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For a periodic system two types of motion are possible. In the first type both coordinates and momenta are periodic functions of time. An example of this type is motion of a simple pendulum. In the second type of motion only the coordinates are periodic functions of time. An example of the second type of motion is the conical pendulum where the angle keeps increasing, but the momentum is a periodic function of time.
One of the problems that will concern us here will be that of determining frequencies of bounded motion. We begin with what is we mean by a bounded motion and types of bounded motion that are possible. Since we will be interested in separable systems, it is sufficient to consider possible types of periodic motion in one dimension. The first kind of periodic motion is one in which both q, p are periodic in time. It is called oscillation. For example \(q, p\) for harmonic oscillator are periodic functions of time. The other kind of motion arises when \(p\) is periodic function of \(q\), but \(q\) may not be periodic function of time. This type of motion is called rotation. For systems with several degrees of freedom, the frequencies associated with different coordinates may not have any relationship and the motion in real time may not appear periodic. The motion in real time will appear periodic only if the frequencies of variations of different coordinates are commensurate.
As examples of oscillation and rotation, consider a simple pendulum, Fig (a), and conical pendulum, Fig(b). Both are examples of periodic motion. In case of simple pendulum, the angle \(\theta\) varies periodically with time. It increases from zero to a maximum value then decreases to become a negative, minimum, value and then it increases. It becomes zero again after a full period. Both \(\theta\) and \(p_\theta\) are periodic functions of time. In the case of a conical pendulum, as the pendulum executes a periodic motion but the time variation of the angle \(\phi\), see Fig.(c), in not periodic. The angle \(\phi\) keeps increasing with time and never comes back to the starting value zero. The angle \(\phi\) increases linearly with time. The momentum conjugate to \(\phi\) remains constant. In general, when both modes are present the momenta will be periodic functions of time.
