A mechanical counterexample to KAM theory with low regularity

We give a mechanical example concerning the fact that some regularity is necessary in KAM theory. We consider the model given by the vertical bouncing motion of a ball on a periodically moving plate. Denoting with f the motion of the plate, some variants of Moser invariant curve theorem apply if f ̇ is small in norm C 5 and every motion has bounded velocity. This is not possible if the function f is only C 1 . Indeed we construct a function f ∈ C 1 with arbitrary small derivative in norm C 0 for which a motion with unbounded velocity exists.