Environmental bias and elastic curves on surfaces

TL;DR

This paper investigates how environmental factors influence the shape and behavior of elastic curves on surfaces, highlighting the roles of geodesic and normal curvatures and deriving equilibrium equations using Lagrange multipliers.

Contribution

It introduces a method to derive shape equations for elastic curves on surfaces considering environmental biases and asymmetries, extending previous models.

Findings

Shape equations for elastic curves on surfaces are derived.

Environmental biases lead to spontaneous curvature values.

Conservation laws are identified for symmetric surface geometries.

Abstract

The behavior of an elastic curve bound to a surface will reflect the geometry of its environment. This may occur in an obvious way: the curve may deform freely along directions tangent to the surface, but not along the surface normal. However, even if the energy itself is symmetric in the curve's geodesic and normal curvatures, which control these modes, very distinct roles are played by the two. If the elastic curve binds preferentially on one side, or is itself assembled on the surface, not only would one expect the bending moduli associated with the two modes to differ, binding along specific directions, reflected in spontaneous values of these curvatures, may be favored. The shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of Lagrange multipliers to the Darboux frame associated with the curve. The forces transmitted to the surface along the surface normal will be determined. Features associated with a number of different energies, both of physical relevance and of mathematical interest, are described. The conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined.