Parametric Resonance: Pumping a Swing

A child on a swing does not push against anything outside the swing, yet the arc keeps growing. The trick is to crouch and stand at the right moments, raising and lowering the center of mass twice per swing, which is the same as changing the pendulum's length, and therefore its natural frequency, periodically in time. This is parametric driving, and it is fundamentally different from pushing a swing at its own frequency. The equation of motion is the Mathieu equation, $\ddot\theta + 2\beta\dot\theta + \omega_0^2(1 + h\cos\omega_d t)\theta = 0$, and the striking result is that when the modulation frequency $\omega_d$ is close to twice the natural frequency (pump twice per period) and the modulation depth $h$ exceeds a damping-set threshold, the amplitude grows exponentially rather than linearly. The scene runs the pendulum with its length pumping; watch the swing build up when you sit on resonance and die away when you do not, with the log-amplitude plot beside it turning a straight rising line into the unmistakable signature of exponential growth. The lower panel is the map that explains it, the Ince-Strutt stability chart of the Mathieu equation: the energy of the system grows inside the shaded resonance tongues, which sprout from $\omega_d = 2\omega_0/n$, and the widest and most easily reached is the first one at $\omega_d = 2\omega_0$. The dot is your current setting, and parametric resonance happens precisely when it sits inside a tongue, far enough in that the growth outruns the damping.