Braids in classical dynamics

Abstract

Point masses moving in 2+1 dimensions draw out braids in space-time. If they move under the influence of some pairwise potential, what braid types are possible? By starting with fictional paths of the desired topology and ``relaxing'' them by minimizing the action, we explore the braid types of potentials of the form V\ensuremath{\propto}${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ from \ensuremath{\alpha}\ensuremath{\le}-2, where all braid types occur, to \ensuremath{\alpha}=2, where the system is integrable. We also discuss issues of symmetry and stability. We propose this kind of topological classification as a tool for extending the ``symbolic dynamics'' approach to many-body dynamics.