Nonlinear Dynamics

, Volume 95, Issue 3, pp 1747–1765 | Cite as

First integrals and analytical solutions of some dynamical systems

  • B. U. Haq
  • I. NaeemEmail author
Original Paper


This article investigates the first integrals and closed- form solutions of some nonlinear first-order dynamical systems from diverse areas of applied mathematics. We use the notion of artificial Hamiltonian, and we show that every first-order system of ordinary differential equations (ODEs) can be written in the form of an artificial Hamiltonian system [see Naz and Naeem (ZNA 73(4):323–330, 2018)]. One can also express the second-order ODE or system of second-order ODEs in the form of system of first-order artificial Hamiltonian system. Then the partial Hamiltonian approach is employed to compute the partial Hamiltonian operators and the corresponding first integrals. The first integrals are utilized to construct the closed-form solutions of two-stream model for tuberculosis and dengue fever, Duffing–van der Pol oscillator, nonlinear optical oscillators under parameter restrictions, nonlinear convection model and the two- dimensional galaxy model. We show that how one can apply the existing “partial Hamiltonian approach” for nonstandard Hamiltonian systems. This study provides a new way of solving the dynamical systems of first-order ODEs, second-order ODE and second-order systems of ODEs which are expressed into the artificial Hamiltonian system.


Artificial Hamiltonian Phase space coordinates Gauge terms First integrals Exact solution 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsLahore University of Management SciencesLahore CantonmentPakistan

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