### Abstract

We introduce a class of symplectic (and in fact also non symplectic) schemes for the numerical integration of highly oscillatory Hamiltonian systems. The bottom line for the approach is to exploit the Hamilton-Jacobi form of the equations of motion. Because we perform a two-scale expansion of the solution of the Hamilton-Jacobi equations itself, we readily obtain, after an appropriate discretization, symplectic integration schemes. Adequate modifications also provide non symplectic schemes. The efficiency of the approach is demonstrated using several variants.

This is joint work with F. Legoll (LAMI-ENPC, France)