730
10 Variational Problems of Functionals with Vector, Tensor …
(∇ × ∇ · ∇ B) × n| S = 0, (∇ × ∇ · ∇ B × ∇) × n| S = 0
(5)
Example 10.7.8 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)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 × ∇| S = 0, ( A × ∇ × ∇) × n| S = 0, A × ∇ × ∇ × ∇| S = 0
(4)
n × B · ∇| S = 0, n × ∇ × ∇ · B| S = 0, n × ∇ × ∇ × ∇ · B| S = 0
(5)
10.7.4 Other Euler Equations of the Functionals
with Series-Type Inner Product Tensors
and the Hamiltonian Operators
Example 10.7.9 Let A be a tensor of order m, B is a tensor of order n, derive the
Euler equations and natural boundary conditions of the functional
J =
V
F(u, A, B, u AB, ∇u AB, ∇ · u AB, ∇ × u AB)dV
(1)
Solution Taking the variation to the functional, we obtain