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Mathematical Problems in Engineering
Volume 2011, Article ID 831647, 14 pages
Research Article

First Integrals for Two Linearly Coupled Nonlinear Duffing Oscillators

1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
2Department of Mathematics, LUMS, School of Science and Engineering, Lahore Cantt 54792, Pakistan
3Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

Received 6 September 2010; Revised 1 December 2010; Accepted 3 January 2011

Academic Editor: G. Rega

Copyright © 2011 R. Naz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate Noether and partial Noether operators of point type corresponding to a Lagrangian and a partial Lagrangian for a system of two linearly coupled nonlinear Duffing oscillators. Then, the first integrals with respect to Noether and partial Noether operators of point type are obtained explicitly by utilizing Noether and partial Noether theorems for the system under consideration. Moreover, if the partial Euler-Lagrange equations are independent of derivatives, then the partial Noether operators become Noether point symmetry generators for such equations. The difference arises in the gauge terms due to Lagrangians being different for respective approaches. This study points to new ways of constructing first integrals for nonlinear equations without regard to a Lagrangian. We have illustrated it here for nonlinear Duffing oscillators.