{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T05:57:18Z","timestamp":1768283838148,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T00:00:00Z","timestamp":1727308800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taif University, Saudi Arabia","award":["TU-DSPP-2024-47"],"award-info":[{"award-number":["TU-DSPP-2024-47"]}]},{"name":"University of Oradea, Romania","award":["TU-DSPP-2024-47"],"award-info":[{"award-number":["TU-DSPP-2024-47"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac\u2019s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler\u2013Lagrange formulation. Focusing on Podolsky\u2019s generalized electrodynamics, the Hamiltonian and corresponding equations of motion are derived. The findings demonstrate that both Hamiltonian and Euler\u2013Lagrange formulations yield equivalent results. This study highlights the Hamiltonian approach as a valuable alternative for understanding the dynamics of second-order systems, validated through a specific application within generalized electrodynamics. The novelty of the research lies in developing advanced theoretical models through Hamiltonian formalism for continuous systems with second-order derivatives. The research employs an alternative method to the Euler\u2013Lagrange formulas by applying Dirac\u2019s theory to study the generalized Podolsky electrodynamics, contributing to a better understanding of complex continuous systems.<\/jats:p>","DOI":"10.3390\/axioms13100665","type":"journal-article","created":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T11:40:27Z","timestamp":1727350827000},"page":"665","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Hamiltonian Formulation for Continuous Systems with Second-Order Derivatives: A Study of Podolsky Generalized Electrodynamics"],"prefix":"10.3390","volume":"13","author":[{"given":"Yazen M.","family":"Alawaideh","sequence":"first","affiliation":[{"name":"MEU Research Unit, Middle East University, Amman 11118, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2855-7535","authenticated-orcid":false,"given":"Alina Alb","family":"Lupas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bashar M.","family":"Al-khamiseh","sequence":"additional","affiliation":[{"name":"MEU Research Unit, Middle East University, Amman 11118, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0206-3828","authenticated-orcid":false,"given":"Majeed A.","family":"Yousif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University Zakho, Duhok 42001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0365-0282","authenticated-orcid":false,"given":"Y. S.","family":"Hamed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/0550-3213(86)90203-8","article-title":"Higher derivative regulation of supersymmetric theories","volume":"268","author":"West","year":"1986","journal-title":"Nucl. Phys. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1007\/BF01018412","article-title":"Infrared asymptotic behavior of gluon Green\u2019s functions in quantum chromodynamics","volume":"52","author":"Alekseev","year":"1982","journal-title":"Theor. Math. 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