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10 Variational Problems of Functionals with Vector, Tensor …
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∂ F
∂u
+ ∇ ·
a · ∇
∂ F
∂∇ · (∇u · ∇v)a
∇v = 0
∂ F
∂v
+ ∇ ·
a · ∇
∂ F
∂∇ · (∇u · ∇v)a
∇u = 0
∂ F
∂ a
− (∇u · ∇v)∇
∂ F
∂∇ · (∇u · ∇v)a
= 0
(6)
The natural boundary conditions are
∂ F
∂∇ · (∇u · ∇v)a
S
= 0, a ·
∂ F
∂∇ · (∇u · ∇v)a
∇u · n| S = 0, a ·
∂ F
∂∇ · (∇u · ∇v)a
∇v · n
S
= 0 (7)
Example 10.4.12 The functional with the divergence of cross product of trivector.
Let the functional
J =
V
F(a, b, c, ∇ · a × (b × c))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
∂ c
· δc +
∂ F
∂∇ · [a × (b × c)]
δ∇ · a × (b × c)
dV (2)
Putting ϕ =
∂ F
∂∇·a×(b×c)
, d = δ[a × (b × c)] = δa × (b × c) + a × (δb × c) + a ×
(b × δc), then according to the parallelepipedal product formula of trivector, there is
ϕδ∇ · [a × (b × c)] = ϕ∇ · δ[a × (b × c)] = ϕ∇ · d = ∇ · ϕd − ∇ϕ · d
= ∇ · ϕd − ∇ϕ · [δa × (b × c) + a × (δb × c) + a × (b × δc)]
= ∇ · ϕd − [(b × c) × ∇ϕ] · δa + a × ∇ϕ · (δb × c) + a × ∇ϕ · (b × δc)
= ∇ · ϕd − [(b × c) × ∇ϕ] · δa + [c × (a × ∇ϕ)] · δb + [(a × ∇ϕ) × b] · δc
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂ a
− (b × c) × ∇ϕ
· δa +
∂ F
∂ b
+ c × (a × ∇ϕ)
· δb
+
∂ F
∂ c
− b × (a × ∇ϕ)
· δc + ∇ · ϕd
dV
(4)
The Euler equations of the functional are