Computer Algebra Recipes

4.3. VARIATIONAL CALCULUS MODELS 191
ray. By plotting a light ray path for physically reasonable parameter values,
discuss how your answer may be related to the phenomenon of mirages.
Problem 4-37: A di®erent refractive index
Making use of Fermat's principle, prove that a light ray will follow a semicircular
path in a medium whose refractive index n(x; y) equals1=y. Plot a typical path.
Problem 4-38: Geodesic
The geodesic between two points is the curve that gives the shortest distance.
Show that the geodesics on the surface of a sphere are the arcs of great circles.
(A great circle is a curve resulting from the intersection of a sphere with a plane
passing through the center of that sphere.) Create a three-dimensional plot of
the geodesic between New York City and Sydney, Australia, assuming that the
Earth is spherical. You will have to look up the necessary parameter values.
4.3.2 Queen Dido Wasn't a Dodo
Mathematics is the queen of the sciences.
Carl Friedrich Gauss, German mathematician (1777{1855)
According to the Aeneid, written by the Roman poet Virgil in the ¯rst century
BC, the Phoenician Queen Dido was able to convince the North African
ruler King Jambas to give her as much land as she could enclose with an oxhide.
Being rather clever, she had the hide cut into very thin strips, the ends
stitched together, and was able to stake out a sizable area along the Mediterranean
coast on which she built the city of Carthage (now Tunis). Queen Dido's
problem was to lay out the joined oxhide strips that had a total ¯xed length in
such a way as to maximize the area enclosed. Can you suggest what shape the
perimeter traced out by the joined strips might take? Problems of this type
that involve maximizing an area enclosed by a perimeter of ¯xed length are
called isoperimetric (constant perimeter) problems.
A recipe is now given to solve Queen Dido's problem. A strip of oxhide of
¯xed length L>ais connected at its ends to the points (x =0,y =0)and
(x = a, y = 0). The area enclosed by a strip of shape y(x) andthex-axis
between x =0andx = a is A = R a
y(x) dx and the length of the strip is given
by L = R a
p 0
1+(y0 ) 2 7 dx: If, say, a = 1 milion and L =1:5 milion, what is
0
the shape y(x) that maximizes the area subject to the length constraint? Plot
y(x) and determine the value of the maximum area.
If the integrands of A and L are labeled as F and G, respectively, the shape
y(x) may be found by solving the Euler{Lagrange equation of the previous
subsection with F replaced by FF = F + ¸G,where¸ is an undetermined
parameter. See, e.g., references [MW71] and [Boa83].
The VariationalCalculus package is loaded,
> restart: with(VariationalCalculus):
7 A milion is a Roman mile, so this recipe involve one very big oxhide!