超细长弹性杆的分析力学问题

超细长弹性杆作为DNA等生物大分子链的力学模型，其平衡和稳定性问题已成为力学与分子生物学交叉的研究热点．虽然在Kirchhoff动力学比拟的基础上，用分析力学方法讨论弹性杆的文章已见诸文献，但尚未形成弹性杆分析力学的严格理论．本文研究了超细长弹性杆分析力学的若干基础性问题．对杆截面的自由度、虚位移、约束方程及约束力等基本概念给出严格的定义和表达式．建立弹性杆平衡的D’Alembert-Lagrange原理、Jourdain原理和Gauss原理；从D’Alembert-Lagrange原理导出Hamilton原理．从变分原理出发导出Lagrange方程、Nielsen方程、Appell方程和Hamilton正则方程；对于受约束的弹性杆，导出了带乘子的Lagrange方程．讨论了Lagrange方程的首次积分．对于杆中心线存在尖点的情形，导出了微段杆平衡的近似方程。

On analytical mechanics for a super-thin elastic rod

The investigations on the problems of equilibrium and stability of a super-thin elastic rod as model of DNA supercoiling have became an interdisciplinary area of classical mechanics and molecular biology with full of activities. Although there are papers using the methods of analytical mechanics to discuses some modeling problem, but the theory on analytical mechanics of a super-thin elastic rod has not formed systematically. In this paper, by applying the theory and method of analytical mechanics to the modeling of a thin-elastic rod, the framework of analytical mechanics is constructed for the equilibrium of a super-thin elastic rod. For the cross section of a rod, concepts such as freedom, constraints and constrained equations and constrained foces are analyzed. And various variational principles of mechanics, such as the D'Alembert-Lagrange principle, the Jourdain principle, and the Gauss principles are established. The principles are applied to derive the Hamilton canonical equation, the Lagrange equation, the Nielsen equation and the Appell equation. For the case that a rod is subjected to constraints, the Lagrange equation with undetermined multiplier is presented. In the neighborhood of a singular point, the equation of equilibrium is transformed into the same form as the one for collisions.