Related papers: Protecting billiard balls from collisions

Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partial T$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed intial point $A\in X$, a…

A body moves in a medium composed of noninteracting point particles; interaction of particles with the body is absolutely elastic. It is required to find the body's shape minimizing or maximizing resistance of the medium to its motion. This…

The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every $n$ there always exist billiard trajectories developing conjugate points at the $n$-th…

We establish a connection between capillary floating in neutral equilibrium and the billiard ball problem. This allows us to reduce the question of floating in neutral equilibrium at any orientation with a prescribed contact angle for…

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures…

A lower bound for the number of 3-periodical billiard trajectories in a manifold embedded in Euclidean space is obtained.

In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism,…

The winning rule of billiards is to drive the billiard ball on the table into the designated holes. We try to study the trajectory of the billiard ball, so that we can predict the direction of the ball. For rational slopes, we got cutting…

Have you ever played or watched a game of pool? If so, you have already seen a billiard system in action. In mathematics and physics, a billiard system describes a ball that moves in straight lines and bounces off walls. Despite these…

We present an open problem about non-colliding freely moving hard disks in the Euclidean plane, together with related positive and negative partial results. The open problem is stated in a non-degenerate form: velocities are required to be…

The well-known three distance theorem states that there are at most three distinct gaps between consecutive elements in the set of the first n multiples of any real number. We generalise this theorem to higher dimensions under a suitable…

The classical Erdos-Szekeres theorem states that a convex $k$-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several…

We consider a mechanical system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points. The ball is reflected elastically when colliding with the rod…

In this article, we consider mechanical billiard systems defined with Lagrange's integrable extension of Euler's two-center problems in the Euclidean space, on the sphere, and in the hyperbolic space of arbitrary dimension $n \ge 3$. In the…

The connected configuration space of a so called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point particle in this configuration space with…

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires…

We prove Poisson limit laws for open billiards where the holes are on the boundaries of billiard tables (rather than some abstract holes in the phase space of a billiard). Such holes are of the main interest for billiard systems, especially…

The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the…

Statistical properties for the recurrence of particles in an oval billiard with a hole in the boundary are discussed. The hole is allowed to move in the boundary under two different types of motion: (i) counterclockwise periodic circulation…

In this work we study the problem of one-dimensional elastic collisions of billiard balls, considered as rigid bodies, in a framework very different from the classical one presented in text books. Implementing the notion of impedance…