The Exact Theory and Numerical Computations of Pendulum Motions on Riemann Surfaces of Genus N with Cut-Structure of Type B

• Main Authors: Chang, Chun-Fu, 張竣富
• Other Authors: Lee, Jong-Eao
• Format: Others
• Language: en_US
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/74514878633551196942

碩士 === 國立交通大學 === 應用數學系所 === 100 === The Sine-Gordon equation u_xx-u_yy+sin⁡u=0 is a well-known Partial differential equation, and there are some special solutions satisfy the nonlinear second-order differential equation (d^2 u)/(dt^2 )+sin⁡u=0 which is the Pendulum motion. As we solving the differential equation (d^2 u)/(dt^2 )+sin⁡u=0. We first replace sin⁡u by the Maclaurin Series of sin⁡u to get the differential equation of the form (d^2 u)/(dt^2 )+P(u)=0 , where P(u) is a polynomial. Solutions of such equations reside in Riemann Surfaces of genus N. We construct these Riemann Surfaces with the correct algebraic structures. So we can perform path integrals on the Riemann Surfaces to get the numerical solution of the equation. Next, we investigate the classical Elliptic functions, and use the Jacobian Elliptic function to analyze this nonlinear pendulum motion. Finally, we derive the exact solutions and the periods of those solutions by the Jacobian Elliptic functions.