Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies

We prove the existence of exact partial solutions in the form of convex quadrilaterals in the general problem of four bodies mutually acting according to an arbitrary law ~1/rк , where k ≥ 2. For each fixed k ≥ 2 the distances between the bodies and their corresponding sets of four masses are found, which determine private solutions in the form of square, rhombus, deltoid and trapezoid. On the basis of the methodology of classical works the equations of motion in Raus - Lyapunov variables in the general problem of four bodies interacting according to a completely arbitrary law are derived, as it took place when Laplace proved the existence of exact partial triangular solutions of the general problem of three bodies with arbitrary masses. An explanation of the problem of existence of this type of solutions is given, due, in particular, to the more complicated geometry of quadrangular solutions in comparison with triangular ones, the existence of which is proved in the general three-body problem by the classics of celestial mechanics. It is suggested that if the arbitrariness of the interaction law is somewhat restricted, it is possible to prove by numerical methods the existence of exact partial solutions at different fixed values k ≥ 2 and unequal values of the masses of the four bodies.