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10 Variational Problems of Functionals with Vector, Tensor …
Putting T =
∂ F
∂∇×(a 1 a 2 ···a n )
, the variation of the last term in Eq. (2) can be written
as
T n
· ∇ × δ(a 1 a 2 · · · a n ) = ∇ · [δ(a 1 a 2 · · · a n ) ×
(n − 1)· T ] + (∇ × T ) n
· δ(a 1 a 2 · · · a n )
= ∇ · [δ(a 1 a 2 · · · a n ) ×
(n − 1)· T ] + (∇ × T ) n
·
(δa 1 a 2 · · · a n + a 1 δa 2 · · · a n + · · · + a 1 a 2 · · · δa n )
= ∇ · [δ(a 1 a 2 · · · a n ) ×
(n − 1)· T ] + [(∇ × T ) n − 1
· a 2 · · · a n ] · δa 1
+ [a 1 · (∇ × T ) n − 2
· a 3 · · · a n ] · δa 2
+ · · · + [a 1 a 2 · · · a n−1 n − 1
· · (∇ × T )] · δa n
(3)
Substituting Eq. (3) into Eq. (2), we obtain
δ J =
V
⎧
⎨
⎩
n
i=1
∂ F
∂ a i
+ a 1 a 2 · · · a i−1 i − 1
· (∇ × T ) n − i
· a i+1 a i+2 · · · a n
· δa i
⎫
⎬
⎭
dV
+
V
{∇ · [δ(a 1 a 2 · · · a n ) ×
(n − 1)· T ]}dV
(4)
The Euler equations of the functional are
∂ F
∂ a i
+ a 1 a 2 · · · a i−1 i − 1
· (∇ × T ) n − i
· a i+1 a i+2 · · · a n = 0 (i = 1, 2, · · · , n)
(5)
The natural boundary conditions are
n ×
∂ F
∂∇ × (a 1 a 2 · · · a n )
S
= 0
(6)
Example 10.6.22 Let a be a vector, A is a tensor of order n, find the Euler equations
and natural boundary conditions of the functional
J =
V
F(a, A, ∇(a × A), ∇ · (a × A), ∇ × (a × A))dV
(1)
where, the inner product of tensors conforms to the rule of parallel operation.
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂ a
· δa +
∂ F
∂ A
· δ A +
∂ F
∂∇(a × A)
n + 1
· δ∇(a × A)