54 Applied calculus of variations for engineers

and

y

− y + z =0.

The elimination of the Lagrange multiplier results in the system of

y − z + z

=0,

and

y

− 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

I(y)=

x

1

x

0

f(x, y, y

,y

)dx = extremum.

with the following boundary conditions given

y(x

0

),y(x

1

),y

(x

0

),y

(x

1

).

By introducing a new function

z(x)=y

(x),

we can reformulate the unconstrained second order variational problem as a

variational problem of the ﬁrst order with multiple functions in the integrand

I(y,z)=

x

1

x

0

f(x, y, z, z

)dx = extremum,

but subject to a constraint involving the derivative

g(x, y, z)=z − y

=0.

Using a Lagrange multiplier function in the form of

h(x, y, z, z

,λ)=f(x, y, z, z

)+λ(x)(z − y

),

Higher order derivatives 55

and following the process laid out in the last section we can produce a system

of three Euler-Lagrange diﬀerential equations.

∂h

∂y

−

d

dx

∂h

∂y

=

∂f

∂y

−

dλ

dx

=0,

∂h

∂z

−

d

dx

∂h

∂z

=

∂f

∂z

+ λ −

d

dx

∂f

∂z

=0,

and

∂h

∂λ

−

d

dx

∂h

∂λ

= z − y

.

This may, of course, be turned into the Euler-Poisson equation by expressing

λ =

d

dx

∂f

∂z

−

∂f

∂z

from the middle equation and diﬀerentiating as

dλ

dx

=

d

2

dx

2

∂f

∂z

−

d

dx

∂f

∂z

.

Substituting this and the third equation into the ﬁrst yields the Euler-Poisson

equation we could have achieved, had we approached the original quadratic

problem directly:

∂f

∂y

−

d

dx

∂f

∂y

+

d

2

dx

2

∂f

∂y

=0.

Depending on the particular application circumstance, the linear system of

Euler-Lagrange equations may be more conveniently solved than the quadratic

Euler-Poisson equation.

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