Number Theory pp 230-258 | Cite as

Mechanics on a surface of constant negative curvature

  • Martin C. Gutzwiller
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1240)


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.


Fundamental Domain Negative Curvature Diophantine Equation Modular Group Commutator Subgroup 
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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Martin C. Gutzwiller
    • 1
  1. 1.IBM T.J. Watson Research CenterYorktown HeightsUSA

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