旋转开普勒问题的特殊凹环面域
The special concave toric domain for the rotating Kepler problem
AI总结：
针对能量≤-3/2的旋转开普勒问题（RKP），通过两种辛正则化识别相空间有界分量，构造特殊凹环面域（SCTD）及组合树结构，推进辛嵌入理解并提供哈密顿动力学研究新工具。
摘要：
旋转开普勒问题（RKP）是天体力学中的基础模型，是圆型限制性三体问题的极限情况。它为研究周期轨道、能级和辛结构提供了一个易于处理且内容丰富的框架。本文研究能量值小于或等于临界阈值-3/2的RKP。通过Ligon-Schaaf和Levi-Civita辛正则化，我们识别出RKP相空间的一个有界分量。在此背景下，我们构造了一个专为RKP设计的特殊凹环面域（SCTD），为计算临界能量以下的嵌入接触同调（ECH）容量提供了具体的几何框架。SCTD能够对动力系统中的辛嵌入问题和能量约束进行严格分析。此外，我们引入了一种受Stern-Brocot树启发的组合树结构，该结构编码能量数据并便于计算有界分量上的ECH容量。这些结果推进了对天体力学中辛嵌入的理解，并为研究旋转系统中的哈密顿动力学提供了新工具。
英文摘要：
The Rotating Kepler Problem (RKP) arises as a fundamental model in celestial mechanics, appearing as a limiting case of the circular restricted three-body problem. It offers a tractable yet rich framework for studying periodic orbits, energy levels, and symplectic structures. In this work, we investigate the RKP for energy values less than or equal to the critical threshold -3/2. Using the Ligon-Schaaf and Levi-Civita symplectic regularizations, we identify a bounded component of the RKP phase space. Within this setting, we construct a special concave toric domain (SCTD) tailored to the RKP, which provides a concrete geometric framework for computing embedded contact homology (ECH) capacities below the critical energy. The SCTD enables a rigorous analysis of symplectic embedding problems and energy constraints in dynamical systems. Furthermore, we introduce a combinatorial tree structure, inspired by the Stern-Brocot tree, that encodes energy data and facilitates the computation of ECH capacities on the bounded component. These results advance the understanding of symplectic embeddings in celestial mechanics and provide new tools for the study of Hamiltonian dynamics in rotating systems.
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