On the dynamics of cranes, or spherical pendula with moving supports

We consider the dynamics of a tower crane: a point mass suspended by a light cable (itself assumed massless) from a horizontally moving support. We derive general equations of motion, and analyse two cases in detail: the linearly accelerating support, and the support describing a circle at constant speed. We show that the former system remains completely integrable, in spite of the fact that energy is no longer conserved. In the second case, while a Hamiltonian is conserved, there is no analogue of angular momentum conservation and the system appears to be non-integrable. We find steadily rotating solutions, discuss their stability and bifurcations, and provide a partial characterisation of global orbit structures.