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.

中文翻译：

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儿童秋千的分段平滑模型

《SIAM 评论》，第 65 卷，第 3 期，第 887-902 页，2023 年 8 月。
展示了一些儿童在操场秋千上荡秋千的简单模型。这些是使用拉格朗日力学的技术进行分析的，但有一点不同：孩子在振荡的关键时刻通过身体的突然运动来改变系统的配置。这可能导致描述系统和/或其速度的广义坐标发生跳跃。跳跃条件可以通过对短时间间隔内的欧拉-拉格朗日方程进行积分，然后在该时间间隔变为零时取极限来确定。这些模型深入了解了挥杆者使用的策略，并回答了诸如挥杆是否有可能在其枢轴上进行完整 360$^\circ$ 转动等棘手的问题。还描述了 Colin Furze 在构造为旋转 360$^\circ$ 的刚性秋千中观察到的枢轴不稳定性模型。它使用了一种新颖的双摆配置，其中摆的两个部件被限制在正交平面内移动。