Elastic Curves via Geometric Mechanics

Abstract

Elastic curves are the mathematical shapes of thin elastic rods in equilibrium, with deep connections to mechanics, geometry, and computer graphics. Traditionally described as stationary points of bending energy under length and torsion constraints, their rich theory admits many equivalent characterizations. We develop a new one from the viewpoint of geometric mechanics. Our main contribution relies on a lesser-known isoperimetric characterization: a curve is elastic if and only if it is a critical point of the length functional under fixed area and volume vectors. We show that these constraints transform naturally under orientation-preserving rigid body motions, identifying them as momentum variables for these symmetries. This structure suggests a new discrete theory. We show that the low-order integral quantities length, area, and volume vectors are all naturally defined for polygonal curves, leaving the same transformation laws exactly satisfied. The resulting definition of discrete elastic curves in terms of the isoperimetric characterization restricted to discrete polygonal curves is variational, structure-preserving, and requires no auxiliary discretizations of curvature or material frames. Finally, the same structure carries the Marsden--Weinstein form, a canonical (pre-)symplectic structure on the space of curves, to polygonal curves. This yields novel approaches to Hamiltonian dynamics on discrete space curves, including tangent, vortex-filament, and modified Korteweg--de Vries flows.