Motion in a symmetric potential on the hyperbolic plane

TL;DR

This paper investigates the dynamics of a particle in a hyperbolic plane under a unidirectional potential, employing symmetry reduction techniques to analyze different force field cases and focusing on linear potentials.

Contribution

It introduces a novel symmetry reduction approach using Poisson varieties for hyperbolic plane motion under various force directions, extending classical methods to non-compact symmetry groups.

Findings

Dynamics characterized for three force field cases

Symmetry reduction via Poisson varieties applied successfully

Special analysis of linear potential cases

Abstract

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 the 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 neces- sarily 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.