On dual-projectively equivalent connections associated to second order superintegrable systems

TL;DR

This paper explores the geometric properties of second order superintegrable systems, revealing that certain affine connections associated with these systems share the same dual-geodesics, thus linking integrable systems with affine differential geometry.

Contribution

It establishes a connection between superintegrable systems and dual-projectively equivalent affine connections through the shared dual-geodesics property.

Findings

Certain torsion-free affine connections are associated with second order superintegrable systems.

These connections share the same dual-geodesics, linking integrable systems to affine geometry.

The work extends geometric understanding of superintegrable systems and their symmetries.

Abstract

Pre-geodesics of an affine connection are the curves that are geodesics after a reparametrization (the analogous concept in K\"ahler geometry is known as J-planar curves). Similarly, dual-geodesics on a Riemannian manifold are curves along which the 1-forms associated to the velocity are preserved after a reparametrization. Superintegrable systems are Hamiltonian systems with a large number of independent constants of the motion. They are said to be second order if the constants of the motion can be chosen to be quadratic polynomials in the momenta. Famous examples include the Kepler-Coulomb system and the isotropic harmonic oscillator. We show that certain torsion-free affine connections which are naturally associated to certain second order superintegrable systems share the same dual-geodesics.