Pavel E. Ryabov

Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field

The problem of motion of the Kovalevskaya top in a double force field is investigated (the integrable case of A.G. Reyman and M.A. Semenov-Tian-Shansky without a gyrostatic momentum). It is a completely integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The critical set of the integral map is studied. The critical subsystems and bifurcation diagrams are described. The classification of all nondegenerate critical points is given. The set of these points consists of equilibria (nondegenerate singularities of rank 0), of singular periodic motions (nondegenerate singularities of rank 1), and also of critical two-frequency motions (nondegenerate singularities of rank 2).

Topological atlas of the Kowalevski–Sokolov top

We investigate the phase topology of the integrable Hamiltonian system on e(3) found by V. V. Sokolov (2001) and generalizing the Kowalevski case. This generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. The 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 isoenergy manifolds of the reduced systems with two degrees of freedom are classified. The set of critical points of the 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 diagram of the momentum map and give a classification of isoenergy and isomomentum diagrams equipped with the description of regular integral manifolds and their bifurcations. We construct the Smale–Fomenko diagrams which, when considered in the enhanced space of the energy-momentum constants and the essential physical parameters, separate 25 different types of topological invariants called the Fomenko graphs. We find all marked loop molecules of rank 0 nondegenerate critical points and of rank 1 degenerate periodic trajectories. Analyzing the cross-sections of the isointegral equipped diagrams, we get a complete list of the Fomenko graphs. The marks on them producing the exact topological invariants of Fomenko–Zieschang can be found from previous investigations of two partial cases with some additions obtained from the loop molecules or by a straightforward calculation using the separation of variables.

Topological Atlas of the Kovalevskaya Top in a Double Field

This article contains a rough topological analysis of the completely integrable system with three degrees of freedom corresponding to the motion of the Kovalevskaya top in a double field. This system is not reducible to a family of systems with two degrees of freedom. We introduce the notion of a topological atlas of an irreducible system. For the Kovalevskaya top in a double field, we complete the topological analysis of all critical subsystems with two degrees of freedom and calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the initial momentum map pointing out all chambers, families of 3-tori, and 4-atoms of their bifurcations. Basing on the ideas of A. T. Fomenko, we define the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all nondegenerate rank 1 singularities.

Phase topology of the Kowalevski–Sokolov top

The phase topology of the integrable Hamiltonian system on

The phase topology of a special case of Goryachev integrability in rigid body dynamics

e(3)

Types of critical points of the Kowalevski gyrostat in double field

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).