{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:58:14Z","timestamp":1762325894838,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,16]],"date-time":"2024-05-16T00:00:00Z","timestamp":1715817600000},"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>The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis.<\/jats:p>","DOI":"10.3390\/axioms13050331","type":"journal-article","created":{"date-parts":[[2024,5,16]],"date-time":"2024-05-16T11:26:17Z","timestamp":1715858777000},"page":"331","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9966-5517","authenticated-orcid":false,"given":"Andronikos","family":"Paliathanasis","sequence":"first","affiliation":[{"name":"Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa"},{"name":"Departamento de Matem\u00e1ticas, Universidad Cat\u00f3lica del Norte, Avda. Angamos 0610, Casilla 1280, Antofagasta 1270709, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,16]]},"reference":[{"key":"ref_1","unstructured":"Lie, S. (1970). Theorie der Transformationsgrupprn: Volume I, Chelsea."},{"key":"ref_2","unstructured":"Lie, S. (1970). Theorie der Transformationsgrupprn: Volume II, Chelsea."},{"key":"ref_3","unstructured":"Lie, S. (1970). Theorie der Transformationsgrupprn: Volume III, Chelsea."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1063\/1.526766","article-title":"First integrals for the modified Emden equation q+  \u03b1(t) q+ qn = 0","volume":"26","author":"Leach","year":"1985","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"159","DOI":"10.2991\/jnmp.2008.15.s1.14","article-title":"A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations","volume":"15","author":"Moyo","year":"2008","journal-title":"J. Nonl. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1006\/jmaa.2000.6748","article-title":"Lie symmetry analysis and approximate solutions for non-linear radial oscillations of an incompressible Mooney\u2013Rivlin cylindrical tube","volume":"245","author":"Mason","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"073507","DOI":"10.1063\/1.2747724","article-title":"Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations","volume":"48","author":"Huang","year":"2007","journal-title":"J. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1139\/p2012-065","article-title":"Symmetries, conservation laws, reductions, and exact solutions for the Klein\u2013Gordon equation in de Sitter space\u2013times","volume":"90","author":"Jamal","year":"2012","journal-title":"Can. J. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.aml.2019.02.028","article-title":"Upper bound on the sum of powers of the degrees of graphs with few crossings per edge","volume":"94","author":"Xin","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.ijnonlinmec.2016.08.005","article-title":"Symmetries of the hyperbolic shallow water equations and the Green\u2013Naghdi model in Lagrangian coordinates","volume":"86","author":"Siriwat","year":"2016","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Muatjetjeja, B., and Khalique, C.M. (2014). Benjamin\u2013Bona\u2013Mahony equation with variable coefficients: Conservation laws. Symmetry, 6.","DOI":"10.3390\/sym6041026"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4679","DOI":"10.1002\/mma.5675","article-title":"Weak shock waves and its interaction with characteristic shocks in polyatomic gas","volume":"42","author":"Zeidan","year":"2019","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2267","DOI":"10.1007\/BF00673841","article-title":"Approximate potential symmetries for partial differential equations","volume":"34","author":"Kara","year":"1995","journal-title":"Int. J. Theor. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1007\/s10808-008-0092-5","article-title":"Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow","volume":"49","author":"Chesnokov","year":"2008","journal-title":"J. Appl. Mech. Technol. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1007\/s11232-007-0070-8","article-title":"Quantizing preserving Noether symmetries","volume":"151","author":"Nucci","year":"2007","journal-title":"Theor. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1142\/S1402925111001374","article-title":"Nonisentropic solutions of simple wave type of the gas dynamics equations","volume":"18","author":"Meleshko","year":"2011","journal-title":"J. Nonl. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Paliathanasis, A. (2019). One-dimensional optimal system for 2D rotating ideal gas. Symmetry, 11.","DOI":"10.3390\/sym11091115"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Ovsiannikov, L.V. (1982). Group Analysis of Differential Equations, Academic Press.","DOI":"10.1016\/B978-0-12-531680-4.50012-5"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bluman, G.W., and Kumei, S. (1989). Symmetries and Differential Equations, Springer.","DOI":"10.1007\/978-1-4757-4307-4"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Olver, P.J. (1993). Applications of Lie Groups to Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_21","unstructured":"Ibragimov, N.H. (2000). CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRS Press LLC."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0022-0396(89)90154-X","article-title":"Linearization of second order ordinary differential equations via Cartan\u2019s equivalence method","volume":"77","author":"Grissom","year":"1989","journal-title":"J. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1023\/B:NODY.0000034645.77245.26","article-title":"Geometric proof of Lie\u2019s linearization theorem","volume":"36","author":"Ibragimov","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1016\/j.jmaa.2005.01.025","article-title":"Linearization of third-order ordinary differential equations by point and contact transformations","volume":"308","author":"Ibragimov","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1080\/16073606.1989.9632170","article-title":"The Lie algebra sl (3, R) and linearization","volume":"12","author":"Mahomed","year":"1989","journal-title":"Quaest. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1088\/0305-4470\/20\/2\/014","article-title":"Generalizations of Noether\u2019s theorem in classical mechanics","volume":"20","author":"Sarlet","year":"1987","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1016\/S0020-7462(00)00032-9","article-title":"Linearization criteria for a system of second-order ordinary differential equations","volume":"36","author":"Wafo","year":"2001","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"591","DOI":"10.2307\/1968307","article-title":"Dynamical trajectories and geodesics","volume":"30","author":"Eisenhart","year":"1928","journal-title":"Ann. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.physletb.2016.11.059","article-title":"Eisenhart lift for higher derivative systems","volume":"765","author":"Galajinsky","year":"2017","journal-title":"Phys. Lett. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"104732","DOI":"10.1016\/j.geomphys.2022.104732","article-title":"Eisenhart lift of Koopman-von Neumann mechanics","volume":"185","author":"Sen","year":"2023","journal-title":"J. Geom. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"065004","DOI":"10.1103\/PhysRevD.103.065004","article-title":"Quantizing the Eisenhart lift","volume":"103","author":"Finn","year":"2021","journal-title":"Phys. Rev. D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1140\/epjc\/s10052-019-6812-6","article-title":"Eisenhart lift of 2-dimensional mechanics","volume":"79","author":"Fordy","year":"2019","journal-title":"Eur. Phys. J. C"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Finn, K. (2021). Geometric Approaches to Quantum Field Theory, Springer. Springer Theses.","DOI":"10.1007\/978-3-030-85269-6"},{"key":"ref_34","unstructured":"Yano, K. (1995). Lie Derivatives and Its Applications, North-Holland Publishing Co."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1063\/1.1664886","article-title":"Curvature Collineations: A Fundamental Symmetry Property of the Space-Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor","volume":"10","author":"Katzin","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1088\/0264-9381\/10\/6\/010","article-title":"Projective collineations in Einstein spaces","volume":"10","author":"Barnes","year":"1993","journal-title":"Class. Quantum Grav."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1995","DOI":"10.1002\/mma.934","article-title":"Symmetry group classification of ordinary differential equations: Survey of some results","volume":"30","author":"Mahomed","year":"2007","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s11071-006-9095-z","article-title":"Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations","volume":"48","author":"Mahomed","year":"2007","journal-title":"Nonlinear Dyn."},{"key":"ref_39","first-page":"163","article-title":"Linearization criteria for two-dimensional systems of third-order ordinary differential equations by complex approach","volume":"8","author":"Dutt","year":"2019","journal-title":"Arab. J. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"793247","DOI":"10.1155\/2014\/793247","article-title":"Linearization from complex Lie point transformations","volume":"2014","author":"Ali","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"126475","DOI":"10.1016\/j.jmaa.2022.126475","article-title":"Symmetry gaps for higher order ordinary differential equations","volume":"516","author":"Kessy","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Tsamparlis, M. (2023). Linearization of Second-Order Non-Linear Ordinary Differential Equations: A Geometric Approach. Symmetry, 15.","DOI":"10.3390\/sym15112082"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"399","DOI":"10.2991\/jnmp.2004.11.3.9","article-title":"Sundman symmetries of nonlinear second-order and third-order ordinary differential equations","volume":"11","author":"Euler","year":"2004","journal-title":"Nonlinear Math. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"402001","DOI":"10.1088\/1751-8113\/41\/40\/402001","article-title":"Conditions for linearization of a projectable system of two second-order ordinary differential equations","volume":"41","author":"Sookmee","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/0022-247X(90)90244-A","article-title":"Symmetry Lie algebras of nth order ordinary differential equations","volume":"151","author":"Mahomed","year":"1990","journal-title":"J. Math. Anal. Appl."},{"key":"ref_46","first-page":"235","article-title":"Invariante Variationsprobleme Koniglich Gesellschaft der Wissenschaften Gottingen Nachrichten","volume":"2","author":"Noether","year":"1918","journal-title":"Math.-Phys. Kl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s11071-010-9710-x","article-title":"Lie symmetries of geodesic equations and projective collineations","volume":"62","author":"Tsamparlis","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1861","DOI":"10.1007\/s10714-011-1166-x","article-title":"The geometric nature of Lie and Noether symmetries","volume":"43","author":"Tsampalis","year":"2011","journal-title":"Gen. Rel. Gravit."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1023\/A:1021041802041","article-title":"Projective Transformations of Pseudo-Riemannian Manifolds","volume":"113","author":"Aminova","year":"2003","journal-title":"J. Math. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1007\/BF00559222","article-title":"Conservation laws for a charged particle moving in gravitational and electromagnetic fields. I. Conserved quantities","volume":"37","author":"Aminova","year":"1994","journal-title":"Russ. Phys. J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1070\/SM2006v197n07ABEH003784","article-title":"Projective geometry of systems of second-order differential equations","volume":"197","author":"Aminova","year":"2006","journal-title":"Sb. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1070\/SM2010v201n05ABEH004085","article-title":"The projective geometric theory of systems of second-order differential equations: Straightening and symmetry theorems","volume":"201","author":"Aminova","year":"2010","journal-title":"Sb. Math."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"025018","DOI":"10.1088\/0143-0807\/36\/2\/025018","article-title":"The Eisenhart lift: A didactical introduction of modern geometrical concepts from Hamiltonian dynamics","volume":"36","author":"Cariglia","year":"2015","journal-title":"Eur. J. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"5333","DOI":"10.1088\/0305-4470\/35\/25\/312","article-title":"A note on the construction of the Ermakov\u2013Lewis invariant","volume":"35","author":"Moyo","year":"2002","journal-title":"J. Phys. A Math. Gen."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/331\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:43:39Z","timestamp":1760107419000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/331"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,16]]},"references-count":54,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["axioms13050331"],"URL":"https:\/\/doi.org\/10.3390\/axioms13050331","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,5,16]]}}}