54 Applied calculus of variations for engineers
− y + z =0.
The elimination of the Lagrange multiplier results in the system of
y − z + z
− y + z =0,
whose solution follows from classical calculus.
4.4 Linearization of second order problems
It is very common in engineering practice that the highest derivative of interest
is of second order. Accelerations in engineering analysis of motion, curvature
in description of space curves, and other important application concepts are
tied to the second derivative.
This speciﬁc case of quadratic problems may be reverted to a linear prob-
lem involving two functions. Consider
f(x, y, y
)dx = extremum.
with the following boundary conditions given
By introducing a new function
we can reformulate the unconstrained second order variational problem as a
variational problem of the ﬁrst order with multiple functions in the integrand
f(x, y, z, z
)dx = extremum,
but subject to a constraint involving the derivative
g(x, y, z)=z − y
Using a Lagrange multiplier function in the form of
h(x, y, z, z
,λ)=f(x, y, z, z
)+λ(x)(z − y