科研之友
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摘要
We propose a two step variational principle to describe helical structures in nature. The first one is governed by an energy action which is a linear function in both curvature and torsion allowing to describe nonclosed structures including elliptical, spherical, and conical helices. These appear as rhumb lines in right cylinders constructed over plane curves. The model is completed with a conformal alternative which, in particular, gives a description of closed structures. The energy action is linear in the curvatures when computed in a conformal spherical metric. Now, helices appear as making a constant angle with a Villarceau flow and so they are loxodromes in surfaces which are stereographic projections of Hopf tubes, in particular, anchor rings, revolution tori, and Dupin cyclides. The model satisfies the requirements of simplicity and beauty as reflected in the three main principles that head its construction: least action, topological, and quantization. According to the latter, the main entities and quantities associated with the model should not be multiplied unnecessarily but they are quantized. In this sense, a quantization principle, a la Dirac, is obtained for closed structures and also for the critical levels of energy.
- 出版日期2009-10
