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

The N-dimensional Hamiltonian H = 1/2f(vertical bar q vertical bar)(2) {p(2)+mu(2)/q(2)+Sigma(N)(i=1) b(i)/q(i)(2)} + U(vertical bar q vertical bar) is shown to be quasi-maximally superintegrable for any choice of the functions f and U. This result is proven by making use of the underlying sl(2, R)-coalgebra symmetry of H in order to obtain a set of (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 U. 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 U, 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 U provide the "intrinsic" oscillator and the KC potentials on these curved manifolds. As a byproduct, the MIC-Keplet, 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. (c) 2009 Elsevier Inc. All rights reserved.