Topological analysis of non-classical integrable problems of rigid body dynamics

Integrable analytic Hamiltonian system with three degrees of freedom of the mechanical origin with compact configuration space is called irreducible if it has no continuous groups of symmetries. Our lecture presents a general algorithm for topological study of such systems. The result of this study can be called a complete topological atlas of an integrable system.

The first example of implementation of the described program is a topological atlas of the irreducible integrable Kowalevski top in a double field. The classical Kowalevski problem in rigid body dynamics is one of the few integrable systems with two degrees of freedom, which can be generalized to irreducible family with three degrees of freedom. This generalization is known as the case of A.G.Reiman and M.A.Semenov-Tian-Shansky [1].

In the work [2], M.P.Kharlamov gave the stratification of the six-dimensional phase space of the generalized Kowalevski top by the rank of the momentum map by pointing out all so-called critical subsystems. A general definition of topological invariants for integrable Hamiltonian systems with many degrees of freedom is given in the works of A.T.Fomenko [3], [4]. M.P.Kharlamov in [2] formulated the problem of describing the analogue of the Fomenko invariant on iso-energy levels for the Kowalevski top in a double field.

We show the result of applying the method of critical subsystems which provides the complete classification of the bifurcation diagrams of the restriction of the momentum map to all energy levels. We give parametric classification of bifurcations, atoms and net topological invariants that determine the phase topology of the system. Thus, we present the topological atlas of the system. The total list of the net topological invariants contain nineteen types of network diagrams on five-dimensional iso-energy levels [5].

Essentially different examples are generated by integrable Hamiltonian systems with three degrees of freedom describing the rotation of a gyrostat about a fixed point in a gravity field. They reduce to a one-parameter family of integrable systems with two degrees of freedom, though the bifurcation diagrams still remain two-dimensional cell complexes in $R^3$. Moreover, the known systems with the conditions of the Kowalevski type (the cases of H.M.Yehia [6] and V.V.Sokolov [7] have additional physical parameters. Therefore, the complete investigation supposes more sophisticated methods of classification.

We prove that the critical points of the momentum map in the case of Kowalevski-Yehia are organized in three critical subsystems, and in the case of Kowalevski-Sokolov, when a homogeneous potential force field is accompanied by gyroscopic forces depending on configuration variables, there exist four critical subsystems. In both cases the critical subsystems are three-dimensional manifolds consisting of special periodic motions being the orbits of the Poisson actions associated with the momentum maps, and their bifurcations. For both systems we give a complete classification of the motions in the critical subsystems by means of constructing the key sets in the planes of the constants of two functionally independent general or partial first integrals.

We describe analytically all Smale-Fomenko diagrams which are the separating sets for one-dimensional topological invariants and explicitly compute the Morse-Bott indices for the cases of Kowalevski-Yehia and Kowalevski-Sokolov. As a result we obtain 29 iso-energy invariants in the case of Kowalevski-Yehia and 25 of invariants in the case of Kowalevski-Sokolov [8], [9].

The work is partially supported by the RFBR (grant No. 15-41-02049).