10.6 Variational Problems of Functionals with Parallel-Type Inner …
691
∂ F
∂∇ × A
n
· ∇ × δ A = ∇ ·
δ A ×
(n − 1)·
∂ F
∂∇ × A
+
∇ ×
∂ F
∂∇ × A
n
· δ A
(10.6.26)
∂ F
∂ A∇
n + 1
· δ A∇ = ∇ ·
δ A n
·
∂ F
∂ A∇
−
∂ F
∂ A∇
· ∇ n
· δ A
(10.6.27)
∂ F
∂ A · ∇
n − 1
· δ A · ∇ = ∇ ·
∂ F
∂ A · ∇
n − 1
· δ A
−
∂ F
∂ A · ∇
∇ n
· δ A
(10.6.28)
∂ F
∂ A × ∇
n
· δ A × ∇ = ∇ ·
∂ F
∂ A × ∇
(n − 1)·
× δ A
+
∂ F
∂ A × ∇
× ∇
n
· δ A
(10.6.29)
Making use of Eqs. (10.6.23)–(10.6.29), the Euler equation and corresponding
natural boundary conditions of the functional with parallel-type inner product tensors
and the Hamiltonian operators-namely with the gradient, divergence and rotation can
be conveniently obtained.
Theorem 10.6.1 Let the functional
J =
V
F( A, A
T
, ∇ A, ∇ · A, ∇ × A, A∇, A · ∇, A × ∇)dV
(10.6.30)
where A is a tensor of order n, then the Euler equation of the functional is
∂ F
∂ A
+
∂ F
∂ A
T
− ∇ ·
∂ F
∂∇ A
− ∇
∂ F
∂∇ · A
+ ∇ ×
∂ F
∂∇ × A
−
∂ F
∂ A∇
· ∇ −
∂ F
∂ A · ∇
∇ +
∂ F
∂ A × ∇
× ∇ = 0
(10.6.31)
The natural boundary conditions of the functional are
n ·
∂ F
∂∇ A
S
= 0, n
∂ F
∂∇· A
S
= 0, n ×
∂ F
∂∇× A
S
= 0,
∂ F
∂ A∇
· n
S
= 0,
∂ F
∂ A·∇
n
S
= 0
∂ F
∂ A,×∇
× n
S
= 0
(10.6.32)
where, for n
∂ F
∂∇· A
S
= 0 and
∂ F
∂ A·∇
n
S
= 0, if they do not appear together with the
other boundary conditions, the vector n in the expressions can be removed.
Proof Taking the variation to Eq. (10.6.30) and making use of Eqs. (10.6.23)–
(10.6.29), there is
δ J =
V
∂ F
∂ A
n
·
δ A +
∂ F
∂ A
n
·
δ A T +
∂ F
∂∇ A
n + 1
·
∇δ A +
∂ F
∂∇ · A
n − 1
·
∇ · δ A +
∂ F
∂∇ × A
n
·
∇ × δ A
dV
+
V
∂ F
∂ A∇
n + 1
·
δ A∇ +
∂ F
∂ A · ∇
n − 1
·
δ A · ∇ +
∂ F
∂ A × ∇
n
·
δ A × ∇
dV
=
V
∂ F
∂ A
+
∂ F
∂ A
T
− ∇ ·
∂ F
∂∇ A
− ∇
∂ F
∂∇ · A
+ ∇ ×
∂ F
∂∇ × A
−
∂ F
∂∇ A
· ∇ −
∂ F
∂ A · ∇
∇ +
∂ F
∂ A × ∇
× ∇
n
·
δ AdV