Elastic curves and surfaces under long-range forces: A geometric approach

Using classical differential geometry, the problem of elastic curves and surfaces in the presence of long-range interactions $\mathrm{\Phi }$, is posed. Starting from a variational principle, the balance of elastic forces and the corresponding projections ${\mathbf{n}}_i\cdot \mathrm{\nabla }\mathrm{\Phi }$, are found. In the case of elastic surfaces, a force coupling the mean curvature with the external potential, $K\mathrm{\Phi }$, appears; it is also present in the shape equation along the normal principal in the case of curves. The potential $\mathrm{\Phi }$ contributes to the effective tension of curves and surfaces and also to the orbital torque. The confinement of a curve on a surface is also addressed, in such a case, the potential contributes to the normal force through the terms $-\kappa\Phi -{\bf n}\cdot \nabla\Phi$. In general, the equation of motion becomes integro-differential that must be numerically solved.