Piecewise Smooth Models of Pumping a Child's Swing

SIAM Review, Volume 65, Issue 3, Page 887-902, August 2023.
Some simple models of a child swinging on a playground swing are presented. These are analyzed using techniques from Lagrangian mechanics with a twist: the child changes the configuration of the system by sudden movements of their body at key moments in the oscillation. This can lead to jumps in the generalized coordinates describing the system and/or their velocities. Jump conditions can be determined by integrating the Euler--Lagrange equations over a short time interval and then taking the limit as this time interval goes to zero. These models give insights into strategies used by swingers, and answer such vexed questions such as whether it is possible for the swing to go through a full 360$^\circ$ turn over its pivot. A model of an instability at the pivot observed by Colin Furze in a rigid swing constructed to rotate through 360$^\circ$ is also described. This uses a novel double pendulum configuration in which the two components of the pendulums are constrained to move in orthogonal planes.