Mikhail Kharlamov
D.Sc., Professor Volgograd Academy of Public Administration
Department of Economics, Chair of Computer Systems and Mathematical Simulation
Publications:
Kharlamov M. P., , Savushkin A. Y.
Topological Atlas of the Kowalevski–Sokolov Top
2016, vol. 21, no. 1, pp. 2465
Abstract
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 oneparameter 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 energymomentum 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 crosssections 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.

Kharlamov M. P.
Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field
2014, vol. 19, no. 2, pp. 226244
Abstract
For the integrable system on $e(3,2)$ found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.

Kharlamov M. P.
Separation of variables in the generalized 4th Appelrot class. II. Real solutions
2009, vol. 14, no. 6, pp. 621634
Abstract
We continue the analytical solution of the integrable system with two degrees of freedom arising as the generalization of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields [Kharlamov, RCD, vol. 10, no. 4]. The separated variables found in [Kharlamov, RCD, vol. 12, no. 3] are complex in the most part of the integral constants plane. Here we present the real separating variables and obtain the algebraic expressions for the initial Euler–Poisson variables. The finite algorithm of establishing the topology of regular integral manifolds is described. The article straightforwardly refers to some formulas from [Kharlamov, RCD, vol. 12, no. 3].

Kharlamov M. P.
Separation of Variables in the Generalized 4th Appelrot Class
2007, vol. 12, no. 3, pp. 267280
Abstract
We consider an analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill a fourdimensional surface $\mathfrak{O}$ in the sixdimensional phase space. The constants of the three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in $\mathbb{R}^3$. We point out a pair of partial integrals to obtain explicit parametric equations of this sheet. The induced system on $\mathfrak{O}$ is shown to be Hamiltonian with two degrees of freedom having a thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on $\mathfrak{O}$ and algebraic formulae for the initial phase variables.

Kharlamov M. P., Shvedov E. G.
On the existence of motions in the generalized 4th Appelrot class
2006, vol. 11, no. 3, pp. 337342
Abstract
The article continues the investigations of bifurcation diagrams of the Kowalevski top in two constant fields started by the first author in [10], [1]. We analyze the admissible values of two almost everywhere independent integrals arising on the invariant submanifold that generalizes the socalled 4th Appelrot class of the Kowalevski top in the field of gravity.

Kharlamov M. P.
Bifurcation diagrams of the Kowalevski top in two constant fields
2005, vol. 10, no. 4, pp. 381398
Abstract
The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in ${\bf R}^3$. A correspondence to the Appelrot classes in the classical Kowalevski problem is established. The admissible regions for the values of the first integrals are found in the form of some inequalities of general character and boundary conditions for the induced diagrams on energy levels.

Kharlamov M. P., Zotev D. B.
Nondegenerate energy surfaces of rigid body in two constant fields
2005, vol. 10, no. 1, pp. 1519
Abstract
The problem of motion of a rigid body in two constant fields is considered. The motion is described by the Hamiltonian system with three degrees of freedom. This system in general case does not have any explicit symmetry groups and, therefore, cannot be reduced to a family of systems with two degrees of freedom. The critical points of the energy integral are found. It appeared that the energy of the system is a Morse function and has exactly four distinct critical points with different critical values and Morse indexes 0,1,2,3. In particular, the body has four equilibria, only one of which is stable. Basing on the Morse theory the smooth type of 5dimensional nondegenerate isoenergetic manifolds is pointed out.

Kharlamov M. P.,
The Bifurcations of the First Integrals in the Case of KowalewskiYehia
1997, vol. 2, no. 2, pp. 2540
Abstract
The bifurcation set in case of KowalewskiYehia integrability has been determined in this paper.
