Phase topology of the Kowalevski -Sokolov top
The phase topology of the integrable Hamiltonian system on e(3) found by V. V. Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force ﬁeld, gyroscopic forces depending on the conﬁgurational variables. Relative equilibria are classiﬁed, their type is calculated and the character of stability is deﬁned. The Smale diagrams of the case are found and the classiﬁcation of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values deﬁning their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).