Computer Algebra Recipes

4.3. VARIATIONAL CALCULUS MODELS 187
The goal is to ¯nd the shape y(x) that minimizes A."
\I know that, Greg," Gino interjects, \but get to the point. As my business
associates from Earth say, time is money. What is the bottom line? How many
square sretem of cloth will I need?"
A little perturbed by this sign of impatience, Greg's dialect has become
stronger and his response is di±cult for us to precisely translate, but the gist of
it goes something like this. Long ago, some ancestors on his mother's side had
adopted a trial-and-error approach to ¯nding y(x). This consisted in choosing
some form for y(x) and performing the integration to calculate A. Then the
venerable ancestors would choose another form and see whether it generated a
smaller value for A. With a touch of hyperbole, we think, Greg said that some
kept repeating this process until either they became residents in the Erehwon
asylum or inductees into the Erehwon Academy of Mathematics. According
to his unbiased (?) version of history, a better approach was discovered long
ago by one of Greg's illustrious ancestors on his father's side, Hil Arious Nerd.
Evidently, Hil independently discovered the same method as developed by Euler
and Lagrange on far-distant Earth. The method is as follows.
Label the integrand of the area integral as F , i.e., F =2¼y p 1+(y0 ) 2 .
The curve y(x) that minimizes the area is the solution of the Euler{Lagrange
equation(knownastheNerdequationonErehwon)
@F d @F
¡ =0: (4.17)
@y dx @y 0
Solving this type of problem by hand has been a standard assignment given in
the past by perverse professors on Erehwon6 to their students.
It appears that Greg has calmed down so that we can understand him better,
so let's pick up Greg's detailed narrative once again.
\It'snowfrownedupontoin°ictmentalpainonstudents(anddressdesigners)
by performing hand calculations when computers can do all the algebra for
you with no errors. At the Erehwon Institute of Technology we use the Elpam
computer algebra system, so let's use it to crack your problem.
I will begin by loading the VariationalCalculus library package which contains
the EulerLagrange command for generating the left-hand side of the
Euler{Lagrange (Nerd) equation (4.17).
> restart: with(VariationalCalculus):
Next let's enter the integrand, F = y(x) p 1+(dy(x)=dx) 2 ,omittingthefactor
of 2 ¼, which would ultimately cancel out of the equation,
> F:=y(x)*sqrt(1+diff(y(x),x)^2);
s
μ
d
F := y(x) 1+
dx y(x)
2
The EulerLagrange command is applied to F , the second argument (x) being
the independent variable, the third argument the dependent variable. The
result is then simpli¯ed.
6 Author's note: This quaint, but archaic, custom is still widely practiced on Earth.