712
10 Variational Problems of Functionals with Vector, Tensor …
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 10.6.20 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 get
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), there are
C n
· δu A m
× B = C n
· A m
× Bδu + (−1)
m uC m
× B n
· δ A + uC (n − m)·
m× A m
· δ B (3)
D n + 1
· ∇δu A m
× B = ∇ · ( D n
· δu A m
× B) − ∇ · D n
· δu A m
× B
= ∇ · ( D n
· δu A m
× B) − ∇ · D n
· ( A m
× Bδu + uδ A m
× B + u A m
× δ B)
= ∇ · ( D n
· δu A m
× B) − ∇ · D n
· A m
× Bδu
− (−1) m u∇ · D m
× B n
· δ A − u∇ · D (n − m)·
m× A m
· δ B
(4)
E n − 1
· ∇ · δu A m
× B = ∇ · (δu A m
× B n − 1
· E) − ∇ E n
· ( A m
× Bδu + uδ A m
× B + u A m
× δ B)
= ∇ · (δu A m
× B n − 1
· E) − ∇ E n
· A m
× Bδu
− (−1) m u∇ E m
× B n
· δ A − u∇ E (n − m)·
m× A m
· δ B
(5)
F n
· ∇ × δu A m
× B = −∇ · [F
×
(n − 1)· δu A m
× B] + ∇ × F n
· ( A m
× Bδu + uδ A m
× B + u A m
× δ B)
= −∇ · [F
×
(n − 1)· δu A m
× B] + ∇ × F n
· A m
× Bδu + (−1)
m u(∇ × F m
× B) n
· δ A
+ u∇ × F (n − m)·
m× A m
· δ B
(6)