Boundary Value Problems for Euler-Bernoulli Planar Elastica. A Solution Construction Procedure

Abstract

We consider the problem of finding a curve minimizing the Bernoulli bending energy among planar curves of the same length, joining two fixed points and possibly carrying orientations at the endpoints (Euler elastica). We focus on the problem of constructing closed form elasticae for given boundary data and show that, rather than employing complicated numerical algorithms, it suffices to use easily available computer algebra systems to implement our procedure. To this end, we first review some fundamental facts about the Euler-Bernoulli variational approach to the elastic rod. Our curves are only assumed to be stationary and not necessarily minimizers. Secondly, the Euler-Lagrange equations are expressed in terms of the curvature of the elasticae, what is used to compute their explicit parametrizations by means of the Jacobi elliptic functions. Lastly, we describe our approach to solving this problem under different boundary conditions, and the procedure is illustrated with numerous examples. We include the numerical code that we use in Appendix B.