Extremals of curvature energy actions on spherical closed curves (Q557232)

Following \textit{J.\ Langer} and \textit{D.\ A.\ Singer} [J. Differ.\ Geom.\ 20, 1--22 (1984; Zbl 0554.53013)], this paper considers the Euler-Lagrange equations of a functional of the form \(\int_\gamma P(\kappa) \,ds\), where \(\gamma(s)\) is an arc length parametrization of a curve with geodesic curvature \(\kappa\) in a space form, and \(P\) is a \(C^2\)-function. It is shown that any critical point \(\gamma\) of such a functional has two commuting Killing vector fields associated with it. In the 3-sphere this allows to relate a cylindrical coordinate system to \(\gamma\). For the cases \(P(\kappa)=\sqrt{\kappa^2+\lambda}\) (\(\lambda>0\)), and \(P(\kappa)=\kappa^2\) (i.e., elastic curves) all periodic solutions are determined and discussed in detail in terms of their natural equation.