Boundedness of bouncing balls in quadratic potentials

Abstract

In this paper, we consider the dynamics of a ball elastically bouncing off an infinitely heavy plate. Suppose the plate periodically moves in vertical direction and the ball between impacts is only subjected to the force with quadratic potential $U\left(z\right)=\frac{\alpha }{2}{z}^2$, where ${\left(4\alpha \right)}^{-1/2}$ is Diophantine. Without imposing any assumption on the motion of plate besides smoothness, we prove that the ball never goes to infinity. Comparing to previous works, we drop certain assumptions which are usually imposed on the motion of the plate to guarantee twist conditions. This result depends on the famed Herman’s Last Geometric Theorem, which is given by Herman no later than 1995 in his “Seminaire de Systemes Dynamiques” at the Universite Paris VII and also in his 1998 ICM address (Herman, 1998 [1]). Its proof is also provided by Fayad and Krikorian (Fayad and Krikorian, 2009 [2]) in 2009 and recently we obtained a slightly different version (Ma and Xu, 2023 [3]), which is more convenient for this physical model.