Minimal Curves of Constant Torsion

The Griffiths formalism is applied to find constant torsion curves which are extremal for arclength with respect to variations preserving torsion, fixing the endpoints and the binormals at the endpoints. The critical curves are elastic rods of constant torsion, which are shown to not realize certain boundary conditions. In the calculus of variations under nonholonomic constraints, one tries to find the least energy trajectory among solutions of a given differential equation. The subject has its roots in the investigations of such classical questions as Pappus’ problem and the Delaunay problem [6]. It is a standard part of optimal control theory, where such problems are investigated using the Pontrjagin maximum principle [12]. In recent years, the subject has come to the attention of geometers again, with the investigations of sub-Riemannian geometry [13], but also with the arrival of a beautiful reformulation, due to Griffiths and his collaborators [3, 9, 10], of the criticality conditions in coordinate-free form. The stated aim in Griffiths’ 1983 book [9] is “to get out formulas” for critical curves. In this note, we carry this out for the problem of length minimization among curves in R of a fixed nonzero constant torsion. This is among the problems to which Hsu [10] applied the Griffiths formalism, but here we go further, giving complete formulas for critical curves, and are able to extract more information about what boundary values can and cannot be achieved. This is due to the observation that the critical curves for this problem coincide exactly with the subset of Kirchhoff elastic rod centerlines having constant torsion, and the detailed description of these centerlines by Langer and Singer [11]. The constant torsion constraint is arrived at by considering the general case. Suppose we adapt an oriented orthornormal frame (T, U, V ) along an oriented curve γ in R, parametrized by arclength s, such that T is the unit tangent. The frame will satisfy generalized Frenet equations d ds   U V   =   0 y p −y 0 r −p −r 0     U V   If these vectors are regarded as attached to a rigid object moving along γ, the functions p, r, y may be visualized as pitch, roll, and yaw, respectively. Suppose two of p, r, y are prescribed constants. Because U and V can be interchanged, we need only consider the p&r case and the p&y case. In the latter case, we can rotate U and V to arrange that p = 0. In either case, the lift (γ(s), T (s), U(s), V (s)) into the oriented 1991 Mathematics Subject Classification. Primary 49K15, 53A04 Secondary 58A17, 58A30, 73C02.