相关论文: A closed contact cycle on the ideal trefoil

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the…

We give an optical physicist view of the problem of the trajectories in a polygonal billiard using only basic facts of Optics and the theory of functions of a complex variable. This approach allow us to stablish a certain correspondence…

In this paper we show that, under certain generic conditions, billiards on ovals have only a finite number of periodic orbits, for each period, all non-degenerate and at least one of them is hyperbolic. Moreover, the invariant curves of two…

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…

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflection from a boundary. For billiards in non-convex areas bounded by segments of confocal quadrics are studied. The topology…

We give the asymptotic growth of the number of primitive periodic trajectories of a two dimensional dispersive billiard, when we prescribe their number of bounces on one of the obstacles.

Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery…

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth…

In this experimental work we study billiard trajectories in triangular pyramids and try to establish conditions that guarantee the existence (or absence) of 4-cycles (there can be not more, than three of them). We formulate conjectures and…

In this article, we define an information-theoretic entropy based on the Ihara zeta function of a graph which is called the Ihara entropy. A dynamical system consists of a billiard ball and a set of reflectors correspond to a combinatorial…

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when $\hbar$ is small in comparison to relevant actions or action differences in the corresponding classical system. In many…

We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the…

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…

A simple relation is developed between elastic collisions of freely-moving point particles in one dimension and a corresponding billiard system. For two particles with masses m_1 and m_2 on the half-line x>0 that approach an elastic barrier…

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…

We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability…

A polygon is called rational if the angle between each pair of sides is a rational multiple of $\pi.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of…

The renowned Fagnano problem asks for the inscribed triangle of minimal perimeter within a given reference triangle. Equivalently, it seeks a billiard trajectory inside the triangle that closes after three reflections. In this note, we…

We describe an exponential Fermi accelerator in a two-dimensional billiard with a moving slit. We have found a mechanism of trapping regions which provides the exponential acceleration for almost all initial conditions with sufficiently…

The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex algebraic version of…