Gravitational billiards -- bouncing inside a paraboloid cavity

TL;DR

This paper analyzes the motion of a point particle inside a paraboloid cavity, proving that the foci of its flight parabola segments lie on a common sphere, with implications for understanding gravitational billiards.

Contribution

It derives the geometric properties of particle trajectories within a paraboloid, establishing that all parabola foci lie on a shared sphere, extending billiard theory to three dimensions.

Findings

Foci of consecutive parabola segments lie on a common sphere.

Results are illustrated in various limiting cases.

Comparison made with two-dimensional billiard systems.

Abstract

In this work the confined domains for a point-like particle propagating within the boundary of an ideally reflecting paraboloid mirror are derived. Thereby it is proven that all consecutive flight parabola foci points lie on the surface of a common sphere of radius $R$. The main results are illustrated in various limiting cases and are compared to its two-dimensional counterpart.