2008/12/10 by Ángel Ballesteros, Angel Ballesteros, Alberto Enciso +3
Mathematics · Physics and Astronomy · #Classical mechanics #Geometry #Gravitational singularity #Hamiltonian (control theory) #Kepler problem #Magnetic monopole #Mathematical physics #Motion (physics) #Nonlinear Waves and Solitons #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Space (punctuation) #Symmetry (geometry) #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1016/j.aop.2009.03.001
published as Annals Phys.324:1219-1233,2009 · 22 pages
arxiv created 2008/12/10 · openalex publication_date 2009/03/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.