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10 Variational Problems of Functionals with Vector, Tensor …
δ J =
V
∂ F
∂u
+ f u a · ∇ × b
δu +
∂ F
∂ a
+ f ∇ × b
· δa − ∇ · (b × c)
dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂u
+ f u (u)a · ∇ ×
∂ F
∂∇ × f (u)a
= 0
∂ F
∂ a
+ f (u)∇ ×
∂ F
∂∇ × f (u)a
= 0
(5)
The natural boundary condition is
∂ F
∂∇ × f (u)a
× n
S
= 0
(6)
Example 10.5.4 The functional with the rotation of cross product of two vectors.
Let the functional
J =
V
F(a, b, a × b, ∇ × (a × b))dV
(1)
Derive its Euler equations and natural boundary condition.
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂ a
· δa +
∂ F
∂ b
· δb +
∂ F
∂(a × b)
· δ(a × b) +
∂ F
∂∇ × (a × b)
· δ∇ × (a × b)
dV
(2)
Putting c =
∂ F
∂(a×b)
, d =
∂ F
∂∇×(a×b)
, the variations of the latter terms in Eq. (2) can
be written as
c · δ(a × b) = c · (δa × b + a × δb) = (b × c) · δa + (c × a) · δb
(3)
d · δ∇ × (a × b) = d · ∇ × δ(a × b) = −∇ · [d × δ(a × b)] + (∇ × c) · δ(a × b)
= −∇ · [d × δ(a × b)] + (∇ × d) · (δa × b + a × δb)
= −∇ · [d × δ(a × b)] − (a × ∇ × d) · δb + (b × ∇ × d) · δa
(4)
Substituting Eqs. (3) and (4) into Eq. (2), we get
δ J =
V
∂ F
∂ a
+ b × (c + ∇ × d)
· δa +
∂ F
∂ b
− a × (c + ∇ × d)
· δb − ∇ · [d × δ(a × b)]
dV
(5)
The Euler equations of the functional are