Pumping a swing by standing and squatting: do children pump time optimally?

Research in biomechanics supports the idea that humans perform skilled tasks as self-optimizing machines. A rider pumping a playground swing can be modeled adequately as a second-order system. The focus of this article is the problem of time-optimal pumping of a swing, where the rider pumps the swing by alternately standing and squatting. The rider-and-swing system is modeled as a pendulum with a bob of mass attached to a fixed support by a rope of variable length. Children playing on a swing generally stand up at the lowest point and squat at the highest. Using the Pontryagin minimum principle and techniques from geometric optimal control, it is shown that this pumping strategy is time-optimal for the problem of maximizing the amplitude of oscillation of a swing.