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.
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December 2015
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December 01 2015
Sagging ropes demonstrate transversality conditions of variational problems
Edward Bormashenko;
Edward Bormashenko
a)
Physics Department,
Ariel University
, Ariel, POB 3 40700, Israel
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Gene Whyman;
Gene Whyman
Physics Department,
Ariel University
, Ariel, POB 3 40700, Israel
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Yelena Bormashenko;
Yelena Bormashenko
Physics Department,
Ariel University
, Ariel, POB 3 40700, Israel
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Roman Grynyov;
Roman Grynyov
Physics Department,
Ariel University
, Ariel, POB 3 40700, Israel
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Evgeny Shulzinger;
Evgeny Shulzinger
Physics Department,
Ariel University
, Ariel, POB 3 40700, Israel
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Alexander Kazachkov
Alexander Kazachkov
School of Physics,
V. N. Karazin Kharkiv National University
, Svobody Sq. 4, 61022 Kharkiv, Ukraine
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a)
Electronic mail: [email protected]
Am. J. Phys. 83, 998–1002 (2015)
Article history
Received:
April 16 2015
Accepted:
September 12 2015
Citation
Edward Bormashenko, Gene Whyman, Yelena Bormashenko, Roman Grynyov, Evgeny Shulzinger, Alexander Kazachkov; Sagging ropes demonstrate transversality conditions of variational problems. Am. J. Phys. 1 December 2015; 83 (12): 998–1002. https://doi.org/10.1119/1.4932160
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