Vasily I. Nikonov

Relative equilibria in the motion of a triangle and a point under mutual attraction

The plane motion of a triangle and a material point under mutual attraction is considered when three mass points are located at the vertices of the triangle. The steady-state configurations are found and several sufficient conditions of their stability are formulated. The applicability of barycentric coordinates is studied for such problems.

On relative equilibria of mutually gravitating massive point and triangular rigid body

Planar motions of a triangular body and a massive point under the action of mutual Newtonian attraction are studied. For the first formulation the triangle is assumed to be composed of three massive points. For the second formulation it is constructed with three homogeneous rods. Some partial solutions are observed within the analysis of the geometry of mass distribution. The investigation is motivated by the problem of motion of spacecrafts near asteroid-like celestial objects possessing irregular mass distribution. Comparison of dynamical effects for two types of mass distribution is another goal of the research. Problems appearing because of irregularities in mass distributions have been known for a long time. Certain approaches to the description of motions under attraction as well as qualitative particulars of dynamics are discussed by Demin ( 1963 ), Burov & Karapetyan ( 1995 ), Buchin & Burov & Troger ( 2008 ), Burov & Guerman & Sulikashvili ( 2010 ), Kholshevnikov & Kuznetsov ( 2011 ), Beletsky & Rodnikov ( 2011 ) and Scheeres ( 2012 ).

An estimate for the number of relative equilibria in the motion of a plane rigid body and a material point under mutual attraction

The plane motion of a rigid body with a discrete mass distribution and a material point under mutual attraction is considered. The stationary configurations of this mechanical system are studied in the case when the mass of the material point can be ignored and the body rotates about its mass center at a nonzero angular velocity and in the general case of mutual interaction between the body and the material point. It is shown that in this mechanical system there always exist at least two different positions of relative equilibrium.

Libration points of a rotating complexified triangle

Similarities and differences between the force fields of a classical real dipole and a complex dipole are analyzed. The complex dipole is a pair of points equipped with complex conjugate masses and situated in a complex domain. The results of this analysis are used in the problem of motion of a material point in the field of attraction of a triangle uniformly rotating in its plane about its center of mass. It is assumed that a complex dipole is assigned to each vertex of the triangle. The existence and stability of libration points are studied. In particular, it is shown that there exist libration points outside the plane of the triangle.

Dynamics of a triple asteroid and a massive point, subjected mutual attraction

Dynamics of a triple asteroid and a massive point under the action of Newtonian attraction is considered. Existence and stability of steady motions are studied within an assumption on approximation of asteroids equation by triangular bodies, in particular, with complexified masses and point distribution.