  This work is devoted to quantifying how periodic perturbation can change the
rate of metastable transition in stochastic mechanical systems with weak
noises. A closed-form explicit expression for approximating the rate change is
provided, and the corresponding transition mechanism can also be approximated.
Unlike the majority of existing relevant works, these results apply to kinetic
Langevin equations with high-dimensional potentials and nonlinear
perturbations. They are obtained based on a higher-order Hamiltonian formalism
and perturbation analysis for the Freidlin-Wentzell action functional. This
tool allowed us to show that parametric excitation at a resonant frequency can
significantly enhance the rate of metastable transitions. Numerical experiments
for both low-dimensional toy models and a molecular cluster are also provided.
For the latter, we show that vibrating a material appropriately can help heal
its defect, and our theory provides the appropriate vibration.
