{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T06:11:46Z","timestamp":1772777506304,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100018711","name":"CIDMA","doi-asserted-by":"crossref","award":["UID\/04106\/2025"],"award-info":[{"award-number":["UID\/04106\/2025"]}],"id":[{"id":"10.13039\/501100018711","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>In this work, we derive necessary optimality conditions for a class of fractional variational problems involving Caputo-type derivatives. We consider functionals defined on appropriate spaces of absolutely continuous functions and study both periodic and antiperiodic boundary conditions, treated in a unified framework. The analysis covers the cases 0&lt;\u03b1&lt;1 and 1&lt;\u03b1&lt;2, leading to fractional Euler\u2013Lagrange equations supplemented by suitable transversality conditions. We further extend the results to problems with integral constraints and holonomic constraints, as well as to a fractional Herglotz variational principle.<\/jats:p>","DOI":"10.3390\/fractalfract10030168","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T13:11:36Z","timestamp":1772629896000},"page":"168","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fractional Euler\u2013Lagrange Equations Under Periodic and Antiperiodic Boundary Conditions"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1305-2411","authenticated-orcid":false,"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-168 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF01911126","article-title":"Fractional integration and differentiation of variable order","volume":"21","author":"Samko","year":"1995","journal-title":"Analysis Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1103\/PhysRevE.53.1890","article-title":"Nonconservative Lagrangian and Hamiltonian mechanics","volume":"53","author":"Riewe","year":"1996","journal-title":"Phys. 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