ИСТИНА

Planar motions of a triangular body and a massive point under the action of mutual Newtonian attraction are studied within two formulations. For the first formulation the triangle is assumed being composed by three massive points. For the second one it is constructed with three homogeneous rods. Some amazing partial solutions are pointed out within the analysis of the geometry of mass distribution. Using barycentric coordinates steady motions are found within the Routh theory. Sufficient conditions of their stability are investigated. Poincaré and Smale bifurcation diagrams are constructed for certain values of constructive parameters. Appropriate regions of possible motions are drawn. The comparative analysis of dynamics for the above mentioned formulations of the problem has been carried out. The investigation is motivated by the problem of motion of spacecrafts near an asteroid-like celestial objects possessing irregular mass distribution. If these gravitating bodies are small in comparison to a distance between them, they are likely to be considered as massive points. Meanwhile, there is another situation when gravitating bodies move at distances comparable to their sizes. It is typical for motion of a spacecraft and an asteroid under the action of the mutual attraction. Understanding why one mass distribution is more preferable in comparison with another one, with the help of an example with two types of triangles, was one of the reasons of this work.