Second Hopf map and Yang-Coulomb system on 5d (pseudo)sphere

TL;DR

This paper uses the second Hopf map to reduce an 8D (pseudo)spherical oscillator to a 5D system with a Yang monopole, leading to new generalizations of the Yang-Coulomb system with conserved quantities.

Contribution

It introduces novel 5D Yang-Coulomb systems on (pseudo)spheres derived via Hopf map reduction, including integrable versions with Stark terms and explicit symmetry generators.

Findings

Derived 5D Yang-Coulomb systems with monopole interactions.

Identified constants of motion and hidden symmetries.

Constructed integrable generalizations with Stark terms.

Abstract

Using the second Hopf map, we perform the reduction of the eight-dimensional (pseudo)spherical (Higgs)oscillator to a five-dimensional system interacting with a Yang monopole. Then, using a standard trick, we obtain, from the latter system, the pseudospherical and spherical generalizations of the Yang-Coulomb system (the five dimensional analog of MICZ-Kepler system). We present the whole set of its constants of motions, including the hidden symmetry generators given by the analog of Runge-Lenz vector. In the same way, starting from the eight-dimensional anisotropic inharmonic Higgs oscillator, we construct the integrable (pseudo)spherical generalization of the Yang-Coulomb system with the Stark term.