On the characterization of integrable systems via the Haantjes geometry

TL;DR

This paper establishes that Haantjes structures are both necessary and sufficient for Hamiltonian systems to be integrable, providing a geometric framework that aids in solving separation of variables and constructing integrable systems.

Contribution

It introduces Haantjes geometry as a unifying framework for integrability, linking geometric structures to classical and new integrable Hamiltonian systems.

Findings

Haantjes structures characterize integrability in Hamiltonian systems.

The theory enables an algorithmic approach to separation of variables.

Classical systems like Gantmacher and Stäckel possess natural Haantjes structures.

Abstract

We prove that the existence of a Haantjes structure is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. This structure, expressed in terms of suitable operators whose Haantjes torsion vanishes, encodes the main features of the notion of integrability, and in particular, under certain hypotheses, allows to solve the problem of determining separation of variables for a given system in an algorithmic way. As an application of the theory, we prove theorems ensuring the existence of a large class of completely integrable systems in the Euclidean plane, constructed starting from a prescribed Haantjes structure. At the same time, we also show that some of the most classical examples of Hamiltonian systems in n dimensions, as for instance the Gantmacher and St\"ackel classes, all possess a natural Haantjes structure.