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

TL;DR

This paper demonstrates that a broad class of N-dimensional curved space Hamiltonians with central potentials, monopoles, and centrifugal terms are quasi-maximally superintegrable, with universal integrals of motion derived from sl(2,R) symmetry.

Contribution

It introduces a universal framework for superintegrability in N-dimensional curved spaces with arbitrary functions, including monopoles and centrifugal effects, unifying several known superintegrable systems.

Findings

Hamiltonian is quasi-maximally superintegrable for any functions f and U

Explicit integrals of motion are derived from sl(2,R) symmetry

Includes known systems like MIC-Kepler and Taub-NUT as special cases

Abstract

The N-dimensional Hamiltonian H formed by a curved kinetic term (depending on a function f), a central potential (depending on a function U), a Dirac monopole term, and N centrifugal terms 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 a function f) under the action of an arbitrary central potental 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-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 Kepler and oscillator potentials on N-dimensional generalizations of the four Darboux surfaces are discussed as well.