Geometric Mechanics and Control Seminar
Contact geometry for simple thermodynamical systems with friction
Speaker: Manuel Lainz (ICMAT-CSIC)Date: Friday, 12 February 2021 - 16:00Place: Online - zoom.us/j/93006808687?pwd=THpXbzhXTGJJY29KeXQxRTUvaGN1QT09 (ID: 930 0680 8687; Access code: 659820)
On a contact manifold $(M, eta)$, given a Hamiltonian function $H$ one can naturally define the Hamiltonian vector field $X_H$. The evolution vector field $E_H = X_H + H R$, where $R$ is the Reeb vector field can also be constructed from the Hamiltonian function and the contact structure alone. This vector field coincides with the Hamiltonian vector field on the zero set of the Hamiltonian but differs from it on the rest of $M$. The evolution vector field is always tangent to the kernel of the contact form. The thermodynamic interpretation of this fact is that its integral curves fulfill the first law of thermodynamics. In addition, with a simple assumption on the Hamiltonian, the second law of thermodynamics is fulfilled. On this talk, we will explain the geometric and dynamical properties of this vector field and its applications to the description of isolated thermodynamic systems. If time permits, we will also show how to generalize this construction to composed systems.  Simoes Alexandre Anahory, de León Manuel, Valcázar Manuel Lainz and de Diego David Martín. 2020, Contact geometry for simple thermodynamical systems with friction. Proc. R. Soc. A. 476:20200244.  A. A. Simoes, D. M. de Diego, M. L. Valcázar, and M. de León, “The geometry of some thermodynamic systems,” arXiv:2012.07404 .