708
10 Variational Problems of Functionals with Vector, Tensor …
The natural boundary conditions are
A · n| S = 0, ∇ · ∇ A · n| S = 0, ∇ · ∇ A · ∇ × n| S = 0, n × A ·∇ × ∇| S = 0 (5)
B × ∇∇| S = 0, n · ∇ B × ∇∇| S = 0, (∇ × B) × n| S = 0, (∇ × B) × ∇| S = 0
(6)
10.6.5 Other Euler Equations of the Functionals
with Parallel-Type Inner Product Tensors
and the Hamiltonian Operators
About all kinds of structured parallel-type inner product tensor functionals with the
Hamiltonian operators, as the vector functional with the Hamiltonian operators, Can
also deduces the corresponding Euler equations and natural boundary conditions can
also be derived.
Example 10.6.17 Let A be a tensor of order n, derive the Euler equations and natural
boundary conditions of the functional
J =
V
F(u, A, ∇u A, ∇ · u A, ∇ × u A)dV
(1)
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂u
δu +
∂ F
∂ A
n
· δ A +
∂ F
∂∇u A
n + 1
· δ∇u A +
∂ F
∂∇ · u A
n − 1
· δ∇ · u A +
∂ F
∂∇ × u A
n
· δ∇ × u A
dV (2)
Let B =
∂ F
∂∇u A
, C =
∂ F
∂∇·u A
, D =
∂ F
∂∇×u A
, for the latter three variations of Eq. (2),
there are
δ J =
V
∂ F
∂u
δu +
∂ F
∂ A
n
· δ A +
∂ F
∂∇u A
n + 1
· δ∇u A +
∂ F
∂∇ · u A
n − 1
· δ∇ · u A +
∂ F
∂∇ × u A
n
· δ∇ × u A
dV (3)
C n − 1
· ∇ · δu A = ∇ · (δu A n − 1
· C) − ∇C n
· ( Aδu + uδ A)
(4)
D n
· ∇ × δu A = −∇ ·
D ×
(n − 1)· δu A
+ ∇ × D n
· ( Aδu + uδ A)
(5)
Substituting Eqs. (3)–(5) into Eq. (2), we get
δ J =
V
∂ F
∂u
− (∇ · B + ∇C − ∇ × D) n
· A
δu +
∂ F
∂ A
− u(∇ · B + ∇C − ∇ × D)
n
· δ A
dV