Brachistochrone curve

Johann Bernoulli's solution

According to Fermat’s principle, the actual path between two points taken by a beam of light is one that takes the least time. In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).

By the conservation of energy, the instantaneous speed of a body v after falling a height y in a uniform gravitational field is given by:

v = 2 g y ,

The speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement.

Bernoulli noted that the law of refraction gives a constant of the motion for a beam of light in a medium of variable density:

sin ⁡ θ v = 1 v d x d s = 1 v m ,

where v_m is the constant and θ represents the angle of the trajectory with respect to the vertical.

The equations above lead to two conclusions:

1. At the onset, the angle must be zero when the particle speed is zero. Hence, the brachistochrone curve is tangent to the vertical at the origin.
2. The speed reaches a maximum value when the trajectory becomes horizontal and the angle θ = 90°.

Assuming for simplicity that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after falling a vertical distance D:

v m = 2 g D .

Rearranging terms in the law of refraction and squaring gives:

v m 2 d x 2 = v 2 d s 2 = v 2 ( d x 2 + d y 2 )

which can be solved for dx in terms of dy:

d x = v d y v m 2 − v 2 .

Substituting from the expressions for v and v_m above gives:

d x = y D − y d y

which is the differential equation of an inverted cycloid generated by a circle of diameter D.

Jakob Bernoulli's solution

Johann's brother Jakob showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements,

d s 2 = d x 2 + d y 2 .

On differentiation with dy fixed we get,

2 d s   d 2 s = 2 d x   d 2 x .

And finally rearranging terms gives,

d x d s d 2 x = d 2 s = v   d 2 t

where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is d²x (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,

d 2 t 1 = 1 v 1 d x 1 d s 1 d 2 x d 2 t 2 = 1 v 2 d x 2 d s 2 d 2 x

For the path of least times these times are equal so for their difference we get,

d 2 t 2 − d 2 t 1 = 0 = ( 1 v 2 d x 2 d s 2 − 1 v 1 d x 1 d s 1 ) d 2 x

And the condition for least time is,

1 v 2 d x 2 d s 2 = 1 v 1 d x 1 d s 1

History

Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, 1696. He published his solution in the journal in May of the following year, and noted that the solution is the same curve as Huygens's tautochrone curve. After deriving the differential equation for the curve by the method given above, he went on to show that it does yield a cycloid. However, his proof is marred by his use of a single constant instead of the three constants, v_m, 2g and D, above.

Bernoulli allowed six months for the solutions but none were received during this period. At the request of Leibniz, the time was publicly extended for a year and a half. On 29 January 1697 the challenge was received by Isaac Newton, who found it in his mail, in a letter from Johann Bernoulli, when he arrived home from the Royal Mint at 4 p.m., and stayed up all night to solve it and mailed the solution anonymously by the next post. Upon reading the solution, Bernoulli immediately recognized its author, exclaiming that he "recognizes a lion from his claw marks." This story gives some idea of Newton's power, since Johann Bernoulli took two weeks to solve it.

At the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital. Four of the solutions (excluding l'Hôpital's) were published in the same edition of the journal as Johann Bernoulli's. In his paper Jakob Bernoulli gave a proof of the condition for least time similar to that above before showing that its solution is a cycloid. According to Newtonian scholar Tom Whiteside, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem. In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called (in 1766) the calculus of variations. Joseph-Louis Lagrange did further work that resulted in modern infinitesimal calculus.

Earlier, in 1638, Galileo had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his Two New Sciences. He draws the conclusion (Third Day, Theorem 22, Prop. 36) that the arc of a circle is faster than any number of its chords,

"From the preceding it is possible to infer that the quickest path of all [lationem omnium velocissimam], from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle. ... Consequently the nearer the inscribed polygon approaches a circle the shorter is the time required for descent from A to C. What has been proven for the quadrant holds true also for smaller arcs; the reasoning is the same."

We are warned earlier in the Two New Sciences (just after Theorem 6) of possible fallacies and the need for a "higher science." In this dialogue Galileo reviews his own work. The actual solution to Galileo's problem is half a cycloid. Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics.