Newton's Off-Center Circular Orbits and the Magnetic Monopole

TL;DR

This paper explores how introducing a magnetic monopole into Newton's off-center circular orbits preserves certain symmetries, deforms the algebra, and reveals zero modes consistent with index theorem predictions.

Contribution

It identifies a unique magnetic monopole field that maintains $E=0$ symmetry and analyzes its algebraic and quantum implications in Newtonian orbits.

Findings

Magnetic monopole preserves $E=0$ symmetry in Newtonian orbits.

Deformation of symmetry algebra by a central extension.

Quantum algebra exhibits zero modes matching index theorem predictions.

Abstract

Introducing a radially dependent magnetic field into Newton's off-center circular orbits potential so as to preserve the $E=0$ dynamical symmetry leads to a unique choice of field that can be identified as the inclusion of a magnetic monopole in the inverse stereographically projected problem. One finds also a phenomenological correspondence with that of the linearly damped Kepler model. The presence of the monopole field deforms the symmetry algebra by a central extension, and the quantum mechanical version of this algebra reveals a number of zero modes equal to that counted using the index theorem of elliptic operators.