Artin billiards

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In mathematics and physics, the Artin billiards are a type of dynamical billiards first studied by Emil Artin in 1924. It is a study of the geodesic motion of a free particle on the non-compact Riemann surface $\mathbb{H}/\Gamma$ where $\mathbb{H}$ is the upper half-plane and $\Gamma=PSL(2,\mathbb{Z})$ is the modular group. That is, Artin's billiards are the study of motion on the fundamental region endowed with the Poincare metric.

The system is notable in that it is an exactly solvable system that is strongly chaotic: it is not only ergodic, but is also strong mixing. As such, it is an example of an Anosov system. Another notable aspect is that Artin's original paper is the first to introduce the important concept of symbolic dynamics in the study of chaotic systems.

The quantum mechanical version of Artin's billiards is also exactly solvable. The eigenvalue spectrum consists of a bound state and a continuous spectrum above the energy $E = 1 / 4$ . The wave functions are given by Bessel functions.

[edit] Exposition

The motion studied is that of a free particle sliding frictionlessly, namely, one having the Hamiltonian

$H(p,q)=\frac{1}{2m} p_i p_j g^{ij}(q)$

where m is the mass of the particle, $q i$ , $i = 1,2$ are the coordinates on the manifold, $p i$ are the conjugate momenta:

$p_i=mg_{ij} \frac{dq^j}{dt}$

and

d s 2 = g i j (q) d q i d q j

is the metric tensor on the manifold. Because this is the free-particle Hamiltonian, the solution to the Hamilton-Jacobi equations of motion are simply given by the geodesics on the manifold.

In the case of the Artin billards, the metric is given by the canonical Poincare metric

$ds^2=\frac{dy^2}{y^2}$

on the upper half-plane. The non-compact Riemann surface $\mathcal{H}/\Gamma$ is a symmetric space, and is defined as the quotient of the upper half-plane modulo the action of the elements of $PSL(2,\mathbb{Z})$ acting as Mobius transforms. The resulting free regular set or fundamental region is given by

$U = \left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}.$

The manifold has, of course, one cusp. This is the same manifold, when taken as the complex manifold, that is the space on which elliptic curves and modular functions are studied.