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10 Variational Problems of Functionals with Vector, Tensor …
+ (−1) m (∇) m
∂ F
∂ A(·∇) m
n
· δ A
(10.7.16)
where, m ≤ n.
In Eq. (10.7.10), change A into A × ∇, in this case, A × ∇ is still the tensor of
order n, there is
∂ F
∂ A × ∇ × ∇
n
· δ A × ∇ × ∇ = ∇ ·
∂ F
∂ A × ∇ × ∇
×
(n − 1)· δ A × ∇
− ∇ ×
∂ F
∂ A × ∇ × ∇
n
· δ A × ∇
(10.7.17)
Making use of Eq. (10.7.10) for the second term on the right side of Eq. (10.7.17),
we obtain
∂ F
∂ A × ∇ × ∇
n
· δ A × ∇ × ∇ = ∇ ·
∂ F
∂ A × ∇ × ∇
×
(n − 1)· δ A × ∇
− ∇ ·
∇ ×
∂ F
∂ A × ∇ × ∇
×
(n − 1)· δ A
+ ∇ × ∇ ×
∂ F
∂ A × ∇ × ∇
n
· δ A
(10.7.18)
Repeat the above process for the last term in Eq. (10.7.18), the recurrence formula
of the right rotation variation of the tensor of order n can be obtained
∂ F
∂ A(×∇) m
n
· δ A(×∇) m =
m
i=1
(−1) i−1 ∇ ·
(∇×) i−1
∂ F
∂ A(×∇) m
×
(n − 1)· δ A(×∇) m−i
+ (−1) m (∇×) m
∂ F
∂ A(×∇) m
n
· δ A
(10.7.19)
10.7.2 The Euler Equations and Natural Boundary
Conditions of the Functionals with Series-Type Inner
Product Tensors and Hamiltonian Operators
Equations (10.7.2), (10.7.3), (10.7.5), (10.7.6), (10.7.9), and (10.7.10) are written
together and rearranged, there are
∂ F
∂∇ A
n + 1
· ∇δ A = ∇ ·
δ A n
·
∂ F
∂∇ A
−
∂ F
∂∇ A
· ∇ n
· δ A
(10.7.20)