Length Minimization for Space Curves Undernonholonomic Constraints : the Constant Torsion

The Griiths formalism is applied to nd constant torsion curves which are extremal for arclength with respect to variations preserving torsion, xing 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 nd the least energy solution among candidates satisfying a given diierential equation. The subject has its roots in the investigations of such classical problems as the Delauney problem 5], where one tries to minimize length among curves in R 3 of xed constant curvature. It has broad applications in optimal control theory, especially in the use of the Pontrjagin maximum principle 10]. In recent years, the subject has come to the attention of geometers again, with the investigations of sub-Riemannian geometry 11], but also with the arrival of a beautiful reformulation of the criticality conditions, in coordinate-free form, due to Griiths and his collaborators 7, 8]. The stated aim in Griiths 1983 book 7] 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 3 of a xed nonzero constant torsion. This is among the examples to which Hsu 8] has applied the Griiths 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 Kirchhoo elastic rod centerlines having constant torsion, and the detailed description of these curves by Langer and Singer 9]. The constant torsion constraint is arrived at by considering of the general case. Suppose we adapt an orthornormal frame (T; U; V) along an oriented curve in R 3 , parametrized by arclength s, such that T is the unit tangent. The frame will satisfy generalized Frenet equations d ds 0 @ T U V 1 A = 0 @ 0 y p ?y 0 r ?p ?r 0 1 A 0 @ T U V 1 A If these vectors are regarded as attached to a rigid object moving along , the functions p; r; y may be visualized as pitch, roll and yaw, respectively. One may always rotate U and V so