10.6 Variational Problems of Functionals with Parallel-Type Inner …
707
Find its Euler equations and natural boundary conditions.
Solution Let the integrand F = A n
· B+∇ A n + 1
· ∇∇× B+∇×∇·∇·∇ A n − 1
· ∇·
B, find the various partial derivatives
∂ F
∂ A
= B,
∂ F
∂∇ A
= ∇∇ × B,
∂ F
∂∇ × ∇ · ∇ · ∇ A
= ∇ · B
(2)
∂ F
∂ B
= A,
∂ F
∂∇∇ × B
= ∇ A,
∂ F
∂∇ · B
= ∇ × ∇ · ∇ · ∇ A
(3)
The Euler equations of the functional are
A − ∇ × ∇ · ∇ A − ∇∇ × ∇ · ∇ · ∇ A = 0
B − ∇ · ∇∇ × B − ∇ · ∇∇∇ × ∇ · B = 0
(4)
The natural boundary conditions are
n · ∇ A| S = 0, n × ∇ · ∇ A| S = 0, ∇ × ∇ · ∇ · ∇ A| S = 0
(5)
n · ∇∇ × B| S = 0, n × ∇ · B| S = 0, ∇ × ∇ · B| S = 0,
∇∇ × ∇ · B| S = 0, n · ∇∇∇ × ∇ · B| S = 0
(6)
Example 10.6.16 Let A be a tensor of order n, B is a tensor of order n − 1, construct
the functional
J =
V
( A n
· B∇ + ∇ · ∇ A n
· B × ∇∇ + ∇ × B n − 1
· A · ∇ × ∇)dV
(1)
Find its Euler equations and natural boundary conditions.
Solution Let the integrand F = A n
· B∇+∇·∇ A n
· B×∇∇+∇× B n − 1
· A·∇×∇,
find the various partial derivatives
∂ F
∂ A
= B∇,
∂ F
∂∇ · ∇ A
= B × ∇∇,
∂ F
∂ A · ∇ × ∇
= ∇ × B
(2)
∂ F
∂ B∇
= A,
∂ F
∂ B × ∇∇
= ∇ · ∇ A,
∂ F
∂∇ × B
= A · ∇ × ∇
(3)
The Euler equations of the functional are
A · ∇ + ∇ · ∇ A · ∇ × ∇ − ∇ × A · ∇ × ∇ = 0
B∇ + ∇ · ∇ B × ∇∇ − ∇ × B × ∇∇ = 0
(4)