Blow-Up, Asymptotics, and Improbability of Collision Singularities in the n-Body Problem
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.
Related
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- Asymptotics for partition functions of random normal matrix models with singularities
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BibTeX
@misc{duignan2026blowup,
title = {{Blow-Up, Asymptotics, and Improbability of Collision Singularities in the n-Body Problem}},
author = {Nathan Duignan and Manuel Quaschner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00793},
eprint = {2609.00793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00793},
}
RIS
TY - UNPB
TI - Blow-Up, Asymptotics, and Improbability of Collision Singularities in the n-Body Problem
AU - Nathan Duignan
AU - Manuel Quaschner
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00793
UR - https://arxiv.org/abs/2609.00793
AB - 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.
ER -