Erratum: constraint algorithm for singular field theories in the k k -cosymplectic framework

We are indebted to Prof. Dieter Van den Bleeken (Bo˘gazi¸ci University) for having drawn our attention to the error that gave rise to this note. We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovaci´on y Universidades project PGC2018-098265-B-C33, and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017-SGR-932.Peer ReviewedPostprint (author's final draft

Skinner–Rusk formalism for k-contact systems

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In previous papers, a geometric framework has been developed to describe non-conservative field theories as a kind of modified Lagrangian and Hamiltonian field theories. This approach is that of k-contact Hamiltonian systems, which is based on the k-symplectic formulation of field theories as well as on contact geometry. In this work we present the Skinner–Rusk unified setting for these kinds of theories, which encompasses both the Lagrangian and Hamiltonian formalisms into a single picture. This unified framework is specially useful when dealing with singular systems, since: (i) it incorporates in a natural way the second-order condition for the solutions of field equations, (ii) it allows to implement the Lagrangian and Hamiltonian constraint algorithms in a unique simple way, and (iii) it gives the Legendre transformation, so that the Lagrangian and the Hamiltonian formalisms are obtained straightforwardly. We apply this description to several interesting physical examples: the damped vibrating string, the telegrapher’s equations, and Maxwell’s equations with dissipation terms.We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades project PGC2018-098265-B-C33 and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932Peer ReviewedPostprint (published version

Constraint algorithm for singular field theories in the k-cosymplectic framework

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of k-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of k-precosymplectic structure, which is a generalization of the k-cosymplectic structure. Next k-precosymplectic Hamiltonian systems are introduced in or- der to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a sub- manifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.Peer ReviewedPostprint (author's final draft

Time-dependent contact mechanics

The version of record is available online at: http://dx.doi.org/10.1007/s00605-022-01767-1Contact geometry allows us to describe some thermodynamic and dissipative systems. In this paper we introduce a new geometric structure in order to describe time-dependent contact systems: cocontact manifolds. Within this setting we develop the Hamiltonian and Lagrangian formalisms, both in the regular and singular cases. In the singular case, we present a constraint algorithm aiming to find a submanifold where solutions exist. As a particular case we study contact systems with holonomic time-dependent constraints. Some regular and singular examples are analyzed, along with numerical simulations.We acknowledge fruitful discussions and comments from our colleague Narciso Román-Roy. MdL acknowledges the financial support of the Ministerio de Ciencia e Innovación (Spain), under grants PID2019-106715GB-C2, “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and EIN2020-112107. JG, XG, MCML and XR acknowledge the financial support of the Ministerio de Ciencia, Innovación y Universidades (Spain), project PGC2018-098265-B-C33.Peer ReviewedPostprint (author's final draft

Constraint algorithm for singular k-cosymplectic field theories

The main objective of this thesis is to develop a constraint algorithm for singular k-cosymplectic field theories

Constraint algorithm for singular field theories in the k-cosymplectic framework

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of k-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of k-precosymplectic structure, which is a generalization of the k-cosymplectic structure. Next k-precosymplectic Hamiltonian systems are introduced in or- der to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a sub- manifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.Peer Reviewe

A K-contact Lagrangian formulation for nonconservative field theories

Dynamical systems with dissipative behaviour can be described in terms of contact manifolds and a modified version of Hamilton's equations. Dissipation terms can also be added to field equations, as showed in a recent paper where we introduced the notion of k-contact structure, and obtained a modified version of the De Donder–Weyl equations of covariant Hamiltonian field theory. In this paper we continue this study by presenting a k-contact Lagrangian formulation for nonconservative field theories. The Lagrangian density is defined on the product of the space of k-velocities times a k-dimensional Euclidean space with coordinates sa, which are responsible for the dissipation. We analyze the regularity of such Lagrangians; only in the regular case we obtain a k-contact Hamiltonian system. We study several types of symmetries for k-contact Lagrangian systems, and relate them with dissipation laws, which are analogous to conservation laws of conservative systems. Several examples are discussed: we find contact Lagrangians for some kinds of second-order linear partial differential equations, with the damped membrane as a particular example, and we also study a vibrating string with a magnetic-like term.Financial support from the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932.Peer ReviewedPostprint (author's final draft