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arXiv · 1405.5118

Haantjes Algebras of Classical Integrable Systems

Piergiulio Tempesta·Giorgio Tondo
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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.

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Piergiulio Tempesta, Giorgio Tondo. 2014-05-20. Haantjes Algebras of Classical Integrable Systems. https://doi.org/10.1007/s10231-021-01107-4

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