**Richard Montgomery** (UCSC)
**Scattering and metric lines**

Tuesday, June 2, 2020, 15:00 GMT

**Abstract:** Scattering in classical mechanics concerns asymptotics of dynamics under forces decaying at infinity rapidly enough so that trajectories escaping to infinity have asymptotically constant velocities. There are many similarities between classical scattering and the study of globally minimizing geodesics, or "metric lines" in noncompact complete metric spaces as initiated largely through the work of Buseman. Solutions to Newton's equations having fixed energy are - for the most part - geodesics for the Jacobi-Maupertuis metric at that energy. This metric is a singular Riemannian metric whose geodesics are solutions to Newton's equations having that energy.

We start off with the classical Rutherford scattering, the scattering for the Kepler problem and look at its geodesic reformulation. The Kepler problem admits no metric lines but it does admit metric rays. Does the planar 3-body problem admit metric lines? We do not know. But by Maderna-Venturelli this 3-body problem does admit metric rays connecting any given finite configuration to any asymptotic configuration at infinity. What do the metric lines look like in subRiemannian Carnot geometries? What are scattering states for the N-body problem? Is there a scattering map? What does it look like?

We will answer some of these questions and point out open problems as we go.

References