Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles

The N-dimensional HamiltonianH=12f(|q|)2p2+μ2q2+∑i=1Nbiqi2+U(|q|)is shown to be quasi-maximally superintegrable for any choice of the functions f   and UU. This result is proven by making use of the underlying sl(2,R)sl(2,R)-coalgebra symmetry of HH in order to obtain a set of (2N-3)(2N-3) functionally independent integrals of the motion, that are explicitly given. Such constants of the motion are “universal” since all of them are independent of both f   and UU. This Hamiltonian describes the motion of a particle on any ND spherically symmetric curved space (whose metric is specified by f  ) under the action of an arbitrary central potential UU, and includes simultaneously a monopole-type contribution together with N   centrifugal terms that break the spherical symmetry. Moreover, we show that two appropriate choices for UU provide the “intrinsic” oscillator and the KC potentials on these curved manifolds. As a byproduct, the MIC–Kepler, the Taub-NUT and the so-called multifold Kepler systems are shown to belong to this class of superintegrable Hamiltonians, and new generalizations thereof are obtained. The KC and oscillator potentials on N-dimensional generalizations of the four Darboux surfaces are discussed as well.