On relaxed elastic lines of second kind on a curved hypersurface in the \(n\)-dimensional Euclidean space (Q2927920)

A relaxed elastic line in a surface is a local minimizer of the total squared curvature among all surface curves with fixed arc-length, prescribed start point and start tangent. Different aspects of relaxed elastic lines have been extensively studied in the past.NEWLINENEWLINEIn this paper, the authors investigate relaxed elastic lines of second kind on oriented hypersurfaces of the Euclidean \(n\)-space. A relaxed elastic line of second kind minimizes the total squared second curvature (torsion).NEWLINENEWLINEThe authors express the squared second curvature of a surface curve in terms of Darboux frame field, second fundamental form, and geodesic curvatures, and use this to derive the Euler-Lagrange equations of the variational problem. These equations simplify a lot if the relaxed elastic line is a geodesic (in general, it is not). The article finishes with an explicit computation of a parametric representation for a relaxed elastic line on a hypercylinder.