Truly nonlinear oscillator with position-dependent mass

In this paper the truly nonlinear oscillator (TNO) with position dependent mass (PDM) is considered. The TNO has no linear term, and the degree of nonlinearity is any integer or non-integer (fractional) power. Based on the Hamiltonian for TNO the Lagrange differential equation of motion is developed. The obtained mathematical model is a strong nonlinear Liénard equation which has the first integral of energy type. Analyzing the first integral it is obtained that the motion of the system is periodic and with the constant amplitude. In the paper a new procedure for determination of the frequency of vibration is introduced. The method is based on the He’s frequency formalism and on the exact solution of the TNO with constant mass. The significance of the obtained analytical solution lies in the fact that it provides an explicit relationship between the frequency, the oscillation amplitude, the TNO and PDM parameters, offering the possibility of frequency control. Conditions for low frequency vibrations are determined. The theoretical consideration is applied for vibration analyzes of a diatomic molecule with PDM function of exponential type. The obtained results are applicable in refining spectroscopy analysis and also in molecular and structural physics. In addition, due to analogy between mechanical and quantum oscillators this research provides guidance for further development in semi-conductors and quantum mechanics.