Parametric Amplification

Parametric amplification is a simple concept that appears both in intuitive, everyday situations, as well as advanced technologies like lasers and quantum computers. From what I can tell, the concept is rarely introduced to undergraduates who are nevertheless well-equipped to understand it. Mathematically, parametric amplification falls under the category of nonlinear dynamics. This field focuses on equations that are famously hard to solve, but parametric amplification can readily be understood in terms of a quite straightforward example. If you've ever swung on a playground swing before (and no one was pushing you!), you were using your legs to parametrically amplify your motion. Let's understand what's going on before we define what "parametric" means. First, you and the swing comprise a pendulum, which has equation of motion \[ \ddot{\theta} = \omega_0 \sin \theta \] where \(\theta\) is the angle you are from hanging vertical and \(\omega_0\) is the typical \(\sqrt{g/l}\), where \(g\) is the gravitational constant and \(l\) is the length of the swing. Let's consider just small oscillations where \(\sin x \approx x\) and let's also add a damping term with damping constant \(\beta\). \[ \ddot{\theta} + 2 \beta \dot{\theta} + \omega_0 \theta = 0 \] In classical mechanics courses, students typically study the this system with a driving term added to the right hand side, often a sinusoidal force. \[ \ddot{\theta} + 2 \beta \dot{\theta} + \omega_0 \theta = f_0 \sin(\omega t) \] But there's no net force or torque on the pendulum - the pendulum can't push against itself! So we are actually not driving the syste So what do you do on a swing when you want to start moving? You start swinging your legs back and forth, extending and tucking them in over and over.