In 1-D, the minimization problem can be stated as
Find
such that
![{\displaystyle U[u(x)]=\int _{x_{0}}^{x_{1}}F(x,u,u^{'})dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24af35cc71f5518e663220f002ddd2165717be48)
is a minimum.
We have seen that the minimization problem can be reduced down to the solution of an Euler equation
![{\displaystyle {\frac {\partial F}{\partial u}}-{\frac {d}{dx}}\left({\frac {\partial F}{\partial u^{'}}}\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fd4a27a0348e35526f7834afc3620c71b6c137)
with the associated boundary conditions
![{\displaystyle \eta (x_{0})=0~{\text{and}}~\eta (x_{1})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81c7809ffeb9be7a61950bbc85f045bad977c34a)
or,
![{\displaystyle \left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{0}}=0~{\text{and}}\left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{1}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2010da8c13149155c3dabe63d24fc693c1c76693)
In 3-D, the equivalent minimization problem can be stated as
Find
such that
![{\displaystyle U[\mathbf {u} (\mathbf {x} )]=\int _{\mathcal {R}}F(\mathbf {x} ,\mathbf {u} ,{\boldsymbol {\nabla }}\mathbf {u} )~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/888bfe30b191a166f5101c98c96b0a308048e6ca)
is a minimum.
We would like to find the Euler equation for this problem and the associated boundary conditions required to minimize
.
Let us define all our quantities with respect to an orthonormal basis
.
Then,
![{\displaystyle \mathbf {x} =x_{i}{\widehat {\mathbf {e} }}_{i}~~;~~~\mathbf {u} =u_{i}{\widehat {\mathbf {e} }}_{i}~~;~~~{\boldsymbol {\nabla }}\mathbf {u} =u_{i,j}{\widehat {\mathbf {e} }}_{i}\otimes {\widehat {\mathbf {e} }}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1401dabcecfceff04eefdab3d743f063f8c24b)
and
![{\displaystyle U[\mathbf {u} (\mathbf {x} )]=\int _{\mathcal {R}}{\tilde {F}}(x_{i},u_{i},u_{i,j})~dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02b5462616ee307cfc7739b622d78afbfd7b5a11)
Taking the first variation of
, we get
![{\displaystyle \delta U=\int _{\mathcal {R}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i}}}\delta u_{i}+{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i,j}\right)dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47d4d8529ab83fad61fbb8ca5e63088d5ddb5c0b)
All the nine components of
are not independent. Why ?
The variation of the functional
needs to be expressed
entirely in terms of
. We do this using the 3-D equivalent of integration by parts - the divergence theorem.
Thus,
![{\displaystyle {\begin{aligned}\int _{\mathcal {R}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i,j}~dV&=\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}\right)dV-\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\delta u_{i}~dV\\&=\int _{\partial {\mathcal {R}}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}~n_{j}~dA-\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}{}{}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\delta u_{i}~dV\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/455d32b06293bff33cb936c0d73cd5b8ff9483d5)
Substituting in the expression for
, we have,
![{\displaystyle {\begin{aligned}\delta U&=\int _{\mathcal {R}}{\frac {\partial {\tilde {F}}}{\partial u_{i}}}\delta u_{i}~dV+\int _{\partial {\mathcal {R}}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}~n_{j}~dA-\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\delta u_{i}~dV\\&=\int _{\mathcal {R}}\left[{\frac {\partial {\tilde {F}}}{\partial u_{i}}}-{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\right]\delta u_{i}~dV+\int _{\partial {\mathcal {R}}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}~n_{j}~dA\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2deeb00c24dd4c82d3079422d52aebccaa924cef)
For
to be minimum, a necessary condition is that
for all variations
.
Therefore, the Euler equation for this problem is
![{\displaystyle {\frac {\partial {\tilde {F}}}{\partial u_{i}}}-{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)=0~~~~\forall ~~\mathbf {x} \in {\mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93f75feb457595b64f055b4f4c6ef43821b62b41)
and the associated boundary conditions are
![{\displaystyle {\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}=0~~~{\text{or,}}~~~\delta u_{i}=0~~~~\forall ~~\mathbf {x} \in \partial {\mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0eea6ed37957f107e492ef3fdf31c4b09a7bc2b)