變分法基本引理

數學裏,特別是在變分法裏,變分法基本引理fundamental lemma of calculus of variations)是一種專門用來變換問題表述的引理,可以將問題從弱版表述weak formulation)(變分形式)改變為強版表述(微分形式)。

敘述

代表阶导数连续(阶光滑)的函数空间,代表无限光滑的函数空间。

變分法基本引理:

若任意 滿足 成立

證明

因為只要存在一個不滿足 ,就可以證明 ,因此我們只須證明其中一個特例。


滿足下列兩個條件:

並且令

可得到

因為 是正值,所以 必須恆等於 0 ,與假設 矛盾。

應用

這引理可用來證明泛函

極值歐拉-拉格朗日方程式

的弱解。

歐拉-拉格朗日方程式在經典力學微分幾何佔有重要的角色。

參閱

參考文獻

  • Leitmann, George. The Calculus of Variations and Optimal Control: An Introduction. Springer. 1981. ISBN 0306407078. 

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