Non-planar elasticae as optimal curves for the magnetic axis of stellarators

The Euler-Lagrange equations are derived for finite length three-dimensional curves that optimize their bending energy while yielding fixed integrated torsion. The obvious translational and rotational symmetry is exploited to express solutions in a preferred cylindrical coordinate system in terms of elliptic Jacobi functions. These solution curves, which, up to similarity transformations, depend on three dimensionless parameters, do not necessarily close. Two closure conditions are obtained for the vertical and toroidal displacement (the radial coordinate being trivially periodic) to yield a countably infinite set of one-parameter families of closed non-planar curves. The behaviour of the integrated torsion (Twist of the Frenet frame), the Linking of the Frenet frame and the Writhe of the solution curves is studied in light of the Caluguareanu theorem. A refreshed interpretation of Mercier's formula for the on-axis rotational transform of stellarator magnetic field-lines is proposed.