General Variation of Integral Functional/Dependent on n Variables

Theorem

Let $\mathbf x$ be an $n$-dimensional vector.

Let $u = \map u {\mathbf x}$ be a real mapping.

Let $J$ be a functional such that:

$\displaystyle J \sqbrk u = \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$

Let $\mathbf x^*, u^*$ be such that:

$\displaystyle \mathbf x^* = \map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \, \map {\boldsymbol \phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } + \map {\mathcal O} {\epsilon^2}$
$\displaystyle u^* = \map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \, \map \psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x } } + \map {\mathcal O} {\epsilon^2}$

where $\boldsymbol \Phi$, $\Psi$ are differentiable with respect to $\epsilon$ and:

$\map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0} = \mathbf x$
$\map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0} = u$
$\displaystyle \map {\boldsymbol \phi} {\mathbf x, u, \frac {\partial u} {\partial \mathbf x} } = \dfrac {\partial \boldsymbol \Phi} {\partial \epsilon} \Bigg \vert_{\epsilon \mathop = 0}$
$\displaystyle \map \psi {\mathbf x, u, \frac {\partial u} {\partial \mathbf x} } = \dfrac {\partial \Psi} {\partial \epsilon} \Bigg \vert_{\epsilon \mathop = 0}$

Then the variation of the functional $J$ due to the original mapping being transformed by the aforementioned transformation reads:

$\displaystyle \delta J = \epsilon \int_R \paren {F_u - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} } \overline \psi \rd x_1 \dotsm \rd x_n + \epsilon \int_R \map {\dfrac {\partial} {\partial \mathbf x} } {F_{u_x} \overline {\boldsymbol \psi} + F \boldsymbol \phi} \rd x_1 \dotsm \rd x_n$

where

$\overline \psi = \psi - u_{\mathbf x} \boldsymbol \phi$

Proof

By definition:

 $\displaystyle \Delta J$ $=$ $\displaystyle J\sqbrk{\map {u^*} {x^*} }-J\sqbrk{\map u x}$ $\displaystyle$ $=$ $\displaystyle \int_{R^*}\map F {\mathbf x^*,u^*,\frac{\partial u^*}{\partial\mathbf x^*} }\mathrm d x_1^*\dots\rd x_n^*-\int_{R}\map F {\mathbf x,u,\dfrac{\partial u}{\partial\mathbf x} }\rd x_1\dots\rd x_n$ $\displaystyle$ $=$ $\displaystyle \int_R\paren{\map F {\mathbf x^*,u^*,\dfrac{\partial u^*}{\partial\mathbf x^*} }\det\paren{J_{\mathbf x^*} }-\map F {\mathbf x,u,\frac{\partial u}{\partial\mathbf x} } }\rd x_1\dots\rd x_n$ Volume element transformation

From the definition of $\mathbf x^*$ a Jacobian matrix can be constructed:

$\displaystyle\dfrac{\partial x_i^*}{\partial x_j}=\delta_i^j+\epsilon\dfrac{\partial\phi_i}{\partial x_j}+\map {\mathcal O} {\epsilon^2}$

By Taylor's theorem, The corresponding Jacobian determinant can be expanded with respect to $\epsilon$.

The $\epsilon^0$ term is obtained from the diagonal product by opening brackets and collecting $\epsilon$-free terms.

The $\epsilon^1$ term is obtained from the same diagonal product by choosing non-constant term once.

$\displaystyle\det\paren{J_{\mathbf x^*} }=1+\epsilon\dfrac{\partial\boldsymbol\phi}{\partial\mathbf x}+\map {\mathcal O} {\epsilon^2}$

$\displaystyle\Delta J=\int_R\sqbrk{\map F {\mathbf x^*,u^*\dfrac{\partial u^*}{\partial\mathbf x^*} }\paren{1+\epsilon\dfrac{\partial\boldsymbol\phi}{\partial\mathbf x} }-\map F {\mathbf x,u,\dfrac{\partial u}{\partial\mathbf x} } }\rd x_1\dots\rd x_n+\map {\mathcal O} {\epsilon^2}$

By definition, the principal part is:

$\displaystyle\delta J=\int_R\sqbrk{F_{\mathbf x}\delta\mathbf x+F_u\delta u+F_{u_{\mathbf x} }\delta u_{\mathbf x}+\epsilon F \dfrac{\partial\boldsymbol\phi}{\partial\mathbf x} }\rd x_1\dots\rd x_n$

Introduce the following differences:

$\Delta u=\map {u^*} {x^*}-\map u x$
$\overline{\Delta u}=\map {u^*} x-\map u x$
$\Delta x=x^*-x$

Then:

$\displaystyle\overline{\Delta u}=\epsilon\overline{\psi}+\map {\mathcal O} {\epsilon^2}$
$\overline{\delta u}=\epsilon\overline{\psi}$
$\displaystyle\Delta u=\dfrac{\partial u}{\partial\mathbf x}\delta\mathbf x+\overline {\delta u}+\map {\mathcal O} {\epsilon^2}$
$\displaystyle\delta u=\dfrac{\partial u}{\partial\mathbf x}\delta\mathbf x+\overline{\delta u}$
$\paren{\delta u}_{x_i}=\paren{\overline{\delta u} }_{x_i}+u_{\mathbf x x_i}{\delta\mathbf x}_{x_i}$
$\Delta\mathbf x=\epsilon\boldsymbol\phi+\map {\mathcal O} {\epsilon^2}$
$\delta\mathbf x=\epsilon\boldsymbol\phi$

Here $\delta$ difference stands for principal part, which is constant or linear with respect to $\epsilon$.

Thus, the variation of $J$ reads:

$\displaystyle\delta J=\int_R\sqbrk{F_{\mathbf x}\delta\mathbf x+F_u\overline{\delta u}+F_uu_{\mathbf x}\delta\mathbf x+F_{u_{\mathbf x} } {\overline{\delta u} }_{\mathbf x}+\sum_{i,j\mathop=1}^n F_{u_{x_i} }u_{x_i x_j}\delta x_j+\map F {\delta\mathbf x}_{\mathbf x} }\rd x_1\dots\rd x_n$

A few terms can be rewritten using:

$\displaystyle\dfrac{\partial}{\partial\mathbf x}\paren{F\delta\mathbf x}=F_{\mathbf x}\delta\mathbf x+\map F {\overline{\delta\mathbf x} }_{\mathbf x}+F_u u_{\mathbf x}\delta\mathbf x+\sum_{i,j\mathop=1}^n F_{u_{x_i} }u_{x_i x_j}\delta x_j$

and:

$\displaystyle \map {F_{u_{\mathbf x} } } {\overline{\delta u} }_{\mathbf x}=\dfrac{\partial}{\partial\mathbf x} {F_{u_{\mathbf x} } \overline{\delta u} }-\dfrac{\partial F_{u_{\mathbf x} } }{\partial\mathbf x}\delta u$

Substitution of previous results into variation of $J$ leads to:

$\displaystyle\delta J=\int_R\paren{F_u-\dfrac{\partial F_{u_{\mathbf x} } }{\partial\mathbf x} }\overline{\delta u}\rd x_1\dots\rd x_n+\int_R\dfrac{\partial}{\partial\mathbf x}\paren{F_{u_{\mathbf x} }\overline{\delta u}+F\delta\mathbf x}\rd x_1\dots\rd x_n$

Substitute expressions for $\overline{\delta u}$ and $\delta\mathbf x$ to obtain the desired result.

$\blacksquare$