Description:
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 E_arc = integral (kappa^2 + lambda tau^2) 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.
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