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Open AccessArticle

First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach

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Department of Basic Sciences & Humanities, College of E & ME, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan
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International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa
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DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Wits, Johannesburg 2050, South Africa
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(10), 1244; https://doi.org/10.3390/sym11101244
Received: 22 July 2019 / Revised: 5 September 2019 / Accepted: 10 September 2019 / Published: 4 October 2019
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a number of forms of arbitrary functions appearing in the set of equations, the Noether-like operators also fulfill the classical Noether symmetry condition for the pairs of real Lagrangians and the generated first integrals are reminiscent of those we obtain from the complex Lagrangian approach. We also investigate the cases in which the underlying systems are reducible via quadrature. We derive some interesting results about the nonlinear systems under consideration and also find that the algebra of Noether-like operators is Abelian in a few cases.
Keywords: complex Lagrangian approach; coupled Lane-Emden systems; Noether-like operators; first integrals complex Lagrangian approach; coupled Lane-Emden systems; Noether-like operators; first integrals
MDPI and ACS Style

Farooq, M.U.; Khalique, C.M.; Mahomed, F.M. First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach. Symmetry 2019, 11, 1244.

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