Brachistochrone & Calculus of Variations
Which curve gets a bead from A to B fastest? The cycloid wins — proved via Euler-Lagrange equations
Euler-Lagrange equation for T=∫√((1+y'²)/(2gy)) dx: d/dx[y'/√(y(1+y'²))] = √((1+y'²)/y)/2. Solution: cycloid x=r(θ−sinθ), y=r(1−cosθ). Bernoulli 1696 challenge; Newton solved it overnight.