catenary problem
/ KAT-uh-ner-ee /
Hold a flexible chain or necklace loosely by its two ends and let it hang. What shape does it settle into? Not a parabola, as people often guess, but a special curve called the catenary (from the Latin for chain). Nature picks this shape automatically because, among all curves of the chain's fixed length joining the two supports, it is the one whose center of mass hangs as low as possible — gravity minimizes the potential energy, and that minimum is a variational problem.
The potential energy of a heavy chain is proportional to integral of y times (arc length element) = integral of y sqrt(1 + y'^2) dx, where y is height. You minimize this functional subject to the constraint that the total length, integral of sqrt(1 + y'^2) dx, is fixed — exactly an isoperimetric (constrained) variational problem, handled with a Lagrange multiplier. The Euler-Lagrange equation for the combined integrand (whose integrand has no explicit x, so the Beltrami identity applies) yields the hyperbolic cosine: y = a cosh(x/a) + constant. The constant a is set by the length and the spacing of the supports.
The catenary is everywhere in engineering. Hanging power lines, suspension-bridge main cables (under their own weight), and chains take this shape; and famously, turning a catenary upside down gives the ideal arch — the inverted catenary carries its load in pure compression, which is why Gaudi and others used hanging-chain models to design self-supporting arches and domes. A clean caveat worth stating: the pure catenary is the shape of a chain loaded only by its own weight. A suspension bridge deck, whose weight is distributed along the horizontal span rather than along the cable, pulls the cable into a parabola instead — a different curve for a different load.
A chain hung from two equal-height posts settles into y = a cosh(x/a), measured from the lowest point. Near the bottom cosh(x/a) ≈ 1 + x^2/(2 a^2), so the catenary looks almost parabolic there — which is why the parabola guess is close but not exact.
The hanging chain is a hyperbolic cosine, close to but not the same as a parabola.
The catenary (cosh) is the chain under its own weight only. A cable loaded by a uniform horizontal deck — a suspension bridge — is a parabola, not a catenary; mixing them up is a classic error.