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Explore a 36-minute mathematics lecture that delves into geometric frameworks for studying dissipative systems, presented at the Erwin Schrödinger International Institute's Thematic Programme on "Infinite-dimensional Geometry." Learn how b-symplectic geometry and the metriplectic (GENERIC) framework overcome the limitations of traditional Hamiltonian mechanics in analyzing non-equilibrium dynamics. Discover how these advanced mathematical approaches incorporate phase space singularities and thermodynamic laws through the combination of symplectic forms, pseudo-Riemannian metrics, and free energy as a dynamical function. Examine practical applications of these methods in fluid systems, particularly in understanding the Navier-Stokes equations and magnetohydrodynamics, supported by recent research publications in Physica D and the Journal of Plasma Physics.
