Construction Of Analytical Approximate Solutions To Relativistic Harmonic Oscillators

The Newton-harmonic balance method is used to construct analytical approximate periods and periodic solutions to the relativistic harmonic oscillator. By introducing new variable and rewriting the control equation in terms of the new variable, we apply the Newton-harmonic balance method to solve the resulted equation. The method yields rapid convergence with respect to exact solution, and the analytical approximations obtained are valid for the whole range of initial oscillation amplitudes. Excellent agreement of the approximate periods and periodic solutions with the exact ones have been demonstrated.In dimensionless form, the relativistic harmonic motion of a rest mass attached to a linear spring on a smooth horizontal plane is governed byIntroducing a new independent variableÏ„=ωt, we can rewrite Eqs. (2) as where ( ' ) denotes differentiation with respect toÏ„and ? =ω4. The new independent variable is chosen such that the solution of Eq. (3) is a periodic function ofÏ„of period 2Ï€. The corresponding period of the nonlinear oscillator is given by .Following the lowest order HB approximation, we set u1 (Ï„)= AcosÏ„. (4)which satisfies the initial conditions in Eq. (3). Substituting Eq. (4) into Eq. (3), expanding the resulting expression in a Fourier series and setting the constant term to zero, yield the first approximate period and periodic solution to Eq. (1)Using u₁ (Ï„) andΩ1 ( A) as initial approximations to the solution of Eq. (3), we combine the Newton's method and the HB method to solve Eq. (3). The first step is the Newton procedure. Let the periodic solution and the quartic of frequency of Eq. (3) can be expressed as whereΩk, u k(Ï„) are the main part of the solution; ?u k, ?? k are the correction part of the solution. Substituting Eq. (6) into Eq. (3) and linearizing with respect to the correction termsΔu kandΔΩk lead to where the correction term ?u k is periodic functions ofÏ„of period 2Ï€, and it can be chosen as which satisfies the initial conditions in Eq. (7). For the second step, we apply harmonic balance method to solve Eq.(7), and obtain the correction termsΔu kandΔΩk. It should be clear how the procedure works for constructing higher order analytical approximate solutions. In fact, only a few iterations are required to provide excellent analytical approximations to period and corresponding periodic solutions for the relativistic nonlinear oscillators.The second and third approximations to period and periodic solution to Eq.(1) are In this thesis, we have constructed new analytical approximate periods and periodic solutions to the relativistic harmonic oscillator by applying the Newton-harmonic balance method to the rewritten governing equation. The procedure yields rapid convergence with respect to exact solution, and the approximations obtained are in good agreement with the exact ones. We can draw a conclusion that the Newton-harmonic balance method has great potential and can flexibly be applied to others strongly nonlinear oscillators with complex elasticity terms.