This research project aims, in the long term, to bridge stochastic thermodynamics and contact Hamiltonian mechanics by exploring their theoretical connections. The internship focuses on two more specific topics: first, determining whether canonical changes of variables and Hamiltonian reductions from a detailed to a standard form commute; and second, developing and testing a conjecture about the Hamilton-Jacobi equations having stable and unstable solutions that intersect on critical submanifolds. The conjecture is expected to hold as an adaptation of the Perron-Frobenius theorem to classical Hamiltonian dynamics. Candidates should possess strong backgrounds in analytical mechanics, symplectic or contact geometry, and mathematical physics to succeed in this position.