Investigation of Manifolds in the CR3BP

Abstract

The first aim of this paper is to find the locations of two Lagrange Points (LP) L1 and L2, calculate the Jacobi constant, to study the motion of a “test particle” inside the Earth-Moon system and then, the aim of this paper is to calculate and construct stable and unstable invariant manifolds through numerical iterative methods. We will present the results graphically and from these results we will give the appropriate explanations as to how a test body moves under the action of the gravitational field of two other massive bodies, such as Earth and Moon. First, two periodic orbits around the two Lagrange points L1 and L2 are calculated. Then, in relation to these orbits, we have constructed the respective manifolds by making the careful choice of the initial conditions. Manifolds are used in different fields and have many important applications. In the field of Mechanics of Celestial Bodies and Astrodynamics, they are mainly used to better understand how the trajectory of a test celestial body in space would be. From many numerical tests, we found the appropriate initial conditions for the problem under consideration and then using the corresponding algorithm and writing the code in the MATLAB software, we have managed to find the Periodic Orbits (PO) and construct the stable and unstable manifolds of these orbits.