Collision of a hard ball with singular points of the boundary.

Abstract

Recently, physical billiards were introduced 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. Therefore, 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. However, a ball could be rough, not smooth. In difference with a smooth ball, a rough ball also acquires rotation after reflection off a point of the boundary, which leads to more complicated dynamics.