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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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References

  1. Aizawa, Y., Saito, N.: On the stability of isolating integrals, I. Effect of the perturbation in the potential function. J. Phys. Soc. Jpn. 32, 1636–1640 (1972)

    Google Scholar 

  2. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Springer, Berlin (1997)

    MATH  Google Scholar 

  3. Benenti, S.: Special symmetric two-tensors, equivalent dynamical systems, cofactor and bi-cofactor systems. Acta Appl. Math. 87, 33–91 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Błaszak, M.: Multi-Hamiltonian Theory of Dynamical Systems, Texts and Monographs in Physics. Springer, Berlin (1998)

    Google Scholar 

  5. Bogoyavlenskij, O.I.: General algebraic identities for the Nijenhuis and Haantjes tensors. Izv. Math. 68, 1129–1141 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Bogoyavlenskij, O.I.: Schouten tensor and bi-Hamiltonian systems of hydrodynamic type. J. Math. Phys. 47(023504), 14 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Brouzet, R.: Systemes bihamiltoniens et complete integrabilite en dimension 4. C. R. Acad. Sci. Paris 311, 895–898 (1990)

    MathSciNet  MATH  Google Scholar 

  8. Brouzet, R.: About the existence of recursion operators for completely integrable Hamiltonian systems near a Liouville torus. J. Math. Phys. 34, 1309–1313 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Chanu, C.M., Rastelli, G.: Block-separation of variables: a form of partial separation for natural Hamiltonians. SIGMA 15, 013 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Daudé, T., Kamran, N., Nicoleau, F.: Separability and symmetry operators for Painlevé metrics and their conformal deformation. SIGMA 15, 069 (2019)

    MATH  Google Scholar 

  11. Falqui, G., Magri, F., Tondo, G.: Bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy. Theor. Math. Phys. 122, 176–192 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Falqui, G., Pedroni, M.: Separation of variables for bi-Hamiltonian systems. Math. Phys. Anal. Geom. 6, 139–179 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Ferapontov, E.V.: On the matrix Hopf equation and integrable Hamiltonian system of hydrodynamic type, which do not possess Riemann invariants. Phys. Lett. A 179, 391–397 (1993)

    MathSciNet  Google Scholar 

  14. Ferapontov, E.V., Marshall, D.G.: Differential-geometric approach to the integrability of hydrodynamics chains: the Haantjes tensor. Mat. Ann. 339, 61–99 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Frolicher, A., Nijenhuis, A.: Theory of vector-valued differential forms. Part I. Indag. Math. 18, 338–359 (1956)

    MATH  Google Scholar 

  16. Gelfand, I.M., Zakharevich, I.: Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures. Selecta Math. 6, 131–183 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Gerdjikov, V.S., Vilasi, G., Yanovski, A.B.: Integrable Hamiltonian Hierarchies. Lecture Notes in Physics, vol. 748. Springer, Berlin (2008)

    MATH  Google Scholar 

  18. Haantjes, J.: On \(X_{n-1}\)-forming sets of eigenvectors. Indag. Math. 17, 158–162 (1955)

    MathSciNet  MATH  Google Scholar 

  19. Kikuchi, K., Takeuchi, T.: Examples of 4-dimensional symplectic-Haantjes manifolds. SUT J. Math. 56, 93-108 (2020)

  20. Kosmann-Schwarzbach, Y.: Beyond recursion operators. In: Kielonowski, P., Odzijewicz, A., Previato, E. (eds.) Trends in Mathematics, Geometric Methods in Physics XXXVI, Workshop and Summer School, Białowieża, Poland, July 2017. Birkhauser, Basel (2019)

  21. Kosmann-Schwarzbach, Y., Magri, F.: Poisson–Nijenhuis structures. Ann. Inst. H. Poincaré Phys. Theor. 53, 3581 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Ibort, A., Magri, F., Marmo, G.: Bi-Hamiltonian structures and Stäckel separability. J. Geom. Phys. 33, 210–228 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Ibort, A., Marmo, G., Rodríguez, M.A., Tempesta, P.: Nilpotent integrability, reduction of dynamical systems and a third-order Calogero–Moser system. Ann. Mat. Pura Appl. (2019). https://doi.org/10.1007/s10231-019-00828-x

    Article  MathSciNet  MATH  Google Scholar 

  24. Landi, G., Marmo, G., Vilasi, G.: Recursion operators: meaning and existence for completely integrable systems. J. Math. Phys. 35, 800–815 (1994)

    MathSciNet  MATH  Google Scholar 

  25. Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(5), 1156–1162 (1978)

    MathSciNet  MATH  Google Scholar 

  26. Magri, F.: A geometrical approach to nonlinear solvable equations. In: Boiti, M., Pempinelli, F., Soliani, G. (eds.) Lecture Notes in Physics, vol. 120, pp. 233–263. Springer, Berlin (1980)

    Google Scholar 

  27. Magri, F.: Geometry and soliton equations. Atti Accad. Sci. Torino Suppl. Phys. 124, 181–209 (1990)

    Google Scholar 

  28. Magri, F.: Lenard chains for classical integrable systems. Theor. Math. Phys. 137, 1716–1722 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Magri, F.: Recursion operators and Frobenius manifolds. SIGMA 8(076), 7 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Magri, F.: Haantjes manifolds. J. Phys. Conf. Ser. 482(012028), 10 (2014)

    Google Scholar 

  31. Magri, F.: WDVV Equations. Theor. Math. Phys. 189, 1486–1499 (2016), arXiv:1510.07950 (2015)

  32. Magri, F.: Haantjes manifolds and Veselov Systems. Theor. Math. Phys. 189, 1486–1499 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Magri, F., Morosi, C.: Characterization of Integrable Systems through the Theory of Poisson–Nijenhuis Manifolds, Quaderno S 19, Università di Milano (1984)

  34. Marsden, J.E., Ratiu, T.: Reduction of Poisson manifolds. Lett. Math. Phys. 11, 161–169 (1986)

    MathSciNet  MATH  Google Scholar 

  35. McKean, H.P.: Compatible brackets in Hamiltonian mechanics. In: Fokas, T., Zakharov, V.E. (eds) Important Development in Soliton Theory. Springer Series in Nonlinear Dynamics, pp. 344–354. Springer, Berlin (1993)

  36. Miller, W., Jr., Post, S., Winternitz, P.: Classical and quantum superintegrability with applications. J. Phys. A 46, 423001 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Morosi, C., Tondo, G.: Quasi-bi-Hamiltonian systems and separability. J. Phys. A 30, 2799–2806 (1997)

    MathSciNet  MATH  Google Scholar 

  38. Morosi, C., Tondo, G.: On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian. Phys. Lett. A 247, 59–64 (1998)

    MathSciNet  MATH  Google Scholar 

  39. Morosi, C., Tondo, G.: The quasi-bi-Hamiltonian formulation of the Lagrange top. J. Phys. A 35, 1741–1750 (2002)

    MathSciNet  MATH  Google Scholar 

  40. Morosi, C., Tondo, G.: Separation of variables in multi-Hamiltonian systems: an application to the Lagrange top. Theor. Math. Phys. 137, 1550–1560 (2003)

    MathSciNet  MATH  Google Scholar 

  41. Nijenhuis, A.: \(X_{n-1}\)-forming sets of eigenvectors. Indag. Math. 54, 200–212 (1951)

    MATH  Google Scholar 

  42. Nijenhuis, A.: Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, II. Indag. Math. 17(390–397), 398–403 (1955)

    MathSciNet  MATH  Google Scholar 

  43. Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras. Birkäuser, Basel (1990)

    MATH  Google Scholar 

  44. Post, S., Winternitz, P.: A non separable quantum superintegrable system in 2D real Euclidean space. J. Phys. A 44(162001), 8 (2011)

    MATH  Google Scholar 

  45. Praught, J., Smirnov, R.G.: Andrew Lenard: a mystery unraveled. SIGMA 1, 005 (2005)

    MathSciNet  MATH  Google Scholar 

  46. Reyes, D., Tempesta, P., Tondo, G.: Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry (2020). arXiv:2012.09819

  47. Stäckel, P.: Ueber quadratische Integrale der Differentialgleichungen der Dynamik. Ann. Math. Pura Appl. 25, 55–60 (1897)

    MATH  Google Scholar 

  48. Tempesta, P., Tondo, G.: Generalized Lenard chains, separation and superintegrability. Phys. Rev. E 85(046602), 11 (2012)

    Google Scholar 

  49. Tempesta, P., Tondo, G.: Haantjes algebras and diagonalization. J. Geom. Phys. 160(103968), 21 (2021). arXiv:1710:04522

    MathSciNet  MATH  Google Scholar 

  50. Tempesta, P., Winternitz, P., Harnad, J., Miller, Jr W., Pogosyan, G., Rodríguez, M.A. (eds.) Superintegrability in Classical and Quantum Systems, Montréal, CRM proceedings and Lecture Notes, AMS, vol. 37 (2004)

  51. Tondo, G.: On the integrability of stationary and restricted flows of the KdV hierarchy. J. Phys. A 28, 5097–5115 (1995)

    MathSciNet  MATH  Google Scholar 

  52. Tondo, G.: Generalized Lenard chains and multi-separability of the Smorodinsky–Winternitz system. J. Phys.: Conf. Ser. 482(012042), 10 (2014)

    Google Scholar 

  53. Tondo, G., Morosi, C.: Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation of variables. Rep. Math. Phys. 44, 255–266 (1999)

    MathSciNet  MATH  Google Scholar 

  54. Tondo, G., Tempesta, P.: Haantjes structures for the Jacobi–Calogero model and the Benenti Systems. SIGMA 12(023), 18 (2016)

    MathSciNet  MATH  Google Scholar 

  55. Tondo, G.: Haantjes Algebras of the Lagrange top. Theor. Math. Phys. 196, 1366–1379 (2018)

    MathSciNet  MATH  Google Scholar 

  56. Tempesta, P., Tondo G.: A New family of higher-order Haantjes Tensors and a Generalized Haantjes Integrability Theorem. arXiv:1809.05908v3 (2021)

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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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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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