AG Mathematische Physik, Manuel Quaschner (FAU): Noncollision singularities with external forces

Manuel Quaschner (FAU)

Noncollision singularities with external forces

Abstract:
The existence of noncollision singularities in the $n$-body problem was already conjectured by Painlevé in 1895. Even before the existence was proven in the 1990s, the question came up, whether the set of all initial conditions leading to noncollision singularities is a set of measure 0. A first result of this kind was proven for $n=4$ particles in $d\geq 2$ dimensions by Saari (1977). Using the so called Poincaré surface method, Fleischer (2018) could improve this for $n=4$ particles in $d \geq 2$ dimensions by extending the result to a wider class of potentials. But the problem is still open for more than four particles.
After an overview of these works, we add some sufficiently small external forces to the well understood case of $n=4$ particles. In order to apply the Poincaré surface method, we need to prove good estimates on the shape of the orbits, which are close to the movement on a straight line. In the end we will show how one could use this results to prove the improbability of larger systems that can be suitably decomposed into diverging systems of up to four particles each.