Elastic Curves with Variable Bending Stiffness

Abstract

We study stationary points of the bending energy of curves $γ\colon[a,b]\to\mathbb{R}^n$ subject to constraints on the arc-length and the curve's holonomy while simultaneously allowing for a variable bending stiffness along the arc-length of the curve. Physically, this can be understood as a model for an elastic wire with isotropic cross-section of varying thickness. We derive the corresponding Euler-Lagrange equations for variations that are compactly supported away from the end points thus obtaining characterizations for elastic curves with variable bending stiffness. Moreover, we provide a collection of alternative characterizations, e.g., in terms of the curvature function. Adding to numerous known results relating elastic curves to dynamics, we explore connections between elastic curves with variable bending stiffness, variable length pendulums and the flow of vortex filaments with finite thickness.