10.3 The Euler Equations of Gradient Type Functionals
647
δ J =
V
∂ F
∂ a
− b × c∇ · d
· δa +
∂ F
∂ b
− c × a∇ · d
· δb
+
∂ F
∂ c
− a × b∇ · d
· δc + ∇ · (dϕ)
dV
(4)
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ F
∂ a
− (b × c)∇ ·
∂ F
∂∇(a · b × c)
= 0
∂ F
∂ b
− (c × a)∇ ·
∂ F
∂∇(a · b × c)
= 0
∂ F
∂ c
− (a × b)∇ ·
∂ F
∂∇(a · b × c)
= 0
(5)
The natural boundary condition is
∂ F
∂∇[a · (b × c)]
· n
S
= 0
(6)
Example 10.3.11 The functional with the gradient of triple scalar product of trivector
composed of the gradients of scalars. Find the Euler equations and natural boundary
condition of the functional
J =
V
F(u, v, w, ∇[∇u · (∇v × ∇w)])dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂v
δv +
∂ F
∂w
δw +
∂ F
∂∇[∇u · (∇v × ∇w)]
· δ∇[∇u · (∇v × ∇w)]
dV
(2)
Putting a =
∂ F
∂∇[∇u·(∇v×∇w)]
, ϕ = δ∇u · (∇v × ∇w) = ∇δu · (∇v × ∇w) +
∇u · (∇δv × ∇w) + ∇u · (∇v × ∇δw), ψ = ∇ · a, using the parallelepipedal product
formula of trivector a · (b × c) = b · (c × a) = c · (a × b), there is
a · ∇δ∇u · (∇v × ∇w) = a · ∇ϕ = ∇ · ϕa − ϕ∇ · a =
∇ · ϕa − ψ[∇δu · (∇v × ∇w) + ∇u · (∇δv × ∇w) + ∇u · (∇v × ∇δw)] =
∇ · ϕa − ψ(∇v × ∇w) · ∇δu − ψ(∇w × ∇u) · ∇δv − ψ(∇u × ∇v) · ∇δw =
∇ · ϕa − ∇ · [ψ(∇v × ∇w)δu] − ∇ · [ψ(∇w × ∇u)δv] − ∇ · [ψ(∇u × ∇v)δw]+
[∇ · ψ(∇v × ∇w)]δu + [∇ · ψ(∇w × ∇u)]δv + [∇ · ψ(∇u × ∇v)]δw
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
+ ∇ · ψ∇v × ∇w
δu +
∂ F
∂v
+ ∇ · ψ∇w × ∇u
δv +
∂ F
∂w
+ ∇ · ψ∇u × ∇v
δw
dV