準座標之高斯原理在非線性動力系統之應用

Abstract

高斯最小約束原理的物理意義就是在加速度空間中，求出約束力的最小值，進而推導出剛體系統的運動方程式，而不需藉由達朗白原理及虛位移，故能應用在求解非線性非完整約束的動力系統上。因此藉由高斯最小約束原理，可以求出線性/非線性、完整/非完整約束剛體系統的運動方程式。

本論文主要提出以角速度為主的準座標系統，並將高斯函數以及高斯最小約束原理重新改寫，將其應用在對稱性高及旋轉的剛體系統上。因此我們特別以實心球體及圓盤為例，探討一球體在一圓盤上及一圓盤在一平面上的運動情形，並針對其物理現象加以説明。

The physical meaning of Gauss’s principle of least constraint is the actual acceleration can be obtained by minimizing the Gaussian function, which is the square of the scaled constraint forces. Thus we can get the solution of equations of motion by solving a minimizing problem without appeal to the use of virtual displacement and the D’Alembert’s principle. This perspective enables Gauss’s principle of least constraint to get the solution of motion for systems having multiple, linear and/or nonlinear, holonomic and/or nonholonomic constraints. In this thesis, we proposed a new form of Gauss’s principle of least constraint, which is based on quasi coordinates of angular velocity, and then applied them to the rotational rigid-body systems with better symmetry. Thus we take the examples of the rigid-body ball and the disk to discuss the behavior of a ball on the surface of a rotating disk and a disk on the plane.