Interacting charged elastic loops on a sphere

A variational approach is used to study the behavior of two closed, inextensible, interacting elastic loops that are constrained to lie on a sphere. In addition to the bending energy of each loop, the total potential energy of the system includes nonlocal contributions that account for intraloop and interloop interactions. Euler–Lagrange equations and energy based stability conditions are derived using the first and second variations of the potential energy functional. As an illustrative application, a problem in which all the interaction potentials are Coulombic and both loops have the same length, bending rigidity, and positive charge density is considered. To ensure the existence of a trivial solution in which the loops are parallel and circular, the length of the loops are taken to be smaller than perimeter of the great circle of the sphere. Detailed bifurcation and linear stability analyses of the trivial solution are conducted. The stability of the trivial solution is governed by three dimensionless parameters a, ζ, and χ, where a is the ratio between of the radius of the loops to radius of the sphere and where ζ and χ encompass information about the ratio of intraloop interaction and interloop interaction to the bending rigidity. While the bending energy and the intraloop interaction energy stabilize the trivial solution, the interloop interaction has a destabilizing influence. Moreover, a cross-over phenomenon associated with the nature of the most destabilizing mode is discovered: for 0 < a < ac, the number of modes represented in the most destabilizing modes varies with ζ and χ; for ac < a < 1, the most destabilizing mode is always the lowest mode in keeping with results for problems involving only bending energy