Mechanics on a surface of constant negative curvature
Chaotic dynamical systems can be studied using the example of a surface of constant negative curvature. Of particular interest are tori with one exceptional point, because the motion of a particle on such a surface is very close to the scattering of an electron on a small molecule. The explicit calculations require a surface which is compatible with the modular group. The construction of the known four cases is carried out with the help of elementary number theory, and the scattering function for the solutions of the Laplace operator is obtained. The geometrical discussion leads to tori with two exceptional points, as well as a special example of the latter which is compatible with the modular group and yet does not belong to a torus with only one exceptional point. A rather unusual representation of the general Fricke-Klein groups in terms of 4 by 4 matrices is also given, which is rational in two of the three traces A, B, and C, and does not use the third one.
KeywordsFundamental Domain Negative Curvature Diophantine Equation Modular Group Commutator Subgroup
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