KNOTTED ELASTIC CURVES IN IR 3
KNOTTED ELASTIC CURVES IN IR 3
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IR 3 中的打结弹性曲线
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
D. Singer
中科院分区：
文献类型：
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作者：
Joel Langer;D. Singer
变分法中最古老的课题之一是研究弹性杆，根据丹尼尔伯努利的理想化，在相同长度和一阶边界数据的曲线之间最小化总平方曲率。经典术语弹性指的是在平面或IR中的曲线，其代表处于平衡的这样的杆。虽然弹性体及其推广长期以来一直（并将继续）在弹性理论的背景下感兴趣，但弹性体作为一个纯粹的几何实体似乎在很大程度上被忽视了。然而，最近，Bryant和Griffiths [1，3]发现了弹性体及其对空间形式的自然推广（其中弧长通常不受约束）是外微分系统一般理论背景下的一个有趣例子。独立地，本作者研究了空间形式中的“自由”弹性曲线，并与微分几何中的著名问题建立了联系[4，5]。从几何的观点来看，封闭的弹性体和它们的整体行为自然是特别感兴趣的。在本文中，我们保持这一重点，但返回到经典的设置的欧氏曲线与固定的弧长。特别地，我们给出了U”中闭合弹性曲线的完整分类，并确定了这些弹性曲线的打结性。我们注意到，可积性方程的经典elasticae已经知道欧拉在平面的情况下，（基本上）拉东的情况下，IR（见布拉施克的Vorlesungen iber Differentialgeometrie我）;以确定封闭elasticae，但是，主要的问题是要了解依赖的椭圆积分的某些参数。由于欧拉已经很好地描述了封闭的平面弹性体，并且由于初值问题的解的唯一性意味着U”中的任何弹性体实际上必须位于U中，因此给出以下内容就足够了。
One of the oldest topics in the calculus of variations is the study of the elastic rod which, according to Daniel Bernoulli's idealization, minimizes total squared curvature among curves of the same length and first order boundary data. The classical term elastica refers to a curve in the plane or IR which represents such a rod in equilibrium. While the elastica and its generalizations have long been (and continue to be) of interest in the context of elasticity theory, the elastica as a purely geometrical entity seems to have been largely ignored. Recently, however, Bryant and Griffiths [1, 3] have found the elastica and its natural generalization to space forms (where arc length is generally not constrained) to be an interesting example in the context of the general theory of exterior differential systems. Independently, the present authors have studied 'free' elastic curves in space forms and have drawn connections to well-known problems in differential geometry [4, 5]. From the geometric point of view, the closed elasticae and their global behaviour are naturally of particular interest. In the present paper we maintain this emphasis but return to the classical setting of Euclidean curves with fixed arc length. Specifically, we give a complete classification of closed elastic curves in U" and determine the knottedness of these elasticae. We note that the integrability of the equations for a classical elastica was known already to Euler in the planar case and (essentially) to Radon in the case of IR (see Blaschke's Vorlesungen iiber Differentialgeometrie I); to determine the closed elasticae, however, the chief problem is to understand the dependence of the resulting elliptic integrals on certain parameters. Since the closed planar elasticae were well described already by Euler, and since uniqueness of solutions in the initial-value problem implies that any elastica in U" must in fact lie in U, it will suffice to present the following.