Transition to chaos with conical billiards

We adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane with tilt angle γ. We explore the dynamics as a function of γ and the cone deficit angle χ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. We find regions of the (γ,χ) plane where, depending on the initial conditions, the trajectories either (a) sample the entire cone base and avoid the apex region, (b) sample only a portion of the base region while again avoiding the apex, or (c) sample the entire cone surface much more uniformly, suggestive of ergodicity. The special case of an untilted cone displays only type (a) trajectories, which form a ring caustic at the distance of closest approach to the apex. However, we observe an intricate transition to chaotic dynamics dominated by type (c) trajectories for sufficiently large χ and γ. A Poincaré map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. We then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps.