固定均匀立方体重力场中的周期轨道
Periodic orbits in the gravity field of a fixed homogeneous cube
摘要：
在目前的研究中，周期轨道的存在性围绕一个固定的均匀立方体进行了调查，其结果有强大的影响，检查周期轨道周围的非球形天体。在固定立方体的两种不同类型的对称平面上，利用Poincaré截面法得到了周期轨道。而在一般的位置，周期轨道的同伦方法找到。结果表明，周期轨道广泛存在于固定立方体的对称平面内，也存在于包含规则Hex截面的非对称平面附近。这些周期轨道的稳定性确定的基础上的monodromy矩阵的特征值。证明了同伦方法在寻找立方体引力场中周期轨道的有效性，为寻找非球形天体的周期轨道提供了一种新的思路。对立方体轨道的研究可以看作是复杂情况下的第一步，有助于理解复杂形状物体的轨道动力学。本文的工作是前人关于简单形体（包括直段、圆环、圆环盘和简单平板）轨道动力学研究工作的推广。
英文摘要：
In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincaré surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.
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