2.3 Variations of the Simplest Functionals and Necessary Conditions …
107
if F ∈ C
1 , y i ∈ C
1 , y
i ∈ C
1
(i = 1, 2, · · · , n), then
δ J =
x 1
x 0
δ Fdx =
x 1
x 0
n
i=1
F y i δy i +
n
i=1
F y
i
δy
i
dx
(2.3.20)
Let F, F 1 and F 2 be the differentiable functions of x, y, y
, …, then the variational
symbol δ has the following fundamental operation properties:
(1) δ(F 1 + F 2 ) = δ F 1 + δ F 2
(2) δ(F 1 F 2 ) = F 1 δ F 2 + F 2 δ F 1
(3) δ(F
n
) = n F
n−1
δ F
(4) δ
F 1
F 2
=
F 2 δ F 1 −F 1 δ F 2
F
2
2
(5) δ[F
(n)
] = (δ F)
(n)
F
(n)
=
d
n F
dx n
(6) δ
x 1
x 0
F(x, y, y
)dx =
x 1
x 0
δ F(x, y, y
)dx.
Proof According to the definition of variational and the property of the sequence of
the variation and derivation can be exchanged, there are
δ(F 1 + F 2 ) =
∂(F 1 + F 2 )
∂ y
δy +
∂(F 1 + F 2 )
∂ y
δy
=
∂ F 1
∂ y
δy +
∂ F 2
∂ y
δy +
∂ F 1
∂ y δy
+
∂ F 2
∂ y δy
=
∂ F 1
∂ y
δy +
∂ F 1
∂ y δy
+
∂ F 2
∂ y
δy +
∂ F 2
∂ y δy
= δ F 1 + δ F 2
δ(F 1 F 2 ) =
∂(F 1 F 2 )
∂ y
δy +
∂(F 1 F 2 )
∂ y δy
=
F 1
∂ F 2
∂ y
+ F 2
∂ F 1
∂ y
δy +
F 1
∂ F 2
∂ y + F 2
∂ F 1
∂ y
δy
= F 1
∂ F 2
∂ y
δy +
∂ F 2
∂ y δy
+ F 2
∂ F 1
∂ y
δy +
∂ F 1
∂ y δy
= F 1 δ F 2 + F 2 δ F 1
δ(F
n
) =
∂(F
n
)
∂ y
δy +
∂(F
n
)
∂ y δy
= n F
n−1 ∂ F
∂ y
δy + n F
n−1 ∂ F
∂ y δy
= n F
n−1
∂ F
∂ y
δy +
∂ F
∂ y δy
= n F
n−1
δ F
δ
F 1
F 2
=
∂
∂ y
F 1
F 2
δy +
∂
∂ y
F 1
F 2
δy
=
F 2
∂ F 1
∂ y
− F 1
∂ F 2
∂ y
F
2
2
δy +
F 2
∂ F 1
∂ y − F 1
∂ F 2
∂ y
F
2
2
δy
=
F 2
∂ F 1
∂ y
δy +
∂ F 1
∂ y δy
− F 1
∂ F 2
∂ y
δy +
∂ F 2
∂ y δy
F
2
2
=
F 2 δ F 1 − F 1 δ F 2
F
2
2
δ(F
(n)
) = F
(n)
− F
(n)
0 = (F − F 0 )
(n)
= (δ F)
(n)
δ
x1
x0
Fdx =
∂
∂ y
x1
x0
Fdx
δy +
∂
∂ y
x1
x0
Fdx
δy
=
x1
x0
F y dx
δy +
x1
x0
F y dx
δy
=
x1
x0
(F y δy + F y δy
)dx =
x1
x0
δ Fdx