Geodesic
Motion in a Symmetric Potential on the Hyperbolic Plane
Canadian journal of mathematics, Tome 67 (2015) no. 2, pp. 450-480

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DOI

We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.
DOI : 10.4153/CJM-2013-026-2
Mots-clés : 37J15, 70H33, 70F99, 37C80, 34C14, 20G20, Hamiltonian systems with symmetry, symmetries, non–compact symmetry groups, singularreduction 
Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian. Motion in a Symmetric Potential on the Hyperbolic Plane. Canadian journal of mathematics, Tome 67 (2015) no. 2, pp. 450-480. doi: 10.4153/CJM-2013-026-2
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     title = {Motion in a {Symmetric} {Potential} on the {Hyperbolic} {Plane}},
     journal = {Canadian journal of mathematics},
     pages = {450--480},
     year = {2015},
     volume = {67},
     number = {2},
     doi = {10.4153/CJM-2013-026-2},
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