  We demonstrate the surprising integrability of the classical Hamiltonian
associated to a spin 1/2 system under periodic external fields. The one-qubit
rotations generated by the dynamical evolution is, on the one hand, close to
that of the rotating wave approximation (RWA), on the other hand to two
different ``average'' systems, according to whether a certain parameter is
small or large. Of particular independent interest is the fact that both the
RWA and the averaging theorem are seen to hold well beyond their expected
region of validity. Finally we determine conditions for the realization of the
quantum NOT operation by means of classical stroboscopic maps.
