The open stadium

A mathematical billiard consists of a point particle moving uniformly except for mirror-like reflections from the boundary. Here we consider a stadium geometry, consisting of two parallel straight segments joined by semicircular arcs. The stadium billiard dynamics is known to have many chaotic properties. An open billiard contains a hole (for example, part of the boundary) at which the particle may escape.

The collision space of the stadium billiard is represented here in canonical coordinates, arc length on the horizontal axis and sine of the angle of reflection on the vertical axis. The colour gives the number of collisions (up to 100) before reaching the hole. The animation shows the dependence of the time to escape on the position of the hole. Parameters: Radius of the semicircles 1, straight segment length 2, hole size 1.

See also research on billiards and open systems.

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