AI 中文总结
本文研究Gutzwiller型各向异性开普勒问题的相对周期轨道，通过对称性约化结合相关公式证明其存在无穷多周期轨道，该模型可应用于多体问题寻找特定相对周期轨道。
AI 中文摘要
我们研究Gutzwiller型各向异性开普勒问题中的相对周期轨道，该问题是从经典Gutzwiller各向异性开普勒问题衍生出的广义模型。通过约化z轴的旋转对称性，我们得到一个具有两个自由度的约化系统，在特定参数范围内其能量曲面构成紧致正则三维球面。结合平面开普勒轨道的Seifert旋转数估计与文献[CHHL23]引入的CHHL公式，我们证明该系统在每个紧致正则能量曲面上都存在无穷多周期轨道。该模型可应用于(1+2n)体问题，以寻找无穷多具有嘻哈对称性的相对周期轨道，以及n角锥问题中的相对周期轨道。
英文摘要
We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the $z$-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in \cite{CHHL23}, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the $(1+2n)$-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the $n$-pyramidal problem.
