Manuel Quaschner - "Non-collision singularities in n-body systems"

Non-collision singularities are particular solutions of the n-body problem that become unbounded within finite time. The first one to conjecture their existence was Painlevé in 1895. It took almost 100 years, until the existence of such orbits was proven rigorously by Xia. We are interested in understanding the set of all these orbits, especially whether or not this is a set of measure zero. A first result of this kind was proven for n=4 particles in d>=2 dimensions by Saari (1977). Using the so called Poincaré surface method, Fleischer (2018) could improve this for n=4 particles in d >=2 dimensions by extending the result to a wider class of potentials. But the problem is still open for more than four particles.

The first part of the talk will be an introduction to the topic and explains these earlier results. Then we will consider a simpler kind of systems, similar to the well-studied billiard trajectories and give some heuristics, why there should be a connection to non-collision singularities. Finally, we will derive some properties for these simpler systems and explain, why the corresponding properties could be interesting for understanding non-collision singularities.