  The dynamics of classical and quantum systems which are driven by a high
frequency ($\omega$) field is investigated. For classical systems the motion is
separated into a slow part and a fast part. The motion for the slow part is
computed perturbatively in powers of $\omega^{-1}$ to order $\omega^{-4}$ and
the corresponding time independent Hamiltonian is calculated. Such an effective
Hamiltonian for the corresponding quantum problem is computed to order
$\omega^{-4}$ in a high frequency expansion. Its spectrum is the quasienergy
spectrum of the time dependent quantum system. The classical limit of this
effective Hamiltonian is the classical effective time independent Hamiltonian.
It is demonstrated that this effective Hamiltonian gives the exact
quasienergies and quasienergy states of some simple examples as well as the
lowest resonance of a non trivial model for an atom trap. The theory that is
developed in the paper is useful for the analysis of atomic motion in atom
traps of various shapes.
