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Systems of Kowalevski type and discriminantly separable polynomials. (English) Zbl 1357.37081

A famous integrable case of the rigid body motion around a fixed point is the Kowalevski top. There is a large literature dedicated to understanding Kowalevski’s original integration procedure. In a recent paper by one of the authors, a new approach to the Kowalevski integration procedure has been suggested. Its novelty lies in the introduction of the new notion of discriminantly separable polynomials. Here, it is shown that by using discriminantly separable polynomials of degree two in each of three base variables of the problem, it is possible to construct a class of integrable dynamical systems so that all main steps of the Kowalevski’s integration procedure follow as easy and transparent logical consequences. Some new examples are discussed.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70E17 Motion of a rigid body with a fixed point
70E40 Integrable cases of motion in rigid body dynamics
14H70 Relationships between algebraic curves and integrable systems
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[1] Appel’rot, G.G., Some Supplements to the Memoir of N.B.Delone, Tr. otd. fiz. nauk, 1893, vol. 6, no. 1, pp. 1–10 (Russian).
[2] Audin, M., Spinning Tops. A Course on Integrable Systems, Cambridge Stud. Adv. Math., vol. 51, Cambridge: Cambridge Univ. Press, 1996. · Zbl 0867.58034
[3] Baker, H. F., Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions, Cambridge: Cambridge Univ. Press, 1995. · Zbl 0848.14012
[4] Bobenko, A. I., Reyman, A.G., and Semenov-Tian-Shansky, M.A., The Kowalewski Top 99 Years Later: A Lax Pair, Generalizations and Explicit Solutions, Comm. Math. Phys., 1989, vol. 122, no. 2, pp. 321–354. · Zbl 0819.58013
[5] Borisov, A. V., Kholmskaya, A. G., and Mamaev, I. S., Kovalevskaya Top and Generalizations of Integrable Systems, Regul. Chaotic Dyn., 2000, vol. 6, no. 1, pp. 1–16. · Zbl 0977.37031
[6] Darboux, G., Principes de géométrie analytique, Paris: Gauthier-Villars, 1917. · JFM 46.0877.14
[7] Delone, N.B., Algebraic Integrals of Motion of a Heavy Rigid Body around a Fixed Point, St. Petersburg: Kirschbaum, 1892 (Russian).
[8] Dragović, V., Poncelet -Darboux Curves, Their Complete Decomposition and Marden Theorem, Int. Math. Res. Not., 2011, no. 15, pp. 3502–3523. · Zbl 1230.14038
[9] Dragović, V., Generalization and Geometrization of the Kowalevski Top, Comm. Math. Phys., 2010, vol. 298, no. 1, pp. 37–64. · Zbl 1252.14021
[10] Dragović, V. and Radnović, M., Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics, Basel: Birkhäuser, 2011. · Zbl 1225.37001
[11] Dragović, V. and Kukić, K., New Examples of Systems of the Kowalevski Type, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 484–495. · Zbl 1309.37052
[12] Dragović, V. and Kukić, K., Discriminantly Separable Polynomials and Quad-Graphs, arXiv:1303.6534v1 (26 Mar 2013). · Zbl 1358.37110
[13] Dubrovin, B.A., Theta-Functions and Nonlinear Equations, Russian Math. Surveys, 1981, vol. 36, no. 2, pp. 11–92; see also: Uspekhi Mat. Nauk, 1981, vol. 36, no. 2, pp. 11–80 (Russian). · Zbl 0549.58038
[14] Gashenenko, I.N., A New Class of Motions of a Heavy Gyrostat, Soviet Phys. Dokl., 1991, vol. 36, no. 5, pp. 375–376; see also: Dokl. Akad. Nauk SSSR, 1991, vol. 318, no. 1, pp. 66–68 (Russian). · Zbl 0738.70003
[15] Golubev, V. V., Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Jerusalem: Israel Program for Scientific Translations, 1960; see also: Moscow: Gostechizdat, 1953 (Russian).
[16] Horozov, E. and van Moerbeke, P., The Full Geometry of Kowalewski’s Top and (1, 2)-Abelian Surfaces, Comm. Pure Appl. Math., 1989, vol. 42, no. 4, pp. 357–407. · Zbl 0689.58020
[17] Jurdjevic, V., Integrable Hamiltonian Systems on Lie Groups: Kowalewski Type, Ann. of Math. (2), 1999, vol. 150, no. 2, pp. 605–644. · Zbl 0953.37012
[18] Komarov, I.V., Kowalewski Basis for the Hydrogen Atom, Theoret. and Math. Phys., 1981, vol. 47, no. 1, pp. 320–324; see also: Teoret. Mat. Fiz., 1981, vol. 47, no. 1, pp. 67–72 (Russian).
[19] Komarov, I. V. and Kuznetsov, V.B., Kowalewski’s Top on the Lie Algebras o(4), e(3) and o(3, 1), J. Phys. A, 1990, vol. 23, no. 6, pp. 841–846. · Zbl 0714.58024
[20] Komarov, I.V., Sokolov, V.V., and Tsiganov, A.V., Poisson Maps and Integrable Deformations of the Kowalevski Top, J. Phys. A, 2003, vol. 36, no. 29, pp. 8035–8048. · Zbl 1073.70005
[21] Kötter, F., Sur le cas traité par Mme Kowalevski de rotation d’un corps solide pesant autour d’un point fixe, Acta Math., 1893, vol. 17, nos. 1–2, pp. 209–263.
[22] Kowalevski, S., Sur le probléme de la rotation d’un corps solide autour d’un point fixe, Acta Math., 1889, vol. 12, pp. 177–232. · JFM 21.0935.01
[23] Kowalevski, S., Sur une propriété du système d’équations différentielles qui définit la rotation d’un corps solide autour d’un point fixe, Acta Math., 1890/1891, vol. 14, pp. 81–93. · JFM 22.0921.02
[24] Mlodzieiowski, B. C., Sur un cas du mouvement d’un corps pesant autour d’un point fixe, Mat. Sb., 1896, vol. 18, no. 1, pp. 76–85 (Russian).
[25] Sokolov, V.V., A New Integrable Case for the Kirchhoff Equation, Theoret. and Math. Phys., 2001, vol. 129, no. 1, pp. 1335–1340; see also: Teoret. Mat. Fiz., 2001, vol. 129, no. 1, pp. 31–37 (Russian). · Zbl 1036.70003
[26] Sokolov, V. V., Generalized Kowalevski Top: New Integrable Cases on e(3) and so(4), in The Kowalevski Property (Leeds, 2000), V. B. Kuznetsov (Ed.), CRM Proc. Lecture Notes, vol. 32, Providence, R.I.: AMS, 2002, pp. 307–313. · Zbl 1015.37044
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