Brachistochrone Curve

The problem Bernoulli set the world

In June 1696, Johann Bernoulli published a challenge in the journal Acta Eruditorum addressed "to the sharpest mathematicians in all the world." Given two points A and B at different heights (but not directly above each other), find the curve down which a frictionless bead, starting from rest at A, slides to B in the shortest time. He named it the brachistochrone, from the Greek brachistos (shortest) + chronos (time).

The naive guess is the straight line — it is the shortest distance, after all. But least distance is not least time. A bead on a curve that plunges steeply at the start gains speed quickly, and that early speed pays dividends across the rest of the trip. The winning shape, Bernoulli already knew, is an inverted cycloid.

Five correct solutions came back: Johann himself, his elder brother and rival Jakob Bernoulli, Gottfried Leibniz, the Marquis de l'Hôpital, and an anonymous English entry that Johann instantly recognized as Newton's — "tanquam ex ungue leonem," he wrote, "as the lion is known by its claw." Newton, by then Master of the Mint, reportedly received the problem at 4 p.m., solved it by 4 a.m., and submitted it without his name. The collision of methods in those five papers is what crystallized a new branch of mathematics: the calculus of variations.

Why distance loses to speed

Energy conservation fixes the speed of the bead at any depth. Starting from rest, if it has dropped a vertical distance y below the start, all that lost potential energy has become kinetic energy:

m·g·y = ½·m·v²    ⟹    v = √(2·g·y)

The speed depends only on how far it has fallen vertically — not on the shape of the path. So a curve that gets the bead deep early lets it run fast for longer. The total descent time is the path length divided element-by-element by this speed:

T = ∫ ds/v = ∫ √(1 + y'²) / √(2·g·y) dx

where y' = dy/dx and ds = √(1 + y'²) dx is the arc-length element. Finding the brachistochrone means finding the function y(x) that makes this integral as small as possible. That is exactly the kind of "minimize an integral over a whole function" problem the calculus of variations was invented to handle.

Solving it: Euler–Lagrange and Beltrami

Write the integrand as L(y, y') = √((1 + y'²) / (2gy)). To minimize ∫ L dx, the optimal curve must satisfy the Euler–Lagrange equation:

d/dx (∂L/∂y') − ∂L/∂y = 0

Here L does not depend on x explicitly, only on y and y'. Whenever that is true, there is a shortcut — the Beltrami identity — which says L − y'·(∂L/∂y') is constant along the curve. Plugging our L in and simplifying collapses everything to a beautifully compact conservation law:

y · (1 + y'²) = C    (a positive constant)

This is a first-order differential equation. Solving it (substitute y = C·sin²(θ/2) = ½C(1 − cos θ)) yields the parametric curve

x = R·(θ − sin θ)
y = R·(1 − cos θ)        with R = C/2

— the equations of a cycloid, the path traced by a fixed point on the rim of a circle of radius R rolling along a line. The single constant R is pinned down by the requirement that the curve pass through the target end point B.

Bernoulli's trick: it's an optics problem in disguise

Johann Bernoulli's own solution was a stroke of genius that sidestepped variational machinery entirely. He noticed that v = √(2gy) looks exactly like how the speed of light changes as it passes through a medium of varying density. By Fermat's principle, light takes the path of least time — and Snell's law tells us that as a ray crosses layers of changing speed, the ratio sin(θ)/v stays constant.

Imagine the bead as a light ray descending through infinitely thin horizontal layers in which the "speed of light" grows as √y. Demanding sin(θ)/v = constant at every layer, and letting the layers shrink to zero thickness, the bending ray traces out — a cycloid. Bernoulli had turned a mechanics problem into Snell's law and read off the answer. The deep lesson is that "least time" governs both the falling bead and the bending photon, which is why the same calculus of variations underlies optics, mechanics, and much of modern physics.

How much does the cycloid actually win?

Take the classic case where B is at horizontal distance equal to π·R and a depth of 2R below A — that is, B sits at the bottom of the first full arch of the cycloid (θ runs from 0 to π). The descent times compare like this:

Path from A to B	Descent time (with R = 1 m, g = 9.81 m/s²)	Relative
Cycloid (brachistochrone)	T = π·√(R/g) ≈ 1.003 s	1.00× (fastest)
Straight line (shortest distance)	≈ 1.19 s	≈ 1.19× slower
Circular arc through A and B	≈ 1.02 s	≈ 1.02× slower
Parabola tuned to A and B	≈ 1.01 s	just behind

The straight ramp loses by roughly 19% even though it is the shortest route. A circular arc is a surprisingly good runner-up — close enough that Galileo, decades earlier, wrongly guessed the arc was the answer. Only the cycloid is exactly optimal, and no other curve can beat it.

The same curve is the tautochrone

The cycloid has a second magical property, discovered by Christiaan Huygens in 1659 — almost 40 years before Bernoulli's challenge. It is the tautochrone (Greek tauto, "same"): a bead released from rest anywhere on an inverted cycloid reaches the bottom in exactly the same time, regardless of where it started. The period of oscillation is

T = π·√(R/g)        (independent of amplitude)

Property	Brachistochrone	Tautochrone
Question asked	Fastest path between two points	Equal arrival time from any start
Curve	Cycloid	Cycloid (same curve)
Discovered	J. Bernoulli, 1696	Huygens, 1659
Use	Optimal descent profiles	Isochronous pendulum clocks

Huygens used the tautochrone property to build a pendulum clock whose period does not drift with the swing amplitude. By hanging the bob from a flexible cord that wraps against two cycloidal "cheeks," the bob itself swings along a cycloid — so a wide swing and a narrow swing take precisely the same time. A simple circular pendulum only fakes this for small angles; the cycloidal pendulum is exactly isochronous at any amplitude.

Computing and comparing descent times

// Cycloid: x = R(θ − sinθ), y = R(1 − cosθ), y measured downward.
// Time to slide from rest at θ=0 to angle θ along the cycloid:
//   T(θ) = θ · √(R/g)   (a clean closed form)
const g = 9.81;

function cycloidTime(R, thetaEnd) {
  return thetaEnd * Math.sqrt(R / g);
}

// Full first arch (θ: 0 → π): brachistochrone / tautochrone time
console.log(`Cycloid bottom: ${cycloidTime(1, Math.PI).toFixed(3)} s`); // ~1.003 s

// Straight ramp from (0,0) to (X, Y) — constant acceleration a = g·(Y/L)
function straightLineTime(X, Y) {
  const L = Math.hypot(X, Y);      // ramp length
  const a = g * (Y / L);           // accel along the ramp
  return Math.sqrt(2 * L / a);     // L = ½at² ⟹ t = √(2L/a)
}

// End point of the first cycloid arch with R = 1:
const X = Math.PI * 1;             // ≈ 3.1416 m across
const Y = 2 * 1;                   // 2 m down
console.log(`Straight line: ${straightLineTime(X, Y).toFixed(3)} s`); // ~1.19 s

// General brachistochrone time between (0,0) and (X, Y):
// 1) solve for R and θ_end so the cycloid hits (X, Y)
function solveCycloid(X, Y) {
  // Find θ from  X/Y = (θ − sinθ)/(1 − cosθ), then R = Y/(1 − cosθ).
  let lo = 1e-6, hi = 2 * Math.PI;
  const ratio = X / Y;
  for (let i = 0; i < 80; i++) {
    const m = (lo + hi) / 2;
    const f = (m - Math.sin(m)) / (1 - Math.cos(m));
    if (f < ratio) lo = m; else hi = m;
  }
  const theta = (lo + hi) / 2;
  const R = Y / (1 - Math.cos(theta));
  return { R, theta };
}

const { R, theta } = solveCycloid(X, Y);
console.log(`Fit: R=${R.toFixed(3)} m, θ=${theta.toFixed(3)} rad`);
console.log(`Brachistochrone: ${cycloidTime(R, theta).toFixed(3)} s`); // ~1.003 s
console.log(`Speedup vs straight: ${(straightLineTime(X, Y) / cycloidTime(R, theta)).toFixed(2)}x`);

Where the brachistochrone shows up

• Calculus of variations. The problem is the founding example of the field. Euler and Lagrange generalized it into the Euler–Lagrange equation, the engine of analytical mechanics.
• Lagrangian & Hamiltonian mechanics. Newton's F = ma is recast as a "minimize the action" principle — the same machinery, applied to the integral of kinetic minus potential energy.
• Optics. Fermat's principle of least time, which Bernoulli exploited, is the optical sibling of the brachistochrone.
• Clockmaking. Huygens' cycloidal pendulum exploited the tautochrone property for amplitude-independent timekeeping.
• Engineering & design. Skate-park transitions, ski-jump in-runs, water-slide profiles and roller-coaster drops all borrow the steep-start logic — though friction and real materials shift the exact optimum.
• Optimal control & trajectory planning. Minimum-time problems in robotics and aerospace are direct descendants, solved by the same variational and Pontryagin techniques.
• General relativity. Particles follow geodesics — paths that extremize proper time — the relativistic echo of "the path that makes an integral stationary."

Common misconceptions

• "The straight line is fastest because it's shortest." Shortest distance is not least time. The cycloid trades extra length for higher average speed and wins by ~16% over the chord.
• "It's a circular arc." Galileo guessed the arc; it is close but not optimal. Only the cycloid exactly minimizes the time.
• "The end point can be anywhere." The cycloid solution requires B to be reachable on a single arch (θ ≤ 2π for that R). If B is much farther across than it is deep, the optimal cycloid first dips below B and climbs back up — the bead momentarily goes lower than its destination.
• "y is plotted upward." In the standard derivation y points downward (it measures depth fallen), which is why v = √(2gy) and the cycloid is "inverted." Mixing the sign flips the equation.
• "Brachistochrone and tautochrone are different curves." They are the same cycloid; one statement is about fastest descent between two points, the other about equal-time descent from any point.
• "Friction doesn't matter." The clean cycloid assumes frictionless sliding. Add friction or air drag and the true minimum-time curve shifts; the ideal result is a frictionless limit.