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    Vasily Nikonov

    Leninskie gory 1, Moscow, 119991, Russia
    Lomonosov Moscow State University


    Burov A. A., Guerman A., Raspopova E., Nikonov V.
    It is well known that several small celestial objects are of irregular shape. In particular, there exist asteroids of the so-called “dog-bone” shape. It turns out that approximation of these bodies by dumb-bells, 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 so-called $K$-mean algorithm proposed by the prominent Polish mathematician H. Steinhaus.
    Keywords: $K$-means algorithm, small celestial bodies, mesh representation of an asteroid’s surface
    Citation: Burov A. A., Guerman A., Raspopova E., Nikonov V.,  On the use of the $K$-means algorithm for determination of mass distributions in dumbbell-like celestial bodies, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp.  45-52
    Burov A. A., Guerman A., Kosenko I., Nikonov V.
    The problem of the motion of a particle in the gravity field of a homogeneous dumbbell-like 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.
    Keywords: generalized planar two-bodies problem, asteroid-like systems, gravitating systems with irregular mass distribution, stability of steady motions, bifurcations of steady motions
    Citation: Burov A. A., Guerman A., Kosenko I., Nikonov V.,  On the gravity of dumbbell-like bodies represented by a pair of intersecting balls, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 2, pp.  243-256
    Burov A. A., Nikonov V.
    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.
    Keywords: generalized planar two-body problem, asteroid-like systems, gravitating systems with irregular mass distribution, stability of steady motions, necessary conditions for stability, gyroscopic stabilization, bifurcations of steady motions, Poincaré bifurcation diagrams
    Citation: Burov A. A., Nikonov V.,  Stability and branching of stationary rotations in a planar problem of motion of mutually gravitating triangle and material point, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 2, pp.  179-196

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