A mechanical counterexample to KAM theory with low regularity

• We consider the model of a bouncing ball on a moving plate.
• If the velocity of the plate is small in norm C5C5 then the ball cannot speed-up.
• We construct movements of the racket with arbitrary small C0C0-norm that allow to speed-up the ball.
• These constitute a mechanical example of the necessity of regularity in KAM theory.

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 ff the motion of the plate, some variants of Moser invariant curve theorem apply if ḟ is small in norm C5C5 and every motion has bounded velocity. This is not possible if the function ff is only C1C1. Indeed we construct a function f∈C1f∈C1 with arbitrary small derivative in norm C0C0 for which a motion with unbounded velocity exists.