Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries

We study the geometry of reflection of a massive point-like particle at conic section boundaries. Thereby the particle is subjected to a central force associated with either a Kepler or Hooke potential. The conic section is assumed to have a focus at the Kepler center, or have its center at the Hookian center respectively. When the particle hits the boundary it is ideally reflected according to the law of reflection. These systems are known to be integrable. We describe the consecutive billiard orbits in terms of their foci. We show that the second foci of these orbits always lie on a circle in the Kepler case. In the Hooke case, we show that the foci of the orbits lie on a Cassini oval. For both systems we analyze the envelope of the directrices of the orbits as well.