Infinitesimal bending influence on the Willmore energy of curves

Abstract

In this paper we study the change of the Willmore energy of curves, as a special case of so-called Helfrich energy, under infinitesimal bending determined by the stationarity of arc length. We examine the variation of the unit tangent, principal normal and binormal vector fields, the curvature and the torsion of the curve. We obtain an explicit formula for calculating the variation of the Willmore energy, as well as the Euler-Lagrange equations describing equilibrium. We find an infinitesimal bending field for a helix and compute the variation of its Willmore energy under such infinitesimal bending.