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

.

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

.