Geodesic
Elliptic hydrodynamics and quadratic algebras of vector fields on a torus
Teoretičeskaâ i matematičeskaâ fizika, Tome 150 (2007) no. 3, pp. 355-370 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin–Feigin–Odesskii algebras. It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the torus.
Mots-clés : Euler hydrodynamic equation, quadratic Poisson algebra.
Keywords: ideal fluid 
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M. A. Olshanetsky. Elliptic hydrodynamics and quadratic algebras of vector fields on a torus. Teoretičeskaâ i matematičeskaâ fizika, Tome 150 (2007) no. 3, pp. 355-370. http://geodesic.mathdoc.fr/item/TMF_2007_150_3_a0/

[1] S. L. Ziglin, Dokl. AN, 250:6 (1980), 1296–1300 | MR | Zbl

[2] B. Khesin, A. Levin, M. Olshanetsky, Commun. Math. Phys., 250 (2004), 581–612 ; nlin.SI/0309017 | DOI | MR | Zbl

[3] E. K. Sklyanin, Funkts. analiz i ego prilozh., 16:4 (1982), 27–34 ; А. В. Одесский, Б. Л. Фейгин, Функц. анализ и его прилож., 23:3 (1989), 45–54 | MR | Zbl | MR | Zbl

[4] M. V. Karasev, Izv. AN SSSR. Ser. matem., 50:3 (1986), 508–538 ; B. Fuchssteiner, Progr. Theoret. Phys., 68 (1982), 1082–1104 | MR | Zbl | DOI | MR | Zbl

[5] A. V. Belavin, V. G. Drinfeld, Funkts. analiz i ego prilozh., 16:3 (1982), 1–29 | MR | Zbl

[6] V. I. Arnold, B. A. Khesin, Topological Methods in Hydrodynamics, Appl. Math. Sci., 125, Springer, N. Y., 1998 | MR | Zbl

[7] V. I. Arnold, UMN, 24:3 (1969), 225–226 | MR | Zbl

[8] F. Magri, J. Math. Phys., 19 (1978), 1156–1162 | DOI | MR | Zbl

[9] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lect. Notes Ser., 124, Cambridge Univ. Press, Cambridge, 1987 ; A. Cannas da Silva, A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Math. Lect. Notes, 10, Amer. Math. Soc., Providence, RI; Berkeley Center for Pure and Appl. Math., Berkeley, CA, 1999 | MR | Zbl | MR | Zbl