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∝${\mathit{r}}^{\mathrm{\alpha }}$ from α≤-2, where all braid types occur, to α=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.