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2014 Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations
Rehana Naz, Igor Leite Freire, Imran Naeem
Abstr. Appl. Anal. 2014(SI37): 1-15 (2014). DOI: 10.1155/2014/978636


Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.


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Rehana Naz. Igor Leite Freire. Imran Naeem. "Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations." Abstr. Appl. Anal. 2014 (SI37) 1 - 15, 2014.


Published: 2014
First available in Project Euclid: 6 October 2014

zbMATH: 07023444
MathSciNet: MR3208578
Digital Object Identifier: 10.1155/2014/978636

Rights: Copyright © 2014 Hindawi

Vol.2014 • No. SI37 • 2014
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