Suppose we have a "conical pendulum" consisting of a bob $P$ of mass $m$ attached to a rigid chord of length $l$ which is attached at its other end to a fixed point $O$. We displace the bob so that it forms an angle $\theta_0$ with the vertical axis and give it some initial velocity $\vec{v_0}$.
What are the possible motions of the pendulum?
We can first consider the easy case where $\vec{v_0}$ is horizontal.
Note: "conical pendulum" just means a pendulum which is allows to move in 3 dimmensions as opposed to the classic planar pendulum motion