Ε 1
Ε 0.5
Ε 0.25
Ε 2
Ε 0
Ε
Ε 3
Ε 0.1
x
0 0.5 1
y
0
0.5
1
a
Ε
Ε 0.2 0.4 0.6 0.8
J
J Ε
2.6
2.65
2.7
b
Figure 1.2. (a) A class of trial functions. (b) J = J( ) on [0.2, 0.8].
J( ∗) 2.58598. Comparing with (1.13), we find that y = y

(x
yields a lower transit time than the circular arc.
But that doesn’t make y = y

(x) the solution of the brachis
tochrone problem, because the true minimizing function may no
belong to the family defined by (1.14). If y =
y∗(x)
is the tru
minimizing curve, then all we have shown is that
(1.16)
J[y∗]
J(
∗)
2.58598.
In other words, we have found an upper bound for the true minimum
It turns out, in fact, that the true minimizer is a cycloid define
parametrically by
(1.17) x =
θ + sin(θ) cos(θ) +
1
2
π
cos2(θ1)
, y = 1
cos(θ)
cos(θ1)
2
for
1
2
π θ θ1, where θ1 −0.364791 is the larger of the only tw
roots of the equation
(1.18) θ1 + sin(θ1) cos(θ1) +
1
2
π =
cos2(θ1)
and
J[y∗]
2.5819045; see Lecture 4, especially (4.26)-(4.27). W
compare
y∗
with y

in Figure 1.3. Both curves are initially vertica
however, the cycloid is steeper (has a more negative slope) than th
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