Trajectories of rolling marbles on various funnels

We investigate the trajectories of a small marble constrained to roll on frictionless surfaces, called funnels, of varying shapes. Actual coin funnels have a hyperbolic surface, and here we disprove the common claim that the orbits of the rolling marble or coin are the same as the Kepler orbits for planets revolving around the sun. In fact, it is straightforward to show that for no funnel surface can the Kepler orbits be recovered. Furthermore, we find that the types of trajectories that can arise depend heavily on the funnel shape.