Stability of periodic orbits in no-slip billiards

TL;DR

This paper investigates the stability and boundedness of periodic orbits in no-slip billiards, revealing conditions for elliptic orbits, non-ergodicity in polygonal cases, and stability thresholds in Sinai-type billiards.

Contribution

It extends previous results by providing new insights into the stability, boundedness, and existence of periodic orbits in no-slip billiard systems, including polygonal and Sinai-type billiards.

Findings

Presence of elliptic period-2 orbits in billiards with corners less than π

Polygonal no-slip billiards have invariant open sets, preventing ergodicity

Existence of linearly stable period-2 orbits in Sinai billiards with a curvature threshold

Abstract

Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards--planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain. In this paper we greatly extend previous results on boundedness of orbits (Broomhead and Gutkin) and linear stability of periodic orbits for a Sinai-type billiard (Wojtkowski) for no-slip billiards. We show among other facts that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than $\pi$, no-slip billiard dynamics will always contain elliptic period-$2$ orbits; (ii) polygonal no-slip billiards always admit small invariant open sets and thus cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the no-slip version of a Sinai billiard must contain linearly stable periodic orbits of period $2$ and, more generally, we provide a curvature threshold at which a commonly occurring period-$2$ orbit shifts from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in a class of polygonal billiards.