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| In [[mathematics]] and [[physics]], the '''Artin billiard''' is a type of a [[dynamical billiards|dynamical billiard]] first studied by [[Emil Artin]] in 1924. It describes the [[geodesic flow|geodesic motion]] of a free particle on the non-compact [[Riemann surface]] <math>\mathbb{H}/\Gamma,</math> where <math>\mathbb{H}</math> is the [[upper half-plane]] endowed with the [[Poincaré metric]] and <math>\Gamma=PSL(2,\mathbb{Z})</math> is the [[modular group]]. It can be viewed as the motion on the [[fundamental domain]] of the modular group with the sides identified.
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| The system is notable in that it is an exactly solvable system that is [[Chaos theory|strongly chaotic]]: it is not only [[ergodic]], but is also [[strong mixing]]. As such, it is an example of an [[Anosov flow]]. Artin's paper used [[symbolic dynamics]] for analysis of the system.
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| The [[quantum mechanical]] version of Artin's billiard is also exactly solvable. The eigenvalue spectrum consists of a bound state and a continuous spectrum above the energy <math>E=1/4</math>. The [[wave functions]] are given by [[Bessel function]]s.
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| ==Exposition==
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| The motion studied is that of a free particle sliding frictionlessly, namely, one having the [[Hamiltonian (quantum mechanics)|Hamiltonian]]
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| :<math>H(p,q)=\frac{1}{2m} p_i p_j g^{ij}(q)</math> | |
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| where ''m'' is the mass of the particle, <math>q^i, i=1,2</math> are the coordinates on the manifold, <math>p_i</math> are the [[conjugate momenta]]:
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| :<math>p_i=mg_{ij} \frac{dq^j}{dt}</math>
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| and
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| :<math>ds^2=g_{ij}(q) \, dq^i \, dq^j</math>
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| 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 [[geodesic]]s on the manifold.
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| In the case of the Artin billiards, the metric is given by the canonical Poincaré metric
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| :<math>ds^2=\frac{dy^2}{y^2}</math>
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| on the upper half-plane. The non-compact Riemann surface <math>\mathcal{H}/\Gamma</math> is a [[symmetric space]], and is defined as the quotient of the upper half-plane modulo the action of the elements of <math>PSL(2,\mathbb{Z})</math> acting as [[Möbius transform]]s. The set
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| :<math>U = \left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}</math>
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| is a [[fundamental domain]] for this action.
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| The manifold has, of course, one [[cusp neighborhood|cusp]]. This is the same manifold, when taken as the [[complex manifold]], that is the space on which [[elliptic curve]]s and [[modular function]]s are studied.
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| ==References==
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| * E. Artin, "Ein mechanisches System mit quasi-ergodischen Bahnen", ''Abh. Math. Sem. d. Hamburgischen Universität'', '''3''' (1924) pp170-175.
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| [[Category:Chaotic maps]]
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| [[Category:Ergodic theory]]
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