幾個變分法的應用

Several applications of calculus of variations

柳銘哲(Ming-Tse Liu)

《加馬》 50期 (2019/12) Pp. 33-44

摘要

變分法不僅在物理、數學或工程學上有充分的應用，很多其他領域的科學家或專家也紛紛用它來提出數學模型以解決他們面臨的問題。本論文先簡介一段變分法由來的歷史，再說明求解最簡泛函極值問題所需的歐拉方程式。然後我們利用它來解決了五個問題：1.彈簧既擺動又振動的力學問題；2.「兩個」幾何光學中的基本實驗定律—彎曲介面的反射定律和折射定律；3.短跑選手如何安排它的速度分布以獲得最佳成績；4.在考慮了生產和儲存兩種費用下，提出一個數學模型使得在限定期限內生產出特定數量的產品並讓成本降至最低。

關鍵字

拉格朗日 ； 變分法 ； 歐拉－拉格朗日方程式 ； 最簡泛函極值 ； 最速陡降線 ； 擺線 ； 最佳化

並列摘要

The calculus of variations has not only made a full application in physics, mathematics or engineering, but many other fields of scientists or experts have also used it solve the problems they face by building mathematical models. This paper first introduces the history of calculus of variations, and then explains the Euler equations which are needed to solve the simplest functional extreme problem. Finally we use it to solve five problems: 1. The mechanical problem of swinging and vibrating of a suspended spring; 2. The ＂two＂ basic experimental law in the geometric optics - law of reflection and the law of refraction both in a curved interface; 3. How to arrange speed distribution of a sprinter for the best running performance; 4. In the consideration of both production and storage costs, a mathematical model is proposed to produce a specific number of products within a deadline and to minimize costs.

並列關鍵字

lagrange ； calculus of variations ； Euler-Lagrange equation ； the simplest functional extreme value ； Brachistochrone curve ； cycloid ； optimization