标题： 空间开普勒问题中具有对称扰动的相对周期解 显示英文标题

标题： Relative periodic solutions in spatial Kepler problem with symmetric perturbation

摘要： 我们研究在满足旋转对称性和相对于垂直于旋转轴的平面的反射对称性的扰动下的空间开普勒问题。 通过应用最近的结果\cite{CHHL23}结合 Franks 定理，在扰动足够小且某些技术条件成立时，我们证明了在紧致能量面上存在无限多个相对周期解。 此外，我们还证明了一个唯一的$z$对称刹车轨道的存在，该轨道与仅需要扰动足够小的紧致能量面上的平面相对周期解形成霍夫链。 我们的结果适用于椭球体周围卫星的运动以及$n$四面体问题，其中一点质量沿$z$轴运动，其他$n$个相等质量始终形成垂直于$z$轴的正$n$边形。 显示英文摘要

摘要： We study the spatial Kepler problem under a perturbation satisfying both rotational symmetry and reflection symmetry with respect to a plane perpendicular to the rotational axis. By applying recent results from \cite{CHHL23} combined with Franks Theorem, we prove the existence of infinitely many relative periodic solutions contained in compact energy surface when the perturbation is sufficiently small and certain technical conditions hold. In addition, we also demonstrate the existence of a unique $z$-symmetric brake orbit which forms a hopf link with a plane relative periodic solutions contained in compact energy surface only requires the perturbation sufficiently small. Our results apply to the motion of a satellite around an ellipsoid with uniform mass distribution and to the $n$-pyramidal problem with one point mass moving along $z$-axis and other $n$ equal masses forming a regular $n$-gon perpendicular to the $z$-axis at all times.