Computation of invariant manifold based on symplectic algorithm of mixed lie operator

Invariant manifolds of periodic orbit near the libration points attract a lot of attentions due to their importance in the low-energy orbits transfer problem. In the process of low-energy orbit design, the energy of the invariant manifolds must be matched, but the energy is dissipated when integrating with traditional numerical integration method. The explicit symplectic algorithm with energy conservation is more efficient than the implicit symplectic algorithm, but it requires the Hamiltonian system to be divided into two integral parts, while the circular restricted three-body problem in the rotating coordinate system being inseparable. It is diffcult to solve the circular restricted three-body problem in the rotating coordinate system by explicit symplectic algorithm. In this paper, the mixed Lie derivative operator of kinetic energy is used to solve the circular restricted three-body problem in the rotating coordinate system, and the effectiveness of this explicit symplectic algorithm with the third derivation in dealing with this problem has been showed. Compared with the Runge-Kutta45 method and Runge-Kutta78 method, the symplectic algorithm with the third-order derivative term not only has high precision but also the smallest energy error and the highest efficiency. Finally, the invariant manifolds are calculated by the symplectic algorithm with the third derivative term, the patched point can match accurately with this method.