INVESTIGATION OF MANIFOLDS IN THE CR3BP

Abstract

Many of the problems facing physicists and applied mathematicians involve difficulties as nonlinear governing equations, variable coefficients, and nonlinear boundary conditions at complex known or unknown boundaries that preclude solving them exactly. Manifolds and optimal control were used to better understand trajectories in the circular restricted three-body problem. Equations
of this problem were used to generate two-dimensional and three-dimensional stable and unstable invariant manifolds. The instability of periodic orbits and similar periodic solutions can be exploited to analyze paths to and from every point on a given orbit. This work
presents a systematic method for the designin of impulsive low-energy transferes between the Earth and the Moon by the explicit using invariant manifold theory. Invariant manifolds are tube-like structures along which a spacecraft may travel using no energy and this techniques usually only provied trajectories for uncontrolled spacecraft. The numerical integration requires an initial position and velocity so that it can generate a trajectory over a specified time interval. A zero initial velocity is required to find the stable manifold to travel to a Lagrange point. The stable manifold can be propagated forward and backward in time so that a spacecraft is able to travel to and from a Lagrange point on the manifold. Finally, this paper discusses the tools used to generate the trajectories in this work. In this study all units of physical quantities are non-dimensional form and the results are plotted using numerical methods in MATLAB.