Numerical Solution of the System of Non-Linear Differential Equations of the Three-Body Problem in a Special Case

The aim of this study is to numerically integrate the system of non-linear differential equations of the three-body problem in a special case. The objective of the present study is to find a lunar trajectory, a manifold and to construct the Poincare section and reconstruction of the phase space in R2 for the Earth-Moon system. Then applying this model to exoplanetary systems, we find different types of orbits for a test astrophysical object in a binary star system. In this paper, we will first present a review of the three-body problem in the context of both historical and modern developments. To achieve the objectives of this study, we have performed a large number of numerical tests in MATLAB® 2024 b software by numerical integration of the nonlinear system equations of motions given by Newton’s and Hamilton’s methods. From these numerical tests, we have obtained some interesting results inside solar system and an exoplanetary system. All the results of this study are original and were calculated in the software MATLAB® 2024 b. This software is very powerful in graphics.