The hanging chain (catenary) simulation

What shape does a chain hanging under its own weight take — the parabola Galileo guessed?

▶ Run the simulationSee the measured result

Units: length units (catenary parameter a for S = 2, L = 2.5; root of L = 2a·sinh(S/2a), Acta Eruditorum June 1691)

How the lab tests it

Hang a flexible chain (a Verlet rope of equal links between two pinned ends, settled with position-based distance constraints) and, as the span slowly breathes, compare its shape to the catenary y = a·cosh(x/a) and to the least-squares best-fit parabola, by the RMS residual of each.

What it checks

the CATENARY, not the parabola: the chain matches y = a·cosh(x/a) to a tiny residual (≈0.006) while even the BEST-FIT parabola is tens of times worse at deep sag — and only catches up when the chain is nearly flat, where cosh ≈ parabola. The cosh wins because the chain has uniform mass per unit LENGTH (a parabola is the cable of a suspension bridge, uniform mass per unit horizontal span); it is also the least-potential-energy shape, which the rope finds on its own (Bernoulli/Huygens/Leibniz, 1691)