The Brachistochrone: Why the Fastest Path Down Is a Cycloid

The challenge Bernoulli posed

In 1696 Johann Bernoulli challenged the mathematicians of Europe: given two points A and B, with B lower than A but not directly below it, what shape of frictionless wire lets a bead released from rest at A reach B in the least time, under gravity alone? The straight line is the shortest path but not the fastest, because a curve that dips more steeply at first lets the bead build speed early, more than compensating for the extra distance travelled. The winning curve is called the brachistochrone, from the Greek for 'shortest time', and the answer - found independently by Newton, Leibniz, L'Hopital and the two Bernoulli brothers - is a cycloid.

Setting up the variational problem

Time is distance over speed, integrated along the path. If the bead has fallen a height y (measuring downward from A), energy conservation gives its speed as v = √(2gy). The total time along a path y(x) is the functional:

T[y] = ∫ √((1 + y'²) / (2gy)) dx

Minimising T over all possible curves y(x) is a problem in the calculus of variations, not ordinary calculus - you are optimising over an infinite-dimensional space of functions, not a finite set of numbers. The Euler-Lagrange equation applied to this functional (or, more directly, Beltrami's identity since the integrand has no explicit x) reduces it to a first-order ODE whose solution, worked out in parametric form, is exactly a cycloid.

Why the answer is a cycloid

A cycloid is the curve traced by a point on the rim of a circle rolling along a straight line - the same family of curve used elsewhere on this site for hypocycloids, just rolled on a flat line instead of another circle. Parametrised by the rolling angle θ and the generating circle's radius r, the brachistochrone connecting the origin to the target point is:

x(θ) = r·(θ - sinθ)
y(θ) = r·(1 - cosθ)

The radius r and the range of θ are fixed by requiring the curve to pass through the target point B - a one-dimensional root-find, since as r grows the curve's endpoint sweeps out a definite path. Remarkably, the optimal curve leaves A vertically, dropping straight down initially no matter how far away B is horizontally - the fastest possible way to convert potential energy into speed before bending toward the target.

Racing four curves: straight line, circular arc, parabola, cycloid

The live demo releases beads simultaneously on a straight line, a circular arc, a parabola and the true cycloid, all constrained to the same start and end points. The straight line is intuitive but wrong - a bead on it accelerates too slowly at first. A circular arc or parabola tuned to dip below the straight line usually beats it but still loses to the cycloid, whose curvature profile is exactly tuned so that every infinitesimal segment trades potential energy for speed at the locally optimal rate. The margin is not huge for short, gently sloped runs, but it grows as the drop becomes steeper or the horizontal distance longer.

The tautochrone: the same curve solves a second puzzle

Remarkably, the cycloid also solves a completely different problem Christiaan Huygens had posed decades earlier: the tautochrone (equal-time) curve, on which a bead released from any starting height reaches the bottom in exactly the same time, regardless of where it starts. Huygens used this property to design a pendulum clock with cycloidal cheeks constraining the pendulum's string, making its period independent of swing amplitude - a large swing traces a longer arc but at correspondingly higher average speed, an early real-world exploitation of the same isochronism that makes the brachistochrone optimal.

Beltrami's identity, the shortcut

Solving the full Euler-Lagrange equation for this functional is unnecessary once you notice the integrand F(y, y') has no explicit x-dependence. Beltrami's identity then gives a first integral directly: F - y'·(∂F/∂y') = constant. Plugging in F = √((1+y'²)/(2gy)) and simplifying yields y·(1 + y'²) = constant, a first-order ODE whose solution - after the standard trigonometric substitution y' = cot(θ/2) - is precisely the cycloid parametrisation above. This is the standard shortcut used whenever a variational problem's integrand does not depend explicitly on the independent variable.

Frequently asked questions

Try it live

Everything above runs in your browser — open Brachistochrone and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.