Seminarium KMMF "Teoria Dwoistości"

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.In recent years there has been a growing interest in studying dissipative mechanical systems from a geometric perspective by using contact structures. In this talk we will review the main results on contact mechanics and work out some examples. We will also give a brief summary of how k-symplectic structures can be used to give a geometric formulation of classical field theories.Furthermore, field theories with damping will be 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.Finally we will study some interesting examples: the damped vibrating string, the Burgers’ equation and Maxwell’s equations of electromagnetism with dissipation. References • J. Gaset, X. Gracia, M. C. Munoz-Lecanda, X. Rivas and N. Roman-Roy. “New contributions to the Hamiltonian and Lagrangian contact formalisms for dissi- pative mechanical systems and their symmetries”. Int. J. Geom. Methods Mod. Phys., 16(6):2050090, 2020. https://doi.org/10.1142/S0219887820500905. • J. Gaset, X. Gracia, M. C. Munoz-Lecanda, X. Rivas and N. Roman-Roy. “A contact geometry framework for field theories with dissipation”. Ann. Phys., 414:168092, 2020. https://doi.org/10.1016/j.aop.2020.168092. • J. Gaset, X. Gracia, M. C. Munoz-Lecanda, X. Rivas and N. Roman-Roy. “A k-contact Lagrangian formalism for nonconservative field theories”. Rep. Math. Phys., 87(3):347–368, 2021. https://doi.org/10.1016/S0034-4877(21)00041-0. • X. Gracia, X. Rivas and N. Roman-Roy. “Skinner–Rusk formalism for k-contact systems”, preprint, 2021. https://arxiv.org/abs/2109.07257. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09