KAM theory for particles in periodic potentials

It is shown that the system of the form x + V (x) = p (t) with periodic V and p and with (p) = 0 is near-integrable for large energies. In particular, most (in the sense of Lebesgue measure) fast solutions are quasiperiodic, provided V C (5) and p L 1; furthermore, for any solution x(t) there exists a velocity bound c for all time: |x(t)| < c for all t R. For any real number r there exists a solution with that average velocity, and when r is rational, this solution can be chosen to be periodic.