10.6 Variational Problems of Functionals with Parallel-Type Inner …
713
Substituting Eqs. (3)–(6) into Eq. (2), we obtain
δ J =
V
∂ F
∂u
+ (C − ∇ · D − ∇ E + ∇ × F) n
· A m
× B
δu
+
∂ F
∂ A
+ (−1) m u(C − ∇ · D − ∇ E + ∇ × F) m
× B
n
· δ A
+
∂ F
∂ B
+ u(C − ∇ · D − ∇ E + ∇ × F) (n − m)·
m× A
m
· δ B
dV
+
V
∇ · ( D n
· δu A m
× B) + ∇ · (δu A m
× B n − 1
· E) − ∇ · [F ×
(n − 1)· δu A m
× B]
dV
(7)
The Euler equations of the functional are
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ F
∂u +
⎛
⎜
⎝
∂ F
∂u A m
×
B
− ∇ ·
∂ F
∂∇u A m
×
B
− ∇
∂ F
∂∇·u A m
×
B
+ ∇ ×
∂ F
∂∇×u A m
×
B
⎞
⎟
⎠ n
· A m
× B = 0
∂ F
∂ A + (−1) m u
⎛
⎜
⎝
∂ F
∂u A m
×
B
− ∇ ·
∂ F
∂∇u A m
×
B
− ∇
∂ F
∂∇·u A m
×
B
+ ∇ ×
∂ F
∂∇×u A m
×
B
⎞
⎟
⎠ m
× B = 0
∂ F
∂ B + u
⎛
⎜
⎝
∂ F
∂u A m
×
B
− ∇ ·
∂ F
∂∇u A m
×
B
− ∇
∂ F
∂∇·u A m
×
B
+ ∇ ×
∂ F
∂∇×u A m
×
B
⎞
⎟
⎠ (n − m)·
m× A = 0
(8)
The natural boundary conditions are
n ·
∂ F
∂∇u A m
× B
S
= 0,
∂ F
∂∇ · u A m
× B
S
= 0, n ×
∂ F
∂∇ × u A m
× B
S
= 0 (9)
Example 10.6.21 The functional with the n vectors and the rotation of their product.
Let the functional
J =
V
F(a 1 , a 2 , · · · , a n , ∇ × (a 1 a 2 · · · a n ))dV
(1)
where, the double inner product of the tensors of order n conforms to the law of the
parallel operation, derive the Euler equations and natural boundary conditions of the
functional.
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂ a 1
· δa 1 +
∂ F
∂ a 2
· δa 2 + · · · +
∂ F
∂ a n
· δa n +
∂ F
∂∇ × (a 1 a 2 · · · a n )
n
· δ∇ × (a 1 a 2 · · · a n )
dV
(2)