In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the figure-eight choreography, and explore some symmetric periodic orbits of this system which turns out to be nonautonomous. We use reversing symmetries to study both theoretically and numerically a certain type of symmetric periodic orbits of this system. The symmetric periodic orbits (initial conditions) were computed by solving some boundary-value problems.

中文翻译：

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八字形编排的受限四体问题

在这项工作中，我们引入了一个平面受限四体问题，其中一个无质量粒子在遵循 8 字形编排的三个物体的引力影响下移动，并探索该系统的一些对称周期轨道，结果证明它是非自治的。我们使用反转对称性从理论上和数值上研究该系统的某种类型的对称周期轨道。对称周期轨道（初始条件）是通过求解一些边值问题来计算的。