734
10 Variational Problems of Functionals with Vector, Tensor …
∂ F
∂∇u A m
× B
· n
S
= 0,
∂ F
∂∇ · u A m
× B
S
= 0,
∂ F
∂∇ × u A m
× B
× n
S
= 0 (9)
When m = n, Eq. (8) can be transformed into the following form
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ F
∂u +
⎛
⎜
⎝
∂ F
∂u A n
× B
−
∂ F
∂∇u A n
× B
· ∇ −
∂ F
∂∇·u A n
× B
∇ +
∂ F
∂∇×u A n
× B
× ∇
⎞
⎟
⎠ n
· A n
× B = 0
∂ F
∂ A + u B n
×
⎛
⎜
⎝
∂ F
∂u A n
× B
−
∂ F
∂∇u A n
× B
· ∇ −
∂ F
∂∇·u A n
× B
∇ +
∂ F
∂∇×u A n
× B
× ∇
⎞
⎟
⎠
T
= 0
∂ F
∂ B + u
⎛
⎜
⎝
∂ F
∂u A n
× B
−
∂ F
∂∇u A n
× B
· ∇ −
∂ F
∂∇·u A n
× B
∇ +
∂ F
∂∇×u A n
× B
× ∇
⎞
⎟
⎠
T
n
× A T = 0
(10)
Example 10.7.11 Let A be a tensor of order n, B is a tensor of order m, with m ≤ n,
derive the Euler equations and natural boundary conditions of the functional
J =
V
F(u, A, B, u A m
× B, ∇u A m
× B, ∇ · u A m
× B, ∇ × u A m
× B)dV (1)
Solution Taking the variation to the functional, we obtain
δ J =
V
⎛
⎜
⎝
∂ F
∂u
δu +
∂ F
∂ A
n
· δ A +
∂ F
∂ B
m
· δ B +
∂ F
∂u A m
× B
n
· δu A m
× B +
∂ F
∂∇u A m
× B
n + 1
· δ∇u A m
× B
+
∂ F
∂∇ · u A m
× B
n − 1
· δ∇ · u A m
× B +
∂ F
∂∇ × u A m
× B
n
· δ∇ × u A m
× B
⎞
⎟
⎠dV
(2)
Let C =
∂ F
∂u A m
×
B
, D =
∂ F
∂∇u A m
×
B
, E =
∂ F
∂∇·u A m
×
B
, F =
∂ F
∂∇×u A m
×
B
, for the latter
four variations of Eq. (2), using Eq. (10.1.14), there are
C n
· δu A m
× B = C n
· A m
× Bδu + (−1) m u(C m m
× B T )C m+1,n n
· δ A + (−1) m u A (n − m)·
m× C m
· δ B
(3)
D n + 1
· ∇δu A m
× B = ∇ · (δu A m
× B n
· D) − D · ∇ n
· δu A m
× B
= ∇ · (δu A m
× B n
· D) − D · ∇ n
· ( A m
× Bδu