平面上部分碰撞收敛至孤立中心配置的无限自转问题
动力系统
2025-04-25 v3 经典分析与常微分方程
摘要
在n体问题中，当一簇物体趋向碰撞时，其归一化形状曲线收敛于归一化中心配置的集合，在平面情况下具有SO(2)对称性。这留下了归一化形状曲线趋向于某一中心配置旋转得到的圆，而而非其上的特定点的可能性。这就是关于n体问题中总碰撞轨道旋转行为的“无限自转问题”。该问题同样适用于部分碰撞。我们证明，如果收敛圆与中心配置集合的其他连通分量孤立，则不存在无限自转。我们的ethods extends recent work for total collision by Moeckel and Montgomery, which was based on a combination of the center manifold theorem with {\L}ojasiewicz 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.
引用
@article{arxiv.2408.16409,
title = {No infinite spin for partial collisions converging to isolated central configurations on the plane},
author = {Anna Gierzkiewicz and Rodrigo G. Schaefer and Piotr Zgliczyński},
journal= {arXiv preprint arXiv:2408.16409},
year = {2025}
}