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10 Variational Problems of Functionals with Vector, Tensor …
= ∇ · [cδ(∇u · ∇v)] − ∇ · (∇ · c∇vδu) + [∇ · (∇ · c∇v)]δu
− ∇ · (∇ · c∇uδv) + [∇ · (∇ · c∇u)]δv
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
+ ∇ · (∇ · c∇v)
δu +
∂ F
∂v
+ ∇ · (∇ · c∇u)
δv
dV
+
V
{∇ · [cδ(∇u · ∇v)] − ∇ · (∇ · c∇vδu) − ∇ · (∇ · c∇uδv)}dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂ F
∂u
+ ∇ ·
∇v∇ ·
∂ F
∂∇(∇u · ∇v)
= 0
∂ F
∂v
+ ∇ ·
∇u∇ ·
∂ F
∂∇(∇u · ∇v)
= 0
(5)
The natural boundary conditions are
∂ F
∂∇(∇u · ∇v)
· n
S
= 0, ∇ ·
∂ F
∂∇(∇u · ∇v)
∇u · n
S
= 0, ∇ ·
∂ F
∂∇(∇u · ∇v)
∇v · n
S
= 0
(6)
Example 10.3.10 The functional with the gradient of triple scalar product of
trivector. Derive the Euler equations and natural boundary condition of the functional
J =
V
F(a, b, c, ∇[a · (b × c)])dV
(1)
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)
Let d =
∂ F
∂∇[a·(b×c)]
, ϕ = δ(a · b × c) = (b × c) · δa + (c × a) · δb + (a × b) · δc,
there is
d · ∇ϕ = d · ∇ϕ = ∇ · (dϕ) − (∇ · d)[(b × c) · δa + (c × a) · δb + (a × b) · δc]
(3)
Substituting Eq. (3) into Eq. (2), we get