Lagrangian Aspects of the Kirchhoff Elastic Rod

Abstract

The relation between Euler–s planar elastic curves and vortex filaments evolving by the localized induction equation (LIE) of hydrodynamics was discovered by Hasimoto in 1971. Basic facts about (an integrable case of) Kirchhoff elastic rods are described here, which amplify the connection between the variational problem for rods and the soliton equation LIE. In particular, it is shown that the centerline of the Kirchhoff rod is an equilibrium for a linear combination of the first three conserved Hamiltonians in the LIE hierarchy.