In this paper, motions around the triangular libration point L4 in the planar elliptic restricted three-body problem are considered. Polar coordinates of the test particle centered at the first primary are adopted. Explicit expansion of the potential function in terms of the polar coordinates is carried out in two ways, by a recursive relation or by a direct expansion. Using the Lindstedt–Poincaré method, the explicit solution based on the polar coordinates is given for the first time. A comparison between the explicit solutions based on the Cartesian coordinates and on the polar coordinates truncated at the same order is made, and the result shows that the latter is at least one order of magnitude better in accuracy than the former. Based on the explicit solution, some analysis on the dynamics around the triangular libration points are made, including the secondary resonance structures on the \(\mu -e\) plane, the practical stable region described by the amplitudes of our explicit solution, and the resonance structures revealed by LCE maps and numerical frequency analysis.

本文考虑了平面椭圆受限三体问题中绕三角形振动点L4的运动。采用以第一初级为中心的测试粒子的极坐标。势函数在极坐标方面的显式展开有两种方式，一种是递归关系，一种是直接展开。利用 Lindstedt-Poincaré 方法，首次给出了基于极坐标的显式解。将基于笛卡尔坐标的显式解与以相同阶数截断的极坐标进行比较，结果表明后者在精度上至少比前者好一个数量级。在显式解的基础上，对三角振动点周围的动力学进行了一些分析，\(\mu -e\)平面，由我们的显式解的幅度描述的实际稳定区域，以及由 LCE 地图和数值频率分析揭示的共振结构。