  We prove a $C^\infty$ global version of Nekhoroshev theorem for time
dependent Hamiltonians in $R^d\times T^d$. Precisely, we prove a result showing
that for all times the actions of the unperturbed systems are bounded by a
constant times $ \langle t\rangle^{\epsilon}$. We apply the result to the
dynamics of a charged particle in $T^d$ subject to a time dependent
electromagnetic field.
