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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 $Φ$, is posed. Starting from a variational principle, the balance of elastic forces and the corresponding projections ${\bf n}_i\cdot \nablaΦ$, are found. In the case of elastic surfaces, a force coupling the mean curvature with the external potential, $KΦ$, appears; it is also present in the shape equation along the normal principal in the case of curves. The potential $Φ$ 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 $-κΦ-{\bf n}\cdot \nablaΦ$. In general, the equation of motion becomes integro-differential that must be numerically solved.