phase portrait

A phase portrait is a picture of a system's state. The horizontal axis is position, the vertical axis is velocity, and every point in the plane corresponds to one possible state of the system. As time passes, the state traces out a curve.

For an undamped oscillator — an ideal pendulum, or a mass on a frictionless spring — the curve is closed: an ellipse or a circle, depending on the units. The system keeps retracing the same orbit forever, because energy is trading back and forth between position and motion without ever leaking away.