In mathematics and physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 1924.
It describes the geodesic motion of a free particle on the non-compact Riemann surface
is the upper half-plane endowed with the Poincaré metric and
It can be viewed as the motion on the fundamental domain of the modular group with the sides identified.
Artin's paper used symbolic dynamics for analysis of the system.
The quantum mechanical version of Artin's billiard is also exactly solvable.
are the conjugate momenta: and 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.
is a symmetric space, and is defined as the quotient of the upper half-plane modulo the action of the elements of