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Haantjes algebras of classical integrable systems

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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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Acknowledgements

The research of Piergiulio Tempesta has been supported by the research project PGC2018-094898-B-I00, Ministerio de Ciencia, Innovación y Universidades and Agencia Estatal de Investigación, Spain, and by the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S), Ministerio de Ciencia, Innovación y Universidades y Agencia Estatal de Investigación, Spain. Piergiulio Tempesta is a member of the Gruppo Nazionale di Fisica Matematica (GNFM) of INDAM.

Funding

Funding was provided by Ministerio de Economía, Industria y Competitividad, Gobierno de España (Grant Nos. FIS2015-63966, MINECO, Spain and SEV-2015-0554).

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Authors and Affiliations

  1. Departamento de Física Teórica II, Facultad de Físicas, Universidad Complutense, 28040, Madrid, Spain

    Piergiulio Tempesta

  2. Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No 13–15, 28049, Madrid, Spain

    Piergiulio Tempesta

  3. Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Piaz.le Europa 1, 34127, Trieste, Italy

    Giorgio Tondo

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  1. Piergiulio Tempesta
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Correspondence to Piergiulio Tempesta.

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The original online version of this article was revised: “Modifications has been made in Page 6 and 30 of the article. Full information regarding the corrections made can be found in the erratum/correction for this article”.

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Tempesta, P., Tondo, G. Haantjes algebras of classical integrable systems. Annali di Matematica 201, 57–90 (2022). https://doi.org/10.1007/s10231-021-01107-4

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