由最速降线问题引出非线性问题的通用解法

The Generalized Solutions to Nonlinear Problems Induced by the Most Rapid Curved Descending Line Problem

摘要: 针对最速降线的问题，首先在不存在摩擦时的情况下推导出适用于求此类非线性微分方程极值的变分法，将其应用于在现实里用拉格朗日乘子法求解库仑摩擦最速降线的过程中，随后在给定的边界条件下应用打靶法结合牛顿法进行快速迭代逼近并利用mathematica建模求得数值解。数值计算在科学研究和工程技术中都起到很重要的作用，而非线性方程组的数值解法是计算数学的一个重要的研究内容。此套方法涉及到以时间和空间等物理量为参数的表达方式，还具有可快速编程代码化的优点。因此可适用于物理实验中求解绝大部分在一定精度要求下的非线性问题。

Abstract: In order to solve the problem of the most rapidly falling line, we first derive the variational method for finding the extrema of such nonlinear differential equations in the absence of friction, apply it to the process of solving the Coulomb frictional most rapidly falling line by Lagrange multiplier method in reality, and then apply the targeting method combined with Newton’s method for fast iterative approximation under the given boundary conditions and use mathematica modeling to find the numerical solution. Numerical computation plays an important role in both scientific research and engineering technology, and the numerical solution of nonlinear systems of equations is an important research element in computational mathematics. This method involves the expression of physical quantities such as time and space as parameters, and also has the advantage of being rapidly programmable and codable. Therefore, it can be applied to solve most of the nonlinear problems in physical experiments under certain accuracy requirements.