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version of Artin's billiard is also exactly solvable. The eigenvalue spectrum consists of a bound state and a continuous spectrum above the energy
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980:{\displaystyle U=\left\{z\in H:\left|z\right|>1,\,\left|\,{\mbox{Re}}(z)\,\right|<{\frac {1}{2}}\right\}}
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The motion studied is that of a free particle sliding frictionlessly, namely, one having the
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In the case of the Artin billiards, the metric is given by the canonical
Poincaré metric
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on the manifold. Because this is the free-particle
Hamiltonian, the solution to the
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The system is notable in that it is an exactly solvable system that is
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E. Artin, "Ein mechanisches System mit quasi-ergodischen Bahnen",
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Type of a dynamical billiard first studied by Emil Artin in 1924
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on the upper half-plane. The non-compact
Riemann surface
453:{\displaystyle H(p,q)={\frac {1}{2m}}p_{i}p_{j}g^{ij}(q)}
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612:{\displaystyle p_{i}=mg_{ij}{\frac {dq^{j}}{dt}}}
279:of the modular group with the sides identified.
701:{\displaystyle ds^{2}=g_{ij}(q)\,dq^{i}\,dq^{j}}
1001:. This is the same manifold, when taken as the
784:{\displaystyle ds^{2}={\frac {dy^{2}}{y^{2}}}}
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1024:Abh. Math. Sem. d. Hamburgischen Universität
264:{\displaystyle \Gamma =PSL(2,\mathbb {Z} )}
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120:Learn how and when to remove this message
275:. It can be viewed as the motion on the
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156:of a free particle on the non-compact
512:are the coordinates on the manifold,
185:{\displaystyle \mathbb {H} /\Gamma ,}
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58:adding citations to reliable sources
866:{\displaystyle PSL(2,\mathbb {Z} )}
717:Hamilton-Jacobi equations of motion
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294:. As such, it is an example of an
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505:{\displaystyle q^{i},i=1,2}
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152:in 1924. It describes the
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532:{\displaystyle p_{i}}
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298:. Artin's paper used
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286:: it is not only
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65:Find sources:
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52:Please help
47:verification
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877:. The set
361:Hamiltonian
296:Anosov flow
134:mathematics
1038:Categories
1017:References
873:acting as
355:Exposition
150:Emil Artin
80:newspapers
902:∈
814:Γ
721:geodesics
230:Γ
177:Γ
539:are the
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288:ergodic
271:is the
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138:physics
94:scholar
463:where
343:. The
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101:JSTOR
87:books
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305:The
222:and
136:and
73:news
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56:by
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