On the elastic closed plane curves (Q1067598)

Studying the variational problem for the functional \(E(C)=(1/2)\int_{C}k^ 2(s)ds\) (C a closed curve, s the arc length parameter of C, k(s) the curvature of C), under the condition \(\int_{C}ds=L=const.\), the author finds the following result: if E(C) is critical for a closed plane curve with \(L=const.\), then the curve C is either the plane circle \(C_ n\) with radius L/2\(\pi\) n or the curve \(D_ m\) which is congruent to \[ x(s)=(2p/\sqrt{R})\cos \phi \] \[ y(s)=(1/\sqrt{R})\int^{\phi}_{-\pi /2}(2\sqrt{1-p^ 2 \sin^ 2 \phi}-1/\sqrt{1-p^ 2 \sin^ 2 \phi})d\phi,\quad \phi \in [-\pi /2,3\pi /2] \] where R, \(p^ 2\) and m are specified in the paper.