The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the n-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It’s conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Łojasiewicz gradient inequality.

无限自旋问题是一个关于n体问题中总碰撞轨道旋转行为的老问题。人们早就知道，当一个解趋于完全碰撞时，其归一化构型曲线必须收敛到归一化中心构型集。在平面n体问题中，每个归一化构型都确定一个旋转等效归一化构型的圆，特别是，存在归一化中心构型的圆。可以想象，通过无限旋转，总碰撞解可以收敛到这样一个圆，而不是收敛到其上的特定点。在这里，我们证明这是不可能的，至少如果中心构型的极限圆与中心构型的其他圆隔离的话。 （据信所有中心配置都是孤立的，但这通常并不为人所知。）我们的证明依赖于将中心流形定理与沃贾谢维奇梯度不等式相结合。