Abstract
We study the shape of a string hanging under its weight and subjected in opposite direction by a force due to a soap film adhered to it. These curves are critical points of an energy \({\mathcal {E}}\) which involves the gravitational potential of the string and the surface area of the soap film. We show that these curves also are critical points of the functional \(\gamma \mapsto \int _\gamma \sqrt{\kappa }+\sigma \), where \(\kappa \) is the curvature of the curve \(\gamma \) and \(\sigma \) is a constant. This gives a relationship between these shapes and the classical Euler’s elastics. Finally we extend the problem assuming that the weight of the soap film is relevant on the physical system.
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Acknowledgements
I thank the anonymous referee for suggestions that helped to improve the exposition and to correct several results. This research has been partially supported by MINECO/MICINN/FEDER grant no. PID2020-117868GB-I00, and by the “María de Maeztu” Excellence Unit IMAG, reference CEX2020-001105- M, funded by MCINN/AEI/10.13039/501100011033/ CEX2020-001105-M.
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López, R. A characterization of the catenary under the effect of surface tension. Rend. Circ. Mat. Palermo, II. Ser 73, 873–885 (2024). https://doi.org/10.1007/s12215-023-00956-7
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DOI: https://doi.org/10.1007/s12215-023-00956-7