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

Abstract

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.

MSC::

• Primary 49K05
• Secondary 49J05

ACKNOWLEDGMENTS

We wish to express our deepest gratitude to Ernst Hairer not only for invaluable comments but also for very kindly providing us with families of functions found by him that outperform the catenary in a large area of (x, y)-space. Two anonymous referees, as well as Arnold Arthurs and Peter Olver, provided precious remarks and advice; let each of them be warmly thanked.

Notes

1 It took more than a century to discover that in this case the second-order Euler equation could be replaced by this more manageable first-order equation; the credit goes to the Italian mathematician Eugenio Beltrami (1835–1900).

Additional information

Notes on contributors

Olivier de La Grandville

Olivier de La Grandville received his Ph.D. in economics from the University of Geneva in 1968, where he was a professor from 1978 to 2007. From 1988 to 2017, as a visiting professor, he taught applications of the calculus of variations to optimal economic growth in the School of Engineering at Stanford University. He is now a senior professor in Frankfurt University. Recently he became interested in deriving the fundamental equations of the calculus of variations and optimal control from economic reasoning. Results appeared in mathematics journals as well as in his last book, Economic Growth: A Unified Approach, 2nd ed., Cambridge Univ. Press (2017). A keen tennis player, he is still waiting to receive his wild card to enter the Wimbledon qualies.