AI 中文总结
研究常曲率二维空间中限制性三体问题相对平衡点的分类与稳定性，结合解析方法和计算机辅助证明为κ>0 情况建立分类结果，证实相关现象并解释不同曲率下平衡点行为差异，指出正曲率对特定平衡点有稳定作用。
AI 中文摘要
我们继续由基林（《规则混沌动力学》4，（1999 年））以及马丁内斯和西莫（《天体力学与动力天文学》128，（2017 年））发起的关于常曲率二维空间中限制性三体问题相对平衡点的分类和稳定性的研究，该研究推广了平面问题的经典拉格朗日点。在将问题表述为具有两个自由度的自治拉格朗日系统（其唯一参数是主天体的曲率κ和质量比μ）后，我们通过结合解析方法和计算机辅助证明，为κ>0 的情况建立了几个分类结果。这些结果为以前只有数值证据的现象提供了严格证实。我们还对正曲率下相对平衡点行为与平面和负曲率情况下行为的定性差异提供了拓扑解释。我们的分析集中在小μ的区域，表明正曲率对三角形平衡点 L₄和 L₅有稳定作用，而负曲率似乎有相反的效果。
英文摘要
We continue the study initiated by Kilin ({\em Reg. Chaot. Dyn.} 4, (1999)) and by Martínez and Simó ({\em Celest. Mech. Dynam. Astronom.} 128, (2017)) on the classification and stability of the relative equilibria of the restricted three-body problem in two-dimensional spaces of constant curvature, which generalize the classical Lagrange points of the planar problem. After formulating the problem as an autonomous Lagrangian system with two degrees of freedom, whose only parameters are the curvature $κ$ and the mass ratio $μ$ of the primaries, we establish several classification results for the case $κ>0$ by combining analytical methods with computer-assisted proofs. These results provide rigorous confirmation of phenomena for which previously only numerical evidence was available. We also provide topological explanations for the qualitative differences between the behavior of relative equilibria in positive curvature and that observed in the planar and negative-curvature cases. Our analysis focuses on the regime of small $μ$ and indicates that positive curvature has a stabilizing effect on the triangular equilibria $\Ll_4$ and $\Ll_5$, whereas negative curvature appears to have the opposite effect.
Comments69 pages, 16 figures
