How does a skipping rope know how many nodes I want? Say I have a skipping rope. I have fixed it in one end and I'm holding it in the other. When I start spinning it, I know that I can make it spin in one large arc, or in two smaller arcs with a stationary node in the middle, or in three arcs with two equally spaced nodes, and so on.
How exactly do I do that? Especially if there's no gravity, I see no reason that the node of spinning is directly tied to the rotational frequency, or to the distance between my hand and the anchor point (the total "slack length" of the rope). At least once it has stabilized. So what is it that I do with my hand that tells the rope which mode to use?
 A: It is based on the initial conditions of the system.
The one large arc mode is the naturally stable configuration because the potential energy of the rope with respect to the rotating reference frame (centripetal force) is at its absolute minimum.
Two arcs with a stationary node is most easily archived by two rope operators. First the rope is put into one arc mode, then one of the operators quickly changes rotational phase by 180degrees. In this fashion one operators hand is now up while the other is down. This accelerates that side of the rope creating the two arc mode.
Two arcs is similarly achievable by a single rope operator by first establishing kinetic energy near the fixed end of the rope in single arc mode, then quickly changing phase 180 degrees to accelerate the operator side of the loop ahead of the fixed end.
The two arc mode is locally stable because the potential energy of the rope with respect to the rotating reference frame is at a local minimum.
This procedure can be repeated for additional arcs/nodes of ever decreasing stability. At some point system noise (rope linear density, air friction, movement consistency, etc) and operator timing ability limit how many nodes can be created.
