古典力學的數學方法之探討

Abstract

本篇論文是我對古典力學的數學方法所做的探討。古典力學由牛頓力學、拉格朗日力學 (Lagrangian mechanics)、漢米爾頓力學 (Hamiltonian mechanics) 所構成。牛頓力學主要受伽利略的相對論原理啟發，因此我從這個原理的數學建構起頭，接著，我推出一個質點在三維中心力場中的運動必維持在某個平面中。在探討拉格朗日力學時，藉由解拉格朗日方程，我求出平面上距離固定之兩質點的運動。透過變分學，我說明拉格朗日力學系統推廣了牛頓位勢力學系統。而藉著勒壤得變換(Legendre transformation)，我推出拉格朗日力學系統其實是漢米爾頓力學系統的特例。

In this thesis, I give a survey of mathematical methods of classical mechanics. Classical mechanics consists of Newtonian, Lagrangian, and Hamiltonian mechanics. Newtonian mechanics is enlightened by Galileo’s principle of relativity, so I give mathematical construction of this principle in the beginning. By methods in Newtonian mechanics, I have shown that every three-dimensional motion in a central force field remains in some plane. By Lagrange’s equations in Lagrangian mechanics, I have solved the motion of two point masses with fixed distance. Through a variational principle, I have shown how a Lagrangian mechanical system generalizes a Newtonian potential mechanical system. Then by Legendre transformation, I have shown how a Lagrangian mechanical system is a particular Hamiltonian mechanical system.