單擺運動之函數理論

Abstract

摘 要 我們研究一個單擺運動。理想的單擺運動 是能量守恆的，因此其數學模型

的運動軌跡被初始總能量決定唯一性，從而，所有的解可以由能量守恆律去分析跟求解。有三種解由初始總能量來區分，即週期解 (the periodic solutions) (時間的週期)，隔間解 (the seperatrices) 和 波動解 (the wavetrains). 在第一部分裡面，從守恆律我們把非線性的ODE問題轉換成所謂的反問題 (積分形式)，然後用古典的橢圓函數將解給表示出來。注意到這些反運算的積分裡面有多值的被積分函數，所以數值的量化計算不可能，例如週期解的週期，等... 在第二部份，我們在genusN的黎曼空間上發展積分技巧來完成對這些積分數值上的計算。並且給一些例子。

Abstract We study the motions of a pendulum. An ideal pendulum motion is energy-conservative, so the traces of the motions of its mathematic model

are uniquely determined by the initial total energys, and, consequently, all solutions are able to be analyzed and solved by the conservation law of energys. There are three kinds of solutions characterized by the initial total energys, namely the periodic solutions (in time), the seperatrices, and the wavetrains. In part I, from the conservation laws, we transferred the nonlinear ODE problem into the so-called inverse problem (in an integral form), and then expressed the solutions in terms of classical elliptic functions. Notice that those integrals for the inverse problem have multi-valued integrands, and it is impossible to do numerical computations for quantities such as periods of periodic solution, etc.. In part II, we developed integral techniques on the Riemann surfaces of genus N to carry out the numerical computations for those integrals. Some examples are given.