單擺運動的函數與擾動理論

Abstract

此篇文章主要在探討單擺運動及其擾動現象。首先介紹偏微分方程以及分類，並給予一些波動方程 ( hyperbolic equation )的實際例子。接著介紹 Weierstrass 及 Jacobian 橢圓函數以及一些可以用它們來描述的物理現象；並用後者來分析理想的單擺運動，比如算出實際解、週期以及畫出相位圖等。最後對理想單擺運動做擾動進行探討，這部分主要以動態系統的理論為工具來對受擾動的單擺進行質的分析( qualitative analysis) ，我們可以發現在相同的系統裡，即使是二個很接近的初始值，在長時間後它們的位置卻是天差地遠，這就是所謂的混沌現象( Chaos)，是個仍然充滿許多未知結果的領域。

The main topic of this article discusses the motion of the ideal pendulum and its perturbation. First, we introduce the partial differential equations and their classification, and we give some practical problems whose mathematical models are systems of linear hyperbolic equations. Next, we study the classical Elliptic functions and one application in solving a nonlinear equation. Moreover, we use the Jacobian Elliptic function to analyze the Sine-Gordon equation to derive the exact solutions, the periods, and to sketch the phase portraits. Finally, we focus on the perturbed pendulum. We do qualitative analysis by using the tools of dynamical system. We find out that even if two initial conditions are close, their behaviors will have big difference in a later time. The phenomenon is called Chaos, a field which still much open.