非线性摆的简单推广
Nonlinear Pendulum: A Simple Generalization
摘要：
本文求解了具有一般初始角位移（$θ(0)=θ_0$）和速度（$\dotθ(0)=ϕ_0$）的单摆的二阶非线性微分方程，得到了Jacobi椭圆函数$\text{sn}(u,k)$和第一类不完全椭圆积分$F(φ,k)$的封闭解。这样一个问题可以用来介绍概念，如椭圆积分和功能，以先进的本科生，以激励使用计算机代数系统来分析所获得的解决方案，并可能作为一个练习，以显示如何进行一个简单的推广，作为一个出发点，该文件的Beléndez \emph{et al} \cite{belendez}，在那里他们认为标准的情况下$\dotθ(0)=0$。
英文摘要：
In this work we solve the nonlinear second order differential equation of the simple pendulum with a general initial angular displacement ($θ(0)=θ_0$) and velocity ($\dotθ(0)=ϕ_0$), obtaining a closed-form solution in terms of the Jacobi elliptic function $\text{sn}(u,k)$, and of the the incomplete elliptical integral of the first kind $F(φ,k)$. Such a problem can be used to introduce concepts like elliptical integrals and functions to advanced undergraduate students, to motivate the use of Computer Algebra Systems to analyze the solutions obtained, and may serve as an exercise to show how to carry out a simple generalization, taking as a starting point the paper of Beléndez \emph{et al} \cite{belendez}, where they have considered the standard case $\dotθ(0)=0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
