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 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.
Hamiltonian systems with symmetry, symmetries, non-compact symmetry groups, singular reduction
37J15 - Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]
70H33 - Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
70F99 - None of the above, but in this section
37C80 - Symmetries, equivariant dynamical systems
34C14 - Symmetries, invariants
20G20 - Linear algebraic groups over the reals, the complexes, the quaternions