Related papers: Track billiards

The notion of a Radon transform is introduced for completely integrable billiard tables. In the case of Liouville billiard tables of dimension 3 we prove that the Radon transform is one-to-one on the space of continuous functions $K$ on the…

In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is…

Recently it was proved that every billiard trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by a $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed $C^3$…

Here we are concerned with a special issue of billiard invisibility, where a bounded set with a piecewise smooth boundary in Euclidean space is identified with a body with mirror surface, and the billiard in the complement of the set is…

Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange…

We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer…

We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal…

In standard (mathematical) billiards a point particle moves uniformly in a billiard table with elastic reflections off the boundary. We show that in transition from mathematical billiards to physical billiards, where a finite size hard…

\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of…

This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under…

We prove some partial results on the periodicity of billiard systems on graphs. The results specialize to the case of $n$ billiards with equal mass on the unit interval or circle traveling at the same speed.

We study pinball billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls are non-elastic: the outgoing angle with the normal vector to the boundary is a uniform factor $\lambda < 1$ smaller than the…

In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that $C^\infty$ generically, every periodic point is either hyperbolic…

This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiard…

We introduce a new class of billiard-like system, ``bouncing outer billiards" which are 3-dimensional cousins of outer billiards of Neumann and Moser. We prove that bouncing outer billiard on a smooth convex body has at least four…

We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift,…

A new type of classical billiard - the Andreev billiard - is investigated using the tangent map technique. Andreev billiards consist of a normal region surrounded by a superconducting region. In contrast with previously studied billiards,…

We define a new class of plane billiards - the `pensive billiard' - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality…

We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these…

We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in…