Helices

Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical center-lines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance.