Rotating Rod and Ball Revisited: Comparison of Billiard and Sliding Motions

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Abstract
We continue the study of the mechanical system composed of a rod (straight line) and a ball (massless point) on the plane, where the rod rotates uniformly about a fixed point and the ball reflects elastically when colliding with the rod and moves freely between successive reflections. A sliding motion of the ball along the rod is also allowed. In a previous paper it was proved that the following regimes of motion are possible: a billiard motion, a sliding motion, and the transient regimes: a billiard motion followed by sliding and sliding followed by a billiard motion. Additionally, the asymptotic behavior of the system was studied. – In this paper we conduct a more detailed study of the dynamics. It is shown, in particular, that billiard motion is asymptotically indistinguishable from sliding one; when the number of iterations is large, the billiard particle and the corresponding sliding particle practically merge, but the billiard particle can be distinguished by vibration with very high frequency and very low amplitude.

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