Analysis of nonlinear oscillations of a punctual charge in the electric field of a charged ring via a Hamiltonian approach and the energy balance method

The method of Hamiltonian approach and the energy balance method are applied to obtain the periodic solutions of nonlinear oscillations of a punctual charge in the electric field of charged ring. The obtained approximate frequencies are accurate for the entire range of oscillation amplitudes. A good agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed. It is also proved that the results of the energy balance method are better than the Hamiltonian approach for solving this equation.