AI 中文总结
本文从几何力学视角提出弹性曲线的等周刻画，建立保结构的离散弹性曲线理论，为离散空间曲线的哈密顿动力学提供新方法。
AI 中文摘要
弹性曲线是平衡态下细弹性杆的数学形状，与力学、几何学、计算机图形学有深刻联系，传统上被描述为长度和扭转约束下弯曲能量的驻点，其丰富理论有多种等价刻画。本文从几何力学视角提出新刻画，核心贡献基于鲜为人知的等周刻画：曲线为弹性曲线当且仅当它是固定面积和体积向量下长度泛函的临界点。研究表明，这些约束在保定向刚体运动下自然变换，可被识别为该对称性的动量变量，该结构催生新的离散理论。长度、面积、体积向量等低阶积分量可自然定义于多边形曲线，且完全满足相同变换规律；由此基于等周刻画限制在离散多边形曲线上得到的离散弹性曲线定义，是变分的、保结构的，无需对曲率或材料标架进行辅助离散。最后，该结构将Marsden–Weinstein形式（曲线上的典范（预）辛结构）传递到多边形曲线，为离散空间曲线的哈密顿动力学提供新方法，包括切线、涡丝及修正Korteweg–de Vries流。
英文摘要
Elastic curves are the mathematical shapes of thin elastic rods in equilibrium, with deep connections to mechanics, geometry, and computer graphics. Traditionally described as stationary points of bending energy under length and torsion constraints, their rich theory admits many equivalent characterizations. We develop a new one from the viewpoint of geometric mechanics. Our main contribution relies on a lesser-known isoperimetric characterization: a curve is elastic if and only if it is a critical point of the length functional under fixed area and volume vectors. We show that these constraints transform naturally under orientation-preserving rigid body motions, identifying them as momentum variables for these symmetries. This structure suggests a new discrete theory. We show that the low-order integral quantities length, area, and volume vectors are all naturally defined for polygonal curves, leaving the same transformation laws exactly satisfied. The resulting definition of discrete elastic curves in terms of the isoperimetric characterization restricted to discrete polygonal curves is variational, structure-preserving, and requires no auxiliary discretizations of curvature or material frames. Finally, the same structure carries the Marsden--Weinstein form, a canonical (pre-)symplectic structure on the space of curves, to polygonal curves. This yields novel approaches to Hamiltonian dynamics on discrete space curves, including tangent, vortex-filament, and modified Korteweg--de Vries flows.
