  First weak solutions of generalized stochastic Hamiltonian systems (gsHs) are
constructed via essential m-dissipativity of their generators on a suitable
core. For a scaled gsHs we prove convergence of the corresponding semigroups
and tightness of the weak solutions. This yields convergence in law of the
scaled gsHs to a distorted Brownian motion. In particular, the results confirm
the convergence of the Langevin dynamics in the overdamped regime to the
overdamped Langevin equation. The proofs work for a large class of (singular)
interaction potentials including, e.g., potentials of Lennard--Jones type.
