Closed generalized elastic curves in \({\mathbf S}^{2}\)(1).
A curvature energy functional of the type \(J (\gamma)= \int_{\gamma}P (k) ds\) defined on the space of closed curves in the two-dimensional sphere \(S^2_r\) is investigated. Here \(k\) is the curvature of a curve \(\gamma\) in \(S^2_r\) and \(P(k)\) is a given function of \(k\). If \(P(k)=k\), then \(J\) is the well-known total curvature functional. If \(P(k)=k^2\) then \(J (\gamma) = \int_{\gamma}k^2 \,ds\), its critical points are classical elastic curves. Closed elastic curves in the two-sphere were intensively studied by \textit{J. Langer} and \textit{D. A. Singer} [J. Differ. Geom. 20, 1--22 (1984; Zbl 0554.53013)]. In the present paper the authors analyze the functional \(J (\gamma)= \int_{\gamma}k^n \,ds\), \(n>2\) and investigate its critical points in \(S^2_r\) called generalized \(n\)-elastic curves. First of all, the corresponding Euler-Lagrange equation, its first integral and second variation formula are discussed. As consequence, it is proved that there are no closed generalized \(n\)-elastic curves in \(S^2_r\) other than geodesics, for arbitrary natural \(n>2\). On the other hand, for \(n=1/2\) the situation is less trivial: in this case the existence of a countable family of non-geodesic closed generalized \((1/2)\)-elastic curves in \(S^2_r\) is demonstrated and a classification of that elastic curves is proposed. Theorem. For any pair of integers \(m\), \(n\) satisfying \(n<2m<\sqrt{2}n\), there exists a convex closed curve \(\gamma_{m,n}\), which is a critical point of \(J(\gamma)=\int_\gamma k^{1/2} \,ds\) in the unit sphere \(S^2_1\). \(\gamma_{m,n}\) closes up after \(n\) periods of its curvature (given by an explicit formula) and \(m\) trips around the equator. Any closed generalized \((1/2)\)-elastic curve is obtained as above. Some particular examples and pictures follow this statement. Finally, functionals of the type \(F_\lambda(\gamma)=\int_\gamma\sqrt{k^2+\lambda}\,ds\), \(\lambda >0\), defined on spaces of curves in \(S^2_r\) are considered. If \(\lambda\) is greater than the radius \(r\) of \(S^2_r\), the only closed critical points of \(F_\lambda(\gamma)\) are geodesics. If \(\lambda=r\), then the circles of \(S^2_r\) are the only closed critical points of \(F_\lambda(\gamma)\). If \(\lambda\) is less than \(r\), then in addition to geodesics there exists a countable family of closed critical points of \(F_\lambda(\gamma)\). They are also classified by the authors.
scientific article; zbMATH DE number 2013010