{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,15]],"date-time":"2026-02-15T21:00:27Z","timestamp":1771189227895,"version":"3.50.1"},"reference-count":105,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,26]],"date-time":"2022-02-26T00:00:00Z","timestamp":1645833600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we will trace the development of the use of symmetry in discussing the theory of motion initiated by Emmy Noether in 1918. Though it started with its use in classical mechanics, and has been heavily used in engineering applications of mechanics, it came into its own in relativity, and quantum theory and their applications in particle physics and field theory. It will be beyond the scope of this article to explain the quantum field theory applications in any detail, but the base for understanding it will be provided here. We will also go on to discuss an insight from some more mathematical developments related to Noether symmetry.<\/jats:p>","DOI":"10.3390\/sym14030476","type":"journal-article","created":{"date-parts":[[2022,2,27]],"date-time":"2022-02-27T20:49:03Z","timestamp":1645994943000},"page":"476","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Physical Significance of Noether Symmetries"],"prefix":"10.3390","volume":"14","author":[{"given":"Asghar","family":"Qadir","sequence":"first","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences (ASSMS), Government College University, 68-B New Muslim Town, Lahore 54600, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3431-2574","authenticated-orcid":false,"given":"Ugur","family":"Camci","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Physics, Roger Williams University, One Old Ferry Road, Bristol, RI 02809, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,26]]},"reference":[{"key":"ref_1","unstructured":"Copernicus, N. (2004). Three Copernican Treatises, Dover Publications."},{"key":"ref_2","unstructured":"Capar, M. (1982). Harmonices Mundi (Harmony of the Worlds), 1619. Johannes Kepler Gesammelte Werke, C. H. Beck. [2nd ed.]."},{"key":"ref_3","unstructured":"Newton, I. (1947). Philosophi Naturalis Principia Mathematica. Sir Isaac Newton\u2019s Mathematical Principles of Natural Philosophy and the System of the World, University of California Press."},{"key":"ref_4","unstructured":"Lagrange, J.L. (1811). Mcanique Analytique, Courcier. [2nd ed.]. Reissued by Cambridge University Press: Cambridge, MA, USA, 2009."},{"key":"ref_5","first-page":"247","article-title":"On a general method in dynamics by which the Study of the motions of all free Systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function","volume":"125","author":"Hamilton","year":"1834","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_6","first-page":"95","article-title":"Second essay on a general method in dynamics","volume":"125","author":"Hamilton","year":"1835","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Euler, L. (2022, January 24). MethoduS Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti. Euler Archive-All Works, 1744. The Latin Title Translates as a Method for Finding Curved Lines Enjoying Properties of Maximum or Minimum, or Solution of Isoperimetric Problems in the Broadest Accepted Sense. Available online: https:\/\/scholarlycommons.pacific.edu\/euler-works\/65.","DOI":"10.5479\/sil.318525.39088000877480"},{"key":"ref_8","first-page":"138","article-title":"Suite des rflexions sur la r\u00e9solution alg\u00e9brique des \u00e9quations, Section troisieme, De la r\u00e9solution des \u00e9quations du cinquieme degr\u00e9 & des degr\u00e9s ult\u00e9rieurs","volume":"202\u2013203","author":"Lagrange","year":"1771","journal-title":"Nouv. Mem. Acad. R. Sci. Berlin"},{"key":"ref_9","unstructured":"Sylow, L., and Lie, S. (1881). M\u00e9moire sur les \u00c9quations Alg\u00e9briques, o\u00f9 on D\u00e9montre l\u2019Impossibilit\u00e9 de la Resolution de l\u2019\u00c9quation G\u00e9n\u00e9rale du Cinqui\u00e9me d\u00e9gr\u00e9. Cuvres Completes de Niels Henrik Abel (in French), Groendahl & Soen. [2nd ed.]. Christiania, De l\u2019imprimerie de Groendahl, 1824."},{"key":"ref_10","first-page":"171","article-title":"Analyse d\u2019un m\u00e9moire sur la r\u00e9solution alg\u00e9brique des \u00e9quations","volume":"13","author":"Galois","year":"1830","journal-title":"Bull. Sci. Math."},{"key":"ref_11","first-page":"187","article-title":"Klassification und Integration von gew\u00f6nlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestaten I, II, III","volume":"8","author":"Lie","year":"1883","journal-title":"Arch. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Qadir, A. (1989). Relativity: An Introduction to the Special Theory, World Scientific.","DOI":"10.1142\/9789814368520"},{"key":"ref_13","unstructured":"Barbour, J.B. (2001). The Discovery of Dynamcis, Oxford University Press."},{"key":"ref_14","unstructured":"Lie, S. (1967). Differential Equations, Chelsea Publ. Co."},{"key":"ref_15","unstructured":"Lie, S. (1891). Lectures on Differential Equations with Known Infinitesimal Transformations, Teubner."},{"key":"ref_16","unstructured":"Lie, S. (1893). Theorie der Transformationsgruppen, III, Teubner."},{"key":"ref_17","unstructured":"Bernoulli, D. (1738). Hydrodynamica, sive de viribus et motibus fluidorum commentarii. Latin, Source ETH-Bibliothek Z\u00fcrich, Rar 5503, Johannis Reinholdi Dulseckeri."},{"key":"ref_18","first-page":"58","article-title":"Masses of the Elementary Molecules of Bodies","volume":"73","author":"Avogadro","year":"1811","journal-title":"J. Phys."},{"key":"ref_19","unstructured":"Richard, E., and John, E. (1855). Experimental Researches in Electricity, Richard Taylor and William Francis. Volume 3."},{"key":"ref_20","first-page":"459","article-title":"A dynamical theory of the electromagnetic field","volume":"155","author":"Maxwell","year":"1865","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_21","unstructured":"Qadir, A. (2020). Einstein\u2019s General Theory of Relativity, Cambridge Scholars Pulishing."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1002\/andp.19053221004","article-title":"Zur elektrodynamik bewegter k\u00f6rper","volume":"17","author":"Einstein","year":"1905","journal-title":"Ann. Phys."},{"key":"ref_23","first-page":"225","article-title":"Entwurf einer verallgemeinerten relativit\u00e4tstheorie und eine theorie der gravitation: I. Physikalischer teil von A. Einstein; II. Mathematischer teil von M. Grossmann","volume":"62","author":"Einstein","year":"1913","journal-title":"Z. Math. Phys."},{"key":"ref_24","first-page":"215","article-title":"Kovarianzeigenschaften der feldgleichungen der auf die verallgemeinerte relativit\u00e4tstheorie gegrndeten gravitationstheorie","volume":"63","author":"Einstein","year":"1914","journal-title":"Z. Math. Phys."},{"key":"ref_25","first-page":"844","article-title":"Zur allgemeinen relativit\u00e4tstheorie","volume":"2","author":"Einstein","year":"1915","journal-title":"Preuss. Akad. Wissen. Sitz."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1002\/andp.19163540702","article-title":"Grundlage der allgemeinen relativit\u00e4tstheorie","volume":"49","author":"Einstein","year":"1916","journal-title":"Ann. Phys."},{"key":"ref_27","first-page":"235","article-title":"Invariante Variationsprobleme","volume":"2","author":"Noether","year":"1918","journal-title":"G\u00f6ttingen Math. Phys. Kl."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Olver, P. (1986). Applications of Lie Groups to Differential Equations, Springer.","DOI":"10.1007\/978-1-4684-0274-2"},{"key":"ref_29","unstructured":"Bluman, G., and Anco, S.C. (2002). Symmetries and Integration Methods for Differential Equations, Springer."},{"key":"ref_30","unstructured":"Ibragimov, N.H. (1994). CRC Handbook of Lie Group Analysis of Differential Equations: Symmetries, Exact Solutions and Conservation Laws, CRC Press."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bluman, G., Cheviakov, A., and Anco, S.C. (2010). Applications of Symmetry Methods to Partial Differential Equations, Springer.","DOI":"10.1007\/978-0-387-68028-6"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1023\/A:1008240112483","article-title":"Lie\u2013B\u00e4cklund and Noether Symmetries with Applications","volume":"15","author":"Ibragimov","year":"1998","journal-title":"Nonlinear Dyn."},{"key":"ref_33","first-page":"26","article-title":"The Noether identity","volume":"Volume 38","author":"Ibragimov","year":"1979","journal-title":"Continuum Dynamics"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02742992","article-title":"N\u00f6ther symmetries in cosmology","volume":"19","author":"Capozziello","year":"1996","journal-title":"Riv. Nuovo C."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1088\/0264-9381\/24\/8\/013","article-title":"Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach","volume":"24","author":"Capozziello","year":"2007","journal-title":"Class. Quantum Grav."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/2296111","article-title":"A Model of General Economic Equilibrium","volume":"13","author":"Neumann","year":"1945","journal-title":"Rev. Econ. Stud."},{"key":"ref_37","first-page":"213","article-title":"Economic implications of Von Neumann\u2019s general equilibrium model","volume":"XIII","author":"Qadir","year":"1975","journal-title":"Pak. Ec. Soc. Rev."},{"key":"ref_38","first-page":"85","article-title":"Inflation in a growing economy","volume":"XIX","author":"Qadir","year":"1981","journal-title":"Pak. Ec. Soc. Rev."},{"key":"ref_39","unstructured":"Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Gravitation, W.H. Freeman."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s11071-006-0729-y","article-title":"The connection between isometries and symmetries of geodesic equations of underlying spaces","volume":"45","author":"Feroze","year":"2006","journal-title":"Nonlinear Dyn."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1142\/S021773231003241X","article-title":"New conserved quantities for the spaces of different curvatures","volume":"25","author":"Feroze","year":"2010","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1016\/j.geomphys.2010.11.015","article-title":"Noether symmetries and conserved quantities for spaces with a section of zero curvature","volume":"61","author":"Feroze","year":"2011","journal-title":"J. Geom. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3329","DOI":"10.1007\/s10773-013-1630-3","article-title":"Classification of Plane Symmetric Static Space-Times According to Their Noether Symmetries","volume":"52","author":"Ali","year":"2013","journal-title":"Int. J. Theor. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1007\/s11232-015-0310-2","article-title":"Complete classification of spherically symmetric static spacetimes via Noether symmetries","volume":"184","author":"Ali","year":"2015","journal-title":"Theor. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Ali, F., and Feroze, T. (2016). Complete classification of cylindrically symmetric static spacetimes and the corresponding conservation laws. Sigma Math., 4.","DOI":"10.3390\/math4030050"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1741006","DOI":"10.1142\/S0218271817410061","article-title":"Conservation laws corresponding to the Noether symmetries of the geodetic Lagrangian in spherically symmetric spacetimes","volume":"26","author":"Jamil","year":"2017","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4384093","DOI":"10.1155\/2017\/4384093","article-title":"Geometrical\/Physical interpretation of the conserved quantities corresponding to Noether symmetries of plane symmetric spacetimes","volume":"2017","author":"Jamil","year":"2017","journal-title":"Adv. Math. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2957","DOI":"10.1007\/s10714-010-1054-9","article-title":"Lie and Noether symmetries of geodesic equations and collineations","volume":"42","author":"Tsamparlis","year":"2010","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_49","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":"Tsamparlis","year":"2011","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1550033","DOI":"10.1142\/S0219887815500334","article-title":"Symmetry analysis of the Klein-Gordon equation in Bianchi I spacetimes","volume":"12","author":"Paliathanasis","year":"2015","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjc\/s10052-016-3903-5","article-title":"A study of positive energy condition in Bianchi V spacetimes via Noether symmetries","volume":"76","author":"Ali","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s10714-017-2228-5","article-title":"Noether symmetries of Bianchi type II spacetimes","volume":"49","author":"Hickman","year":"2017","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"002","DOI":"10.1088\/1475-7516\/2014\/07\/002","article-title":"Symmetries of geodesic motion in G\u00f6del-type spacetimes","volume":"2014","author":"Camci","year":"2014","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1560072","DOI":"10.1142\/S2010194515600721","article-title":"Noether gauge symmetries of geodesic motion in stationary and nonstatic G\u00f6del-type spacetimes","volume":"38","author":"Camci","year":"2015","journal-title":"Int. J. Mod. Phys. Conf. Ser."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1550120","DOI":"10.1142\/S0219887815501200","article-title":"Noether gauge symmetry classes for pp-wave spacetimes","volume":"12","author":"Camci","year":"2015","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/BF02741372","article-title":"The classification of spherically symmetric spacetimes","volume":"110","author":"Qadir","year":"1995","journal-title":"Nuovo C. B"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Stephani, H. (1989). Differential Equations: Their Solution Using Symmetries, Cambridge University Press.","DOI":"10.1017\/CBO9780511599941"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/0375-9601(85)90262-2","article-title":"Carter\u2019s fourth invariant of the motion","volume":"109","author":"Qadir","year":"1985","journal-title":"Phys. Lett. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1142\/S0218271807009280","article-title":"Similarities between classical timelike geodesics in a naked Reissner\u2013Nordstrom singularity background and the behaviour of electrons in quantum theory","volume":"15","author":"Qadir","year":"2007","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_60","unstructured":"Landau, L.D., and Lifshitz, E.M. (1975). The Classical Theory of Fields, Elsevier. [4th ed.]."},{"key":"ref_61","unstructured":"Jackson, J.D. (1998). Classical Electrodynamics, John Wiley and Sons. [3rd ed.]."},{"key":"ref_62","unstructured":"Sakurai, J.J., and Neapolitino, J. (2021). Modern Quantum Mechanics, Cambridge University Press. [3rd ed.]."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1103\/RevModPhys.52.525","article-title":"Gauge unification of fundamental forces","volume":"52","author":"Salam","year":"1980","journal-title":"Rev. Mod. Phys."},{"key":"ref_64","unstructured":"Gell-Mann, M., and Ne\u2019eman, Y. (1964). The eightfold way. The Eightfold Way, W.A. Benjamin Inc."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1103\/PhysRevLett.13.508","article-title":"Broken symmetries and the masses of gauge bosons","volume":"13","author":"Higgs","year":"1964","journal-title":"Phys. Rev. Lett."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1103\/PhysRevD.10.275","article-title":"Lepton number as the fourth \u201ccolor\u201d","volume":"10","author":"Pati","year":"1974","journal-title":"Phys. Rev. D"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1103\/PhysRevLett.32.438","article-title":"Unity of all elementary particle forces","volume":"8","author":"Georgi","year":"1974","journal-title":"Phys. Rev. Lett."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/0003-4916(75)90211-0","article-title":"Unified interactions of leptons and hadrons","volume":"93","author":"Fritzsch","year":"1975","journal-title":"Ann. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/0550-3213(74)90355-1","article-title":"Supergauge transformations in four dimensions","volume":"70","author":"Wess","year":"1974","journal-title":"Nucl. Phys. B"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/0370-2693(74)90226-3","article-title":"Super-symmetry and non-Abelian gauges","volume":"51","author":"Salam","year":"1974","journal-title":"Phys. Lett. B"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/0370-1573(81)90157-5","article-title":"Supergravity","volume":"68","year":"1981","journal-title":"Phys. Rep."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0550-3213(95)00267-V","article-title":"Superstrings and supermembranes in the doubly supersymmetric geometrical approach","volume":"446","author":"Bandos","year":"1995","journal-title":"Nucl. Phys. B"},{"key":"ref_73","unstructured":"Bastin, T. (1971). Angular momentum: An approach to combinatorial spacetime. Quantum Theory and Beyond, Cambridge University Press."},{"key":"ref_74","unstructured":"Penrose, R. (1967). An Analysis of the Structure of Space-Time, Cambridge University. Adams Prize Essay."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1063\/1.1705200","article-title":"Twistor algebra","volume":"8","author":"Penrose","year":"1967","journal-title":"J. Math. Phys."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1063\/1.1664756","article-title":"Solution of sero rest-mass field equations","volume":"10","author":"Penrose","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_77","unstructured":"Penrose, R., and Rindler, W. (1988). Spinors and Space-Time: Vol. 2, Spinor and Twistor Methods in Space-Time Geometry, Cambridge University Press. [1st ed.]."},{"key":"ref_78","unstructured":"Qadir, A. (1971). Twistor Fields. [Ph.D. Thesis, London University]."},{"key":"ref_79","first-page":"74","article-title":"An interesting representation of Lie algebras of linear groups","volume":"14","author":"Qadir","year":"1976","journal-title":"Int. J. Theor. Phys."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/0370-1573(73)90008-2","article-title":"Twistor theory: An approach to the quantisation of fields and space-time","volume":"6","author":"Penrose","year":"1973","journal-title":"Phys. Rep."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/0370-1573(78)90185-0","article-title":"Penrose graphs","volume":"39","author":"Qadir","year":"1978","journal-title":"Phys. Rep."},{"key":"ref_82","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_83","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02418270","article-title":"Sur les Invariants Diff\u00e9rentiels des Groupes Continus de Transformations","volume":"18","author":"Tresse","year":"1894","journal-title":"Acta Math."},{"key":"ref_84","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_85","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1142\/S1402925109000236","article-title":"Invariant linearization criteria for systems of cubically semi-linear second order ordinary differential equations","volume":"16","author":"Mahomed","year":"2009","journal-title":"J. Nonlin. Math. Phys."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1016\/j.cpc.2008.04.001","article-title":"Constructing a space from the system of geodesic equations","volume":"179","author":"Fredericks","year":"2008","journal-title":"Comp. Phys. Commun."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"3335","DOI":"10.1016\/j.nonrwa.2008.07.011","article-title":"Complex Lie symmetries for scalar second-order ordinary differential equations","volume":"10","author":"Ali","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_88","unstructured":"Ali, S. (2009). Complex Lie Symmetries for Differential Equations. [Ph.D. Thesis, National University of Sciences and Technology]."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s11071-010-9912-2","article-title":"Linearizability criteria for systems of two second order ordinary differential equations by complex methods","volume":"66","author":"Ali","year":"2011","journal-title":"Nonlinear Dyn."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"171834","DOI":"10.1155\/2011\/171834","article-title":"Linearizability of systems of ordinary differential equations obtained by complex symmetry analysis","volume":"2011","author":"Safdar","year":"2011","journal-title":"Math. Probl. Eng."},{"key":"ref_91","first-page":"923","article-title":"Inequivalence of classes of linearizable systems of cubically semi-linear ordinary differential equations obtained by real and complex symmetry analysis","volume":"16","author":"Ali","year":"2011","journal-title":"Math. Comput. Appl."},{"key":"ref_92","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_93","doi-asserted-by":"crossref","first-page":"25","DOI":"10.2991\/jnmp.2008.15.s1.2","article-title":"Complex Lie symmetries for variational problems","volume":"15","author":"Ali","year":"2008","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1016\/j.cnsns.2010.08.007","article-title":"Invariants of two-dimensional systems via complex Lagrangians with applications","volume":"16","author":"Farooq","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_95","doi-asserted-by":"crossref","unstructured":"Safdar, M., Qadir, A., and Farooq, M.U. (2019). Comparison of Noether symmetries and first integrals of two-dimensional systems of second order ordinary differential equations by real and complex methods. Symmetry, 11.","DOI":"10.3390\/sym11091180"},{"key":"ref_96","doi-asserted-by":"crossref","unstructured":"Hall, G.S. (2004). Symmetries and Curvature Structure in General Relativity, World Scientific.","DOI":"10.1142\/1729"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"1707","DOI":"10.1063\/1.528668","article-title":"Ricci collineation vectors in fluid space-times","volume":"31","author":"Tsamparlis","year":"1990","journal-title":"J. Math. Phys."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"3543","DOI":"10.1063\/1.530043","article-title":"Collineations of the Ricci tensor","volume":"34","author":"Bokhari","year":"1993","journal-title":"J. Math. Phys."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/BF02106969","article-title":"Ricci and matter collineations in space-time","volume":"28","author":"Hall","year":"1996","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"3639","DOI":"10.1063\/1.532058","article-title":"Classification of spherically symmetric static space-times by their curvature collineations","volume":"38","author":"Bokhari","year":"1997","journal-title":"J. Math. Phys."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1088\/0264-9381\/19\/2\/312","article-title":"Ricci collineations in Friedmann-Robertson-Walker spacetimes","volume":"19","author":"Camci","year":"2002","journal-title":"Class. Quantum Grav."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1023\/A:1024068901739","article-title":"A Complete Classification of Curvature Collineations of Cylindrically Symmetric Static Metrics","volume":"35","author":"Bokhari","year":"2003","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1142\/S021827180500695X","article-title":"Weyl Collineations that are not Curvature Collineations","volume":"14","author":"Hussain","year":"2005","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1063\/1.527594","article-title":"Symmetries of static, spherically symmetric space-times","volume":"28","author":"Bokhari","year":"1987","journal-title":"J. Math. Phys."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"2547","DOI":"10.1063\/1.531994","article-title":"Homothetic motions of spherically symmetric space-times","volume":"38","author":"Ahmad","year":"1997","journal-title":"J. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/476\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:28:04Z","timestamp":1760135284000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/476"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,26]]},"references-count":105,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["sym14030476"],"URL":"https:\/\/doi.org\/10.3390\/sym14030476","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,26]]}}}