Xavier Rivas Guijarro defended his PhD thesis Geometrical Aspects of Contact Mechanical Systems and Field Theories (pdf) on December 17th, 2021. The thesis was produced within the UPC doctoral program in Applied Mathematics and was supervised by Professors Xavier Gràcia Sabaté and Narciso Román Roy.
Thesis summary
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 [2], and develop ing the Lagrangian-Hamiltonian mixed formalism (Skinner–Rusk formalism) for these systems [5].
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 [1].
Furthermore, field theories with damping are described through a modification of the De Donder–Weyl Hamiltonian field theory [3]. 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 [4]. 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 [6].
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.
Highlighted publications
References
[1] X. Gràcia, X. Rivas and N. Román-Roy. “Constraint algorithm for singular field theories in the k-cosymplectic framework”. Journal of Geometric Mechanics, 12:1-23, 2020. doi.
[2] J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas and N. Román-Roy. “New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries”. International Journal of Geometric Methods in Modern Physics, 16(6):2050090, 2020. doi.
[3] J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas and N. Román-Roy. “A contact geometry framework for field theories with dissipation”. Annals of Physics, 414:168092, 2020. doi.
[4] J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas and N. Román-Roy. “A k-contact Lagrangian formalism for nonconservative field theories”. Reports on Mathematical Physics, 87(3):347-368, 2021. doi.
[5] M. de León, J. Gaset, M. Lainz-Valcázar, X. Rivas and N. Román-Roy. “Unified Lagrangian-Hamiltonian formalism for contact systems”. Fortschritte der Physik/Progress of Physics, 68(8):2000045, 2020. doi.
[6] X. Gràcia, X. Rivas and N. Román-Roy. “Skinner-Rusk formalism for k-contact systems”, Journal of Geometry and Physics 172:104429, 2022. doi.