论文标题
在受限制的三体问题和希尔的月球问题中确定双对称周期性轨道
Determination of the doubly-symmetric periodic orbits in the restricted three-body problem and Hill's lunar problem
论文作者
论文摘要
我们回顾了关于N体问题的周期性轨道研究的最新进展，并提出了一个数值方案，以确定空间双对称周期性轨道（简称SDSP）。计算了圆形限制的三体问题中的彗星和月球型SDSP，以及Hill的月球问题中的Hill型SDSP。利用双重对称性，以便可以有效地计算SDSP。可以通过第四个时期的信息来计算单构矩阵。周期性条件通过Broyden的方法通过线路进行了搜索来解决，并审查了算法。一些数值示例表明该方案非常有效。对于固定的周期比和给定的急性角，存在16例初始值。对于受限制的三体问题，考虑了“哥本哈根问题”和Sun-Jupiter-Asteroid模型的病例。在希尔的月球问题中也发现了新的SDSP。尽管从理论上讲，周期比应该很小，但是当比率不太小时，发现一些新的周期轨道，并且大多数搜索的SDSP是线性稳定的。
We review some recent progress on the research of the periodic orbits of the N-body problem,and propose a numerical scheme to determine the spatial doubly-symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill's lunar problem. Doubly symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden's method with a line-search, and the algorithm is reviewed. Some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of "Copenhagen problem" and the Sun-Jupiter-asteroid model are considered. New SDSPs are also numerically found in Hill's lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and most of the searched SDSPs are linearly stable.
