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Approximate partial Noether operators and first integrals for coupled nonlinear oscillators

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Abstract

We study the construction of approximate first integrals of approximate partial Euler–Lagrange equations via approximate partial Noether operators corresponding to a partial Lagrangian. An approximate Noether-like theorem which gives the approximate first integrals for the perturbed equations without regard to a standard Lagrangian is deduced. These approximate partial Noether operators, in general, do not form an approximate Lie algebra. The results are applied to a system of two coupled nonlinear oscillators. The approximate first integrals are obtained for both the resonant and nonresonant cases with the help of approximate partial Noether operators associated with a partial Lagrangian. This approach can give rise to further studies in the construction of approximate first integrals for perturbed equations without a variational principle.

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Authors and Affiliations

  1. School of Computational and Applied Mathematics, Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits, 2050, South Africa

    I. Naeem & F. M. Mahomed

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  1. I. Naeem
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  2. F. M. Mahomed
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Correspondence to I. Naeem.

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Naeem, I., Mahomed, F.M. Approximate partial Noether operators and first integrals for coupled nonlinear oscillators. Nonlinear Dyn 57, 303–311 (2009). https://doi.org/10.1007/s11071-008-9441-4

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  • DOI: https://doi.org/10.1007/s11071-008-9441-4

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