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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
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
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Publisher Resources

ISBN: 9781482253597