Solutions of three-body problem based on an equivalent system approach

Generalised reduced masses with a set of equations governing the three relative motions between two of 3-bodies in their gravitational field are established, of which the dynamic characteristics of 3-body dynamics, fundamental bases of this paper, are revealed. Based on these findings, an equivalent system is developed, which is a 2-body system with its total mass, constant angular momentum, kinetic and potential energies same as the total ones of three relative motions, so that it can be solved using the well-known theory of the 2-body system. From the solution of an equivalent system with the revealed characteristics of three relative motions, the general theoretical solutions of the 3-body system are obtained in the curve-integration forms along the orbits in the imaged radial motion space. The possible periodical orbits with generalised Kepler’s law are presented. Following the description and mathematical demonstrations of the proposed methods, the examples including Euler’s/Lagrange’s problems, and a reported numerical one are solved to validate the proposed methods. The methods derived from the 3-body system are extended to N-body problems.