On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine

In the present note we study the motion of a particle under uniform gravity on a tilted smooth cone. We point out some integrable cases of that problem. In particular, we establish that motion on the cone with semi-vertical angle π/6 and one generator vertical is integrable. Moreover, we show that the swinging Atwood machine is equivalent to a special version of the motion on the tilted cone. This analogy makes it easier to analyze motion of SAM from a geometrical point of view.