Haantjes Algebras of Classical Integrable Systems

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 $\omega \mathcal{H}$ 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.

32 pages

References in corpus (3)

• Classical and Quantum Superintegrability with Applications
• Haantjes Manifolds and Veselov Systems
• Haantjes Algebras of the Lagrange Top

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• On the Jordan-Chevalley decomposition problem for operator fields in small dimensions and Tempesta-Tondo conjecture
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• Higher Haantjes Brackets and Integrability
• Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian
• Integrable sigma models with Haantjes structure on $H_4$ Lie group