Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{so}(3,1)$

D. V. Novikov

Sbornik. Mathematics, Tome 205 (2014) no. 8, pp. 1107-1132

Résumé

The integrable Sokolov case on $\mathrm{so}(3,1)^{\star}$ is investigated. This is a Hamiltonian system with two degrees of freedom, in which the Hamiltonian and the additional integral are homogeneous polynomials of degrees 2 and 4, respectively. It is an interesting feature of this system that connected components of common level surfaces of the Hamiltonian and the additional integral turn out to be noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed, and the topology of noncompact level surfaces is determined, that is, the closures of solutions of the Sokolov system on $\mathrm{so}(3,1)$ are described. Bibliography: 24 titles.

Détail

Keywords: integrable Hamiltonian systems, complete vector fields, bifurcation diagram, noncompact singularities.
Mots-clés : moment map

Bibliographie

[1] H. Poincaré, “Sur la précession des corps déformables”, Bull. Astron., 27 (1910), 321–356

[2] A. V. Borisov, I. S. Mamaev, Dinamika tverdogo tela, NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevsk, 2001, 384 pp. | MR | Zbl

[3] A. V. Borisov, I. S. Mamaev, V. V. Sokolov, “A new integrable case on $\mathrm{so}(4)$”, Dokl. Phys., 46:12 (2001), 888–889 | DOI | MR

[4] V. V. Sokolov, “One class of quadratic $\mathrm{so}(4)$ Hamiltonians”, Dokl. Math., 69:1 (2004), 108–111 | MR | Zbl

[5] V. V. Sokolov, “A new integrable case for the Kirchhoff equation”, Theoret. and Math. Phys., 129:1 (2001), 1335–1340 | DOI | DOI | MR | Zbl

[6] V. V. Trofimov, A. T. Fomenko, Algebra i geometriya integriruemykh gamiltonovykh sistem differentsialnykh uravnenii, Faktorial, M.; Izd-vo Udmurt. un-ta, Izhevsk, 1995, 448 pp. | MR | Zbl

[7] 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

[8] G. Haghighatdoost, “The bifurcation diagram of a class of Hamiltonians on the algebra $\mathrm{so}(4)$”, Moscow Univ. Math. Bull., 60:6 (2005), 1–8 | MR | Zbl

[9] Gh. Haghighatdoost, “The topology of isoenergetic surfaces for the Sokolov integrable case in the Lie algebra $\mathrm{so}(4)$”, Dokl. Math., 71:2 (2005), 256–259 | MR | Zbl

[10] D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$”, Sb. Math., 202:5 (2011), 749–781 | DOI | DOI | MR | Zbl

[11] D. V. Novikov, “The topology of isoenergy surfaces for the Sokolov integrable case on the Lie algebra $\mathrm{so}(3,1)$”, Moscow Univ. Math. Bull., 66:4 (2011), 181–184 | DOI | MR

[12] D. V. Novikov, Topologiya nekotorykh novykh integriruemykh sluchaev na algebrakh Li $\mathrm{so}(4)$, $\mathrm{so}(3,1)$ i $\mathrm{e}(3)$, Diplomnaya rabota, 2007

[13] A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Russian Math. Surveys, 44:1 (1989), 181–219 | DOI | MR | Zbl

[14] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[15] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658 | DOI | MR | Zbl

[16] A. T. Fomenko, H. Zieschang, “On the topology of three-dimensional manifolds arising in Hamiltonian mechanics”, Soviet Math. Dokl., 35:2 (1987), 529–534 | MR | Zbl

[17] 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

[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, “Symmetry groups of nice Morse functions on surfaces”, Dokl. Math., 86:2 (2012), 691–693 | DOI | MR | Zbl

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

[21] A. V. Bolsinov, A. T. Fomenko, “Orbital classification of geodesic flows on two-dimensional ellipsoids. The Jacobi problem is orbitally equivalent to the integrable Euler case in rigid body dynamics”, Funct. Anal. Appl., 29:3 (1995), 149–160 | DOI | MR | Zbl

[22] A. A. Oshemkov, “Vychislenie invariantov Fomenko dlya osnovnykh integriruemykh sluchaev dinamiki tverdogo tela”, Trudy seminara po vekt. i tenz. analizu, 25 (1993), 23–109 | Zbl

[23] P. E. Ryabov, “Bifurcations of first integrals in the Sokolov case”, Theoret. and Math. Phys., 134:2 (2003), 181–197 | DOI | DOI | MR | Zbl

[24] 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