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 phi, is posed. Starting from a variational principle, the balance of elastic forces and the corresponding projections (sic), are found. In the case of elastic surfaces, a force coupling the mean curvature with the external potential, K Phi, appears; it is also present in the shape equation along the normal principal in the case of curves. The potential 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 -k Phi -n. (sic)phi. In general, the equation of motion becomes integro-differential that must be numerically solved.