Collision of a Hard Ball with Singular Points of the Boundary

TL;DR

This paper analyzes how a hard ball collides with boundary singularities in physical billiards, showing that such collisions can be modeled as elastic reflections off spheres centered at the singular points.

Contribution

It demonstrates that collisions with boundary singularities in physical billiards are equivalent to elastic reflections off spheres centered at the singular points.

Findings

Collision with singular points modeled as elastic reflection off spheres

Motion after collision matches physical billiards dynamics

Provides a geometric interpretation of singular point collisions

Abstract

Recently were introduced physical billiards where a moving particle is a hard sphere rather than a point as in standard mathematical billiards. It has been shown that in the same billiard tables the physical billiards may have totally different dynamics than mathematical billiards. This difference appears if the boundary of a billiard table has visible singularities (internal corners if the billiard table is two-dimensional), i.e. the particle may collide with these singular points. Here, we consider the collision of a hard ball with a visible singular point and demonstrate that the motion of the smooth ball after collision with a visible singular point is indeed the one that was used in the studies of physical billiards. So such collision is equivalent to the elastic reflection of hard ball's center off a sphere with the center at the singular point and the same radius as the radius of the moving particle.