Murai, Minoru; Matsumoto, Waichiro; Yotsutani, Shoji Representation formula for the plane closed elastic curves. (English) Zbl 1310.53004 Discrete Contin. Dyn. Syst. 2013, Suppl., 565-585 (2013). Summary: Let \(\Gamma\) be a plane closed elastic curve with length \(L>0.\) Let \(M\) be the signed area of the domain bounded by \(\Gamma\). We are interested in the following variational problem. Find a curve \(\Gamma\) (the curvature \(\kappa(s)\)) which minimizes the elastic energy subject to \(L^{2}-4 \pi M >0\) and \( L^{2} \neq 4 \pi \omega M\), where \(\omega\) is the winding number. This variational problem was first studied in the case \(\omega=1\) and the Euler-Lagrange equation was derived. The existence of the minimizer was shown and the profile near the disk was investigated by using the Euler-Lagrange equation. As the first step to investigate the structure of solutions of this equation, we show all the solutions to an auxiliary second order boundary value problem. Moreover, we obtain the representation of the integral of \(\kappa(s)\). Cited in 3 Documents MSC: 53A04 Curves in Euclidean and related spaces 34A05 Explicit solutions, first integrals of ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 49Q10 Optimization of shapes other than minimal surfaces Keywords:variational problem; Euler-Lagrange equation; elliptic functions; complete elliptic integrals; exact solution PDFBibTeX XMLCite \textit{M. Murai} et al., Discrete Contin. Dyn. Syst. 2013, 565--585 (2013; Zbl 1310.53004) Full Text: Link