標題: 在C 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計算
The Exact Theory and Numerical Computations of Pendulum Motions on Riemann Surface of Genus N with Cut-Structure of Type C
作者: 范名宏
Fan, Ming-Hong
李榮耀
Lee, Jong-Eao
應用數學系所
關鍵字: 黎曼空間;單擺運動;Riemann Surface;Pendulum Motions
公開日期: 2012
摘要: u′′ + sin u = 0 在數學的歸類中是一個二階常微分方程,同時也是單擺運動的數學模型。 應用大學課程所學到的微積分技術,可以得到 ∫1 /√2(E + cos u)du =∫dt; 其中E 是積分常數,並且u 是時間t 的函數。 以大學課程的能力,上述的積分是難以處理的,因為√2(E + cos u) 不是單值函數。因此我們先探討此常微分方程的解u(t) 所處的空間,並且討論sin u 的非線性逼近在此空間上的運算情形,此空間就是N 相黎曼空間。 除此之外,我們研究橢圓函數,並應用雅可比橢圓函數來分析理想單擺運動的數學模型,也就是我們在摘要開始時所提到的微分方程u′′ + sin u = 0 ,並且確實的求解,以及討論解的週期性及相關性質。
u′′ + sin u = 0 is a second order differential equation, which is a pendulum motion. In the process of solving the O.D.E., we have the integral form ∫1 /√2(E + cos u)du =∫dt where E is the integration constant (a parameter), and u is a function of time t. The integration is noway to solve due to that √2(E + cos u) is not a singlevalued function. So we study the space where the solution u(t) resides, which is a Riemann Surface of genus N when sin u is replaced by the N-th partial sum of its Taylor series (which is a polynomial). And we study the corresponding O.D.E. under this Riemann Surface with the help of Mathematica computations. Next, we study the classical elliptic function to solve the exact O.D.E. u′′ + sin u = 0 and analyze the associated properties.
URI: http://140.113.39.130/cdrfb3/record/nctu/#GT079922513
http://hdl.handle.net/11536/72095
Appears in Collections:Thesis