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10 Variational Problems of Functionals with Vector, Tensor …
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂ a
+ (b × c) × ∇ × ∇ × d
· δa +
∂ F
∂ b
− c × (a × ∇ × ∇ × d)
· δb
+
∂ F
∂c
+ b × (a × ∇ × ∇ × d)
· δc − ∇ · (d × ∇ × e) − ∇ · [(∇ × d) × e]
dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ F
∂ a
+ (b × c) × ∇ × ∇ ×
∂ F
∂∇ × ∇ × [a × (b × c)]
= 0
∂ F
∂ b
− c ×
a × ∇ × ∇ ×
∂ F
∂∇ × ∇ × [a × (b × c)]
= 0
∂ F
∂c
+ b ×
a × ∇ × ∇ ×
∂ F
∂∇ × ∇ × [a × (b × c)]
= 0
(5)
The natural boundary conditions are
∂ F
∂∇ × ∇ × [a × (b × c)]
× n
S
= 0, ∇ ×
∂ F
∂∇ × ∇ × [a × (b × c)]
× n
S
= 0
(6)
Corollary Let the functional
J =
V
F(a, b, c, ∇
n
× [a × (b × c)])dV
(1)
The variation of the functional is
δ J =
V
∂ F
∂ a
· δa +
∂ F
∂ b
· δb +
∂ F
∂ c
· δc +
∂ F
∂∇ n × [a × (b × c)]
· δ∇ n × [a × (b × c)]
dV (2)
Putting d =
∂ F
∂∇ n ×[a×(b×c)]
, e = δ[a × (b × c)] = δa × (b × c) + a × (δb × c) +
a × (b × δc), there is
δ J =
V
∂ F
∂ a
+ (b × c) × ∇
n
× d
· δa +
∂ F
∂ b
− c × (a × ∇
n
× d)
· δb
+
∂ F
∂ c
+ b × (a × ∇
n
× d)
· δc −
n−1
i=0
∇ · (∇
i
× d) × ∇
n−1−i
× e
dV
(3)
The Euler equations of the functional are