416
7 Variational Principles
δ
2 J =
ε
2
2
x 1
x 0
F yy −
d
dx
F yy
η
2
− η
d
dx
F y y
d
dx
η
dx
=
ε
2
2
x 1
x 0
η
F yy −
d
dx
F yy
−
d
dx
F y y
d
dx
ηdx
=
ε
2
2
x 1
x 0
η
S −
d
dx
R
d
dx
ηdx
=
ε
2
2
[η, T (y)η] =
ε
2
2
[T (y)η, η]
where, S = F yy −
d
dx
F yy , R = F yy , T (y) = S−
d
dx
R
d
dx
, thus the second derivative
of the functional J [y] is obtained
J
[y] = T (y) = S −
d
dx
R
d
dx
(7.5.4)
From the expression (7.3.4) it follows that the second derivative of the functional
J [y] is an operator.
The third variation of the functional is
δ
3 J =
1
3!
x 1
x 0
[F yyy (δy)
3
+ 3F yyy (δy)
2
δy
+ 3F yy y δy(δy
)
2
+ F y y y (δy
)
3
]dx
ε
3
3!
x 1
x 0
(F yyy η
3
+ 3F yyy η
2
η
+ 3F yy y ηη
2
+ F y y y η
3
)dx
Using integration by parts to the second term, we get
x 1
x 0
3F yyy η
1
2
(η
2
)
]dx =
x 1
x 0
3
2
F yyy η
d
dx
η
2 dx =
x 1
x 0
η
3
2
F yyy
d
dx
η
2 dx
Using integration by parts to the third term, we get
x 1
x 0
3F yy y ηη
2 dx =
x 1
x 0
3F yy y ηη
dη = 3F yy y ηη
η
x 1
x 0
−
x 1
x 0
η
3
2
d
dx
[F yy y (η
2
)
]dx
= −
x 1
x 0
η
3
2
d
dx
F yy y
d
dx
η
2
dx = −
x 1
x 0
η
3
2
d
dx
F yy y
d
dx
η
2 dx
Using integration by parts to the fourth term, we get
x 1
x 0
F y y y η
3 dx =
x 1
x 0
F y y y η
2 dη = F y y y η
2
η
x 1
x 0
−
x 1
x 0
η
d
dx
(F y y y η
2
)dx
−
x 1
x 0
η
η
2 d
dx
F y y y + F y y y 2η
η
dx