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10 Variational Problems of Functionals with Vector, Tensor …
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∂ F
∂u
+ ∇ × a · ∇ ×
∂ F
∂∇ × u∇ × a
= 0
∂ F
∂ a
+ ∇ × u∇ ×
∂ F
∂∇ × u∇ × a
= 0
(5)
The natural boundary conditions are
∂ F
∂∇ × u∇ × a
× n
S
= 0, n · u∇ ×
∂ F
∂∇ × u∇ × a
S
= 0
( 6 )
Example 10.5.8 The functional with the rotation of quotient of two scalars
multiplied by the rotation of vector. Let the functional
J =
V
F
u, v, a, ∇ ×
u
v
∇ × a
dV
(1)
Derive its Euler equations and natural boundary conditions.
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂v
δv +
∂ F
∂ a
· δa +
∂ F
∂∇ ×
u
v
∇ × a
· δ∇ ×
u
v
∇ × a
dV (2)
Let b = ∇ ×
u
v
∇ × a, c = δ
u
v
∇ × a =
∇×a
v
δu +
u
v
∇ × δa −
u∇×a
v 2 δv, there is
b · ∇ × c = −∇ · (b × δc) + ∇ × b ·
∇ × a
v
δu +
u
v
∇ × δa −
u∇ × a
v 2 δv
= −∇ · (b × δc) +
∇ × b · ∇ × a
v
δu +
u∇ × b
v
· ∇ × δa
−
u∇ × b · ∇ × a
v 2
δv
= −∇ · (b × δc) +
∇ × b · ∇ × a
v
δu − ∇ ·
u∇ × b
v
× δa
+
∇ ×
u∇ × b
v
· δa −
u∇ × b · ∇ × a
v 2
δv
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
+
∇ × b · ∇ × a
v
δu +
∂ F
∂v
−
u∇ × b · ∇ × a
v 2
δv
+
∂ F
∂ a
+ ∇ ×
u∇ × b
v
· δa
dV
−
V
∇ · (b × δc) + ∇ ·
u∇ × b
v
× δa
dV
(4)