The topology of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)

I. K. Kozlov

Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 532-572

Résumé

In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra so(4), which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank 0 are determined, the bifurcations of Liouville tori are described, and the loop molecules are computed for all singular points of the bifurcation diagrams. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra so(4) by taking a natural limit. Bibliography: 21 titles.

Détail

Keywords: integrable Hamiltonian systems, Kovalevskaya case, bifurcation diagram, topological invariants.
Mots-clés : Liouville foliation

Bibliographie

[1] S. Kowalevski, “Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe”, Acta Math., 14:1 (1890), 81–93 | DOI | MR | Zbl

[2] S. Kowalevski, “Sur le probleme de la rotation d'un corps solide autour d'un point fixe”, Acta Math., 12:1 (1889), 177–232 | DOI | MR | Zbl

[3] I. V. Komarov, “Kowalewski basis for the hydrogen atom”, Theoret. and Math. Phys., 47:1 (1981), 320–324 | DOI | MR

[4] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | MR | MR | Zbl | Zbl

[5] A. A. Oshemkov, “The topology of surfaces of constant energy and bifurcation diagrams for integrable cases of the dynamics of a rigid body on $\operatorname{SO}(4)$”, Russian Math. Surveys, 42:6 (1987), 241–242 | DOI | MR | Zbl

[6] G. Haghighatdoost, A. A. Oshemkov, “The topology of Liouville foliation for the Sokolov integrable case on the Lie algebra $\mathrm{so}(4)$”, Sb. Math., 200:6 (2009), 899–921 | DOI | DOI | MR | Zbl

[7] Kh. Khorshidi, “The topology of an integrable Hamiltonian system for the Steklov case on the Lie algebra $\mathrm{so}(4)$”, Moscow Univ. Math. Bull., 61:5 (2006), 40–44 | MR | Zbl

[8] A. S. Mishchenko, A. T. Fomenko, “Euler equations on finite-dimensional Lie groups”, Math. USSR-Izv., 12:2 (1978), 371–389 | DOI | MR | Zbl

[9] A. T. Fomenko, “Topological invariants of Liouville integrable Hamiltonian systems”, Funct. Anal. Appl., 22:4 (1988), 286–296 | DOI | MR | Zbl

[10] M. P. Kharlamov, Topologicheskii analiz integriruemykh zadach dinamiki tverdogo tela, Izd-vo LGU, L., 1988, 200 pp. | MR

[11] M. P. Kharlamov, “Bifurcation of common levels of first integrals of the Kovalevskaya problem”, J. Appl. Math. Mech., 47 (1983):6 (1985), 737–743 | DOI | MR | Zbl

[12] M. P. Kharlamov, “Topological analysis of classical integrable systems in the dynamics of a rigid body”, Soviet Math. Dokl., 28:3 (1983), 802–805 | MR | Zbl

[13] A. V. Bolsinov, P. H. Richter, A. T. Fomenko, “The method of loop molecules and the topology of the Kovalevskaya top”, Sb. Math., 191:2 (2000), 151–188 | DOI | DOI | MR | Zbl

[14] I. K. Kozlov, T. S. Rat'yu, “A bifurcation diagram for the Kovalevskaya case on the Lie algebra $\mathrm{so(4)}$”, Dokl. Math., 86:3 (2012), 827–830 | DOI | MR | Zbl

[15] A. V. Borisov, I. S. Mamaev, Sovremennye metody teorii integriruemykh sistem, Sovremennaya matematika, Institut kompyuternykh issledovanii, M.–Izhevsk, 2003, 296 pp. | MR | Zbl

[16] A. T. Fomenko, “Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body”, Differentsialnaya geometriya, gruppy Li i mekhanika. 15–2, Zap. nauchn. sem. POMI, 235, POMI, SPb., 1996, 104–183 ; J. Math. Sci. (New York), 94:4 (1999), 1512–1557 | MR | Zbl | DOI

[17] E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko, “Maximally symmetric cell decompositions of surfaces and their coverings”, Sb. Math., 199:9 (2008), 1263–1353 | DOI | DOI | MR | Zbl

[18] A. T. Fomenko, A. Yu. Konyaev, “New approach to symmetries and singularities in integrable Hamiltonian systems”, Topology Appl., 159:7 (2012), 1964–1975 | DOI | MR | Zbl

[19] E. A. Kudryavtseva, A. T. Fomenko, “Symmetries groups of nice Morse functions on surfaces”, Dokl. Math., 86:2 (2012), 691–693 | DOI | MR | Zbl

[20] A. T. Fomenko, E. A. Kudryavtseva, “Each finite group is a symmetry group of some map (an “Atom”-bifurcation)”, Moscow Univ. Math. Bull., 68:3 (2013), 148–155 | DOI

[21] P. V. Morozov, Tonkaya liuvilleva klassifikatsiya nekotorykh integriruemykh sluchaev mekhaniki tverdogo tela, kandidatskaya dissertatsiya, Mosk. gos. un-t, mekh.-matem. f-t, M., 2006, 170 pp.