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Brachistochrone Part I
Brachistochrone Part II
Brachistochrone Part III

Brachistochrone Part IV

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The Brachistochrone
The Famous Problem of Fastest Descent
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Deposit #77

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." Greek name

Descent graphWhat 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.

  Descent animation
Brachistochrone values are plotted on the x-axis.
Linear values appear on the y-axis.

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 . . . .

Animated Cycloid

Play this animation.

and its equations in parametric form are . . . .

Cycloid animation.

Law for the Refraction
of Light

is often applied in
deriving the equations.
Snell's Law for Reflection of Light
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]];
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 R-1}}, PlotStyle→{RGBColor[0,0,1], RGBColor[0,1,0]},
    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.

Johann Bernoulli
John (Jean, Johann)

Leibniz stamp

Huygens stamp

Newton stamp

Beckmann, Petr, A History of  π (PI), St. Martin's Press, 1971, pp. 139-140.
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. 192-197.  
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 Euler-Lagrange 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. 547-549, 562.
Katz, Victor J., A History of Mathematics, Brief ed., Pearson Addison Wesley, 2004, pp. 250, 331-332.
Simmons, George F., Calculus Gems: Brief Lives and Memorable Mathematics, McGraw-Hill,1992, pp. 308-313.
"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."
"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,
< >
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. 60-66 and 385-389.   ISBN 0-7167-2229-1  or  ISBN 0-7167-2202-X (pbk.)
Wagon, Stan, MATHEMATICA® IN ACTION, 2nd ed., Springer-Verlag, 2000.  ISBN 0-387-98684-7 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. 68-69.

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