Effective Hamiltonians for periodically driven systems

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.