  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.
