Hamiltonian mechanics, while effective for studying conservative systems, does not account for dissipation, limiting its application to non-equilibrium dynamics. In this presentation, we explore two geometric frameworks that address this limitation. The first, b-symplectic geometry, allows for singularities in phase space, enabling the study of dissipative systems [1]. The second, the metriplectic (GENERIC) framework, augments the symplectic form with a pseudo-Riemannian metric and free energy as a dynamical function, naturally incorporating the laws of thermodynamics [2]. In this talk, we will provide generic methods to use these methods to track dissipative dynamics with geometry. These approaches provide powerful tools for analyzing dissipative dynamics, particularly in fluid systems like the Navier-Stokes equations and magnetohydrodynamics.
[1] Baptiste Coquinot, Pau Mir, Eva Miranda, Singular cotangent models and complexity in fluids with dissipation, Physica D (2023)
[2] Baptiste Coquinot, Philip J. Morrison, A General Metriplectic Framework With Application To Dissipative Extended Magnetohydrodynamics, Journal of Plasma Physics (2020)