Abstract. 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 ( Keywords. celestial mechanics, generalized two-body problem, relative equilibria, stability, Poincaré bifurcation diagram, barycentric coordinates Consider a motion of ΔP 1 P 2 P 3 and the point mass P in the fixed plane under the action of mutual attraction. The triangular object ΔP 1 P 2 P 3 is assumed to be composed of masses m 1 , m 2 and m 3 located at its vertices ("point triangle", PT), or by homogeneous rods P 1 P 2 , P 2 P 3 and P 1 P 3 of masses m 3 , m 1 and m 2 respectively ("wire triangle", WT). Let P have mass m.Denote M = m 1 + m 2 + m 3 , |P 1 P 2 | = 3 > 0 (1,2,3), where (1,2,3) is the cyclic permutation of indices. Let C and S be the centers of mass of the triangle and the whole system respectively, and C f be the point where the gravitational forces generated by the the triangle vanish. Then, the following statement is true.Assertion. If C = C f , then for any value of m there exist steady motions of the triangle with an arbitrary angular velocity, such that C = C f = P .For PT the condition C = C f requires C to be center of the circumscribed circle, i.e. (1, 2, 3),For WT this condition is true if and only if m 1 = m 2 = m 3 .In case of PT, positivity of masses implies that all angles of the triangle are acute. Using the Routh method (see Routh (1877)), one can investigate in general steady motions of the systems and their degrees of instability. If (x 1 , x 2 , x 3 ) are barycentric coordinates (BC) of the point P with respect to the triangle P 1 P 2 P 3 , then the Routh 170 available at https://www.cambridge.org/core/terms. https://doi
