In mathematics, the art of questioning often is more important than the art of solving a problem. —Georg Cantor, 1845-1918Measuring a series of values in the field makes sense only if something actually happens, i.e., if there is some change to be seen. In a pendulum that has come to rest, there is not much to be measured. To start some action, we need a force; for example, someone giving the pendulum a push. After that, there are two independent forces that keep the pendulum swinging: inertia keeps it moving, while gravity pulls it down.
Only if two such forces act on the same body (thereby acting on each other in an opposing way) can cyclic motion occur. When the pendulum is at its largest displacement from its initial (at rest) position, it is at rest again for a short moment (velocity = 0 in Figure 3). All the energy from the initial push happens to be conserved in potential energy.
During each cycle, all the potential energy is first transformed completely into kinetic energy (at the bottom of the pendulum's arc) and then back again into potential energy. Instead of graphing each of the two independent variables (displacement and velocity) over time, it is sometimes advantageous to graph them against each other, thereby eliminating the time axis (Figure 1) and revealing the cyclic nature of the process. This is derivative phase space.
What is phase space good for? If you know the values of both variables (displacement and velocity) at any time, you can make a pretty good guess about future values simply by following the closed curve in Figure 1. In case of noisy measurements (as in Figure 2), take the average motion.
Most of the time, we can measure only one of the variables (displacement), losing the other one. Hence, we need to somehow reconstruct the lost variable (velocity) from the measured one. This can be done by graphing the measured variable against a delayed copy of itself, i.e., the displacement at time t on one axis and displacement at time t-tau on the other axis. This works surprisingly well if the delay is chosen properly, as in Figure 5.