10.3 The Euler Equations of Gradient Type Functionals
649
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂u
+ k cos kua · ∇∇ ·
∂ F
∂∇∇ · sin kua
= 0
∂ F
∂ a
+ sin ku∇∇ ·
∂ F
∂∇∇ · sin kua
= 0
(5)
The natural boundary conditions are
∂ F
∂∇∇ · sin kua
· n
S
= 0, ∇ ·
∂ F
∂∇∇ · sin kua
S
= 0
( 6 )
Example 10.3.13 The functional with the gradient of product of a scalar and the
cross product of two vectors. Let the function
J =
V
F(u, a, b, ∇(ua × b))dV
(1)
where, the double inner product of tensors of second order conforms to the law of
parallel operation, derive the Euler equations and natural boundary condition.
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂ a
· δa +
∂ F
∂ b
· δb +
∂ F
∂∇(ua × b)
: δ∇(ua × b)
dV (2)
Let T =
∂ F
∂∇(ua×b)
, c = δ(ua × b), and using Eq. (10.2.1), there is
T : δ∇(ua × b) = T : ∇c = ∇ · (T · c) − (∇ · T ) · c
= ∇ · (T · c) − (∇ · T ) · (a × bδu + uδa × b + ua × δb)
= ∇ · (T · c) − [(a × b) · ∇ · T ]δu + (ub × ∇ · T ) · δa
+ (ua × ∇ · T ) · δb
(3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂u
− (a × b) · ∇ · T
δu +
∂ F
∂ a
− ub × ∇ · T
· δa
+
∂ F
∂ b
+ ua × ∇ · T
· δb + ∇ · (T · c)
dV
(4)
The Euler equations of the functional are