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Torsion and semi-degeneracy of second-order maximally superintegrable systems

Jeremy Nugent, Andreas Vollmer

TL;DR

This work identifies a geometric obstruction, the tensor $N_{ijk}$, that governs whether a $(n+1)$-parameter, second-order maximally superintegrable system extends to a non-degenerate system. The authors prove that extendability is equivalent to the vanishing of $N$, and they show this condition remains valid under conformal transformations, yielding a conformal extension criterion. They reinterpret the criterion in terms of statistical manifolds with torsion, linking $N$ to vectorial torsion of a dual connection and providing a clean, covariant framework for classifying semi-degenerate versus extendable systems. The results provide both a practical test for extension and a conceptual bridge between superintegrability, Wilczynski-type equations, and torsionful geometric structures, with potential applications to higher-dimensional integrable systems and conformal variants.

Abstract

The isotropic harmonic oscillator and the Kepler-Coulomb system are pivotal models in the Sciences. They are two examples of second-order (maximally) superintegrable (Hamiltonian) systems. These systems are classified in dimension two. A partial classification exists in dimension three. In this paper, our focus is on second-order superintegrable systems with a $(n+1)$-parameter potential with $n\geq3$. We find that these systems are underpinned by an information-geometric structure, namely the structure of a statistical manifold with torsion. We obtain a necessary and sufficient condition for such systems to extend to non-degenerate systems, i.e. to admit a maximal family of compatible potentials. The condition is geometric: we show that a $(n+1)$-parameter potential is the restriction of a non-degenerate potential if and only if a certain trace-free tensor field vanishes. We interpret this condition as the requirement that a certain affine connection has vectorial torsion. We also show that the condition for a system to be extendable is conformally invariant, allowing us to extend our results to second-order conformally superintegrable systems with a $(n+1)$-parameter potential.

Torsion and semi-degeneracy of second-order maximally superintegrable systems

TL;DR

This work identifies a geometric obstruction, the tensor , that governs whether a -parameter, second-order maximally superintegrable system extends to a non-degenerate system. The authors prove that extendability is equivalent to the vanishing of , and they show this condition remains valid under conformal transformations, yielding a conformal extension criterion. They reinterpret the criterion in terms of statistical manifolds with torsion, linking to vectorial torsion of a dual connection and providing a clean, covariant framework for classifying semi-degenerate versus extendable systems. The results provide both a practical test for extension and a conceptual bridge between superintegrability, Wilczynski-type equations, and torsionful geometric structures, with potential applications to higher-dimensional integrable systems and conformal variants.

Abstract

The isotropic harmonic oscillator and the Kepler-Coulomb system are pivotal models in the Sciences. They are two examples of second-order (maximally) superintegrable (Hamiltonian) systems. These systems are classified in dimension two. A partial classification exists in dimension three. In this paper, our focus is on second-order superintegrable systems with a -parameter potential with . We find that these systems are underpinned by an information-geometric structure, namely the structure of a statistical manifold with torsion. We obtain a necessary and sufficient condition for such systems to extend to non-degenerate systems, i.e. to admit a maximal family of compatible potentials. The condition is geometric: we show that a -parameter potential is the restriction of a non-degenerate potential if and only if a certain trace-free tensor field vanishes. We interpret this condition as the requirement that a certain affine connection has vectorial torsion. We also show that the condition for a system to be extendable is conformally invariant, allowing us to extend our results to second-order conformally superintegrable systems with a -parameter potential.

Paper Structure

This paper contains 23 sections, 14 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Main results.
  3. Extendability of a (n+1)-parameter superintegrable system
  4. Interpretation in terms of statistical manifolds with torsion
  5. Structure of the paper
  6. Prerequisites
  7. Young symmetrisers
  8. Review of non-degenerate systems
  9. The structure tensor of a (n+1)-parameter system
  10. Extendable (n+1)-parameter systems
  11. Necessity of the condition in the main theorem
  12. The restricting condition
  13. Sufficiency: Extendability of a (n+1)-parameter system
  14. Extendability of the superintegrable potential
  15. The associated Killing tensors
  16. ...and 8 more sections

Key Result

Lemma 1

In a second-order superintegrable system, for a function eq:F:

Theorems & Definitions (37)

  • Remark 1
  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Example 1
  • Example 2
  • Remark 3
  • Theorem 1
  • ...and 27 more