The Fermi–Ulam problem and sticking mode

Analysis of the dynamics of a point classical particle upon its elastic reflection from a single periodically oscillating wall and in the scheme of dynamic billiards with reflections from the stationary and oscillating wall is carried out. It is demonstrated that, in the case of a single wall, the particle can stick to the wall being virtually localized on it during the half-period of oscillation and periodically reflected therefrom. It is shown that, upon varying the parameters of the problem in the range corresponding to the variation of the consecutive number of reflections from one and the same wall, the dependence of the particle velocity on these parameters has discontinuities of the derivative. For the scheme of dynamic billiard, we discuss stable regimes of various types with a constant kinetic energy of the particle and the modes of deterministic chaos. In the latter case, the presence of aforementioned discontinuities is also substantial.