  In this paper, we generalize the classical Freidlin-Wentzell's theorem for
random perturbations of Hamiltonian systems. In stead of the two-dimensional
standard Brownian motion, the coefficient for the noise term is no longer the
identity matrix but a state-dependent matrix plus a state-dependent matrix that
converges uniformly to 0 on any compact sets as $\epsilon$ tends to 0. We also
take the drift term into consideration where the drfit term also contains two
parts, the state-dependent mapping and a state-dependent mapping that converges
uniformly to 0 on any compact sets as $\epsilon$ tends to 0. In the proof, we
use the result of generalized differential operator. We also adapt a new way to
prove the weak convergence inside the edge by constructing an auxiliary process
and apply Girsanov's theorem in the proof of gluing condition.
