Infinitesimal bending influence on the Willmore energy of curves

Abstract

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