Hinz, Andreas, Kolokvij april 2014

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We survey the history of the problem to describe the shape of an ideal heavy chain, and assuming an inverse square law of central gravitational force we show that the Euler-Lagrange equation admits an explicit and unique solution for any physically admissible set of data. This opens a way to prove that these solutions do actually minimize potential energy. For the corresponding upright arches special cases occur where the curves are segments of hyperbolic or logarithmic spirals. Numerical simulations show for which dimensions of chains and arches the difference from the classical catenary becomes significant. It turns out that in some cases even the parabola is a better approximation for the true catenary than the hyperbolic cosine.

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