Superintegrable harmonic oscillator and Kepler-Coulomb potentials on spaces and spacetimes of constant curvature
Abstract:
A family of classical integrable systems defined on the 3D sphere,
Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter,
Minkowskian and de Sitter spacetimes is constructed. The resulting expressions
cover the six spaces in a unified way as these are parametrised by two
contraction parameters that govern the curvature and the signature of the
metric for each space. Next two maximally superintegrable Hamiltonians
are identified within the initial integrable family. The former potential
is interpreted as the superposition of a central harmonic oscillator
potential with other three oscillators or centrifugal barriers (depending
on each specific space). The latter is a superposition of the Kepler-Coulomb
potential with another two oscillators or centrifugal barriers. For both
Hamiltonians, the four functionally independent integrals or motion are
explicitly given. The corresponding generalization to arbitrary dimension
is also presented.