This article is cited in 12 papers

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

D. V. Novikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The Sokolov integrable case on $\mathrm{e}(3)^{\star}$ is investigated. This is a Hamiltonian system with $2$ degrees of freedom in which the Hamiltonian and the additional integral are homogeneous polynomials having degree $2$ and $4$, respectively. This system is of interest because connected joint level surfaces of the Hamiltonian and the additional integral are noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed and the Liouville foliation of the system is described. The Hamiltonian vector fields corresponding to the Hamiltonian and the additional integral are proved to be complete.
Bibliography: 22 titles.

Keywords: integrable Hamiltonian systems, completeness of vector fields, bifurcation diagram, moment map, noncompact singularities.

UDC: 517.938.5

MSC: Primary 37J35; Secondary 70E40

Received: 23.11.2010

DOI: 10.4213/sm7823

 English version: Sbornik: Mathematics, 2011, 202:5, 749–781

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