A transformation is employed to obtain expressions for the decay of the displacement, the velocity, and the energy for various forms of nonlinear oscillators. The equation of motion of the nonlinear oscillator is transformed into a first-order decay term plus an energy term, where this transformed equation can be decoupled into a set of two analytically solvable equations. The decoupled equations can be solved for the decay formulas. Unlike other methods in the literature, this transformation method is directly applied to the equation of motion, and an approximate solution is not required to be known a priori. The method is first applied to a purely nonlinear oscillator with a non-negative, real-power restoring force to obtain the decay formulas. These decay formulas are found to behave similarly to those of a linear oscillator. In addition, these formulas are employed to obtain an accurate formula for the frequency decay. Based on this result, the exact frequency formula given in the literature for this oscillator is generalized by substituting the initial values of the envelopes for the actual initial conditions. By this modification, the formulas for the initial and time-varying frequencies become valid for any combination of the initial displacement and velocity. Furthermore, a generalized nonlinear oscillator for which the transformation is always valid is introduced. From this generalized oscillator, the proposed transformation is applied to analyze various types of oscillators.
Quantum equation of motion for a particle in the field of primordial fluctuations
