{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T11:54:37Z","timestamp":1770378877102,"version":"3.49.0"},"reference-count":61,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T00:00:00Z","timestamp":1627516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020 (CIDMA)"],"award-info":[{"award-number":["UIDB\/04106\/2020 (CIDMA)"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The calculus of variations is a field of mathematical analysis born in 1687 with Newton\u2019s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler\u2013Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton\u2019s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler\u2013Lagrange type. The new calculus of variations complements the standard one in a nontrivial\/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.<\/jats:p>","DOI":"10.3390\/axioms10030171","type":"journal-article","created":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T10:47:46Z","timestamp":1627555666000},"page":"171","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["On a Non-Newtonian Calculus of Variations"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,29]]},"reference":[{"key":"ref_1","unstructured":"Grossman, M., and Katz, R. (1972). Non-Newtonian Calculus, Lee Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"203","DOI":"10.5269\/bspm.39444","article-title":"Solvability of bigeometric differential equations by numerical methods","volume":"39","author":"Boruah","year":"2021","journal-title":"Bol. Soc. Parana. Mat."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s11766-011-2767-6","article-title":"On modeling with multiplicative differential equations","volume":"26","author":"Bashirov","year":"2011","journal-title":"Appl. Math. J. Chin. Univ. Ser. 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