conditions. Suppose our initial condition consists in the pendulum being held up at 70 degrees from its vertical position and released. Think about the shape that the pendulum will trace through its state space as it swings. At first, the angular velocity will be zero (as the pendulum is held ready). As the pendulum falls, its position will change in an arc, so its angular displacement will approach zero until it hits the vertical position, where its angular velocity will peak. The pendulum is now one-quarter of the way through a full period, and begins its upswing. Now, its angular displacement starts to increase (it gets further way from vertical), while its angular momentum decreases (it slows down). Eventually, it will hit the top of this upswing, and pause for a moment (zero angular velocity, high angular displacement), and then start swinging back down. If the pendulum is a real-world one (and isn’t being fed by some energy source), it will repeat this cycle some number of times. Each time, though, its maximum angular displacement will be slightly lower—it won’t make it quite as high—and its maximum angular velocity (when it is vertical) will be slightly smaller as it loses energy to friction. Eventually it will come to rest.
If we plot behavior in a two-dimensional state space (with angular displacement on one axis and angular momentum on the other), we will see the system trace a spiral-shaped trajectory ending at the origin. Angular velocity always falls as angular displacement grows (and vice-versa), so each full period will look like an ellipse, and the loss of energy to friction will mean that each period will be represented by a slightly smaller ellipse as the system spirals toward its equilibrium position of zero displacement and zero velocity: straight up and down, and not moving. See Figure 5.1 for a rough plot of what the graph of this situation would look like in a state-space for the pendulum.