notes on “application of the hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms”

 

very recently, yildirim et al.applied the so-called hamiltonian approach (ha) to obtain analytical approximatesolutions for three well-known nonlinear oscillators. the authors mentioned that this approach is a kind of energy methodwith a vast application in conservative oscillatory systems, and they applied the approach to nonlinear oscillators withrational and irrational elastic terms. they also pointed out that a comparison of the approximate solutions and the exactones proves that the ha is quite accurate in nonlinear analysis of dynamical systems. their results are based on a newmethod that has been developed which can be applied to conservative nonlinear oscillators with odd elastic terms. inthis paper, we will demonstrate that, when the trial function u(t) = a cos ωt is used in the ha, the results obtained are thesame as those one can obtain using the known first-order harmonic balance method (hbm), and that the ha can be derivedfrom the equations obtained when the first-order hbm is considered. therefore, the application of the ha in could beconsidered as a corollary of the first-order hbm, and all the results obtained are the same as those obtained by applying thehbm. finally, we include additional comments about the analytical approximate expressions for the frequency given in as well as a general expression for this frequency for an extensive set of conservative nonlinear oscillators.