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We address the properties of the catenary in its role of generating a minimal rotational area around the abscissa. We first correct a small flaw in an excellent, highly respected text on the calculus of variations. We then highlight the fact that, for numerous boundary conditions, there is an infinitely large number of continuously differentiable functions that outperform the catenary. We will show this thanks to beautiful parametric expressions found and provided to us by Ernst Hairer.

On a Classic Problem in the Calculus of Variations: Setting Straight Key Properties of the Catenary