642
10 Variational Problems of Functionals with Vector, Tensor …
δ J =
V
n
i=1
∂ F
∂ a i
−
a i
|a i |
2
n
i=1
|a i |∇ · c
· δa + ∇ ·
cδ
n
i=1
|a i |
dV
(4)
The Euler equations of the functional are
∂ F
∂ a i
−
a i
|a i |
2
n
i=1
|a i |∇ ·
∂ F
∂∇
n
i=1
|a i |
= 0
(5)
The natural boundary condition is
∂ F
∂∇
n
i=1
|a i |
· n
S
= 0
( 6 )
Example 10.3.5 The functional with the gradient of quotient of two scalars. Find
the Euler equations and natural boundary condition
J =
V
F
u, v, ∇
u
v
dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂v
δv +
∂ F
∂∇(u/v)
· δ∇
u
v
dV
(2)
Putting a =
∂ F
∂∇(u/v)
, the integral of the latter term in Eq. (2) can be written as
a · δ∇
u
v
= a · ∇δ
u
v
= ∇ ·
aδ
u
v
− (∇ · a)
vδu − uδv
v 2
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
−
∇ · a
v
δu +
∂ F
∂v
+
u
v 2 ∇ · a
δv + ∇ ·
aδ
u
v
dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂u
−
1
v
∇ ·
∂ F
∂∇(u/v)
= 0
∂ F
∂v
+
u
v 2 ∇ ·
∂ F
∂∇(u/v)
= 0
(5)