Topology of the Liouville foliation on a 2-sphere in the Dullin-Matveev integrable case

Abstract

The paper is concerned with the study of the topology of the Liouville foliations of the Dullin-Matveev integrable case. The critical point set of the Hamiltonian is found, the types of isoenergy surfaces are calculated, the non-degeneracy conditions are verified, the types of non-degenerate points of the Poisson action are determined, the moment map is investigated and the bifurcation diagram is constructed. A test for the Bott property is verified by numerical simulation. The indices of critical circles, the bifurcation types and the rough molecules are found. The rough Liouville classification of this integrable case is virtually accomplished as a result.Bibliography: 24 titles.

Smale-Fomenko diagrams and rough topological invariants of the Kowalevski-Yehia case

“…Note that in this problem we find quite a short list of basic atoms. If more complicated atoms appear, the transformations of molecules can take place on the same isoenergetic manifold without crossing degenerate points (see, for example, the results of numerical modeling in the work [25]). Nevertheless, to-day we cannot say whether it is possible, using only the local analysis of singularities, to predict such transformations and add the corresponding separating set to the Smale -Fomenko diagram.…”