Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

In this paper, an analytical approximation of damped oscillations of some strongly non-linear, planar Hamiltonian systems is considered. To apply the Krylov–Bogoliubov–Mitropolsky method in this strongly non-linear case, we mainly provide the formal and exact solutions of the homogeneous part of the variational equations with periodic coefficients resulting from the Hamiltonian systems. It is shown that these are simply expressed in terms of the partial derivatives of the solutions, written in action-angle variables, of the Hamiltonian systems. Two examples, including a non-linear harmonic oscillator and the Morse oscillator, are presented to illustrate this extension of the method. The approximate first order solution obtained in each case is observed to be quite satisfactory.