
8 Applied calculus of variations for engineers
Here the newly introduced second variation is
δI
2
=
2
2
x
1
x
0
(η
2
(x)
∂
2
f(x, y, y
)
∂y
2
+
2η(x)η
(x)
∂
2
f(x, y, y
)
∂y∂y
+
η
2
(x)
∂
2
f(x, y, y
)
∂y
2
)dx.
We now possess all the components to test for the existence of the extremum
(maximum or minimum). The Legendre test in [7] states that if indepen-
dently of the choice of the auxiliary η(x) function
- the Euler-Lagrange equation is satisfied,
- the first variation vanishes (δI
1
=0),and
- the second variation does not vanish (δI
2
=0)
over the interval of integration, then the functional has an extremum. This
test manifests the necessary conditions for the existence of the