A short remark on the nonlinear oscillator with a damping term

A nonlinear oscillator with a damping term can model many nonlinear vibration problems. This short remark insights into its physical understanding by the variational principle, which is established by the semi-inverse method. The dissipative energy involved in the variational formulation can be explained by the two-scale thermodynamics. Taylor series method is used to solve its frequency-amplitude relation