平面上收敛到孤立中心构型的部分碰撞不存在无限自旋
No infinite spin for partial collisions converging to isolated central configurations on the plane
AI总结：
本文证明在平面n体问题中，若部分碰撞的极限中心构型圆与其它分量隔离，则不存在无限自旋，扩展了Moeckel-Montgomery方法并引入阴影结果。
摘要：
在$n$体问题中，当一组天体趋向于碰撞时，其归一化形状曲线收敛到归一化中心构型的集合，该集合在平面情形下具有$SO(2)$对称性。这留下了可能性：归一化形状曲线趋向于通过旋转某个中心构型得到的圆，而不是该圆上的特定点。这就是无限自旋问题，它关注$n$体问题中总碰撞轨道的旋转行为。该问题对于部分碰撞同样有意义。我们证明，如果极限圆与归一化中心构型集合的其他连通分量隔离，则无限自旋不可能发生。我们的方法扩展了Moeckel和Montgomery最近关于总碰撞的工作中的方法，该方法基于中心流形定理与Łojasiewicz不等式的结合。在此基础上，我们增加了关于法向双曲流形附近伪轨道的阴影结果，以及对其他天体对碰撞天体团影响的仔细估计。
英文摘要：
In the $n$-body problem, when a~cluster of bodies tends to a collision, then its normalized shape curve converges to the set of normalized central configurations, which has $SO(2)$ symmetry in the planar case. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation of some central configuration instead of a particular point on it. This is the \emph{infinite spin problem} which concerns the rotational behavior of total collision orbits in the $n$-body problem. The question also makes sense for partial collision. We show that the infinite spin is not possible if the limiting circle is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on a combination of the center manifold theorem with Łojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.
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