First integrals for the damped Helmholtz oscillator

Abstract

Recently, the modified Prelle–Singer method for finding general solutions of second-order nonlinear ordinary differential equations has attracted considerable attention. Many researchers used this method to derive the first integrals of various dynamical systems. In this article, we are concerned with the first integrals of the damped Helmholtz oscillator under certain parametric conditions. Our analysis indicates that there exist some errors on the first integrals of the damped Helmholtz oscillator and the Duffing–van der Pol oscillator in the literature. We clarify the errors, and instead give a refined result in a simple and straightforward manner with much less calculations. Finally, two independent first integrals of the Helmholtz oscillator under different parametric conditions are obtained by the Lie symmetry method, and a class of exact solutions in terms of the hyperbolic and the Jacobian elliptic functions is presented through these two first integrals.

Keywords:

• first integral
• autonomous system
• Lie symmetry
• infinitesimal generator
• elliptic function
• total derivative

2000 AMS Subject Classifications :

• 34C14
• 34A05

Acknowledgements

This manuscript was initially submitted to Proc. R. Soc. A (London) on 26 February 2008, with Manuscript ID: RSPA-2008-0084. Main contents have been presented at the Seventh AIMS International Conference on Dynamical Systems, Differential Equations and Applications, University of Texas, Arlington, 18–21 May 2008. We have noticed that the online version of 7 is different from its printed version (hard copy) (A Notice of Correction was attached after the References of 7). Major changes were made in September 2008, for instance, for the Helmholtz oscillator on page 2464 of the online version of 7. However, the errors pointed in Section 3.3 are not corrected yet, in both online and printed versions. The work is supported by NSF Grant CCF–0514768 and UTPA FRC Grant 119100.