The Brachistochrone: Finding the Fastest Way Down

Release a bead at point A and let it slide without friction along a wire to a lower point B. What shape of wire gets it there fastest? The straight line is the shortest path, but it is not the quickest. The answer, a curve called the cycloid, was the prize in one of the most famous challenges in the history of mathematics. Solving it helped start a new branch of mathematics, the calculus of variations Brachistochrone curve, Wikipedia^[1] Calculus of variations, Wikipedia^[2].

What is the brachistochrone problem?

The brachistochrone, from the Greek for “shortest time”, is the curve along which a bead sliding without friction under uniform gravity travels from a given start point to a given end point in the least time Brachistochrone curve, Wikipedia^[1]. The end point is lower than the start but not directly below it.

Who posed the challenge, and who solved it?

Johann Bernoulli posed the problem to the readers of Acta Eruditorum in June 1696 Brachistochrone curve, Wikipedia^[1]. He allowed six months for solutions, but none arrived in that time. At Leibniz's request the deadline was publicly extended for a year and a half Brachistochrone curve, Wikipedia^[1].

Isaac Newton, by then working at the Royal Mint, found the challenge in a letter from Bernoulli when he came home at 4 p.m. on 29 January 1697 Brachistochrone curve, Wikipedia^[1]. According to his niece Catherine Conduitt, he had solved it by 4 a.m. the next morning Brachistochrone curve, Wikipedia^[1]. His solution was published anonymously in the Philosophical Transactions Brachistochrone curve, Wikipedia^[1] Newton 1697^[3]. It was correct, but it did not show his method Brachistochrone curve, Wikipedia^[1]. Bernoulli is reported to have recognized the author anyway, as one knows a lion by its claw Brachistochrone curve, Wikipedia^[1]. Our Isaac Newton sticker honours the night's work; for Newton's larger legacy, see how Newton built on Galileo and Kepler.

In the end five mathematicians sent solutions: Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital Brachistochrone curve, Wikipedia^[1]. Johann had his own. The Bernoulli brothers often worked on the same problems, and their most bitter dispute was over this one Johann Bernoulli, Wikipedia^[4].

Why isn't the straight line fastest?

A frictionless bead that has dropped a height y below its start has speed Brachistochrone curve, Wikipedia^[1]

v = √(2g·y)

where v is the speed in m/s, g ≈ 9.81 m/s² is the gravitational acceleration and y is the vertical drop in m, measured downward. Speed depends only on depth, not on the path taken to get there. So it pays to drop steeply at first and build up speed early, even if the path gets longer. The total time along a curve y(x) is

T = ∫ √(1 + (dy/dx)²) / √(2g·y) dx

where the numerator is the arc length element per unit dx and x is the horizontal distance in m. The problem is to choose the whole function y(x) that makes T smallest. That is a question about functions, not numbers, and it is what the calculus of variations was built for Calculus of variations, Wikipedia^[2].

How did Bernoulli solve it with light?

Johann Bernoulli's indirect method used optics Brachistochrone curve, Wikipedia^[1]. By Fermat's principle, light travelling between two points takes the path of least time, and that path obeys Snell's law of refraction Brachistochrone curve, Wikipedia^[1]. Imagine the bead's region sliced into thin horizontal layers, each with a speed √(2gy), like layers of glass with a changing index of refraction. Snell's law then says that sin θ / v stays constant along the path, where θ is the angle between the path and the vertical Brachistochrone curve, Wikipedia^[1]. Bernoulli showed that the curve satisfying this condition is a cycloid Brachistochrone curve, Wikipedia^[1]. The same least-time idea bends light in a mirage; see our post on thermal mirages and our Refracta: Fermat's Principle sticker.

What is a cycloid?

A cycloid is the curve traced by a point on the rim of a wheel rolling along a straight line Cycloid, Wikipedia^[5]. Turned upside down, with y measured downward, it is

x = r(θ − sin θ), y = r(1 − cos θ)

where r is the radius in m of the generating circle and θ is the angle in radians through which it has rolled. Galileo coined the name cycloid and was the first to study the curve seriously Cycloid, Wikipedia^[5]. Along this curve the travel time from the start to the point at angle θ_B is

T = θ_B·√(r/g)

(our derivation, checked numerically below).

How much faster is the cycloid? A worked race

Take B 1 m to the right of A and 1 m below it, with g = 9.81 m/s². Our numerical integration gives these times:

• Straight line (length 1.414 m): 0.639 s.
• Quarter circle (vertical at A, horizontal at B, length 1.571 m): 0.592 s.
• Cycloid (r = 0.573 m, θ_B = 2.412 rad, length 1.474 m): 0.583 s.

The circle beats the line by 7%, and the cycloid beats the circle by about 1.5%. Galileo had made a similar comparison. In his 1638 Two New Sciences, working on a related problem, he concluded that an arc of a circle is faster than any number of its chords, and inferred that the quickest path is not the straight line but the circular arc Brachistochrone curve, Wikipedia^[1]. The first part was right. The second was not, because the cycloid is faster still.

Can the fastest path go below the finish?

Yes. On this cycloid the lowest point is at θ = π. If B lies more than π/2 times its drop to the side, the fastest path dips below B and climbs back up to it (our derivation).

What does the cycloid have to do with clocks?

The same curve solves a second problem. On an upside-down cycloid, a bead released from rest reaches the bottom in the same time wherever it starts. This is the tautochrone. Christiaan Huygens solved it in 1659 and proved it in his Horologium Oscillatorium of 1673 Tautochrone curve, Wikipedia^[6]. When Johann Bernoulli published his own solution in May 1697, he noted that the brachistochrone is the same curve as Huygens' tautochrone Brachistochrone curve, Wikipedia^[1]. More pendulum physics is in our Classical Mechanics collection.

What came out of the problem?

Leonhard Euler first elaborated the general subject, beginning in 1733. After he saw the 1755 work of the 19-year-old Joseph-Louis Lagrange, Euler adopted Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum Calculus of variations, Wikipedia^[2]. The Euler–Lagrange equation that came out of this work now underlies the principle of least action across physics.

Quick FAQ

Does friction change the answer?

Yes. With friction, or with a nonzero starting speed, the time-minimizing curve is no longer the tautochrone Brachistochrone curve, Wikipedia^[1].

Is the brachistochrone the shortest path?

No. It is longer than the straight line but quicker, because the bead gains speed early.

Did Newton really solve it overnight?

That account comes from his niece, Catherine Conduitt Brachistochrone curve, Wikipedia^[1].

References

1. Wikipedia, “Brachistochrone curve”. https://en.wikipedia.org/wiki/Brachistochrone_curve
2. Wikipedia, “Calculus of variations”. https://en.wikipedia.org/wiki/Calculus_of_variations
3. Isaac Newton (published anonymously) (1697), “III. Epistola missa ad prænobilem virum D. Carolum Mountage Armigerum, Scaccarii Regii apud Anglos cancellarium, & Societatis Regiae præsidem, in qua solvuntur duo problemata Mathematica à Johanne Barnoullo Mathematico celeberrimo proposita,” Philosophical Transactions of the Royal Society of London 19, 384–389 (DOI 10.1098/rstl.1695.0064; metadata verified via Crossref). https://doi.org/10.1098/rstl.1695.0064
4. Wikipedia, “Johann Bernoulli”. https://en.wikipedia.org/wiki/Johann_Bernoulli
5. Wikipedia, “Cycloid”. https://en.wikipedia.org/wiki/Cycloid
6. Wikipedia, “Tautochrone curve”. https://en.wikipedia.org/wiki/Tautochrone_curve

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.