arXiv subjects

Manuel Quaschner

Publications and source records attributed to Manuel Quaschner.

4 recordsLinked to original sources

arXiv2026

Blow-Up, Asymptotics, and Improbability of Collision Singularities in the n-Body Problem

We study finite-time singularities in $n$-body systems with pair potentials that are homogeneous of degree $-\alpha$, where $0<\alpha<2$, 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 Painlev\'e 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.

math-ph↗

arXiv2026

A general improbability result for non-collision singularities as subsystems with at most four particles

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.

math-ph↗

arXiv2026

Jet-Density of Finite-Gap Solutions for Classes of BKM Systems

We show that jets of initial data can be approximated up to arbitrary order by finite-gap solutions for classes of so-called BKM systems of PDEs introduced by Bolsinov--Konyaev--Matveev, which include classical PDEs such as KdV, Kaup--Boussinesq and Camassa--Holm. Finite-gap solutions are obtained via a finite-reduction map, defined algebraically, which sends solutions of a St\"ackel system to solutions of the BKM PDE. For the classes containing KdV and Kaup--Boussinesq we obtain full jet-surjectivity via a triangular structure, whereas for the class containing Camassa--Holm we establish jet-surjectivity on an open set of initial data over $\mathbb{R}$ and a Zariski-open (dense) set over $\mathbb{C}$.

math.AP↗

arXiv2024

Nondeterministic particle systems

We consider systems of n particles that move with constant velocity between collisions. Their total momentum but not necessarily their kinetic energy is preserved at collisions. As there are no further constraints, these systems are nondeterministic. In particular we examine trajectories with infinitely many collisions.

math-ph↗