Computational Physics - Department of Physics

8.7 Physics Project: the pendulum 265

such a curve in the limitτ→∞. It is called periodic, since it exhibits periodic motion in
time, as seen from Fig. 8.6. In addition, we should note that this periodic motion shows
what we call resonant behavior since the the driving frequency of the force approaches the
natural frequency of oscillation of the pendulum. This is essentially due to the fact that we
are studying a linear system, yielding the well-known periodic motion. The non-linear system
exhibits a much richer set of solutions and these can only be studied numerically.

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

vˆ

θ

Fig. 8.7Phase-space curve of a linear damped pendulum withQ= 2 ,ωˆ= 2 / 3 andAˆ= 0. 5. The inital velocity
isvˆ 0 = 0 andθ 0 = 0. 01.

In order to go beyond the well-known linear approximation wechange the initial conditions
to sayθ 0 = 0. 3 but keep the other parameters equal to the previous case. Thecurve forθ
is shown in Fig. 8.8. The corresponding phase-space curve isshown in Fig. 8.9. This curve
demonstrates that with the above given sets of parameters, after a certain number of periods,
the phase-space curve stabilizes to the same curve as in the previous case, irrespective of
initial conditions. However, it takes more time for the pendulum to establish a periodic motion
and when a stable orbit in phase-space is reached the pendulum moves in accordance with
the driving frequency of the force. The qualitative pictureis much the same as previously.
The phase-space curve displays again a final periodic attractor.
If we now change the strength of the amplitude toAˆ= 1. 35 we see in Fig. 8.10 thatθas
function of time exhibits a rather different behavior from Fig. 8.8, even though the initial
conditions and all other parameters exceptAˆare the same. The phase-space curve is shown
in Fig. 8.11.
We will explore these topics in more detail in Exercise 8.2 below, where we extend our
discussion to the phenomena of period doubling and its link to chaotic motion.

8.7.2 The pendulum code

The program used to obtain the results discussed above is presented here. The enclosed code
solves the pendulum equations for any angleθwith an external forceAcos(ωt). It employes