Gravitational billiards bouncing inside general domains -- foci curves and confined domains

TL;DR

This paper analyzes the trajectories of a particle bouncing under gravity inside various mirror shapes, deriving focus curves and confined domains, with visualizations for specific mirror surfaces.

Contribution

It introduces the concept of foci curves for gravitational billiards and derives the associated flight parabola envelopes for general mirror shapes.

Findings

Foci points of trajectories lie on specific curves for given initial conditions.

Confined domains are characterized by the envelope of flight parabolas and mirror surface.

Visualizations demonstrate the theoretical results for particular mirror geometries.

Abstract

A massive particle under the influence of a constant gravitational force that is bouncing inside an ideal reflecting mirror described by some function $f(x)$ is considered. For the associated flight trajectories we derive the parametric curves, named foci curves. All foci points of the parabolas for a given initial position and energy lie on these curves. From these foci curves the associated flight parabola envelopes are derived resulting, together with the mirror surface, in a confined domain for all possible particle trajectories in the non-periodic orbit case. The general results are briefly discussed and visualized for three concrete mirror surfaces.