Minimizing a functional in 1-D
edit
In 1-D, the minimization problem can be stated as
Find
u
(
x
)
{\displaystyle u(x)}
U
[
u
(
x
)
]
=
∫
x
0
x
1
F
(
x
,
u
,
u
′
)
d
x
{\displaystyle U[u(x)]=\int _{x_{0}}^{x_{1}}F(x,u,u^{'})dx}
is a minimum.
We have seen that the minimization problem can be reduced down to the solution of an Euler equation
∂
F
∂
u
−
d
d
x
(
∂
F
∂
u
′
)
=
0
{\displaystyle {\frac {\partial F}{\partial u}}-{\frac {d}{dx}}\left({\frac {\partial F}{\partial u^{'}}}\right)=0}
with the associated boundary conditions
η
(
x
0
)
=
0
and
η
(
x
1
)
=
0
{\displaystyle \eta (x_{0})=0~{\text{and}}~\eta (x_{1})=0}
or,
∂
F
∂
u
′
|
x
0
=
0
and
∂
F
∂
u
′
|
x
1
=
0
{\displaystyle \left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{0}}=0~{\text{and}}\left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{1}}=0}
Minimizing a Functional in 3-D
edit
In 3-D, the equivalent minimization problem can be stated as
Find
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )}
U
[
u
(
x
)
]
=
∫
R
F
(
x
,
u
,
∇
u
)
d
V
{\displaystyle U[\mathbf {u} (\mathbf {x} )]=\int _{\mathcal {R}}F(\mathbf {x} ,\mathbf {u} ,{\boldsymbol {\nabla }}\mathbf {u} )~dV}
is a minimum.
We would like to find the Euler equation for this problem and the associated boundary conditions required to minimize
U
{\displaystyle U}
Let us define all our quantities with respect to an orthonormal basis
(
e
^
i
)
{\displaystyle ({\widehat {\mathbf {e} }}_{i})}
Then,
x
=
x
i
e
^
i
;
u
=
u
i
e
^
i
;
∇
u
=
u
i
,
j
e
^
i
⊗
e
^
j
{\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}}
and
U
[
u
(
x
)
]
=
∫
R
F
~
(
x
i
,
u
i
,
u
i
,
j
)
d
V
{\displaystyle U[\mathbf {u} (\mathbf {x} )]=\int _{\mathcal {R}}{\tilde {F}}(x_{i},u_{i},u_{i,j})~dV}
Taking the first variation of
U
{\displaystyle U}
δ
U
=
∫
R
(
∂
F
~
∂
u
i
δ
u
i
+
∂
F
~
∂
u
i
,
j
δ
u
i
,
j
)
d
V
{\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}
All the nine components of
δ
u
i
,
j
{\displaystyle \delta u_{i,j}}
The variation of the functional
U
{\displaystyle U}
δ
u
i
{\displaystyle \delta u_{i}}
divergence theorem .
Thus,
∫
R
∂
F
~
∂
u
i
,
j
δ
u
i
,
j
d
V
=
∫
R
∂
∂
x
j
(
∂
F
~
∂
u
i
,
j
δ
u
i
)
d
V
−
∫
R
∂
∂
x
j
(
∂
F
~
∂
u
i
,
j
)
δ
u
i
d
V
=
∫
∂
R
∂
F
~
∂
u
i
,
j
δ
u
i
n
j
d
A
−
∫
R
∂
∂
x
j
(
∂
F
~
∂
u
i
,
j
)
δ
u
i
d
V
{\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}}}
Substituting in the expression for
δ
U
{\displaystyle \delta U}
δ
U
=
∫
R
∂
F
~
∂
u
i
δ
u
i
d
V
+
∫
∂
R
∂
F
~
∂
u
i
,
j
δ
u
i
n
j
d
A
−
∫
R
∂
∂
x
j
(
∂
F
~
∂
u
i
,
j
)
δ
u
i
d
V
=
∫
R
[
∂
F
~
∂
u
i
−
∂
∂
x
j
(
∂
F
~
∂
u
i
,
j
)
]
δ
u
i
d
V
+
∫
∂
R
∂
F
~
∂
u
i
,
j
δ
u
i
n
j
d
A
{\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}}}
For
U
{\displaystyle U}
δ
U
=
0
{\displaystyle \delta U=0}
δ
u
{\displaystyle \delta \mathbf {u} }
Therefore, the Euler equation for this problem is
∂
F
~
∂
u
i
−
∂
∂
x
j
(
∂
F
~
∂
u
i
,
j
)
=
0
∀
x
∈
R
{\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}}}
and the associated boundary conditions are
∂
F
~
∂
u
i
,
j
=
0
or,
δ
u
i
=
0
∀
x
∈
∂
R
{\displaystyle {\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}=0~~~{\text{or,}}~~~\delta u_{i}=0~~~~\forall ~~\mathbf {x} \in \partial {\mathcal {R}}}
![{\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)
![{\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)
![{\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)
![{\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)
