The elastica: a mathematical history

This report traces the history of the elastica from its first precise formulation by James Bernoulli in 1691 through the present. The complete solution is most commonly attributed to Euler in 1744 because of his compelling mathematical treatment and illustrations, but in fact James Bernoulli had arrived at the correct equation a half-century earlier. The elastica can be understood from a number of different aspects, including as a mechanical equilibrium, a problem of the calculus of variations, and the solution to elliptic integrals. In addition, it has a number of analogies with physical systems, including a sheet holding a volume of water, the surface of a capillary, and the motion of a simple pendulum. It is also the mathematical model of the mechanical spline, used for shipbuilding and similar applications, and directly insipired the modern theory of mathematical splines. More recently, the major focus has been on efficient numerical techniques for computing the elastica and fitting it to spline problems. All in all, it is a beautiful family of curves based on beautiful mathematics and a rich and fascinating history. This report is adapted from a Ph. D. thesis done under the direction of Prof. C. H. Séquin.