Relaxed elastic line on a curved surface

In an effort to begin to understand the mechanics of various forms of biologically packaged DNA, we develop the Euler—Lagrange equations for the equilibrium path of an elastic line constrained to a surface but otherwise relaxed. We find, in contrast to a statement by Hilbert and Cohn-Vossen [1], that whether or not the solutions are geodesic curves of the surface depends on the boundary conditions and on the surface. Not surprisingly, the relaxed elastic line on a plane or a sphere is always a geodesic (straight line and great circle, respectively). On a cylinder and a “pseudotorus,” however, the relaxed line is a geodesic only if both ends are free. For example, a relaxed line on a cylinder, with fixed initial point and oblique tangent, does not wind on the corresponding geodesic (helix).