Abstract
A family of perturbative Lagrangians that describe approximate and multidimensional Klein-Gordon equations are studied. We probe the existence of approximate Noether symmetries via generalized geometric conditions for a perturbation of any order. The knowledge of the geometric conditions uncovers that unlike exact symmetries, the approximate Noether symmetries of the Lagrangian which describes the motion of a particle in n-dimensional space under the action of an autonomous force, is inequivalent to the Noether symmetries admitted by the Klein-Gordon Lagrangian in general.
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Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Approximate symmetries of equations with a small parameter, Matematicheskiǐ. Sbornik 136, 435–450 (1988). (English Transl. in Math. USSR Sb. 64(1989) 427)
Fushchych, W.I., Shtelen, W.H.: On approximate symmetry and approximate solution of the non-linear wave equation with a small parameter. J. Phys. A: Math. Gen. 22, 887 (1989)
Pakdemirli, M., Yürüsoy, M., Dolapçi, T.: Comparison of approximate symmetry methods for differential equations. Acta Appl. Math. 80, 243–271 (2004)
Govinder, K.S., Heil, T.G., Uzer, T.: Approximate noether symmetries. Phys. Lett. A 240, 127–131 (1998)
Euler, N., Shulga, M.W., Steeb, W.-H.: Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation. J. Phys. A: Math. Gen. 25, L1095–L1103 (1992)
Kara, A.H., Mahomed, F.M., Qadir, A.: Approximate symmetries and conservation laws of the geodesic equations for the Schwarzschild metric. Nonlinear Dyn. 51, 183–188 (2008)
Hénon, M., Heiles, C.: The applicability of the third integral of motion, some numerical experiments. Astrophysics J. 69, 73–79 (1964)
Nucci, M.C.: Interactive REDUCE programs for calculating Lie point, non-classical, Lie-Bäcklund, and approximate symmetries of differential equations: Manual and floppy disk. In: Ibragimov, N.H. (ed.) CRC Handbook of Lie Group Analysis of Differential Equations, New Trends, III, p 415. CRC Press, Boca Raton (1996)
Jefferson, G.F., Carminati, J.: ASP: Automated Symbolic computation of approximate symmetries of differential equations. Comp. Phys. Comm. 184, 1045–1063 (2013)
Bozhkov, Y., Freire, I.L.: Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation. J. Diff. Eq. 249, 872 (2010)
Jamal, S.: A group theoretical application of SO(4,1) in the de Sitter universe. Gen. Relativ. Grav. 49, 1–14 (2017)
Jamal, S., Paliathanasis, A.: Group invariant transformations for the Klein-Gordon equation in three dimensional flat spaces. J. Geom. Phys. 117, 50–59 (2017)
Jamal, S., Shabbir, G.: Geometric properties of the Kantowski-Sachs and Bianchi-type Killing algebra in relation to a Klein-Gordon equation. Eur. Phys. J. Plus 132, 70 (2017)
Tsamparlis, M., Paliathanasis, A.: Lie and Noether symmetries of geodesic equations and collineations. Gen. Relat. Grav. 42, 2957 (2010)
Bluman, G.W.: Simplifying the form of Lie groups admitted by a given differential equation. J. Math. Anal. Appl. 145, 5262 (1990)
Noether, E.: Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-Phys. Klasse 235 (1918)
Hussain, I., Qadir, A.: Approximate Noether symmetries of Lagrangian for plane symmetric gravitational wave-like spacetimes. Nuovo Cim. B122, 593–597 (2007)
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I acknowledge financial support from the National Research Foundation of South Africa (99279).
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Jamal, S. nth-Order Approximate Lagrangians Induced by Perturbative Geometries. Math Phys Anal Geom 21, 25 (2018). https://doi.org/10.1007/s11040-018-9283-3
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DOI: https://doi.org/10.1007/s11040-018-9283-3