在A型代數結構下之N相黎曼空間的單擺運動之確切理論與數值運算

Abstract

此篇文章我們主要在探討單擺運動，內容上可分為兩個部分。第一個部分，藉由多項式去逼近，並探討相對應之方程式，接著介紹如何造出相對應的黎曼空間，並利用Mathematica 幫助我們去計算相對應的黎曼空間上的路徑積分及方程式上相關之性質。第二個部分，首先介紹橢圓函數的基本性質，再利用橢圓函數解出原微分方程的實際解、週期及相關性質。

In this paper, we mainly investigate the pendulum motion, the content is divided into two parts. The first part, we use the polynomials to approach and the corresponding ode. And introduce that how to create the corresponding Riemann surfaces where the solutions of the approximate ode reside. And we use Mathmetica to help us to compute the path integrals corresponding to the Riemann surface and certain properties of ode. The second part, we introduce basic properties of the elliptic functions and use them to solve the exactly solutions, periods and certain properties for the exact ode.