空间限制三体问题中截面的整体超曲面
Global hypersurfaces of section in the spatial restricted three-body problem
摘要：
我们提出了一个接触拓扑的方法来解决空间圆形限制三体问题，能量低于和略高于第一临界能量值。证明了正则化动力学的整体截面超曲面的圆族的存在性。在第一个临界值以下，这些超曲面与$2$ -球面的单位圆盘余切丛是同构的，并且它们的内部带有辛形式，每个辛形式都是与标准辛形式等价的变形。整体截面超曲面的边界是正则化动力学的不变集，它等于描述正则化平面问题的哈密顿量的水平集。第一个返回映射是哈密尔顿的，并限制到边界的时间- $1$映射的Reeb流的一个积极的重新参数化的平面问题。这种结构适用于任何质量比的选择，因此是非微扰的。我们说明的技术在完全可积的情况下，旋转开普勒问题，返回地图可以明确研究。
英文摘要：
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.
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