Closed generalized elastic curves in \({\mathbf S}^{2}\)(1).
type: Item
instance of: scholarly article
title: Closed generalized elastic curves in \({\mathbf S}^{2}\)(1).
zbMATH Open document ID: 1045.53002
DOI: 10.1016/S0393-0440(03)00047-0
published in: Journal of Geometry and Physics
publication date: 2003-12-04T00:00:00Z
review text: 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 cons…
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 49J05
Mathematics Subject Classification ID: 74K10
zbMATH DE Number: 2013010
zbMATH Keywords: elastic curves
zbMATH Keywords: curvature
zbMATH Keywords: curvature energy functional
zbMATH Keywords: variation formula
author: Josu Arroyo
author: Oscar J. Garay
author: José J. Mencía
reviewed by: Vasyl Gorkaviy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2013010