Arc Geometry II: Variational Structures and Energies of Arc Helices

Arc Geometry II develops the variational framework of Arc Geometry and continues the program initiated in Arc Geometry I: Differential and Topological Structure. Arc Geometry studies a class of spatial curves called Arc Helices, defined by a two-frequency helical modulation. In Arc Geometry I it was shown that these curves naturally embed in a torus and realize torus knots when the frequency ratio is rational. The present work establishes the variational structure associated with Arc Helices. Its main contributions include explicit formulas for curvature and torsion, the introduction of the Arc energy functional Eₐrc = integral (kappa² + lambda tau²) ds, the discrete energy spectrum of closed Arc Helices indexed by torus knot type, a small-modulation asymptotic formula, a stability criterion under perturbations, and numerical illustrations of curvature modulation, torsion modulation, and torus-knot realizations.