This talk considers highly oscillatory Hamiltonian systems with a slowly varying, state-dependent high frequency. The long-time behaviour of the St"ormer--Verlet--leapfrog method is studied using an extension of the technique of modulated Fourier expansions from a constant high frequency to a state-dependent high frequency.

It is proved that the St"ormer--Verlet method approximately conserves a modified action and a modified total energy over a long time interval that covers a negative integer power of the small parameter. This power depends on the size of the product of the stepsize with the high frequency.

This is joint work with Christian Lubich.