Stability of periodic orbits in no-slip billiards

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 π, 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.We are aware of only two articles on this subject prior to our [7,8]: one by Broomhead and Gutkin [2] showing that no-slip billiard orbits in an infinite strip are bounded, and another by Wojtkowski, characterizing linear stability for a special type of period-2 orbit. In this paper we extend their results as will be detailed shortly, and develop the basic theory of no-slip billiards in a more systematic way. In this section we explain the organization of the paper and highlight our main new results.Section 2 gives preliminary information and sets notation and terminology concerning rigid collisions. It specializes the general results from [7] (stated in that paper in arbitrary dimension for bodies of general shapes and mass distributions) to discs in the plane with rotationally symmetric mass distributions. The main fact is briefly summarized in Proposition 2.1. Although the classification into specular and no-slip collisions is the same as in [2], our approach is more differential geometric in style and may have some conceptual advantages. For example, we derive this classification (in [7]) from an orthogonal decomposition of the tangent bundle T M restricted to the boundary ∂M (orthogonal relative to the kinetic energy Riemannian metric in the system's configuration manifold M ) into physically meaningful subbundles. This orthogonal decomposition is explained here only for discs in the plane.By a planar no-slip billiard system we mean a mechanical system in R 2 in which one of the colliding bodies, which may have arbitrary shape, is fixed in place, whereas the second, moving body is a disc wi...

Rolling and no-slip bouncing in cylinders

“…(The 3-dimensional circular cylinder will have some prominence here, but we also consider other domains.) In dimension 2, [3] showed that the no-slip billiard motion in an infinite strip is bounded, and in [6,7] we extended and refined this observation in a way that provides some insight into the dynamics of general polygonal no-slip billiards. As might be expected, the higher dimensional story is more subtle; we describe in this paper some of the new phenomena that arise beyond dimension 2.…”

“…Before stating the definition, we recall the set-up of no-slip billiard systems. (The reader is referred to [5] for a detailed account of what is briefly skimmed over below, and to [7] for some dynamical results for 2-dimensional systems.) In a billiard system in B, the motion of the particle (of radius r and spherically symmetric mass distribution of total mass m and distribution parameter γ) consists of a sequence of flight segments in the interior of M ⊂ SE(n) separated by instantaneous collisions with the boundary ∂M .…”

“…In dimension 3, transversal period 2 orbits are very ubiquitous and easy to obtain. (See Section 3 of [7].) Figure 2 shows the conditions under which they arise.…”

“…However, in what follows, it will be convenient to reserve the notation S ⊥ q for the orthogonal vectors to S q contained in T q (∂M ). Thus defined, we have (7) S ⊥ (A,a) =…”

“…See also [10,12], as an example of how no-slip collision models are used in situations where exchange of linear and angular momentum of particles is desired. Recently, the authors have begun to pursue a more systematic investigation of the dynamics of no-slip billiards; see [5,6,7]. Except for [5]-a differential geometric study of rigid collisions in R n in which no-slip billiards arise in a natural way-we are only aware of studies that are restricted to dimension 2.…”