Bounds on the growth of energy for particles on the torus with unbounded time dependent perturbations

We prove a C∞ version of Nekhoroshev theorem for time dependent Hamiltonians in Rd×Td. Precisely, we prove a result showing that for all times the energy of the system is bounded by a constant times ⟨t⟩ɛ. We apply the result to the dynamics of a charged particle in Td subject to a time dependent electromagnetic field.