Second Order Infinitesimal Bending of Curves

Abstract

We investigate a second order infinitesimal bending of curves in a three-dimensional Euclidean space in this paper. We give the necessary and sufficient conditions for the vector fields to be infinitesimal bending fields of the corresponding
order, as well as explicit formulas which determine these fields. We examine the
first and the second variation of some geometric magnitudes which describe a
curve, specially a change of the curvature. Two illustrative examples (a circle
and a helix) are studied not only analytically but also by drawing curves using
computer program Mathematica.