  This work focuses on topics related to Hamiltonian stochastic differential
equations with L\'{e}vy noise. We first show that the phase flow of the
stochastic system preserves symplectic structure, and propose a stochastic
version of Hamilton's principle by the corresponding formulation of the
stochastic action integral and the Euler-Lagrange equation. Based on these
properties, we further investigate the effective behaviour of a small
transversal perturbation to a completely integrable stochastic Hamiltonian
system with L\'{e}vy noise. We establish an averaging principle in the sense
that the action component of solution converges to the solution of a stochastic
differential equation when the scale parameter goes to zero. Furthermore, we
obtain the estimation for the rate of this convergence. Finally, we present an
example to illustrate these results.
