Sagging ropes demonstrate transversality conditions of variational problems

We present an account of transversality conditions of variational problems and show how they give rise to essential results in the analysis of different physical phenomena. We find such conditions to be a powerful and elegant approach to a diverse number of variational problems with free endpoints....

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Bibliographic Details
Published in:American journal of physics 2015-12, Vol.83 (12), p.998-1002
Main Authors: Bormashenko, Edward, Whyman, Gene, Bormashenko, Yelena, Grynyov, Roman, Shulzinger, Evgeny, Kazachkov, Alexander
Format: Article
Language:English
Subjects:
Asymmetry
Differential equations
Friction
Physics
Rope
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Summary:We present an account of transversality conditions of variational problems and show how they give rise to essential results in the analysis of different physical phenomena. We find such conditions to be a powerful and elegant approach to a diverse number of variational problems with free endpoints. In this work, we illustrate such condition with the study of a heavy inextensible rope sagging both symmetrically and asymmetrically between two variously shaped, steering-guide wires without friction. In such a situation, the transversality conditions lead to the orthogonality of the rope to the wires at endpoints of the rope, which we confirm experimentally. Freeing the endpoints of the rope yields exact analytical equations that predict the tension of the rope. Heavy ropes whose endpoints are free to slip between arbitrarily shaped, steering wires are discussed.
ISSN:0002-9505
1943-2909
DOI:10.1119/1.4932160