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 epsilon-C(5) and p epsilon-L1; furthermore, for any solution x(t) there exists a velocity bound c for all time: \x(t) < c for all t epsilon-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.