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...

The purpose of this paper is to compare a classical non-holonomic system-a sphere rolling against the inner surface of a vertical cylinder under gravity-and a class of discrete dynamical systems known as no-slip billiards in similar configurations. A well-known notable feature of the non-holonomic system is that the rolling sphere does not fall; its height function is bounded and oscillates harmonically up and down. The central issue of the present work is whether similar bounded behavior can be observed in the no-slip billiard counterpart. Our main results are as follows: for circular cylinders in dimension 3, the no-slip billiard has the bounded orbits property, and very closely approximates rolling motion, for a class of initial conditions which we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to an overall downward acceleration. Considering cylinders with different cross-section shapes, we show that no-slip billiards between two parallel hyperplanes in Euclidean space of arbitrary dimension are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept that depends on a factorization of the motion into transversal and longitudinal components) has period two-a very common occurrence in planar no-slip billiards-the motion in the longitudinal direction, under no forces, is generically not bounded. This is shown using a formula for a longitudinal linear drift that we prove in arbitrary dimensions. While the systems for which we can prove the existence of bounded orbits have relatively simple transverse dynamics, we also briefly explore numerically a noslip billiard system, namely the stadium cylinder billiard, that can exhibit chaotic transversal dynamics.

The question as to whether the shape of a drum can be heard has existed for around fifty years. The simple answer is ‘no’ as shown through the construction of isospectral domains. Isospectral domains are non-isometric domains that display the same spectra of frequencies of sound. These frequencies, deduced from the eigenvalues of the Laplacian, are determined by solving the wave equation in a domain omega , where alpha-omega is subject to Dirichlet boundary conditions. This paper presents methods to expand the already existing two dimensional transplantation proof into Euclidean 3-space and, through these means, provides a number of three dimensional isospectral domains.

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