在D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

Sine-Gordorn方程之某特殊解滿足二階微分方程，此為單擺運動方程式。 此論文主要探討兩部份，我們利用代數與幾何方法定義一個新曲面黎曼面替代複數平面，使得 在此黎曼面上是單值且可解析的函數。在黎曼面上對封閉曲線的基底 a,b-cycles 積分可以解決許多微分方程的問題。我們可以找到 a,b-cycles 的等價路徑，由Cauchy Integral Theorem得a,b-cycles 積分值與它們的等價路徑積分值相等。利用Mathematica，等價路徑的積分值可以被正確的求出。其次我們學習了橢圓函數並分析其性質，尤其是Weierstrass和Jacobian 這兩個代表性的橢圓函數。利用這些古典函數分析 ，求其正解及相關性質。

The well-known sine-Gordorn equation has certain particular solution satisfying the 2nd-order ordinary equation.In the paper we analyze the nonlinear approximation and the exact theory of the pendulum motion .In the process of solving the motion , the function occur, which are two-valued. In order to analyze it, we develop new surface to replace complex plane for such that is a single-valued function. We call the Riemann surface of genus N. With proper cut-structure, we can compute integrals of a,b-cycles and any curves on ,and analyze certain properties of. Furthermore,we study classical elliptic functions. From which , we can solve the exact solution. In terms of those functions. And we are also able to analyze certain properties of this equation.