Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms

Very recently, Yildirim et al. [1] applied the so-called Hamiltonian approach (HA) to obtain analytical approximate solutions for three well-known nonlinear oscillators. The authors mentioned that this approach is a kind of energy method with a vast application in conservative oscillatory systems, and they applied the approach to nonlinear oscillators with rational and irrational elastic terms. They also pointed out that a comparison of the approximate solutions and the exact ones proves that the HA is quite accurate in nonlinear analysis of dynamical systems. Their results are based on a new method that has been developed [2] which can be applied to conservative nonlinear oscillators with odd elastic terms. In this paper, we will demonstrate that, when the trial function u(t) = A cosωt is used in the HA, the results obtained are the same as those one can obtain using the known first-order harmonic balance method (HBM), and that the HA can be derived from the equations obtained when the first-order HBM is considered. Therefore, the application of the HA in [1] could be considered as a corollary of the first-order HBM, and all the results obtained are the same as those obtained by applying the HBM. Finally, we include additional comments about the analytical approximate expressions for the frequency given in [1] as well as a general expression for this frequency for an extensive set of conservative nonlinear oscillators.