Maximal superintegrability of the generalized Kepler–Coulomb system onN-dimensional curved spaces

Abstract

The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper we present the generalization of this result to the N D spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature parameter. The resulting Hamiltonian, formed by the (curved) Kepler-Coulomb potential together with N centrifugal terms, is shown to be endowed with 2N − 1 functionally independent integrals of the motion: one of them is quartic and the remaining ones are quadratic. The transition from the proper Kepler-Coulomb potential, with its associated quadratic Laplace-Runge-Lenz N -vector, to the generalized system is fully described. The role of spherical, nonlinear (cubic), and coalgebra symmetries in all these systems is highlighted.

Quadratic algebra structure and spectrum of a new superintegrable system inN-dimension

“…Let us mention the possible generalizations to monopole interaction and their dual based on [17,18]. Moreover, the classification of certain families of superintegrable systems with quadratic integrals of motion in N-dimensional curved spaces have been done and their quadratic algebra structures should be studied [29]. In recent a paper [30] a superintegrable system with spin has been obtained.…”

An algebraic geometric classification of superintegrable systems in the Euclidean plane

“…Above dimension three, only sporadic families of second-order superintegrable systems are known, such as the harmonic oscillator, a generalisation of the Kepler system [PP90,BH09], respectively the hydrogen atom [Nie79], the Smorodinsky-Winternitz system I [FMS + 65], also referred to as the generic system, and the Smorodinsky-Winternitz system II [KKMP07], which can be interpreted as an anisotropic oscillator model [RTW08]. 1 From these, further n-dimensional families can be obtained through Bôcher contractions [Bôc94,KKMP07] or coupling constant metamorphosis (aka Stäckel transforms) [Pos10].…”

Classical ladder functions for Rosen–Morse and curved Kepler–Coulomb systems

“…In the classical context, the ladder functions supply information on the motion of the system. The aim of this paper is to find the ladder functions of two families of one dimensional systems known as Rosen-Morse II (RMII) [4,5] and curved Kepler-Coulomb (KC) [6,7,8,9]. Ladder functions for other one dimensional systems have been computed in previous works [10,11], but they were still missing for the RMII and curved KC systems, in particular.…”