Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions

Theorem
Let $J[y_1,~y_2,~...,y_n]$ be a functional of the form

$\displaystyle J[y_1,~y_2,~...,y_n]=\int_{a}^{b}F\left(x,~y_1,~y_2,...,~y_n,~y_1',~y_2',...,~y_n'\right)\mathrm{d}{x}$

Let $y_1,~y_2,~...,y_n$ be functions in $C^1\left[{{a}\,.\,.\,{b}}\right]$, and satisfy the boundary conditions $y_i(a)=A_i$ and $y_i(b)=B_i$, for $i=(1,~2,...,~n)$.

Then a necessary condition for $J[y_1,~y_2,~...,y_n]$ to have an extremum (strong or weak) for a given functions $y_1,~y_2,~...,y_n$ is that they satisfy Euler's equations:

$\displaystyle F_{y_i}-\frac{\mathrm{d}{}}{\mathrm{d}{x}}F_{y_i'}=0,~i=(1,~2,~...,~n)$

Proof
From Condition for Differentiable Functional of N Functions to have Extremum we have


 * $\displaystyle\delta J[y_1,~y_2,~...,y_n; h_1,~h_2,~...,h_n]\bigg\rvert_{y_i=\hat{y}_i,~i=(1,~2,~...,~n)}=0$

For the variation to exist it has to satisfy the requirement for a differential functional.

Note that the endpoints of $y_i(x)$ are fixed. $h_i(x)$ is not allowed to change values of $y_i(x)$ at those points.

Hence $h_i(a)=0$ and $h_i(b)=0$.

We will start from the increment of a functional:



Using multivariate Taylor's theorem, one can expand $F\left(x,y_1+h_1,...,y_n+h_n,y_1'+h_1',...,y_n'+h_n'\right)$ with respect to functions $h_i(x)$ and $h_i'(x)$:


 * $\displaystyle

F\left(x,y_i+h_i,y_i'+h_i'\right)=F\left(x,y_i+h_i,y_i'+h_i'\right)\bigg\rvert_{h_i=0,~h_i'=0 }+ \frac{ \partial{F\left(x,~y_i+h_i,y_i'+h_i'\right)} }{ \partial{{y} } }\bigg\rvert_{h_i=0,~h_i'=0 } {h_i} +\frac{ \partial{F\left(x,{y}+{h},f{y}'+{h}'\right)} }{ \partial{{y}'} }\bigg\rvert_{{h}={0},~{h}'={0}} {h}'+\mathcal{O}\left({h}^2, {h}\dot{h}', {h}'^2\right) $

We can substitute this back into the integral. Note that the first term in the expansion and the negative one in the integral will cancel out.


 * $\displaystyle\Delta J[y;h]=\int_{a}^{b}\left[F(x,y,y')_y h + F(x,y,y')_{y'} h' + \mathcal{O}\left(h^2, hh', h'^2\right)\right]\mathrm{d}{x}$

Terms in $\mathcal{O}\left(h^2,h'^2\right)$ represent terms of order higher than 1 with respect to $h$ and $h'$.

Now, suppose we expand $\int_{a}^{b}\mathcal{O}\left(h^2, hh', h'^2\right)\mathrm{d}{x}$.

Every term in this expansion will be of the form


 * $\displaystyle\int_{a}^{b}A\left(m, n\right)\frac{\partial^{m+n} F\left(x, y, y'\right)}{\partial{y}^m\partial{y'}^n}h^m h'^n \mathrm{d}{x}$

where $m,~n\in\N$ and $m+n\ge 2$

By definition, the integral not counting in $\mathcal{O}(h^2, hh', h'^2)$ is a variation of functional:


 * $\displaystyle \delta J[y;h]=\int_{a}^{b}\left[F_y h+F_{y'}h'\right]\mathrm{d}{x}$

According to lemma, this implies that


 * $\displaystyle F_y-\frac{\mathrm{d}{}}{\mathrm{d}{x}}F_{y'}=0$