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

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