General Variation of Integral Functional/Dependent on N Functions

Theorem
Let $J$ be a functional of the form

$\displaystyle J[...y_i...]=\int_{x_0}^{x_1} F\left({x, ... y_i ..., ...y_i'...}\right)\mathrm{d}{x},~i=\left({1, ..., n}\right)$

Then

$\displaystyle \delta J=\int_{x_0}^{x_1}\sum_{i=1}^n\left(F_{y_i}-\frac{ \mathrm{d} }{ \mathrm{d}{x} }F_{y_i'}\right)h_i\left(x\right)+\sum_{i=1}^n F_{y_i'}\delta y_i\bigg\rvert_{x=x_0}^{x=x_1}+\left(F-\sum_{i=1}^n y_i'F_{y_i'}\right)\delta x\bigg\rvert_{x=x_0}^{x=x_1}$

Proof
Let ${y=y(x), y=y^*(x)}$ be smooth real functions.

Let $h(x)=y^*(x)-y(x)$

Let the endpoints of the curve $y_i=y_i(x), ~i=\left( 1, ..., n \right)$ be


 * $P_0=\left( {x_0, ... y_i^0 ...} \right),~ P_1=\left( {x_1, ... y_i^1 ...} \right)$

Let the endpoints of the curve $y_i=y_i^*=y_i(x)+h_i(x), ~i=\left( 1, ..., n \right)$ be


 * $P_0^*=\left( {x_0+\delta x_0, ... y_i^0+\delta y_i^0 ...} \right),~ P_1^*=\left( {x_1+\delta x_1, ... y_i^1+\delta y_i^1 ...} \right)$

Note that the endpoints of both functions may not necessarily match, thus making the functions defined on different intervals.

We choose to extend both curves in such a way, that if there is a difference between original endpoints at the same end,

then the curve that is not defined in the given interval is extended linearly by drawing a straight line along the tangent of the curve at the point, closest to that interval.

Now both functions $y(x)_i$ and $y^*_i(x)$ are defined in $\left[{{x_0}\,.\,.\,{x_1+\delta x_1}}\right]$.

The corresponding variation $\delta J$ of $J[{...y_i...}]$ is defined as the expression which is linear in ${\delta x_0, \delta x_1}$ and ${h_i, h_i', y_i^0, y_i^1}$ for $i=\left({1, ..., n}\right)$,

and which differs from the increment by a quantity of order higher than 1 relative to

$\displaystyle\sum_{i=1}^n \rho\left({y_i, y_i^*}\right)$

Here $\rho\left({y_i, y_i^*}\right)$ is defined as


 * $\rho\left({y_i, y_i^*}\right)=\mathrm{max}\left\vert y_i-y_i^* \right\vert+\mathrm{max}\left\vert y_i'-{y_i^*}' \right\vert+$$d_2\left({P_0, P_0^*}\right)$$+d_2\left({P_1, P_1^*}\right)$

By using Taylor's theorem, $\Delta J$ can be rewritten as

$h_i$ terms were integrated by parts.

Also, $h_i\left(x_0\right)\sim\delta y_1^0-y_i'\left(x_0\right)\delta x_0,~h_i\left(x_1\right)\sim\delta y_1^1-y_i'\left(x_1\right)\delta x_1$.

Thus,

where $\delta x\rvert_{x=x_j}=\delta x_j,~\delta y_i\rvert_{x=x_j}=\delta y_i^j,~j=\left({0, 1}\right)$.