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

TL;DR

This paper employs classical differential geometry to analyze elastic curves and surfaces influenced by long-range forces, deriving equations that incorporate external potentials into the shape and force balance conditions.

Contribution

It introduces a geometric framework for elastic structures under long-range interactions, deriving shape equations with potential-dependent terms and addressing confinement effects.

Findings

Force coupling with mean curvature appears in shape equations.

Potential influences effective tension and orbital torque.

Equations of motion are integro-differential and require numerical solutions.

Abstract

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 ${\bf n}_i\cdot \nabla\Phi$, 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 $-\kappa\Phi -{\bf n}\cdot \nabla\Phi$. In general, the equation of motion becomes integro-differential that must be numerically solved.