Abstract and Applied Analysis

Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Rehana Naz, Igor Leite Freire, and Imran Naeem

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Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 978636, 15 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412605852

Digital Object Identifier
doi:10.1155/2014/978636

Mathematical Reviews number (MathSciNet)
MR3208578

Zentralblatt MATH identifier
07023444

Citation

Naz, Rehana; Freire, Igor Leite; Naeem, Imran. Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 978636, 15 pages. doi:10.1155/2014/978636. https://projecteuclid.org/euclid.aaa/1412605852


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