Towards a classification of Euler–Kirchhoff filaments

Abstract

Euler–Kirchhoff filaments are solutions of the static Kirchhoff equations for elastic rods with circular cross sections. These equations are known to be formally equivalent to the Euler equations for spinning tops. This equivalence is used to provide a classification of the different shapes a filament can assume. Explicit formulas for the different possible configurations and specific results for interesting particular cases are given. In particular, conditions for which the filament has points of self-intersection, self-tangency, vanishing curvature or when it is closed or localized in space are provided. The average properties of generic filaments are also studied. They are shown to be equivalent to helical filaments on long length scales.