Notes on ‘ Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms ’

In a recent paper [A. Yıldırım, Z. Saadatnia, H. Askari, “Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms”, Mathematical and Computer Modelling 54 (2011) 697-703] the so-called Hamiltonian approach (HA) was applied to obtain analytical approximate solutions for conservative nonlinear oscillators with certain elastic terms. In this paper we demonstrate that the approach proposed is equivalent to the well known harmonic balance method (HBM) and that the equations they obtained using the HA can be easily derived from the well known first-order HBM applied to conservative nonlinear oscillators with odd nonlinear elastic terms. This implies that the approximate frequency and periodic solution obtained using the HA with the trial function proposed in that paper are the same as those obtained using the first-order HBM. We think the comments presented here could be useful for people working in approximate analytical methods for nonlinear oscillators and they would have to be taken into account in the developing and application of some approximate techniques.