Related papers: Projective Integrable Mechanical Billiards
Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $\nu$, of the eigenfunctions are considered. The billiards for which the time-independent Schr\"odinger equation (Helmholtz equation) is…
This paper deals with the so-called Boltzmann billiard, that is, a billiard subjected to a central force of the type $V(r)=-\alpha/r-\beta/r^2$, $\alpha$ and $\beta$ being positive constants, and with a straight reflection table. In the…
We investigate chaotic scattering on an attractive step potential with a quadrupolar deformation. The phase space features of the bound billiard are studied by using the notion of symmetry lines to find periodic orbits. We show that the…
We identify the hyperbolic Kac Moody algebras for which there exists a Lagrangian of gravity, dilatons and $p$-forms which produces a billiard that can be identified with their fundamental Weyl chamber. Because of the invariance of the…
Dynamical billiards consist of a particle on a two-dimensional table, bouncing elastically off a boundary curve. The state of the system is given by two numbers: one describing the location along the curve where the bounce occurs, and…
We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic…
We report on the experimental study of the spectral properties of quantum systems consisting of two quantum billiards (QBs), one with chaotic, the other one with integrable classical dynamics, that are coupled to each other via an opening…
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…
We introduce the spherical wedge billiard, a dynamical system consisting of a particle moving along a geodesic on a closed non-Euclidean surface of a spherical wedge. We derive the analytic form of the corresponding Poincar\'e map and find…
We derive Cayley's type conditions for periodical trajectories for the billiard within an ellipsoid in the Lobachevsky space. It appears that these new conditions are of the same form as those obtained before for the Euclidean case. We…
This work addresses the Hamiltonian dynamics of the Kepler problem in a deformed phase space, by considering the equatorial orbit. The recursion operators are constructed and used to compute the integrals of motion. The same investigation…
The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase space. The structure and topological features of typical billiard phase portrait are defined. The connection between geometric,…
We discuss the interplay between the piece-line regular and vertex-angle singular boundary effects, related to integrability and chaotic features in rational polygonal billiards. The approach to controversial issue of regular and irregular…
The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gaz, is a discontinuous map. Assuming finite horizon and another condition we introduce -- namely \emph{negligible singularities} -- we prove that the metric pressure…
The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta.…
A certain class of partial differential equations possesses singular solutions having discontinuous first derivatives ("peakons"). The time evolution of peaks of such solutions is governed by a finite dimensional completely integrable…
In this paper, we study an integrable system with both quadratic and cubic nonlinearity: $m_t=bu_x+1/2k_1[m(u^2-u^2_x)]_x+1/2k_2(2m u_x+m_xu)$, $m=u-u_{xx}$, where $b$, $k_1$ and $k_2$ are arbitrary constants. This model is kind of a cubic…
In this work, we introduce a novel concept of magic billiards, which can be seen as an umbrella, unifying several well-known generalisations of mathematical billiards. We analyse properties of magic billiards in the case of elliptical…
Billiard systems offer a simple setting to study regular and chaotic dynamics. Gravitational billiards are generalizations of these classical billiards which are amenable to both analytical and experimental investigations. Most previous…
The subject of this paper is degenerate integrability in Hamiltonian mechanics. It starts with a short survey of degenerate integrability. The first section contains basic notions. It is followed by a number of examples which include the…