How do you pump a playground swing higher without anyone pushing you?

Settled in 1996 by William Case (seated 1990, standing 1996).

Kapitza's problem #46 asks how children increase a swing's amplitude through body movement alone. William Case's 1990 and 1996 American Journal of Physics papers modeled the seated and standing pumping techniques as, respectively, a driven oscillator arising from angular-momentum conservation about the pivot, and a torque-driven oscillator for the standing case, correcting earlier claims that standing-pumping was a pure parametric-resonance effect. Later work (e.g., a 2023 Physical Review E paper on phase and frequency modulation) has refined the timing strategy used by real children.

A move every child discovers by trial and error on a playground turns out to require distinguishing two genuinely different flavors of nonlinear oscillator physics.

Explanations that circulated

• Swing-pumping is purely a parametric oscillator effect, like changing a pendulum's length periodically (Case showed the standing-position case is actually dominated by a direct driving torque term, not the classic parametric mechanism, contradicting some earlier textbook treatments)

References

• The pumping of a swing from the seated position, William B. Case, American Journal of Physics, 1990
• The pumping of a swing from the standing position, William B. Case, American Journal of Physics, 1996