Introducing symplectic billiards

Introducing projective billiards

We introduce and study a new class of dynamical systems, the projective billiards, associated with a smooth closed convex plane curve equipped with a smooth field of transverse directions. Projective billiards include the usual billiards along with the dual, or outer, billiards.

Track Billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the seg...

Quantum Billiards

Rigid-body motion, interacting billiards, and billiards on curved manifolds.

It is shown that the free motion of any three-dimensional rigid body colliding elastically between two parallel, at walls is equivalent to a three-dimensional billiard system. Depending upon the inertial parameters of the problem, the billiard system may possess a potential energy eld and a non-Euclidean connguration space. The corresponding curvilinear motion of the billiard ball does not nece...

Symplectic subspaces of symplectic Grassmannians

Let V be a non-degenerate symplectic space of dimension 2n over the field F and for a natural number l < n denote by Cl(V ) the incidence geometry whose points are the totally isotropic l-dimensional subspaces of V . Two points U, W of Cl (V ) will be collinear when W ⊂ U⊥ and dim(U ∩ W ) = l − 1 and then the line on U and W will consist of all the l-dimensional subspaces of U + W which contain...