# Hamiltonian Systems with L\'evy Noise: Symplecticity, Hamilton's Principle and Averaging Principle

## TL;DR

This paper studies Hamiltonian systems influenced by Le9vy noise, demonstrating symplecticity, formulating a stochastic Hamilton's principle, and establishing an averaging principle with convergence rates for small perturbations.

## Contribution

It introduces a stochastic Hamilton's principle for systems with Le9vy noise and proves an averaging principle with convergence rates for perturbed stochastic Hamiltonian systems.

## Key findings

- Phase flow preserves symplectic structure.
- Stochastic Hamilton's principle formulated with action integral.
- Convergence of action component to a stochastic differential equation.

## Abstract

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.