Chapter 7.6, Problem 22E

Interpretation Introduction

Interpretation:

To find the first few terms in the expansion for the solution of ${\ddot{\text{x}}\text{+x+εx}}^{\text{2}}\text{ = 0}$ by using Poincaré-Lindstedt method, with initial conditions $\text{x}\left(\text{0}\right)\text{ = a}$, $\dot{\text{x}}\left(\text{0}\right)\text{ = 0}$ and show that the center of oscillation is at $\text{x}\approx \frac{\text{1}}{\text{2}}{\text{εa}}^{\text{2}}$.

Concept Introduction:

In perturbation theory, when regular perturbation approach fails the technique of uniformly approximating periodic solution to ordinary differential equations is known as Poincaré-Lindstedt method; by this method the secular terms are removed.

The equation of nonlinear system is

$\ddot{\text{x}}\text{ + x + εh}\left(x,\dot{x}\right)\text{= 0}$

In Poincaré-Lindstedt method $\ddot{\text{x}}\text{ and }\dot{\text{x}}$ are expressed as follows.

$\ddot{\text{x}}\text{ = }\frac{{\text{d}}^{\text{2}}\text{x}\left(\text{τ}\left(\text{t}\right)\right)}{\text{dt}}{\text{= ω}}^{\text{2}}\frac{{\text{d}}^{\text{2}}\text{x}}{{\text{dτ}}^{\text{2}}}$

$\dot{\text{x}}\text{ = ω}\frac{\text{d x}\left(\text{τ}\left(\text{t}\right)\right)}{\text{dτ}}$

$\text{x = x}\left(\text{τ}\right)$

The equation for time period is $\text{τ = ωt}$