论文标题
空间限制的三体问题的全局曲面
Global hypersurfaces of section in the spatial restricted three-body problem
论文作者
论文摘要
我们建议对空间循环限制的三体问题提出一种接触式方法，以使能量以下和略高于第一个临界能量值。我们证明了针对正规动力学的截面全球超曲面的圆家族。低于第一个临界值，这些超曲面与$ 2 $ -sphere的单位磁盘cotangent束相差，并且它们在其内部具有符号形式，每个变形均等效于标准符号形式。截面全局超出表面的边界是适用于正规化动力学的一个不变集，该集合等于汉密尔顿级的水平集，描述了正则化的平面问题。第一个返回地图是哈密顿式的，并将边界限制为时间-1 $ $ 1 $的映射。这种结构可用于任何质量比的选择，因此是非扰动的。我们在旋转开普勒问题的完全可集成的情况下说明了该技术，可以在其中明确研究返回图。
We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of section for the regularized dynamics. Below the first critical value, these hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the $2$-sphere, and they carry symplectic forms on their interior, which are each deformation equivalent to the standard symplectic form. The boundary of the global hypersurface of section is an invariant set for the regularized dynamics that is equal to a level set of the Hamiltonian describing the regularized planar problem. The first return map is Hamiltonian, and restricts to the boundary as the time-$1$ map of a positive reparametrization of the Reeb flow in the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.
