10.7 Variational Problems of Functionals with Series-Type …
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Example 10.7.6 Let A and B be both the tensors of order n, construct the functional
J =
V
( A n
· B + ∇ · A n − 1
· ∇ × ∇ · B)dV
(1)
Find its Euler functions and the natural boundary conditions.
Solution Let the integrand F = A n
· B + ∇ · A n − 1
· ∇ × ∇ · B, find the various
partial derivatives
∂ F
∂ A
= B,
∂ F
∂∇ · A
= ∇ × ∇ · B,
∂ F
∂ B
= A,
∂ F
∂∇ × ∇ · B
= ∇ · A
(2)
The Euler equations of the functional are
A + ∇ · A × ∇∇ = 0
B − ∇ × ∇ · B∇ = 0
(3)
The natural boundary conditions are
∇ × ∇ · B| S = 0, (∇ · A) × n| S = 0, (∇ · A) × ∇| S = 0
(4)
Example 10.7.7 Let A and B be both the tensors of order n, construct the functional
J =
V
( A n
· B + ∇ × ∇ × A n
· ∇ × ∇ · ∇ B)dV
(1)
Find its Euler functions and the natural boundary conditions.
Solution Let the integrand F = A n
· B + ∇ × ∇ × A n
· ∇ × ∇ · ∇ B, find the various
partial derivatives
∂ F
∂ A
= B,
∂ F
∂∇ × ∇ × A
= ∇ × ∇ · ∇ B,
∂ F
∂ B
= A,
∂ F
∂∇ × ∇ · ∇ B
= ∇ × ∇ × A
(2)
The Euler equations of the functional are
A − ∇ × ∇ × A × ∇∇ · ∇ = 0
B + ∇ × ∇ · ∇ B × ∇ × ∇ = 0
(3)
The natural boundary conditions are
(∇ × ∇ × A) × n| S = 0, (∇ × ∇ × A) × ∇| S = 0, (∇ × ∇ × A × ∇∇) · n| S = 0 (4)