  We study the dynamics of a class of Hamiltonian systems with dissipation,
coupled to noise, in a singular (small mass) limit. We derive the homogenized
equation for the position degrees of freedom in the limit, including the
presence of a {\em noise-induced drift} term. We prove convergence to the
solution of the homogenized equation in probability and, under stronger
assumptions, in an $L^p$-norm. Applications cover the overdamped limit of
particle motion in a time-dependent electromagnetic field, on a manifold with
time-dependent metric, and the dynamics of nuclear matter.
