Related papers: A Three Dimensional Gravitational Billiard in a Co…
We analyze the impact of two equal billiard balls in three ideal situations: when the balls freely slide on the plane of the billiard, when they roll without sliding and when one of them freely slides and the other rolls. In all the cases…
Motivated by the high-energy limit of the $N$-body problem we construct non-deterministic billiard process. The billiard table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a…
We apply periodic orbit theory to a two-dimensional non-integrable billiard system whose boundary is varied smoothly from a circular to an equilateral triangular shape. Although the classical dynamics becomes chaotic with increasing…
We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of…
Systems of pinned billiard balls serve as simplified models of collisions, where all particles remain fixed in their positions while their (pseudo-)velocities evolve in accordance with the laws of conservation of energy and momentum. For…
We investigate the semiclassical energy spectrum of quantum elliptic billiard. The nearest neighbor spacing distribution, level number variance and spectral rigidity support the notion that the elliptic billiard is a generic integrable…
The paper gives a review of very recent results related to the Poncelet Theorem, on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of Geometry, recalling for the general mathematical…
The most general solution to the Einstein equations in $4=3+1$ dimensions in the asymptotical limit close to the cosmological singularity under the BKL (Belinski-Khalatnikov-Lifshitz) hypothesis, for which space gradients are neglected and…
For two-dimensional quantum billiards we derive the partial Weyl law, i.e. the average density of states, for a subset of eigenstates concentrating on an invariant region $\Gamma$ of phase space. The leading term is proportional to the area…
Generic one-parameter billiards are studied both classically and quantally. The classical dynamics for the billiards makes a transition from regular to fully chaotic motion through intermediary soft chaotic system. The energy spectra of the…
We study the "fermionic billiards", i.e. the chaotic dynamics of the gravitino, that arise in the near-spacelike-singularity limit of eleven-dimensional supergravity and of its dimensional truncations (notably four-dimensional simple…
We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards.…
We introduce a class of billiards with chaotic unidirectional flows (or non-chaotic unidirectional flows with "vortices") which go around a chaotic or non-chaotic "core", where orbits can change their orientation. Moreover, the…
We show that wave functions in planar rational polygonal billiards (all angles rationally related to Pi) can be expanded in a basis of quasi-stationary and spatially regular states. Unlike the energy eigenstates, these states are directly…
We study dynamics of a ball moving in gravitational field and colliding with a moving table. The motion of the limiter is assumed as periodic with piecewise constant velocity - it is assumed that the table moves up with a constant velocity…
We consider a polygon in a two-dimensional plane with a homogeneous constant magnetic field orthogonal to such plane, but inside the polygon, the magnetic field is zero. We study the dynamics of an electron with an initial velocity in this…
\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…
We study classical and quantum dynamics of a particle in a circular billiard with a straight cut. This system can be integrable, nonintegrable with soft chaos, or nonintegrable with hard chaos, as we vary the size of the cut. We use a…
We consider a bimodal light field envelope propagating in a bulk medium characterized by competing cubic and quintic nonlinearities. The subfields are coupled by a cross-phase modulation term and experience effective attraction. We find…
Consider a strictly convex set $\Omega$ in the plane, and a homogeneous, stationary magnetic field orthogonal to the plane whose strength is $B$ on the complement of $\Omega$ and $0$ inside $\Omega$. The trajectories of a charged particle…