100
2 Variational Problems with Fixed Boundaries
δ
dy
dx
=
d
dx
δy
(2.2.12)
namely the variation of derivative of a function is equal to the derivative of
variation of the function. In other words, the two kinds of operation orders of the
derivation and variation can be exchanged. During the derivation of the variational
methods this property of variation is often used.
The above property can be generalized to the variational situation of higher
derivative, namely
δy
= (δy)
, δy
= (δy)
, . . . , δy
(n)
= (δy)
(n)
From Eqs. (2.2.7) and (2.2.11), we obtain
δy
= (δy)
= εη
(x)
(2.2.13)
For the variation of higher derivative, there is
δy
(n)
= (δy)
(n)
= εη
(n)
(x)
(2.2.14)
Because the Hamiltonian operator is a kind of operator of derivation operation,
there is
δ∇ϕ = δ
∂
∂ x i
e i ϕ =
∂
∂ x i
e i δϕ = ∇δϕ
(2.2.15)
δ∇ · a = δ
∂
∂ x i
e i · a j e j =
∂
∂ x i
e i · δa j e j =
∂
∂ x i
e i · δa = ∇ · δa
(2.2.16)
δ∇ × a = δ
∂
∂ x i
e i × a j e j =
∂
∂ x i
e i × δa j e j =
∂
∂ x i
e i × δa = ∇ × δa (2.2.17)
namely the operation orders of the Hamiltonian operator and variational symbol can
be exchanged.
Similarly, for the Laplace operator, there is
δδϕ = δ
∂ 2 ϕ
∂ x 2 +
∂ 2 ϕ
∂ y 2 +
∂ 2 ϕ
∂z 2
=
∂ 2 δϕ
∂ x 2 +
∂ 2 δϕ
∂ y 2 +
∂ 2 δϕ
∂z 2 =
∂ 2
∂ x 2 +
∂ 2
∂ y 2 +
∂ 2
∂z 2
δϕ = δϕ (2.2.18)
namely the operation orders of the Laplace operator and variational symbol can also
exchanged.