发表机构
Institut für Mathematik, Friedrich Schiller Universität Jena（耶拿弗里德里希·席勒大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对至多含四个粒子的子系统，通过最终聚类分解方法，证明了n体问题中满足特定条件的非碰撞奇异轨道集合概率极低，为解决Barry Simon提出的相关数学物理问题提供了途径。
AI 中文摘要
n体问题的非碰撞奇点是指不存在全局解但极限中不发生碰撞的初始条件，即系统的转动惯量发散。n体问题中导致非碰撞奇点的初始条件集合是否概率极低这一问题尚未解决，它是Barry Simon在1984年提出的15个数学物理问题列表中的第一个问题。迄今为止，仅在n=4的情况下该问题得到肯定回答。通过将系统最终分解为子系统，其中子系统内的粒子间相互作用强，而不同子系统的粒子间仅施加微小作用力，我们可以证明满足以下条件的所有奇异轨道集合的概率极低：- 可以有任意多的子系统在其质心处发生完全碰撞；- 可以有表现出非碰撞奇点的子系统，即该子系统各自的转动惯量发散，这些子系统每个恰好由四个粒子组成，且它们的渐近方向必须不同（这是一个温和条件）。该结果涵盖了所有已知结论，并为解决一般问题指明了方向。
英文摘要
Non-collision singularities of the $n$-body problem are initial conditions for which no global solution exists and which, however, do not lead to a collision in the limit; i.e., the moment of inertia of the system diverges. The question of whether the set of initial conditions leading to non-collision singularities in the $n$-body problem is improbable is open and is the first on Barry Simon's list of fifteen problems in mathematical physics from the year 1984. So far, this question has only been answered positively in the case $n=4$. Using a final cluster decomposition into subsystems whose particles interact strongly with each other and exert only small forces on particles from different subsystems, we can prove the improbability of the set of all singular orbits with the following conditions: -There may be arbitrarily many subsystems undergoing total collision at their center of mass. - There may be subsystems exhibiting a non-collision singularity, i.e. the respective moment of inertia of the subsystem diverges. These subsystems consist of exactly four particles each and the asymptotic directions of these subsystems have to differ (which is only a mild condition). This result includes all known statements and indicates a way how the general problem could be solved.
Comments55 pages, 7 figures
