  We consider a modulated magnetic field, $B(t) = B_0 +\varepsilon f(\omega
t)$, perpendicular to a fixed plane, where $B_0$ is constant, $\varepsilon>0$
and $f$ a periodic function on the torus ${\mathbb T}^n$. Our aim is to study
classical and quantum dynamics for the corresponding Landau Hamiltonian. It
turns out that the results depend strongly on the chosen gauge. For the Landau
gauge the position observable is unbounded for "almost all" non resonant
frequencies $\omega$. On the contrary, for the symmetric gauge we obtain that,
for "almost all" non resonant frequencies $\omega$, the Landau Hamiltonian is
reducible to a two dimensional harmonic oscillator and thus gives rise to
bounded dynamics. The proofs use KAM algorithms for the classical dynamics.
Quantum applications are given. In particular, the Floquet spectrum is
absolutely continuous in the Landau gauge while it is discrete, of finite
multiplicity, in symmetric gauge.
