Numerical Computation of Quasi-Periodic Solutions in the Circular Restricted Three-Body Problem Based on Percival's Variational Principle

Abstract There are various solutions to the circular restricted three-body problem, which is used as a model to describe the movement of spacecraft. One of the most famous solutions is the periodic orbits and quasi-periodic solutions that are located near the equilibrium point on the same line as the main and secondary celestial bodies. These solutions are advantageous as they can help reduce fuel consumption while maintaining the spacecraft's orbit. It is expected that these solutions will be applied in many missions. In this paper, we will first explain a functional that uses Percival's variational principle to apply an invariant torus to the circular restricted three-body problem. We will then use a method to obtain an approximate quasi-periodic solution by minimizing this functional using the steepest descent method. Next, we will compare the results obtained from the proposed method with the ones obtained by numerically integrating the circular restricted three-body problem. Finally, we will discuss the prospects of these solutions.