Geometrical aspects of contact mechanical systems and field theories

Abstract

Many important theories in modern physics can be stated using differential geometry. Symplectic geometry is the natural framework to deal with autonomous Hamiltonian mechanics. This admits several generalizations for nonautonomous systems, both regular and singular. Some of these extensions are the subject of this thesis. Recently there has been a growing interest in studying dissipative mechanical systems from a geometric perspective using contact geometry. In this thesis we review what has been done in this topic and go deeper, studying symmetries and dissipated quantities of contact systems, and developing the 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 these systems. Field theories with damping are described through a modification of the De Donder-Weyl Hamiltonian field theory. This is achieved by combining contact geometry and k-symplectic structures, resulting in the k-contact formalism. We introduce two notions of dissipation laws, generalizing the concept of dissipated quantity. These developments are also applied to Lagrangian field theory. The Skinner-Rusk formulation for k-contact systems is described in detail and we show how to recover the Lagrangian and Hamiltonian formalisms from it. Throughout the thesis we present several examples in mechanics and field theory. The most remarkable mechanical examples are the damped harmonic oscillator, the motion in a gravitational field with friction, the parachute equation and the damped simple pendulum. In field theory, we study the damped vibrating string, the Burgers' equation, the Klein-Gordon equation and its relation with the telegrapher's equation, and the Maxwell's equations with dissipation.