On levitation by blowing

Anyone who has visited a science museum has seen the demonstration where a beach (or ping-pong) ball is suspended in mid-air at a fixed position by constant blowing from below. After a while, the ball inevitably tumbles to the ground but can easily be rebalanced, by hand, again at the suspension point. Here, we ask a different more delicate question. Can we blow the ball from rest, starting at the nozzle opening (x = 0), moving it up to the suspension point x = x* above the nozzle? We show that it is not possible to do this using constant blowing because the point at which the downward gravitational force balances the upward blowing force is an elliptic fixed point of the governing equations, so there is no transfer trajectory that connects the origin to x*. To overcome this problem, we design time-dependent blowing schedules that achieve the transfer, making use of orbit transfer ideas developed in the orbital mechanics literature. Then, we ask which of these time-dependent schedules are optimal? We show that, generally, it is bang–bang (on–off) blowing schedules that achieve the transfer in minimal time, using minimal energy (action) and minimal air volume. For certain parameter values, however, there are more complicated blowing schedules that are optimal (with respect to energy), which can be designed using the Pontryagin Maximum Principle (PMP) and singular control. We use elementary concepts from mechanics, nonlinear dynamics, and control theory and challenge the inclined experimentalist to try to implement some of these nonconstant blowing schedules in the lab.