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.

Infinitesimal bending of curves on the ruled surfaces

“…Infinitesimal bending of surfaces and manifolds was widely studied in (Aleksandrov, 1936;Alexandrov, 2010;Efimov, 1948;Hinterleitner et al, 2008;Ivanova-Karatopraklieva & Sabitov, 1995;Kon-Fossen, 1959;Najdanović, 2014;Vekua, 1959;Velimirović, 2009). Infinitesimal bending of curves was considered in (Efimov, 1948;Najdanovic, 2015;Najdanovic & Velimirovic, 2017a,b;Rancic et al, 2009;Velimirović, 2001aVelimirović, ,b, 2009Velimirović et al, 2010;Yano et al, 1946).…”

Infinitesimal bending of knots and energy change

“…The theory of infinitesimal bending is in close connection with thin elastic shell theory and leads to major mechanical applications. Infinitesimal bending of curves and surfaces is studied, for instance, in (A. D. Aleksandrov 1936) [1], (N. V. Efimov 1948) [8], M. Najdanović [15,16], I. Vekua [21], Velimirović et al [22]- [26]. Infinitesimal bending in generalized Riemannian space was studied at Velimirović et al [27].…”

Total normalcy of knots

“…It is necessary to find the variations of the curvature and the torsion under infinitesimal bending. Below we will do that according to [11].…”