Manifolds with a commutative and associative product structure that encodes superintegrable Hamiltonian systems

TL;DR

This paper demonstrates that certain algebraic structures on Euclidean spaces precisely encode superintegrable Hamiltonian systems, establishing a deep connection between algebraic and dynamical system frameworks.

Contribution

It shows that commutative, associative product structures satisfying specific conditions characterize all abundant superintegrable Hamiltonian systems in Euclidean spaces.

Findings

Equivalence between Manin-Frobenius manifolds and abundant structures.

Construction of superintegrable systems from algebraic structures.

Example provided with the Smorodinski-Winternitz system.

Abstract

We show that two natural and a priori unrelated structures encapsulate the same data, namely certain commutative and associative product structures and a class of superintegrable Hamiltonian systems. More precisely, consider a Euclidean space of dimension at least three, equipped with a commutative and associative product structure that satisfies the conditions of a Manin-Frobenius manifold, plus one additional compatibility condition. We prove that such a product structure encapsulates precisely the conditions of a so-called abundant structure. Such a structure provides the data needed to construct a family of second-order (maximally) superintegrable Hamiltonian systems of second order. We prove that all abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise in this way. As an example, we present the Smorodinski-Winternitz Hamiltonian system.