SOLVABLE CASES FOR THE SIMPLIFIED SYSTEMS IN THE PROBLEM OF THE MOTION OF FOUR BODIES IN THE PLANE

Abstract

The introduction contains the object of investigation the system consisting of N ordinary differential equations, which is a mathematical model of the motion of N bodies in the plane. The basic concepts are: motion of four bodies, interparticle interaction constant, Painleve property, simplified system.
The purpose of this study is to establish the analytic properties of solution of simplified systems for a system of nonlinear differential equations describing the motion of the four bodies.
The main part deals with the study of simplified systems of a system describing the planar motion of the four bodies. These systems consist of nonlinear differential equations, each of which is of second order.
A set of interparticle interaction constants in the two cases of the problem under investigation in the plane is found. The general solution at these constants can be written in closed (rather simple) form.
Obtained are 15 nonlinear autonomous differential equations of first order with respect to one of the components of the system, whose general solution is an integer, i.e., these ordinary differential equations possess the Painleve property, as well as 23 autonomous nonlinear differential equations of first order, whose general solution contains a logarithm. Therefore, these equations are not the equations of Painleve type.
Necessary and sufficient conditions are established for the existence of the Painleve property of the studied system that reveal 56 cases in the problem of four bodies in the plane when trajectories of the given bodies in the plane can be described.
The results can be applied in the analytical theory of differential equations, as well as in celestial mechanics theory.