Billiards in a gravitational field: A particle bouncing on a parabolic and right angle mirror

TL;DR

This paper develops geometric tools to analyze billiard trajectories in a gravitational field within specific boundary shapes, deriving equations for trajectory envelopes and revealing new geometric properties.

Contribution

It introduces novel geometric methods for studying billiards in force fields and provides explicit equations for trajectory envelopes in parabolic and right angle boundaries.

Findings

Derived equations for trajectory envelopes that match previous results.

Identified new geometric properties of billiard trajectories.

Showed impact points can be constructed via ellipse reflections in parabolic cases.

Abstract

We propose geometric tools that are suitable for studying the behavior of a billiard trajectory in a homogeneous force field. Two examples are considered: a vertical plane with an open top and with a parabolic or right angle boundary at the bottom. In either case, we obtain equations that describe the envelope of a trajectory. These equations are in good agreement with those found earlier with the use of other calculation methods. In addition, we present some new geometric properties of trajectories. We show that in the case of a parabolic boundary the sequence of the trajectory impact points can be easily constructed by multiple reflections of a single ellipse.