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Motion in a Symmetric Potential on the Hyperbolic Plane

Published online by Cambridge University Press:  20 November 2018

Manuele Santoprete ,
Jürgen Scheurle  and
Sebastian Walcher
Manuele Santoprete
Affiliation:
Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, CanadaN2L 3C5. e-mail: msantopr@wlu.ca
Jürgen Scheurle
Affiliation:
Zentrum Mathematik, TU München, Boltzmannstr. 3, 85747 Garching, Germany. e-mail: scheurle@ma.tum.de
Sebastian Walcher
Affiliation:
Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany. e-mail: walcher@matha.rwth-aachen.de
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Abstract

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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.

Keywords

37J1570H3370F9937C8034C1420G20Hamiltonian systems with symmetrysymmetriesnon–compact symmetry groupssingular reduction
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 450 - 480
Copyright
Copyright © Canadian Mathematical Society 2015

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