| Kathleen A. Hoffman
An Extended Conjugate Point Theory with Application to the Stability of Planar Buckling of an Elastic Rod Subject to a Repulsive Self-potential
K.A. Hoffman and R.Manning
Abstract:The theory of conjugate points in the classic calculus of variations allows, for a certain class of functionals, the characterization of a critical point as stable (i.e., a local minimum) or not. In this work, we generalize this theory to more general functionals, assuming certain generic properties of the second variation operator. The extended conjugate point theory is then applied to a two-dimensional elastic rod subject to pointwise self-repulsion. The critical points are computed by numerically solving first-order integro-differential equations using a finite difference scheme. The stability of each critical point is then computed by determining conjugate points of the second variation operator. In addition, the generalized theory requires the numerical evaluation of the crossing velocity of the zero eigenvalue of the second variation operator at each conjugate point, a feature not present in the classic case (where the crossing velocity can be shown to always be negative). Results demonstrate that the repulsive potential has a stabilizing influence on some branches of critical points.