  We apply Bogolyubov's averaging theorem to the motion of an electron of an
atom driven by a linearly polarized laser field in the Kramers-Henneberger
frame. We provide estimates of the differences between the original
trajectories and the trajectories associated with the averaged system as a
function of the parameters of the laser field and the region of phase space. We
formulate a modified Bogolyubov averaging theorem based on the Hamiltonian
properties of the system, and show that this version is better suited for these
systems. From these estimates, we discuss the validity of the
Kramers-Henneberger approximation.
