Contact Hamiltonian Dynamics and Structure-Preserving Numerical Methods
Abstract:Â Contact geometry naturally arises in a wide range of contexts, including the study of boundaries of symplectic manifolds, isoenergetic hypersurfaces in Hamiltonian mechanics, as well as in control theory, Riemannian geometry, and the modeling of dissipative systems. A key feature of contact manifolds is that any one-parameter group of transformations preserving the underlying geometric structure is necessarily Hamiltonian, in the sense that it can be generated by a smooth function describing the dynamics. This correspondence provides a powerful bridge between geometry and dynamics, and forms the basis for the construction of structure-preserving numerical methods. In this talk, after reviewing the main features of contact Hamiltonian systems, we present geometric numerical schemes for approximating their flows. In particular, we discuss splitting methods based on the decomposition of the Hamiltonian, as well as variational integrators derived from a Lagrangian formulation. Furthermore, we show how these approaches can be extended to the numerical integration of first-order partial differential equations arising from the underlying geometric structure.