Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Duignan, Nathan, Quaschner, Manuel
Institutions: The University of Sydney, Friedrich Schiller University Jena
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206159401