Extension of the Hamiltonian approach with general initial conditions

In this paper, the Hamiltonian approach is extended for solving vibrations of nonlinear
conservative oscillators with general initial conditions. Based on the assumption that the
derivative of Hamiltonian is zero, the frequency as a function of the amplitude of vibration
and initial velocity is determined. A method for error estimation is developed and the accuracy
of the approximate solution is treated. The procedure is based on the ratio between the
average residual function and the total energy of the system. Two computational algorithms
are described for determining the frequency and the average relative error. The extended
Hamiltonian approach presented in this paper is applied for two types of examples: Duffing
equation and a pure nonlinear conservative oscillator.