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10 Variational Problems of Functionals with Vector, Tensor …
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∂ F
∂u
+ ∇ ·
(a × b) × ∇
∂ F
∂∇ · ∇u × (a × b)
= 0
∂ F
∂ a
+ b ×
∇u × ∇
∂ F
∂∇ · ∇u × (a × b)
= 0
∂ F
∂ b
− a ×
∇u × ∇
∂ F
∂∇ · ∇u × (a × b)
= 0
(5)
The natural boundary conditions are
∂ F
∂∇ · ∇u × (a × b)
S
= 0,
(a × b) × ∇
∂ F
∂∇ · ∇u × (a × b)
· n
S
= 0 (6)
Example 10.4.14 The functional with the divergence of cross product of trivector,
where, the vector in the middle is the gradient of a scalar. Let the functional
J =
V
F(u, a, b, ∇ · a × (∇u × b))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
∂ a
· δa +
∂ F
∂ b
· δb +
∂ F
∂∇ · a × (∇u × b)
δ∇ · a × (∇u × b)
dV (2)
Putting ϕ =
∂ F
∂∇·a×(∇u×b)
, c = δ[a × (∇u × b)] = δa × (∇u × b) + a × (∇δu ×
b) + a × (∇u × δb), and using the parallelepipedal product formula of trivector,
there is
ϕδ∇ · a × (∇u × b) = ϕ∇ · δ[a × (∇u × b)] = ϕ∇ · c = ∇ · ϕc − c · ∇ϕ
= ∇ · ϕc − ∇ϕ · [δa × (∇u × b) + a × (∇δu × b) + a × (∇u × δb)]
= ∇ · ϕc − [(∇u × b) × ∇ϕ] · δa − (∇ϕ × a) · (∇δu × b) − (∇ϕ × a) · (∇u × δb)
= ∇ · ϕc − [(∇u × b) × ∇ϕ] · δa + [b × (a × ∇ϕ)] · ∇δu − [∇u × (a × ∇ϕ)] · δb
= ∇ · ϕc − [(∇u × b) × ∇ϕ] · δa + ∇ · [b × (a × ∇ϕ)δu]
− [∇ · b × (a × ∇ϕ)]δu − [∇u × (a × ∇ϕ)] · δb
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
− ∇ · b × (a × ∇ϕ)
δu +
∂ F
∂ a
− (∇u × b) × ∇ϕ
· δa
+
∂ F
∂ b
− ∇u × (a × ∇ϕ)
· δb
dV +
V
[∇ · ϕc + ∇ · b × (a × ∇ϕ)δu]dV (4)
The Euler equations of the functional are