Exact shape of an ideal heavy spring suspended at both ends

Cyril Belardinelli

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arXiv · 2509.22829

Exact shape of an ideal heavy spring suspended at both ends

Cyril Belardinelli

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Abstract

In the present article we determine the exact shape of an ideal spring suspended at both ends. Historically, this goes under the name of Elastica problem. The total energy of the suspended spring is a functional depending on two independent functions: on one hand the function describing the shape of the hanging spring, on the other hand a non-negative function describing the mass distribution along the elongated spring. Calculus of Variations provides a rather short and elegant solution. This method does not appear to be widely known in the literature.

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Cyril Belardinelli. 2025-09-26. Exact shape of an ideal heavy spring suspended at both ends. https://arxiv.org/abs/2509.22829

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