從微積分的發展，到伯努利兄弟的出現，更因為他們兄弟彼此在數學學數上的一些紛爭與較量，以及弟弟約翰在當時提出了一個問題想挑戰當代的大師牛頓之後，進而使得數學的發展上有了一些新的想法，也就是泛函的概念以及變分法的出現，而在變分法當中則以『Euler-Lagrange Equation』為主要核心，以此作為變分的基礎模型來解決與討論一些在物理上應用的問題。

New mathematical ideas have been evolved in calculus since the Bernoulli brothers. The brothers competed and argued in academic mathematics. The younger even raised questions to challenge Newton, who was the authority at that time. These events led to the orientation of functional idea and calculus of variations. And the Euler-Lagrange equation is the core of calculus of variations. Using it as the base model of variation, we raised a few more discussions and proposed some possible solutions for the application in Physics.

Introduction ……………………………………………2
1.Background of Variations ……………………………………2
2.Euler-Lagrange Equation ………………………………9
3.Poisson-Boltzmann Equations …………………………11
4.Modified Poisson-Boltzmann Equations……………………12
5.Charge Conserving Poisson-Boltzmann Equations…………16
References ………………………………………………22

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