n体问题中碰撞奇点的爆破、渐近行为与碰撞的不可能性
Blow-Up, Asymptotics, and Improbability of Collision Singularities in the n-Body Problem
- University of Sydney（悉尼大学）
- Friedrich Schiller Universität Jena（耶拿弗里德里希·席勒大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对带时变外力的n体系统，推广von Zeipel和Painlevé结果，实现能量不守恒时仍有效的McGehee爆破，证明碰撞初始条件测度为零。
AI中文摘要：
我们研究具有度数为-α的齐次对势的n体系统中的有限时间奇点，其中0<α<2，这些对势不一定是吸引的，且带有有界的随时间变化的外力。我们开发了一种系统方法，用于分析系统中其他地方存在同时碰撞或非碰撞奇点时的碰撞子系统。我们将von Zeipel和Painlevé的经典结果推广到这类子系统，确定了总碰撞附近子系统内部能量的界，并实现了在能量不守恒时仍有效的McGehee爆破。该爆破给出了团簇大小的渐近行为，在吸引势的情况下，还给出了向中心构型集合的收敛性以及相对位置、相对速度和内部角动量的渐近行为。作为渐近结果的应用，我们构造了一个简化证明，证明导致碰撞的初始条件构成测度为零的集合。
英文摘要：
We study finite-time singularities in $n$-body systems with pair potentials that are homogeneous of degree $-α$, where $0<α<2$, that are not necessarily attractive and with bounded time-dependent external forces. We develop a systematic method to analyse a colliding subsystem in the presence of simultaneous collision or non-collision singularities elsewhere in the system. We extend classical results of von Zeipel and Painlevé to such subsystems, establish bounds on their internal energy near total collision, and implement a McGehee blow-up that remains effective without conservation of energy. The blow-up yields the asymptotics of the cluster size, and in the case of attractive potentials, the convergence towards the set of central configurations and asymptotics of relative position, relative velocities, and internal angular momentum. As an application of the asymptotic results, we construct a simplified proof that the initial conditions leading to collision form a set of measure zero.
