Model the swinger as a point mass on a pendulum whose length L(t) is prescribed by the rider standing up (shortening L, raising their centre of mass) or crouching down (lengthening L) — an external, rheonomic constraint, not a free coordinate. With θ measured from the vertical, the Lagrangian is:
𝓛 = ½m(L²θ̇² + L̇²) + m g L cos θ
Applying the Euler–Lagrange equation to θ alone (L is driven, not dynamical) and dividing by mL² gives the governing ODE actually integrated below, with an added linear damping term c for air drag and chain friction:
θ̈ + 2(L̇/L)θ̇ + (g/L)sin θ + c θ̇ = 0
- 2(L̇/L)θ̇ is the parametric-pumping term: it behaves like time-varying damping. Shortening L (L̇<0) while moving toward the bottom injects energy; the classic technique is to stand at the bottom and crouch at the extremes, i.e. pump at twice the pendulum's own frequency.
- Auto mode reproduces that real technique directly: the model toggles stand/crouch every time θ crosses zero.
- Fixed-frequency drive instead forces L(t) = L₀ + ΔL·cos(ωpumpt) explicitly, with ωpump = 2·ratio·ω₀ (ω₀ = √(g/L₀)). Resonant growth appears only in a narrow band around ratio = 1 — the hallmark 2:1 parametric-resonance condition (a Mathieu-equation instability tongue).
- Growth saturates into a steady limit cycle because sin θ detunes the resonance as amplitude grows — the same nonlinearity that keeps a real swing from flipping over the bar.
- The three panels on the right show the same live state three ways: the swing itself, the phase portrait (θ vs θ̇) spiraling outward during growth and settling onto a closed loop at the limit cycle, and a scrolling time strip of θ(t) and L(t) so you can see the 2:1 phase-locking directly.