Related papers: A survey on Polynomial in momenta integrals for bi…
We consider a multi-dimensional billiard system in an (n+1)-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and…
A new algorithm for determining the eigenstates of n-dimensional billiards is presented. It is based on the application of the Cauchy theorem for the determination of the null space of the boundary overlap matrix. The method is free from…
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.
The constrained Dirichlet boundary value problem $\ddot x=f(t,x)$, $x(0)=x(T)$, is studied in billiard spaces, where impacts occur in boundary points. Therefore we develop the research on impulsive Dirichlet problems with state-dependent…
We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case,…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
We consider the problem on the existence of two dimensional superintegrable systems in the presence of a magnetic field in the two dimensional Euclidean space. We assume the existence of two integrals of motion, besides the Hamiltonian,…
In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent. In this paper we focus on the problem of recurrence for elements of smooth curves in the moduli space. We…
Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two and three-dimensional billiards. It is shown that the time-averaged Euler characteristic of the probability…
This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding…
We study two-dimensional Riemannian metrics which are superintegrable in the class of polynomial in momenta integrals. The study is based on our main technical result, Theorem 3, which states that the Poisson bracket of two polynomial in…
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…
In this paper we establish a kind of bijection between the orbits of a polygonal outer billiards system and the orbits of a related (and simpler to analyze) system called the pinwheel map. One consequence of the result is that the outer…
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…
In this note, we use the concept of a polynomial ring to give an elementary proof to Cayley-Hamilton Theorem. We also give an elementary proof to Birkhoff theorem on Bi-stochastic matrices.
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets…
E. Gutkin found a remarkable class of convex billiard tables in the plane which have a constant angle invariant curve. In this paper we prove that in dimension 3 only round sphere has such a property. For dimension greater than 3 it must be…
For a given rotation number we compute the Hausdorff dimension of the set of well approximable numbers. We use this result and an inhomogeneous version of Jarnik's theorem to show strong recurrence properties of the billiard flow in certain…
We establish a duality between the quantum wave vector spectrum and the eigenmodes of the classical Liouvillian dynamics for integrable billiards. Signatures of the classical eigenmodes appear as peaks in the correlation function of the…
In this paper we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal{M}_\mathcal{B}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal{B}$ bounded by…