10.6 Variational Problems of Functionals with Parallel-Type Inner …
715
+
∂ F
∂∇ · (a × A)
n − 1
· δ∇(a × A) +
∂ F
∂∇(a × A)
n
· δ∇ × (a × A)
dV (2)
Let B =
∂ F
∂∇(a× A)
, C =
∂ F
∂∇·(a× A)
, D =
∂ F
∂∇×(a× A)
, for the rear three variations of
expression (2), there are
B n + 1
· δ∇(a × A) = ∇ · [B n
· δ(a × A)] − ∇ · B n
· δ(a × A)
= ∇ · [B n
· δ(a × A)] − ∇ · B n
· (δa × A + a × δ A)
= ∇ · [B n
· δ(a × A)] − [A ×
(n − 1)· ∇ · B] · δa + a × ∇ · B n
· δ A
(3)
C n − 1
· δ∇(a × A) = ∇ · [δ(a × A) n − 1
· C] − ∇C n
· δ(a × A)
= ∇ · [δ(a × A) n − 1
· C] − ∇C n
· (δa × A + a × δ A)
= ∇ · [δ(a × A) n − 1
· C] − [A ×
(n − 1)· ∇C] · δa + a × ∇C n
· δ A
(4)
D n
· δ∇ × (a × A) = ∇ · [δ(a × A) ×
(n − 1)· D] + ∇ × D n
· δ(a × A)
= ∇ · [δ(a × A) ×
(n − 1)· D] + ∇ × D n
· (δa × A + a × δ A)
= ∇ · [δ(a × A) ×
(n − 1)· D] + [A ×
(n − 1)· ∇ × D] · δa − a × ∇ × D n
· δ A (5)
Substituting Eqs. (3), (4), and (5) into expression (2), we get
δ J =
V
∂ F
∂ a
− A ×
(n − 1)· ∇ · B − [A ×
(n − 1)· ∇ C] + [A ×
(n − 1)· ∇ × D]
· δa
+
∂ F
∂ A
+ a × ∇ · B + a × ∇C − a × ∇ × D
· δ A
dV
+
V
∇ · [B n
· δ(a × A)] + ∇ · [δ(a × A) n − 1
· C] + ∇ · [δ(a × A) ×
(n − 1)· D]
dV
(6)
The Euler equations of the functional are
⎧
⎨
⎩
∂ F
∂ a − A ×
(n − 1)· ∇ ·
∂ F
∂∇(a× A) − A ×
(n − 1)· ∇
∂ F
∂∇·(a× A) + A ×
(n − 1)· ∇ ×
∂ F
∂∇×(a× A) = 0
∂ F
∂ A + a × ∇ ·
∂ F
∂∇(a× A) + a × ∇
∂ F
∂∇·(a× A) − a × ∇ ×
∂ F
∂∇×(a× A) = 0
(7)
The natural boundary conditions are
n ·
∂ F
∂∇(a × A)
S
= 0, n
∂ F
∂∇ · (a × A)
S
= 0, n ×
∂ F
∂∇ × (a × A)
S
= 0 (8)