变弯曲刚度的弹性曲线
Elastic Curves with Variable Bending Stiffness
AI总结：
该研究针对受弧长和完整约束、弯曲刚度沿弧长变化的曲线，推导其弯曲能平稳点的Euler-Lagrange方程，给出曲率函数等替代刻画，并探究其与变长度摆、有限厚度涡丝流的动力学联系。
摘要：
我们研究曲线$\gamma\colon[a,b]\to\mathbb{R}^n$的弯曲能的平稳点，该曲线受弧长和曲线完整约束，同时允许沿曲线弧长的弯曲刚度可变。从物理角度，这可被理解为具有各向同性变厚度横截面的弹性丝的模型。我们推导了对应于在端点外紧支撑变分的Euler-Lagrange方程，从而得到变弯曲刚度弹性曲线的刻画。此外，我们给出了一系列替代刻画，例如用曲率函数表示的形式。在众多将弹性曲线与动力学联系起来的已知结果基础上，我们探究了变弯曲刚度弹性曲线、变长度摆与有限厚度涡丝流之间的联系。
英文摘要：
We study stationary points of the bending energy of curves $γ\colon[a,b]\to\mathbb{R}^n$ subject to constraints on the arc-length and the curve's holonomy while simultaneously allowing for a variable bending stiffness along the arc-length of the curve. Physically, this can be understood as a model for an elastic wire with isotropic cross-section of varying thickness. We derive the corresponding Euler-Lagrange equations for variations that are compactly supported away from the end points thus obtaining characterizations for elastic curves with variable bending stiffness. Moreover, we provide a collection of alternative characterizations, e.g., in terms of the curvature function. Adding to numerous known results relating elastic curves to dynamics, we explore connections between elastic curves with variable bending stiffness, variable length pendulums and the flow of vortex filaments with finite thickness.
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