Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry

TL;DR

This paper introduces a geometric framework using Haantjes algebras on symplectic manifolds to understand multiseparable and superintegrable Hamiltonian systems, enabling coordinate separation and revealing multiple Haantjes structures.

Contribution

It formulates classical Hamiltonian systems with separation variables within the ($,$) structures, linking Haantjes algebras to multiseparability and superintegrability.

Findings

Multiseparable systems admit multiple $$ structures.

Large class of superintegrable systems possess multiple Haantjes structures.

Constructs Darboux-Haantjes coordinates for separation of variables.

Abstract

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ($\omega, \mathscr{H}$) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra $\mathscr{H}$, namely an algebra of (1,1)-tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We shall prove that a multiseparable system admits as many $\omega\mathscr{H}$ structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.