The NCB invites
you to join Galileo, Newton, Leibniz, Huygens, l'Hôpital, and two
Bernoullis in thinking about one of the most celebrated problems of
17th century mathematics. The word
"brachistochrone" is from the Greek meaning "shortest" and "time."



What is the path
 curve  producing the shortest possible
time for a particle to
descend from a given point to another point not
directly below the start? Will it be a
straight line, an arc of a
circle, or just what? Will it be a minimum of a function?
The shortest distance between two points is
a line, but the descent of a weighted particle is acted upon, in the
very least, by gravity. A bead descending a wire is often
used to depict the pathway, but investigators usually ignore
friction.
The correct answer
is that a body takes less time to fall along the arc of a circumference
than to fall along the "line" of a corresponding chord.
The cycloid path
allows the particle to move rapidly at first, while
in steep descent, and thus build up sufficient speed to overcome the
greater distance the particle must travel. Thus, the speed of the
descending particle is
accelerated by gravity.
We introduce a MATHEMATICA^{®} animation.

Brachistochrone
values
are plotted on the xaxis.
vs.
Linear values appear
on
the yaxis.
The brachistochrone is an inverted cycloid.

Our
animation features . . .
two
particles "racing,"
linear descent vs. the brachistochrone.
You also need
to be familiar with the cycloid. A cycloid is
the locus of a point on the circumference of a circle rotating along a
fixed line . . . .

Snell's
Law for the
Refraction
of Light
is often applied in
deriving the equations.



A lengthy discussion of the equations
is found on two other pages.
Brachistochrone
Part II
Brachistochrone
Part IV
A reproduction of the original Acta figures (1697) is found on Brachistochrone
Part III
A model from the Istituto e Museo di
Storia della Scienza
in Florence, Italy is at
Brachistochrone
Part V

Suggestions
for additional code are on the other brachistochrone links.
Abel Qawasmi's MATHEMATICA^{®} Code
MATHEMATICA^{®}
Code: Linear vs. Brachistochrone Descent of a Particle
brach[t_] : = {Sqrt[R
g] t  R Sin[Sqrt[g/R]t], R(1  Cos[Sqrt[g/R]t])};
straightline[t_] : = {(1/2) g (Sin[θ] Cos[θ]) t^2, (1/2) g (Sin[θ]^2) t^2};
θ=ArcSin[2 / Sqrt[Pi^2+2^2]];
g=9.8;
R=4;
T=Pi (Sqrt[R /g ]);
Do [curves=ParametricPlot [{brach[t], straightline[t]}, {t,0,Pi
R}, AspectRatio→Automatic, PlotRange→{{.5, Pi
R+1}, {.5, 2
R1}}, PlotStyle→{RGBColor[0,0,1], RGBColor[0,1,0]},
DisplayFunction→Identity];
beads=ListPlot[{brach[t], straightline[t]}, PlotRange→{{0,Pi
R},{0,2
R}}, PlotStyle→PointSize[0.02], DisplayFunction→Identity];
Show [curves, beads, ImageSize→4*72, DisplayFunction→$DisplayFunction];, {t, 0, (5/4)T, T/50}];

Note: Spaces have been added
to the text of this code to make it easier to read.
MATHEMATICA^{®} does not permit spaces.
[ Please see Stan Wagon, (1991) p. 63, p. 386, for additional
discussion of the comparison
of the brachistochrone's descent to other
curves of degree 1,2,3,6, and 10.]

Useful Links and Books
The
brachistochrone and cycloid have a very rich math and physics
literature.
The National Curve Bank also has MAPLE
animations of the cycloid family of curves.

Newton

Beckmann, Petr, A History of π (PI), St.
Martin's Press, 1971, pp. 139140.

Burton, David M., The History of Mathematics, 5th
ed., McGraw Hill, 2003, p. 446.
The
brachistochrone is for "the shrewdest
mathematicians of all the world."
John
Bernoulli, June, 1696

Eves, Howard, AN
INTRODUCTION TO THE HISTORY OF MATHEMATICS, 6th ed.,
Saunders College Publishing, 1992, p. 426.
Leibniz:
"Splendid problem."

Johnson, Nils P., The
Brachistochrone Problem, The College
Mathematics Journal, vol. 35 (3), May 2004, pp.
192197.
A cycloid has the
noteworthy property that the time it takes from any point along it to
its lowest point does not depend on the starting point.
Johnson and others
suggest the EulerLagrange formula and boundary conditions applied to
the brachistochrone will define a differential equation whose solution
is similar to finding critical points (usually a maximum or minimum) in
ordinary calculus. "Thus a global problem  an integral over a
path  has a local constraint  a differential equation which dictates
the solution at each point along the way."


Lockwood, E. H., A Book of CURVES, Cambridge
University Press, 1961, p. 88.

Katz, Victor J., A
History of Mathematics, 2nd ed., Addison Wesley Longman, 1998,
pp. 547549, 562.
Katz, Victor J., A History of
Mathematics, Brief ed., Pearson Addison Wesley, 2004, pp. 250,
331332.

Simmons, George F., Calculus Gems: Brief Lives and Memorable
Mathematics, McGrawHill,1992, pp. 308313.
"Bernoulli's
vivid, enthusiastic, personal style is in sharp contrast to the dead,
gray, impersonal style of most of the writing in scientific journals
nowadays."
Simmons

"With justice
we admire Huygens because he first discovered that a heavy particle
slides down to the bottom of a cycloid in the same time, no matter
where it starts. But you will be petrified with astonishment when
I
say that this very same cycloid, the tautochorone of Huygens is also
the brachistochrone we are seeking."
Johann
Bernoulli


Szapiro, Ben, Revisiting A Classic Least Time Problem,
< http://www.sewanee.edu/physics/TAAPT/TAAPTTALK.html
> 
Stewart, James, Calculus, 5th ed.,
THOMSONBrooks/Cole,
2003, p. 691.
"One of the first people to study the
cycloid was Galileo, who proposed that bridges be built in the shape of
cycloids and who tried to find the area under one arch of a cycloid."


Wagon, Stan, MATHEMATICA^{®}IN
ACTION, W. H. Freeman and Co., 1991, pp. 6066 and 385389.
ISBN
0716722291 or ISBN 071672202X
(pbk.)
Wagon, Stan, MATHEMATICA® IN ACTION, 2nd ed.,
SpringerVerlag, 2000. ISBN 0387986847 for other animations.

Weisstein, Eric W., CRC Concise Encyclopedia of MATHEMATICS,
CRC Press, 1999.
"The brachistochrone
was one of the earliest problems posed in the Calculus of Variations."


Yates, Robert C., Curves and Their Properties,
NCTM, 1952, pp. 6869.

