Contact hamiltonian systems

Resumen

Contact Hamiltonian systems are a generalization of the Hamiltonian systems of classical mechanics. The action is added as an extra variable in phase space, and symplectic geometry is changed by contact geometry. In this way, we are able to model a large new class of Hamiltonian systems. Indeed, symplectic geometry is unable to deal with dynamics where there is energy dissipation. Nonetheless, this is possible in the contact world. In the recent years, there is a growing interest in the applications of contact Hamiltonian systems, expanding the classical ones in equilibrium thermodynamics. Several problems in physics, from areas including dissipative mechanical systems, electromagnetism, non-equilibrium thermodynamics, geometric optics, celestial mechanics, cosmology and quantum mechanics can be studied in the framework of contact Hamiltonian systems. Applications do not end here, but they cover other fields such as information geometry, control theory and optimization. The aim of this PhD thesis is to provide a comprehensive theory of contact Hamiltonian systems, hoping that this will aid other scientists and mathematicians doing research in this topic. This dissertation incorporates the Hamiltonian and Lagrangian formalism, including the variational formulation through the Herglotz principle. We study and classify the symmetries and the dissipated quantities they are related to. Means to deal with different kinds of constrained systems are provided. We develop alternative formulations of the dynamics, such as the Hamilton-Jacobi theory and the Tulczyjew triples. A theory of variational integrators adapted to contact systems is also included. Apart from this theoretical research, we also discuss novel applications in control theory and non-equilibrium thermodynamics, using the recently introduced evolution vector field.