Investigation of manifolds and optimized trajectories in the three-body problem

Manifolds and optimal control were used to better understand trajectories in the circular restricted three-body problem (CR3BP). CR3BP equations were used to generate two-dimensional stable and unstable manifolds. Optimized trajectory solutions were found using the Hamilton-Jacobi-Bellman equation applied to the third body traveling from L 1 toward m2. Three sets of optimal trajectories with various fixed final positions were compared to the L 1 manifold. The cases with final positions closest to the manifold remained close and had lower costs. Trajectories with low time allowances took more direct paths to their final positions, leaving the manifold and resulting in higher costs. Large time allowances caused increased trajectory length and early departure from the manifold, resulting in increased cost. For the intermediate time constrained cases, the trajectories stay longer on the manifold and cost less. From this investigation, optimized trajectories were shown to use manifolds when finding the optimal trajectory in the CR3BP.