Chapter 4.1, Problem 27P

The Linear Versus the Nonlinear Pendulum.

Convert Eq. $\left(23\right)$ and Eq. $\left(24\right)$ to planar systems using the state vector $\text{x}=x_1\text{i}+x_2\text{j}=\theta \text{i}+\theta \text{'}\text{j}$ for each question. For each of the following cases of parameter values and initial conditions, use an IVP solver to draw, on the same set of co-ordinate axes, plots of the first component of both the non-linear and linearized system.

(a) $\gamma =0,{\omega }^2=1,\theta \left(0\right)=\frac{\pi }{8},\theta \text{'}\left(0\right)=0$

(b) $\gamma =0,{\omega }^2=1,\theta \left(0\right)=\frac{\pi }{4},\theta \text{'}\left(0\right)=0$

Based on the graphical output, discuss the dependence of accuracy of the linear approximation on size of initial displacement and operating time. Are the periods of the nonlinear pendulum and the linear pendulum identical ?