Exercise 3.2: Large oscillations

The pendulum responds to the equation of motion (40) only in the limit of small angles. In the case of large oscillations, the equation has to ve modified becoming

$$\frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin{\theta}.$$

The enrgy of the pendulum is then given by

$$E=\frac{1}{2}mL^2(\frac{d\theta}{dt})^2+mgL(1-\cos{\theta})$$

1. Modify your program to simulate large amplitude oscillations in a pendulum. Set $g/L=9$ and choose $\Delta t$ so that the numerical solution is stable, i.e. it does no diverge with time from the ``true'' solution. Check the stability by clculating the total energy and ensuring thatt it does not drift from its initial value.

2. Set $d\theta /dt\vert _{t=0}=0$ and make plots of $\theta (t)$ and $d\theta /dt(t)$ for the initial conditions $\theta(t=0)=0.1$, 0.2, 0.4, 0.8, 1.0. Describle the qualitative behavior of $\theta$ and $d\theta /dt$. What is the period $T$ and the maximum amplitude $\theta _max$ in each case? Plot $T$ versus $\theta _max$ and discuss the qualitative dependence of the period on the amplitude. How do the results compare in the linear and non-linear cases, e.g. which period is larger? Explain the relative values of $T$ in physical terms.