Alexandr Burov
Publications:
Burov A. A., Guerman A., Raspopova E., Nikonov V.
On the use of the $K$means algorithm for determination of mass distributions in dumbbelllike celestial bodies
2018, Vol. 14, no. 1, pp. 4552
Abstract
It is well known that several small celestial objects are of irregular shape. In particular, there exist asteroids of the socalled “dogbone” shape. It turns out that approximation of these bodies by dumbbells, as proposed by V.V. Beletsky, provides an effective tool for analytical investigation of dynamics in vicinities of such bodies. There remains the question of how to divide reasonably a “dogbone” body into two parts using available measurement data. In this paper we introduce an approach based on the socalled $K$mean algorithm proposed by the prominent Polish mathematician H. Steinhaus. 
Burov A. A., Kosenko I.
Motion of a satellite with a variable mass distribution in a central field of Newtonian attraction
2017, Vol. 13, No. 4, pp. 519–531
Abstract
Within the framework of the socalled satellite approximation, configurations of the relative equilibrium are built and their stability is analyzed. In this case the elliptic Keplerian motion of the satellite/the spacecraft tight group mass center is predefined. The attitude motion of the system does not influence its orbital motion. The principal central axes of inertia are assumed to move as a rigid body. Simultaneously masses of the body can redistribute in a way such that the values of moments of inertia can change. Thus, all configurations can perform pulsing motions changing it own dimensions. One obtains a system of equations of motion for such a compound satellite. It turns out that the resulting system of equations is similar to the wellknown equation of V.V.Beletsky for the satellite in elliptic orbit planar oscillations. We use true anomaly as an independent variable as it is in the Beletsky equation. It turned out that there are planar pendulumlike librations of the whole system which may be regarded as perturbations of the mathematical pendulum. One can introduce actionangle variables in this case and can construct the dynamics of mappings over the nonautonomous perturbation period. As a result, one is able to apply the wellknown Moser theorem on an invariant curve for twisting maps of annulus. After that one can get a general picture of motion in the case of the system planar oscillations. So, the whole description in the paper splits into two topics: (a) general dynamical analysis of the satellite planar attitude motion using KAM theory; (b) construction of periodic solutions families depending on the perturbation parameter and rising from equilibrium as the perturbation value grows. The latter families depend on the parameter of the perturbation and are absent in the nonperturbed problem. 
Burov A. A., Guerman A., Kosenko I., Nikonov V.
On the gravity of dumbbelllike bodies represented by a pair of intersecting balls
2017, Vol. 13, No. 2, pp. 243256
Abstract
The problem of the motion of a particle in the gravity field of a homogeneous dumbbelllike body composed of a pair of intersecting balls, whose radii are, in general, different, is studied. Approximation for the Newtonian potential of attraction is obtained. Relative equilibria and their properties are studied under the assumption of uniform rotation of the dumbbells.

Burov A. A., Nikonov V.
Stability and branching of stationary rotations in a planar problem of motion of mutually gravitating triangle and material point
2016, Vol. 12, No. 2, pp. 179196
Abstract
The planar motion of an equilateral triangle with equal masses at vertices and of a point subjected to mutual Newtonian attraction is considered. Necessary conditions for the stability of “straight”, axial steady configurations, when the massive point is located on one of the symmetry axes of the triangle, are studied. The generation of other, “oblique”, steady configurations is discussed in connection with the variation, for certain parameter values, of the degree of instability of some “straight” steady configurations.
