Superintegrable harmonic oscillator and Kepler-Coulomb potentials on spaces and spacetimes of constant curvature
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.