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

Contact Lie systems

We define and analyse the properties of contact Lie systems, namely systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of Hamiltonian vector fields relative to a contact structure. As a particular example, we study families of conservative contact Lie systems. Liouville theorems, contact reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, etcetera.Comment: 29 pp, 4 figures. New version of the manuscript with Sections 4, 5.4, and 6 added. Many new results included and typos correcte

On Darboux theorems for geometric structures induced by closed forms

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe regular and singular mechanical systems), and certain cases of multisymplectic manifolds, and extends it in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.Comment: improved and extended proofs. 33 p

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