New numerical methods for determining periodic orbits in the circular restricted three-body problem

The generation of periodic orbits in the context of the circular restricted three-body problem has been pursued with different approaches, for decades. The equations of motion for the problem at hand are not amenable to an analytical solution, and therefore past studies are mostly focused on analytical approximations, shooting algorithms, and generating functions. These techniques exhibit limitations due to the high sensitivity to the starting guess or to the reduced spatial range of applicability. This paper formulates the problem of determining periodic orbits as an optimal control problem, and applies two distinct methods to its solution. The first method is the particle swarm optimization technique, which is a stochastic, population-based methodology inspired by the unpredictable motion of bird flocks. The second method is represented by the direct transcription with nonlinear programming algorithm, which is based on the conversion of the original problem into a nonlinear programming problem. This research describes these two novel approaches to determining periodic orbits, their respective favorable features, and their drawbacks.