Haantjes Algebras of Classical Integrable Systems

TL;DR

This paper introduces a tensorial geometric framework using Haantjes tensors to characterize and generate integrable Hamiltonian systems, providing new insights and models in classical integrability theory.

Contribution

It develops the concept of symplectic-Haantjes manifolds and establishes Haantjes algebras as a criterion for Liouville integrability, also deriving new integrable models.

Findings

Haantjes algebras characterize integrability in Hamiltonian systems.

New integrable models are constructed from Haantjes geometry.

Application to Post-Winternitz system and KdV hierarchy flows.

Abstract

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathscr{H}$ manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes algebras of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. We also show that new integrable models arise from the Haantjes geometry. Finally, we present an application of our approach to the study of the Post-Winternitz system and of a stationary flow of the KdV hierarchy.