Math Fest 2008
The Brachistochrone Revisited: A Timely Consideration
Thursday, July 31, 2008
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Abstract:
One of the origins of the Calculus of Variations is the brachistochrone
problem posed in 1696 by Johann Bernoulli. The problem is to determine
for a frictionless bead accelerated from rest the path that would minimize
the time of descent between two specified points. The well-known solution
is an inverted cycloid. The fundamental independent variable of the problem
is r, the ratio of the horizontal to vertical
displacement between the two points. Properties of the cycloid solution
as r varies, including its uniqueness, are elaborated.
In particular, if r exceeds 
, then the least time path attains an absolute minimum below the
terminal point of the trajectory. Explicit asymptotic expansions for the
time of descent for both large and small r are
developed. Relatively simple and computationally efficient formulas that
allow for an accurate graph of the minimizing cycloid for any value of r
are derived. In addition, variational calculations that minimize the time
of descent for trial function trajectories are presented. The first trajectories
considered were piecewise linear segments. Their ability to approximate
the solution of the brachistochrone is summarized. Finally, a rather thorough
analysis of a minimizing parabola is given. Despite the fact that having
y descend as a quadratic function of x between the two points
leaves only one free parameter, such a trajectory is able to come very close
in matching the time of descent of the minimizing cycloid.
An article detailing the work is available at http://faculty.matcmadison.edu/alehnen/brach/brach.pdf
.
A parallel html version of the article is at http://faculty.matcmadison.edu/alehnen/brach/brachchistochrone.html
.
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In June 1696 Johann Bernoulli published as a challenge
the following problem.
"Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time."
Figure 1
We could imagine that we have a bead that slides from rest down a "frictionless" wire. The challenge then amounts to finding the shape of the wire that minimizes the time of travel. Five correct solutions were written: one by Johann Bernoulli himself, one by Gottfried von Leibniz, one by Isaac Newton, one by Jacob Bernoulli (Johann’s brother) and a final solution (seemingly lost) by the Marquis de L’Hospital.
Johann was jealous of his Brother’s solution, so he returned to the problem in 1718. His new solution along with work on related isoperimetric problems formed the basis of the subject now known as the Calculus of Variations, a subject brought to greater maturity through the work of Leonhard Euler and Joseph-Louis Lagrange.
II. The Solution of the Brachistochrone Problem
Let v represent the bead’s speed with 
. If M is the mass of the bead, then from Newton’s
second law the coordinates of the bead must satisfy the second order
differential equations
subject to the initial conditions that at t = 0, (x, y) = (0, L) and v = 0. Using the chain rule and the condition between the derivative of y and the components of the normal force gives the following coupled equations:
.
So, 
. The initial conditions therefore require that 
, which is just the statement of energy conservation since
there are no frictional losses in the wire. Solving for speed as a function
of y gives
. (1)
The time of travel between Point A and Point B is given by the following integral.
.
To simplify the analysis the following "normalized" or scaled variables are introduced.
Here 
is the time to drop from rest through a vertical distance
L and r is the ratio of the net horizontal
displacement to the net vertical displacement.
The problem of finding Y(X) such that 
is an extremum can be solved by the Euler-Lagrange equation,
with the boundary conditions that Y(0) = 1 and Y(r) = 0 .
The solution of the Brachistochrone problem is an inverted cycloid with the bead released from the top left cusp. The constant k is the diameter of the “generating circle” of the cycloid.
(2)
The bottom of the cycloid must extend below Y = 0, so the minimum
value of k is 1. The boundary conditions determine the value of
k and the maximum value of theta,
.
(3)
These conditions lead to the following two equations:
(4)
. (5)
are functions of r. Thus, r
is the fundamental independent variable of the Brachistochrone.
The normalized least time is given by
. (6)
Theorem : If 
, 
is in the interval 
, if 
,
is in the interval 
, and if 
, 
.
From Equation (4) the constant k is given in terms of
by 
. From the results stated in the Theorem and equation (5) this
can be inverted.
. (7)
Combining this result with Equation (3) gives two different formulas for r in terms of k. This of course means that r is not a function of k .
(8)
A parametric plot of Equation (8) is shown in Figure 4.
An interesting feature of the minimizing cycloid occurs when
and
is in
. The bottom of the cycloid at 
is part of the minimizing trajectory. Thus, the curve “over
shoots” y = 0 and approaches the terminal point of the trajectory
from below. If
, the coordinates of this lowest point are given by the following
equations:
. (9)This result is really not surprising. As r increases,
the horizontal component of the trajectory dominates the curve. In order
to travel this horizontal distance as quickly as possible the vertical drop
distance increases to “build up speed”. A typical solution with
is shown in Figure 3.
Figure 3
III. Explicit Expansions as a Function r for the Cycloid Solution
Various graphical methods have long been used to solve the Brachistochrone
problem. For example, the parametric plot of k versus r shown in Figure 2 provides a “complete solution”
of the Brachistochrone problem. For any positive value of r the graph gives the corresponding value of k.
This value through Equation (7) and the cycloid formulas of Equation
(2) determines the precise form of the least time curve.
Asymptotic series as 
Probably the domain of greatest interest is when r is large. In this case, from Equation (8),
.
, 
, and since k > 1, 
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It is reasonable therefore to assume the following expansion for k:
.
, 
. This yields the following results: 
, 
. From the binomial expansion,
. It will be seen in the next section that time of descent
increasing like the square root of a is a feature common to a
variety of minimum time curves. However, the factor of 
associated with the minimizing cycloid is the smallest possible
coefficient. For large r this lowest
point has a horizontal component that is about halfway between the
initial and terminal points of the trajectory and is approximately 
times the horizontal displacement below the terminal point.
Asymptotic series as 
The small r limit of the Brachistochrone
problem is of some interest since in many textbooks the solutions,
or at least the graphs of the solutions, usually have 
.
As
. This reflects the fact that the least time path for a purely
vertical displacement is just a vertical line segment. As r approaches zero no other path than the minimizing
cycloid has an expansion in r for the
time of descent smaller than 
Expansion about 
with 
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Accuracy and Applications of the r
Expansions
The accuracy of the various expansions is displayed in Table 1 for
the four values of r stated.
Here Relative Error is defined as (Explicit r
Expansion Answer – Newton’s Method Answer)/ Newton’s Method Answer.
Table 1: Relative Error of the Explicit r
Expansions
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| Small r Expansions | k | 
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--------- | --------- | --------- |
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--------- | --------- | --------- | |
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--------- | --------- | --------- | |
Taylor
Series about
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k | 
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--------- |
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--------- | |
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--------- | |
| Large r Expansions | k |
--------- | 
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--------- | 
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--------- |
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Taken together with Equation (3) the formulas for 
and 
allow for a very accurate graph of the minimizing cycloid for any
value of r. In fact, they have been used to generate
computer animations that display the shape of minimizing cycloid “dynamically”
as r changes.
IV. Variational Solutions:
A Linear Piece-Wise Trajectory
Consider the path from (0, L) to (a, 0) made up of
the following three line segments:
| 1. (0, L) to (b,
-D) |
| 2. (b, -D)
to (b + h, -D) |
| 3. (b + h, -D)
to (a, 0) |
Figure 6
The variables b and h are constrained
to be non-negative and satisfy the inequality that 
. The variable D must be greater than –L . Let
T1 be the time on the first segment, T2
the time on the second segment and T3 be the time on
the third segment. These times are computed as follows:
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Let T represent the total time along the path, i.e., T = T1 + T2+ T3 . To simplify the analysis introduce the following "normalized" or scaled variables.
As in the analysis of the minimizing cycloid,
is just the total time measured in units of the time required
for a "vertical" fall from rest through a displacement of L.
The total vertical displacement along the first segment of the path
is given by
. Both 
are non-negative and must satisfy the constraint that
. In terms of these scaled variables the total time along
the path is given by
When 
and 
, the path is the straight line from (0, L) to (a,
0) shown in Figure 7.
Figure 7
This path has a total time of
.For small r this can be expanded as 
which is of course greater than the corresponding time on
the cycloid solution to the Brachistochrone which has the expansion
.
For large r the time on the straight-line path grows nearly linearly in r in contrast to the square root growth of the minimizing path.
An "L" Trajectory (One Free Parameter)
When
and 
is allowed to vary, the path follows an "L" shaped trajectory
as shown in Figure 8.
Figure 8
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the minimum total time of the "L" trajectory is actually longer
than the time along the straight-line path! For large r it does much better,
having a square root dependence on r just like the cycloid solution to the Brachistochrone. Of course,
2, the coefficient on the square root of r is larger than 
. "Left Ramp" Trajectory (Two Free Parameters)
When E is fixed at zero but both 
and
are allowed to vary, the path follows a trajectory which "slants"
down from the left as illustrated in Figure 9.
Figure 9
The scaled time along this path is given by
.
This function needs to be minimized subject to the
constraints that
and 
. We deduce the following relationships:
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"Two Ramp" Trajectory (Three Free Parameters)
are allowed to vary the path has two "ramps" one on the "left"
side and a second on the "right" side. It soon becomes evident that a
"full case" analysis of this problem is extremely complicated by the constraint
that 
. Therefore, to simplify the analysis we will only consider
the large r asymptotic limit. The
total time is given by 
.
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The value of the coefficient of
the square root of r for the total time is 1.8612097182… . This essentially doubles
the improvement of the "Left Ramp" path over the "L" Trajectory, but
of course falls short (actually falls longer!) than the result for the
minimizing cycloid,
.
Given the long history of the Brachistochrone and the even longer
history of the conic sections the least-time parabolic trajectory is an inherently
interesting problem. An arbitrary parabola in the plane has four degrees
of freedom: the coordinates of the focus and the placement of the directrix.
Thus, determining the least-time parabola between two points means solving
a problem with two free parameters. A simpler, but still challenging, problem
with a single parameter results if the directrix is constrained to be parallel
to the x axis. For convenience, the solution of this problem will still be
called the minimizing parabola, while the designation, least-time parabola
will be reserved for the solution of the problem with a variable orientation
of the directrix. Obviously, the general parabolic path includes the zero-slope
directrix as a special case. Hence the time of descent along the least-time
parabola will always be shorter than that along the minimizing parabola.
Consider as in Figure 10 the path of “quickest descent” from (0, L) to (a,
0) when y is a quadratic function of x. Since the given two
points must lie on the parabola, there is only one degree of freedom available.
This can be taken as the coefficient on x2, i.e.,
.
Figure 10
In terms of the scaled variables X = x/L
and Y = y/L, with m = ac.
,The parabola has its vertex at
.
Making the change of variables
,
gives the result that
.
For fixed r the minimizing time is found by determining the value of m
that minimizes H(m, r). For small r
the value of m, m0 , that minimizes
H(m, r) has the following asymptotic form
, Iterations using Newton’s method determines that
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For small r this compares well with the absolute minimum
time of the cycloid solution where the coefficient on r2
is 0.375 and it certainly “beats” the straight line trajectory where
the coefficient is 0.5 .
The asymptotic behavior of m0(r) and 
for large values of r is also
amenable to analysis.
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The coefficient of the first term of this result when compared to coefficient
of the first term of the large r asymptotic time of the
minimizing cycloid is too big by 3.48%. However, this is closer than the
corresponding result for the “two ramp” trajectory, which is too big by 5.00%.
For large r the least-time parabola descends about 20%
lower than the absolute minimum curve. One could imagine that in the parabola’s
“race” with the minimizing cycloid it lowers the minimum so as to pick up
more speed. Unfortunately, it also picks up more arc length, so in the end
it still loses, but not by much. Thus, a parabola can closely approximate
the minimum time of descent without being able to closely match the shape
of the minimizing cycloid. This appreciable difference in appearance between
the two minimizing curves is illustrated in Figure 11 when r
= 6.
The following interpolating polynomial approximates m0(r)
to within 0.2% for 1.25 < r < 5.25.
The following rather simple representations of m0(r) enable one to generate a very accurate graph of the minimizing parabola for any positive value of r.
Figure 12
Figure 13
Animation of the Minimizing Cyloid and Least-time Parabola a function of r
Figure 14
Animation of the Minimizing Cyloid and Least-time Variations as a function of r
T(1, r, 0, r) is the straight line trajectory, T(gamma1(r), 0, 0, r) is the “L” trajectory and T(gamma2(r), 0, 0, r) is the “Left Ramp” trajectory. For the least-time parabolic trajectory the discrete points represent the approximate solutions obtained by Newton’s method and for r > 3.5, the large r asymptotic expression is also plotted. Given the close agreement between the times of descent of the minimizing cycloid and the minimizing parabola, the solution of the least-time parabola problems now seems even more interesting. Its time of descent must lie between these two.
Figure 15
Table 2 gives a summary for large r of various quantities for the cycloid solution as well as all of the variational solutions.
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