Minimal curves of constant torsion
Minimal curves of constant torsion
复制标题
恒定扭转的最小曲线
DOI：
10.1090/s0002-9939-00-05526-x
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
T. Ivey
中科院分区：
文献类型：
--
作者：
T. Ivey
Griffiths形式主义被应用到寻找恒定的扭转曲线，这是极值弧长相对于保持扭转的变化，固定的端点和在端点处的副法线。临界曲线是常扭弹性杆，这是不实现某些边界条件。在非完整约束下的变分法中，人们试图在给定的微分方程的解中找到能量最小的轨迹。这个问题的根源在于对帕普斯问题和德劳内问题等经典问题的研究[6]。它是最优控制理论的标准部分，其中使用Pontrjagin最大值原理研究此类问题[12]。近年来，随着亚黎曼几何[13]的研究，这个问题再次引起了几何学家的注意，但也因为格里菲斯及其合作者[3，9，10]以坐标自由形式对临界条件进行了漂亮的重新表述。Griffiths在1983年出版的书[9]中所述的目标是“给出临界曲线的公式”。本文对[R3]中具有固定非零常数挠率的曲线间的长度极小化问题进行了研究。这是Hsu [10]应用Griffiths形式主义的问题之一，但在这里我们更进一步，给出了临界曲线的完整公式，并且能够提取更多关于边界值可以和不能实现的信息。这是由于观察到该问题的临界曲线与具有恒定扭转的Kirchhoff弹性杆中心线的子集完全一致，以及Langer和Singer [11]对这些中心线的详细描述。通过考虑一般情况得到常扭约束。假设我们调整一个定向正交标架（T，U，V）沿着R3中的一条定向曲线-y，由弧长s参数化，使得T是单位切线。该框架将满足广义Frenet方程ds（V）（prO（V））如果这些矢量被认为是附着到沿着沿着-y运动的刚性物体上，则函数p、r、y可以分别被可视化为俯仰、滚转和偏航。编辑于1998年9月2日收到。2000年数学学科分类。小学49 K15、53 A04;中学58 A17、58 A30、73 C 02。
The Griffiths formalism is applied to find constant torsion curves which are extremal for arclength with respect to variations preserving torsion, fixing 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 find the least energy trajectory among solutions of a given differential equation. The subject has its roots in the investigations of such classical questions as Pappus' problem and the Delaunay problem [6]. It is a standard part of optimal control theory, where such problems are investigated using the Pontrjagin maximum principle [12]. In recent years, the subject has come to the attention of geometers again, with the investigations of sub-Riemannian geometry [13], but also with the arrival of a beautiful reformulation, due to Griffiths and his collaborators [3, 9, 10], of the criticality conditions in coordinate-free form. The stated aim in Griffiths' 1983 book [9] 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 fixed nonzero constant torsion. This is among the problems to which Hsu [10] applied the Griffiths 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 Kirchhoff elastic rod centerlines having constant torsion, and the detailed description of these centerlines by Langer and Singer [11]. The constant torsion constraint is arrived at by considering the general case. Suppose we adapt an oriented orthornormal frame (T, U, V) along an oriented curve -y in R3, parametrized by arclength s, such that T is the unit tangent. The frame will satisfy generalized Frenet equations ds (V ) ( prO(V) If these vectors are regarded as attached to a rigid object moving along -y, the functions p, r, y may be visualized as pitch, roll, and yaw, respectively. Received by the editors September 2, 1998. 2000 Mathematics Subject Classification. Primary 49K15, 53A04; Secondary 58A17, 58A30, 73C02.