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 引用本文: 郑丹丹, 罗建军, 张仁勇, 刘磊. 基于混合Lie算子辛算法的不变流形计算[J]. 力学学报, 2017, 49(5): 1126-1134.
Zheng Dandan, Luo Jianjun, Zhang Renyong, Liu Lei. COMPUTATION OF INVARIANT MANIFOLD BASED ON SYMPLECTIC ALGORITHM OF MIXED LIE OPERATOR[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1126-1134.
 Citation: Zheng Dandan, Luo Jianjun, Zhang Renyong, Liu Lei. COMPUTATION OF INVARIANT MANIFOLD BASED ON SYMPLECTIC ALGORITHM OF MIXED LIE OPERATOR[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1126-1134.

## COMPUTATION OF INVARIANT MANIFOLD BASED ON SYMPLECTIC ALGORITHM OF MIXED LIE OPERATOR

• 摘要: 平动点附近周期轨道的不变流形因其在低能轨道转移中起着重要作用而受到广泛关注.在设计低能轨道过程中不变流形要实时进行能量匹配，但利用传统数值积分方法进行积分时能量会耗散.显式辛算法具有比隐式辛算法计算效率高的优势，但其要求Hamilton系统必须分成两个可积的部分，而旋转坐标系下的圆型限制性三体问题是不可分的，因而显式辛算法难以用于求解旋转坐标系下的圆型限制性三体问题.本文通过引入混合Lie算子，成功实现了带三阶导数项的力梯度辛算法对圆型限制性三体问题的求解，并将基于混合Lie算子的带三阶导数项的辛算法与Runge-Kutta78算法和Runge-Kutta45算法进行仿真对比，仿真结果表明基于混合Lie算子的含有三阶导数项的辛算法位置精度高、能量误差小且计算效率高.利用基于混合Lie算子的带三阶导数项的辛算法计算不变流形，可以实现低能轨道转移过程中轨道拼接点的能量精准匹配.

Abstract: 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 difficult 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.

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