Catenary & Calculus of Variations

A hanging chain minimizes gravitational potential energy — Euler-Lagrange gives y = a·cosh(x/a)

Chain parameter a

Half-span (px)

Beads

Catenary y=a·cosh(x/a)

Parabola (wrong)

Chain beads

Euler-Lagrange equation:
d/dx(y'/√(1+y'²)) = c

Solution:
y = a·cosh(x/a) + b

The catenary is the curve formed by a flexible chain hanging between two points under gravity. Galileo guessed it was a parabola — he was wrong. The true shape comes from calculus of variations: minimize ∫y·√(1+y'²) dx (potential energy ∝ height × arc length element). The Euler-Lagrange equation yields y = a·cosh(x/a), where a depends on the chain's weight-to-tension ratio. Catenaries appear in suspension bridges, power lines, and the Gateway Arch (inverted).