A geometric variational problem: the elastic curves of J. Bernoulli (Q2912637)

In 1732 L.~Euler began to publish his investigations of the statics of thin elastic bands or laminae initiated by Jacob Bernoulli. Euler's investigation centered on determining the shapes such laminae assumed in equilibrium when subject to various loadings. In the case of principal interest, the ''elastica'' or ''elastic curve'', external forces are assumed to act at the ends of the lamina while its weight was itself regarded as negligible. The foundations for Euler's investigation were very much established by the scientists who had preceded him. Thus the equation of the elastica was derived by Jakob Bernoulli, the problem of determining its bent forms was formulated by Huygens, the theme of classification originated with Newton, and the possibility of a variational treatment was suggested by Daniel Bernoulli, J.~Bernoulli's nephew.NEWLINENEWLINEIn this paper, the author presents a modern approach to these problems by applying variational methods and using the energy functional \({\mathcal F}(\gamma)=\int_\gamma f(\kappa)\), where \(\kappa\) is the curvature of a curve \(\gamma\). Various applications in physics, biophysic, and geometry are shown.NEWLINENEWLINEFor the entire collection see [Zbl 1243.00023].