Classical Mechanics
Brachistochrone Curve
Solve Johann Bernoulli's 1696 challenge problem: What curve allows a bead to slide under gravity between two points in the shortest time? Compare Cycloid, Straight Line, Circular Arc, and Parabola.
Cycloid Time
0.000 s
Straight Line Time
0.000 s
Circular Arc Time
0.000 s
Parabola Time
0.000 s
Simulation Params
Visual Overlay
Controls
Euler-Lagrange Solution for Brachistochrone
Minimize time functional: T[y] = ∫ √(1 + y'²) / √(2gy) dx
Solution is the Cycloid parametric equations:
x(θ) = R(θ - sin θ)
y(θ) = R(1 - cos θ)
Solution is the Cycloid parametric equations:
x(θ) = R(θ - sin θ)
y(θ) = R(1 - cos θ)
Developer Reference
Core Algorithm & Standalone Script
// Brachistochrone Cycloid solver & particle race engine
const canvas = document.getElementById('canvas');
const ctx = canvas.getContext('2d');
// Cycloid parametric: x = R(theta - sin(theta)), y = R(1 - cos(theta))