(swinging the other way). This gives us two points on the vertical axis. Those four points in phase space, which we have marked in figure 6-7, represent the extreme states of the pendulum—maximum elongation and maximum velocity. The exact location of these points will depend on our units of measurement.
If we were to go on and mark the points corresponding to the states of motion among the four extremes, we would find that they lie on a closed loop. We could make it a circle by choosing our units of measurement appropriately, but in general it will be some kind of an ellipse (figure 6-8). This loop is called the pendulum’s trajectory in phase space. It completely describes the system’s motion. All the variables of the system (two in our simple case) are represented by a single point, which will always be somewhere on this loop. As the pendulum swings back and forth, the point in phase space will go around the loop. At any moment we can measure the two coordinates of the point in phase space, and we will know the exact state—angle and velocity—of the system. Note that this loop is not in any sense a trajectory of the ball on the pendulum. It is a curve in an abstract mathematical space, composed of the system’s two variables.
So this is the phase-space technique. The variables of the system are pictured in an abstract space, in which a single point describes