Brachistochrone: Fastest Descent

The cycloid minimizes travel time — Euler-Lagrange calculus of variations

Race: Ball Rolling Down Four Curves

Cycloid
—
Straight Line
—
Parabola
—
Circle Arc
—

Cycloid Generator

Theory — Euler-Lagrange

T = ∫ ds/v = ∫ √(1+y'²)/√(2gy) dx
Euler-Lagrange: d/dx(∂L/∂y') − ∂L/∂y = 0
Solution: x=R(θ−sinθ), y=R(1−cosθ)
T_cycloid = π√(R/g)

Johann Bernoulli posed the brachistochrone problem in 1696. Solutions arrived from Newton, Leibniz, L'Hopital, Jakob Bernoulli, and Johann himself. All found the cycloid — the curve traced by a point on a rolling circle.


The variational approach: extremize the functional T[y(x)] by requiring δT = 0. The integrand does not depend explicitly on x (Beltrami identity), yielding y(1+y'²) = const, which is the cycloid.


Tautochrone property: the cycloid is also the tautochrone — a ball released from any point reaches the bottom in the same time T = π√(R/g).