Catenaria vera -- the true catenary
One might have thought that everything that could be said about the question of the shape of a hanging chain had long ago been said. After all, the problem was posed over 300 years ago. In fact, there was more to say and the paper under review says it, and says it in an interesting manner. In 1638, Galileo Galilei asserted that a chain hangs in the shape of a parabola. This assertion was refuted by Huygens in 1646, but not until infinitesimal methods were available could the actual shape be found. The classical solution assumes that the gravitational field is constant, and the authors call the resulting shape the classical catenary. Of course, the gravitational field near the surface of the earth is more accurately modeled by a \(1/r\) potential, and the authors solve the hanging chain problem assuming such a \(1/r\) potential. They go through the complete argument including showing that the shape they obtain, which they call the true catenary, is the unique minimizer of the potential energy. With both the classical catenary and the true catenary in hand, the authors can compare the two. Not surprisingly the classical catenary provides a good approximation to the true catenary, even for chains that one would consider very large. The authors tell us that for a chain (near the surface of the earth) spanning 50 kilometers and hanging down about 1 kilometer the difference between the classical catenary and the true catenary is a mere 2.7 centimeters. The accuracy of the approximation deteriorates for truly huge chains. The authors give a figure showing that for a truly huge chain the parabolic shape claimed by Galileo is actually a better approximation to the true catenary than is the classical catenary; though the difference between the parabola and the classical catenary is less than the difference of either from the true catenary.
scientific article; zbMATH DE number 1310242