In this lecture we discussed harmonic motion, which turned out to pose certain numerical problems in its solution. We found out that the standard Euler method used up to now must be replaced by better, more suitable methods. Therefore, the Euler-Cromer method has been presented and analysed. We found that this method conserves energy over a period of the pendulum. The same would also hold true for a Runge-Kutta method.
We then used our numerical method to analyse various aspects of the linear equation of motion of the mathematical pendulum, by adding dissipation and a driving force to the game. In principle, we could immediately have set out to leave the approximation of small deflection angles underlying the mathematical pendulum, and embark on a discussion of a more realistic version of the pendulum. There, however, completely new effects such as the emergence of deterministic chaos kick in.
Anyway, by now we have an instrument at hand to study chaos, which will be the subject of the homework exercise and of the next lecture.