A mechanical counterexample to KAM theory with low regularity

Abstract

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 $\dot{f}$ 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 $f\in C^1$ with arbitrary small derivative in norm $C^0$ for which a motion with unbounded velocity exists.