Geometrical equilibrium of curves: A showcase of helical numerical solutions

In this work we plot loop configurations that minimize energy functionals depending on geometrical invariants of the loop itself. In particular, we consider a family of functionals including curvature and torsion terms, both linear and quadratic, such that their combinations produce geometrical invariants. In [1], Noether's theorem was advantageously used to identify the constants of integration of the Euler-Lagrange equations describing the equilibrium of loops for this family of energy functionals. Those authors demonstrated the integrability of these functionals by means of quadratures. In this work we follow that approach to realize numeric calculations and to show plots of numerical solutions of the relevant equations for equilibrium of loops, as these were presented but not studied in [1]. We then show as the main result of this work a representative catalogue of such solutions in Euclidean three-dimensional space.