Time-optimal control of a swing

It is well-known that some form of optimization takes place in skilled human tasks. Using largely numerical techniques researchers in the field of biomechanics have optimized different cost functions to predict human motion for tasks like walking and rowing. In a similar vein in this paper we show, using Pontryagin's Minimum Principle and after some simplification, that the typical pumping strategy used by children on a swing is time-optimal. We present results for the two special cases: When the swing is to be taken from small initial oscillations to a specific final angle in minimum time, and when it is to be brought from an initial angle to rest in minimum time. The method outlined is extendable to other initial and final conditions, though the results obtained for other boundary conditions are not as elegant and intuitive as in the cases discussed.