Description of the buckling of a chain of hard spheres in terms of Jacobi functions

A linear chain of hard spheres, confined by a transverse harmonic potential, buckles under compression between two hard walls. Jacobi functions provide exact analytic solutions of a differential equation (related to the Duffing equation) for the displacement profile of this chain, within a continuum approximation. Here we explore these solutions, describing their forms and the way in which they vary as system parameters are changed. This is illustrated by reference to two-dimensional diagrams in which each such solution is represented by a point and contour plots illustrate various characteristics of the solutions (period, compression, localization, etc.). Our findings enrich the study of the buckling instability for a linear chain of particles. The approach presented here has the advantage of being based on a simple chain of hard spheres and is straightforward in its interpretation. As such these results may provide insight into more complex experiments, such as those involving the buckling of ion chains.