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Functional Analysis and Its Applications

, Volume 52, Issue 4, pp 316–320 | Cite as

Integrable Systems of Algebraic Origin and Separation of Variables

  • O. K. SheinmanEmail author
Article
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Abstract

A plane algebraic curve whose Newton polygon contains d integer points is completely determined by d points in the plane through which it passes. Its coefficients regarded as functions of sets of coordinates of these points commute with respect to the Poisson bracket corresponding to the pair of coordinates of any of these points. This observation was made by Babelon and Talon in 2002. A result more general in some respects and less general in others was obtained by Enriquez and Rubtsov in 2003. As a particular case, we obtain that the coefficients of the Lagrange interpolation polynomial commute with respect to a Poisson bracket on the set of interpolation data. We prove a general assertion in the framework of the method of separation of variables which explains all these facts. This assertion is as follows: Any (nondegenerate) system of n smooth functions in n+2 variables generates an integrable system with n degrees of freedom. In addition to those mentioned above, the examples include a version of the Hermite interpolation polynomial and systems related to Weierstrass models of curves (= miniversal deformations of singularities). The integrable system related to the Lagrange interpolation polynomial has recently arisen as a reduction of rank-2 Hitchin systems (and, thereby, it gives particular solutions of such systems; see the author’s paper in Doklady Mathematics), and it is also closely related to the integrable systems on universal bundles of symmetric powers of curves introduced by Buchstaber and Mikhailov in 2017.

Key words

plane algebraic curve Poisson bracket Lagrange interpolation polynomial integrable system the method of separation of variables hyperelliptic Hitchin systems quantum analogue 

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References

  1. [1]
    Jacobi’s Lectures on Dynamics: Delivered at the University of Königsberg in the Winter Semester 1842–1843 and According to the Notes Prepared by C. W. Brockard (ed. A. Clebsch), Hindustan Book Agency, New Delhi, 2009.Google Scholar
  2. [2]
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, in: Dynamical Systems IV. Encyclopaedia Math. Sci., vol. 4, Springer-Verlag, Berlin–Heidelberg, 1990, 177–332.Google Scholar
  3. [3]
    E. K. Sklyanin, Prog. Theor. Phys. Suppl., 118:1 (1995), 35–60; http://arxiv.org/abs/ solv-int/9504001.CrossRefGoogle Scholar
  4. [4]
    V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1997.Google Scholar
  5. [5]
    S. P. Novikov and I. A. Taimanov, Modern Geometric Structures and Fields [in Russian], MTsNMO, Moscow, 2005.Google Scholar
  6. [6]
    V. M. Buchstaber and A. V. Mikhailov, Funkts. Anal. Prilozhen, 51:1 (2017), 4–27; English transl.: Functional Anal. Appl., 51:1 (2017), 2–21.CrossRefGoogle Scholar
  7. [7]
    V. M. Buchstaber and A. V. Mikhailov, Uspekhi Mat. Nauk, 73:6 (2018) (to appear).Google Scholar
  8. [8]
    V. M. Buchstaber, V. Z. Enolski, and D. V. Leykin, Multi-dimensional sigma-functions, http://arxiv.org/abs/1208.0990.Google Scholar
  9. [9]
    O. Babelon, D. Bernard, and M. Talon, Introduction to Classical Integrable Systems, Cambridge University Press, Cambridge, 2003.CrossRefzbMATHGoogle Scholar
  10. [10]
    O. Babelon and M. Talon, Phys. Lett. A, 312:1–2 (2003), 71–77; http://arxiv.org/abs/ hep-th/0209071.MathSciNetCrossRefGoogle Scholar
  11. [11]
    B. Enriquez and V. Rubtsov, Duke Math. J., 119:2 (2003), 197–219.MathSciNetCrossRefGoogle Scholar
  12. [12]
    D. Talalaev, Riemann bilinear form and Poisson structure in Hitchin-type systems, ITEP-TH- 22/03, Institute for Theoretical and Experimental Physics, Moscow, 2003.Google Scholar
  13. [13]
    K. Takasaki, in: Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, vol. 17, Amer. Math. Soc., Providence, RI, 2004; http://arxiv.org/abs/nlin/0211021.Google Scholar
  14. [14]
    O. K. Sheinman, Dokl. Ross. Akad. Nauk, Ser. Mat., 479:3 (2018), 254–256; English transl.: Russian Acad. Sci. Dokl. Math., 97:2 (2018), 144–146.Google Scholar
  15. [15]
    O. K. Sheinman, Some reductions of rank 2 and genera 2 and 3 Hitchin systems, http:// arxiv.org/abs/1709.06803.Google Scholar
  16. [16]
    O. K. Sheinman, Spectral curves of the hyperelliptic Hitchin systems, http://arxiv.org/ abs/1806.10178.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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