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10 Variational Problems of Functionals with Vector, Tensor …
or
∂ F
∂ a
+ ∇∇ ·
∂ F
∂∇ × ∇ × a
−
∂ F
∂∇ × ∇ × a
= 0
(6)
The natural boundary conditions are
n ×
∂ F
∂∇ × ∇ × a
S
= 0, n × ∇ ×
∂ F
∂∇ × ∇ × a
S
= 0
(7)
From Eq. (3), there is b · δ(∇
n
× a) = (∇
n
× b) · δa+ the divergence terms.
Corollary Let the functional
J =
V
F(a, ∇
n
× a)dV
(1)
The Euler equation of the functional is
∂ F
∂ a
+ ∇
n
×
∂ F
∂∇ n × a
= 0
(2)
The natural boundary conditions are
n × (∇×)
i−1
∂ F
∂∇ n × a
S
= 0 (i = 1, 2, · · · , n)
(3)
Example 10.5.2 The functional with the secondary rotation of a scalar multiplied by
a vector. Derive the Euler equations and natural boundary conditions of the functional
J =
V
F(u, a, ∇ × ∇ × ua)dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂ a
· δa +
∂ F
∂∇ × ∇ × ua
· δ∇ × ∇ × ua
dV
(2)
Putting b =
∂ F
∂∇×∇×ua
, Using the result of the above example, δua = aδu + uδa,
there is
δ J =
V
∂ F
∂u
δu +
∂ F
∂ a
· δa − ∇ · (b × δ∇ × ua) − ∇ · (∇ × b × δua) + (∇ × ∇ × b) · (aδu + uδa)
dV
(3)
The Euler equations of the functional are