Abstract:

The Schrödinger analogy of thin elatic rod is studied. Compared with the Kirchhoff dynamics analogy, the Schrödinger analogy is proposed. By the new analogy, the Kirchhoff equation of elastic rod can be written as curvature equation which is similar to nonlinear Schrödinger equation. Thus, the Jacobi elliptic function solution of Schrödinger equation can be taken into elastic rod equation. The space configurations of the elastic rod described by the solution are given. Schrödinger analogy reveals the relations between the quantum state of wave function and the geometry configuration of elastic rod, and gives a new way to solve the Kirchhoff equation.

Key words: stationary differential equation of thin elastic rod, dynamics analogy, Schrödinger equation

摘要：

研究弹性细杆静力学的薛定谔粒子波动比拟。类似于Kirchhoff动力学比拟, 依据弹性细杆曲率平衡微分方程与一维定态非线性薛定谔方程数学形式的相似性, 给出两者的动力学比拟关系, 称为Schrödinger粒子波动比拟。基于比拟关系, 给出弹性细杆方程的Jacobi椭圆函数解, 并画出此解所描述的弹性细杆的空间位形。Schrödinger粒子波动比拟建立了波函数的量子态与弹性细杆的几何构型的对应关系,给予波函数的量子态直观的几何图像, 为弹性细杆方程的求解提供了新的途径。

关键词: 弹性细杆平衡微分方程, 动力学比拟, 薛定谔方程