Here two remarks are in order. First, multiplication by a constan
of a quantity to be optimized has no effect on the
optimizer.4
So, from
(1.8) and (1.9), the brachistochrone problem is equivalent to that o
finding y to minimize J[y]. Second, from (1.11) and (1.13), the bea
travels faster down a circular arc than down a straight line: whateve
the optimal curve is, it is not a straight line. But is there a curv
that yields an even lower transit time than the circle?
One way to explore this question is to consider a one-paramete
family of trial curves satisfying (1.2), e.g., the family defined by
(1.14) y = y (x) = 1 x
for 0. Note the contrast with (1.1). Now each different tria
function y is distinguished by its value of ; y is used only to denot
the ordinate of its graph, as illustrated by Figure 1.2(a). When (1.14
is substituted into (1.9), J becomes a function of : we obtain
J( ) = J[y ] =
1
0
1 + {y
(x)}2
1
2
1 y (x)
dx
=
1
0
x−
2
{1 +
2x2 −2}2
1
dx
(1.15)
after simplification. This integral cannot be evaluated analyticall
(except when = 1), but is readily evaluated by numerical mean
with the help of a software package such as Maple, Mathematica o
MATLAB. Because, from Figure 1.2(a), the curve is too steep initiall
when is very small, is too close to the line when is close to 1 an
bends the wrong way for 1, let us consider only values between
say, = 0.2 and = 0.8. A table of such values is
0.2 0.3 0.4 0.5 0.6 0.7 0.8
J( ) 2.690 2.634 2.602 2.587 2.589 2.608 2.647
and the graph of J over this domain is plotted in Figure 1.2(b
We see that J( ) achieves a minimum at = 0.539726 wit
4For
example, the polynomials x(2x 1) and 3x(2x 1) both have minimiz
x =
1
4
, although in the first case the minimum is
1
8
and in the second case th
minimum is
3
8
.
Previous Page Next Page