General Variation of Integral Functional/Dependent on N Functions

Theorem
Let $J$ be a functional of the form

$\displaystyle J \sqbrk {\ldots y_i \ldots} = \int_{x_0}^{x_1} \map F{x, \ldots y_i \ldots, \ldots y_i' \ldots} \rd x, i = \openint 1 n$

Then:

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

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

Let $\map h x=\map {y^*} x-\map y x$

Let the endpoints of the curve $y_i=\map {y_i} x, ~i=\openint 1 n$ be


 * $P_0=\paren {x_0, ... y_i^0 ...},~ P_1=\paren {x_1, ... y_i^1 ...}$

Let the endpoints of the curve $y_i=y_i^*=\map {y_i} x+\map {h_i} x, ~i=\openint 1 n$ be


 * $P_0^*=\paren {x_0+\delta x_0, ... y_i^0+\delta y_i^0 ...},~ P_1^*=\paren {x_1+\delta x_1, ... y_i^1+\delta y_i^1 ...}$

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 of intervals 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 $\map {y_i} x$ and $\map {y^*_i} x$ are defined in $\closedint {x_0} {x_1+\delta x_1}$.

The corresponding variation $\delta J$ of $J\sqbrk{...y_i...}$ 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 $i=\openint 1 n$,

and which differs from the increment by a quantity of order higher than 1 relative to $\displaystyle\sum_{i=1}^n \map \rho {y_i,y_i^*}$.

Here $\map {\rho} {y_i,y_i^*}$ is defined as


 * $\map {\rho} {y_i,y_i^*}=$ $\mathrm{max}$$\size {y_i-y_i^*}$ $+\mathrm{max}\size {y_i'-{y_i^*}'}+$$\map {d_2} {P_0, P_0^*}$$+\map {d_2} {P_1, P_1^*}$

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

where $h_i$ terms were integrated by parts.

In the same manner, $\map h x$ can be expanded as


 * $\displaystyle \map {h_i} {x_0}=\delta y_1^0-\map {y_i'} {x_0}\delta x_0+$$\map {\mathcal{O} } {\map {\rho} {y, y+h} }$


 * $\map {h_i} {x_1}=\delta y_1^1-\map {y_i'} {x_1}\delta x_1+\map {\mathcal{O} } {\map {\rho} {y,y+h} }$

Thus,

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