{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T15:38:49Z","timestamp":1770565129642,"version":"3.49.0"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,3,22]],"date-time":"2017-03-22T00:00:00Z","timestamp":1490140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Understanding advanced physical phenomena such as vertically hanging elastic column, soap bubbles, crystals and cracks demands expressing and manipulating a system\u2019s potential energy under equilibrium conditions. However, students at schools and universities are usually required to consider the forces acting on a system under equilibrium conditions, rather than taking into account its potential energy. As a result, they find it difficult to express the system\u2019s potential energy and use it for calculations when they do need to do so. The principle of least potential energy is a powerful idea for solving static equilibrium physics problems in various fields such as hydrostatics, mechanics, and electrostatics. In the current essay, the authors describe this principle and provide examples where students can apply it. For each problem, the authors provide both the force consideration solution approach and the energy consideration solution approach.<\/jats:p>","DOI":"10.3390\/sym9030045","type":"journal-article","created":{"date-parts":[[2017,3,22]],"date-time":"2017-03-22T12:04:00Z","timestamp":1490184240000},"page":"45","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Interweaving the Principle of Least Potential Energy in School and Introductory University Physics Courses"],"prefix":"10.3390","volume":"9","author":[{"given":"Yuval","family":"Ben-Abu","sequence":"first","affiliation":[{"name":"Department of Physics and Project Unit, Sapir Academic College, Sderot, Hof Ashkelon 79165, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haim","family":"Eshach","sequence":"additional","affiliation":[{"name":"Department of Science and Technology Education, Ben-Gurion University of Negev, P.O.B. 653, Beer-Sheva 8410501, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hezi","family":"Yizhaq","sequence":"additional","affiliation":[{"name":"Department of Solar Energy and Environmental Physics, Jacob Blaustein Institute for Desert Research, Ben-Gurion University of Negev, P.O.B. 653, Beer-Sheva 8410501, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hilderbrandt, S., and Tromba, A. (1996). The Persimmons Universe, Springer.","DOI":"10.1007\/978-1-4612-2424-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1080\/0950069870090202","article-title":"Should energy be illustrated as something quasi-material?","volume":"9","author":"Duit","year":"1987","journal-title":"Int. J. Sci. Educ."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lemons, D.S. (1997). Perfect Form, Princeton University Press.","DOI":"10.1515\/9780691214825"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1119\/1.1528915","article-title":"Simple derivation of Newtonian mechanics from the principle of least action","volume":"71","author":"Hanc","year":"2003","journal-title":"Am. J. Phys."},{"key":"ref_5","unstructured":"Glass, B., Temkin, O., and Straus, V. (1959). Forerunners of Darwin, Johns Hopkins University Press."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hilderbrandt, S., and Tromba, A. (1996). The Persimmons Universe, Springer.","DOI":"10.1007\/978-1-4612-2424-2"},{"key":"ref_7","unstructured":"Meriam, J.L., and Kraige, L.G. (2002). Engineering Mechanics, Statics, John Wiley & Sons, Inc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/PhysRevLett.114.037801","article-title":"Orogin of metastable knot in single flexible chain","volume":"114","author":"Dai","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1080\/00029890.1980.11995034","article-title":"The equations for large vibrations of strings","volume":"87","author":"Antman","year":"1980","journal-title":"Amer. Math. Mon."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1119\/1.19539","article-title":"Motion of a hanging chain after the free end is given an initial velocity","volume":"68","author":"Bailey","year":"2000","journal-title":"Amer. J. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"114301","DOI":"10.1103\/PhysRevLett.87.114301","article-title":"Dynamic patterns and self-knotting of a driven hanging chain","volume":"87","author":"Belmonte","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Feynman, R.P., Leighton, R.B., Sands, M., and Lindsay, R.B. (1964). The Feynman Lectures on Physics, Addison-Wesley. Chapter 19.","DOI":"10.1063\/1.3051743"},{"key":"ref_13","unstructured":"Wolfram, S. (1996). The Mathematica Book, Wolfram Media\/Cambridge University Press. [3rd ed.]."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1088\/0143-0807\/30\/1\/010","article-title":"The stability of the catenary shapes for a hanging cable of unspecified length","volume":"30","author":"Mareno","year":"2008","journal-title":"Eur. J. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1119\/1.18505","article-title":"Hanging shapes of nonuniform cables","volume":"65","author":"Fallis","year":"1997","journal-title":"Am. J. Phys."},{"key":"ref_16","first-page":"117","article-title":"Catenaria Vera\u2014The True Catenary","volume":"17","author":"Denzler","year":"1999","journal-title":"Expo. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s11044-007-9097-8","article-title":"Least action principles and their application to constrained and task-level problems in robotics and biomechanics","volume":"19","author":"Khatib","year":"2008","journal-title":"Multibody Syst. Dyn."},{"key":"ref_18","unstructured":"Principle Of Least Action Interactive. Available online: http:\/\/www.eftaylor.com\/software\/ActionApplets\/LeastAction.html."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1119\/1.1555874","article-title":"A call for action","volume":"71","author":"Taylor","year":"2003","journal-title":"Am. J. 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