{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:28:23Z","timestamp":1760059703986,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T00:00:00Z","timestamp":1751846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study explores the dynamics of particle motion in pseudo-Galilean 3\u2212space G31 by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler\u2013Lagrange equations corresponding to this functional under some boundary conditions in G31. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in G31 and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications.<\/jats:p>","DOI":"10.3390\/axioms14070520","type":"journal-article","created":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T06:03:13Z","timestamp":1751868193000},"page":"520","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Geometric Variational Problem for Pseudo-Galilean Particles"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8051-2879","authenticated-orcid":false,"given":"Ay\u015fe Y\u0131lmaz","family":"Ceylan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Akdeniz University, Antalya 07070, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9632-2180","authenticated-orcid":false,"given":"Tunahan","family":"Turhan","sequence":"additional","affiliation":[{"name":"Department of Division of Elementary Mathematics Education, S\u00fcleyman Demirel University, Isparta 32200, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1800-5718","authenticated-orcid":false,"given":"G\u00f6zde \u00d6zkan","family":"T\u00fckel","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Isparta University of Applied Sciences, Isparta 32200, T\u00fcrkiye"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5065","DOI":"10.1088\/0264-9381\/18\/23\/304","article-title":"Frenet\u2013Serret dynamics","volume":"18","author":"Arreaga","year":"2001","journal-title":"Class. Quantum Gravity"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"6571","DOI":"10.1088\/0305-4470\/35\/31\/304","article-title":"Hamiltonians for curves","volume":"35","author":"Capovilla","year":"2002","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2277","DOI":"10.1088\/0264-9381\/19\/8\/315","article-title":"Hamiltonian Frenet\u2013Serret dynamics","volume":"19","author":"Capovilla","year":"2002","journal-title":"Class. Quantum Gravity"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1666","DOI":"10.1016\/j.geomphys.2005.09.004","article-title":"Particles with curvature and torsion in three-dimensional pseudo-Riemannian space forms","volume":"56","author":"Guerrero","year":"2006","journal-title":"J. Geom. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2124","DOI":"10.1016\/j.geomphys.2007.05.006","article-title":"Relativistic particles and the geometry of 4-D null curves","volume":"57","author":"Lucas","year":"2007","journal-title":"J. Geom. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0550-3213(93)90290-6","article-title":"The model of the relativistic particle with curvature and torsion","volume":"389","author":"Kuznetsov","year":"1993","journal-title":"Nucl. Phys. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/0550-3213(91)90555-C","article-title":"The model of the relativistic particle with torsion","volume":"362","author":"Plyushchay","year":"1991","journal-title":"Nucl. Phys. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2931","DOI":"10.3906\/mat-1906-1","article-title":"A variational study on a natural Hamiltonian for curves","volume":"43","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF02783425","article-title":"Closed free hyperelastic curves in the hyperbolic plane and Chen-Willmore rotational hypersurfaces","volume":"138","author":"Arroyo","year":"2003","journal-title":"Isr. J. Math."},{"key":"ref_10","first-page":"43","article-title":"Elastic Curves in the Galilean plane","volume":"20","author":"Bilir","year":"2021","journal-title":"G\u0130DB J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1063\/1.2918095","article-title":"Lectures on elastic curves and rods","volume":"1002","author":"Singer","year":"2008","journal-title":"AIP Conf. Proc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2623","DOI":"10.1007\/s40840-018-0622-0","article-title":"Elastic strips with spacelike directrix","volume":"42","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_13","first-page":"711","article-title":"Kirchhoff Elastic Rods in Minkowski 3-space","volume":"17","author":"Soylu","year":"2017","journal-title":"J. Sci. Arts"},{"key":"ref_14","first-page":"419","article-title":"Elastica in Galilean 3-Space","volume":"8","author":"Turhan","year":"2020","journal-title":"Konuralp J. Math."},{"key":"ref_15","unstructured":"Turhan, T., and T\u00fckel, G.\u00d6. (2023, January 20\u201321). A Natural Hamiltonian in Galilean 3-Space. Proceedings of the 4th International Conference on Engineering and Applied Natural Sciences, Konya, T\u00fcrkiye."},{"key":"ref_16","unstructured":"Sa\u011fl\u0131ker, H. (2023). On the Geometry of Natural Hamiltonians for Curves in Pseudo Galilean 3-Space. Academic Research and Reviews in Science and Mathematics, Platanus Publishing."},{"key":"ref_17","unstructured":"Y\u00fccesan, A. (2024). Hyperelastic Curves in Galilean 3-Space. Geometry, Algorithms and Variations: Modern Mathematical Theories, BZTTuran Publishing House."},{"key":"ref_18","first-page":"44","article-title":"Motion of Galilean particles with curvature and torsion","volume":"1","author":"Turhan","year":"2025","journal-title":"Calculation"},{"key":"ref_19","unstructured":"Cartan, \u00c9., Glazebrook, J., and Hermann, R. (1983). Geometry of Riemannian Spaces, Math Sci Press."},{"key":"ref_20","unstructured":"Cartan, \u00c9. (1986). On Manifolds with an Affine Connection and the Theory of General Relativity, Bibliopolis. English Translation."},{"key":"ref_21","unstructured":"Schouten, J.A. (1951). Tensor Analysis for Physicists, Clarendon Press."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4757-2063-1"},{"key":"ref_23","first-page":"119","article-title":"Curves in pseudo-Galilean geometry","volume":"41","author":"Divjak","year":"1998","journal-title":"Ann. Univ. Sci. Budapest"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"39","DOI":"10.5539\/jmr.v6n3p39","article-title":"On generalization of helices in the Galilean and the pseudo-Galilean space","volume":"6","author":"Erjavec","year":"2014","journal-title":"J. Math. Res."},{"key":"ref_25","first-page":"199","article-title":"Inelastic Admissible Curves in the Pseudo-Galilean Space G31","volume":"4","year":"2011","journal-title":"Int. J. Open Probl. Compt. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"150685","DOI":"10.1155\/2015\/150685","article-title":"Motions of curves in the pseudo-Galilean space G31","volume":"2015","author":"Cengiz","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/j.joems.2015.09.001","article-title":"Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space","volume":"24","author":"Saad","year":"2016","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"102","DOI":"10.28924\/2291-8639-21-2023-102","article-title":"Geometry of Admissible Curves of Constant-Ratio in Pseudo-Galilean Space","volume":"21","author":"Saad","year":"2023","journal-title":"Int. J. Anal. Appl."},{"key":"ref_29","first-page":"2079","article-title":"Position vectors of curves with recpect to Darboux frame in the Galilean space G3","volume":"68","year":"2017","journal-title":"Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat."},{"key":"ref_30","unstructured":"Kele\u015f, Y. (2014). Galilean And Pseudo-Galilean Space Curves. [Master\u2019s Thesis, Karadeniz Technical University the Graduate School of Natural and Applied Sciences]."},{"key":"ref_31","unstructured":"Turhan, T., and T\u00fckel, G.\u00d6. (2021, January 1\u20132). Elastic Curves in pseudo-Galilean 3-space. Proceedings of the International Asian Congress on Contemporary Sciences\u2014V, Nakhchivan, Azerbaijan."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/520\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:05:45Z","timestamp":1760033145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/520"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,7]]},"references-count":31,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070520"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070520","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,7]]}}}