Spinning particles on 2-sphere in accord with the Bianchi classification

TL;DR

This paper constructs superintegrable models of spinning particles on a 2-sphere, incorporating Lie algebraic spin structures, and discusses generalizations involving monopoles, group manifolds, and scalar potentials.

Contribution

It introduces new superintegrable models with Lie algebraic spin degrees of freedom and provides a framework for extending these models to higher-dimensional Lie algebras.

Findings

Models exhibit superintegrability with quadratic constants of motion

Extensions include monopole fields and group manifold dynamics

Framework for higher-dimensional Lie algebra extensions provided

Abstract

Motivated by recent studies of superconformal mechanics extended by spin degrees of freedom, we construct minimally superintegrable models of spinning particles on 2-sphere, the spin degrees of freedom of which are represented by a 3-vector obeying the structure relations of a 3d real Lie algebra. Generalisations involving an external field of the Dirac monopole, or the motion on the group manifold of SU(2), or a scalar potential giving rise to two quadratic constants of the motion are discussed. A procedure how to build similar extensions, which rely upon d=4,5,6 real Lie algebras, is elucidated.