A catenary is formed when a rope or chain hangs freely in the effect of gravity supported by two ends. The shape is a result of optimisation of potential energy along the length of the rope. This kind of curve is generally represented in a static case in equilibrium.
What happens when the supporting ends move (maybe slowly)? I believe now that more kinetic energy is added to the system. Can the equations of motion of points along a catenary be derived based on minimization of potential energy and kinetic energy ? Can this kind of formulation work for describing a motion of any point in a catenary ? I would like to know your suggestions and views on this fascinating problem and approach.