potential well

A potential well is a dip in the potential energy landscape where a system can be trapped. Think of a marble in a bowl: it rolls back and forth, always returning to the bottom. The shape of the bowl determines the character of the oscillation.

For a pendulum, the potential energy is U(θ) = mgl(1 − cos θ). Near the bottom this looks like a parabola — U ≈ ½mglθ² — and the motion is simple harmonic. But farther out the well flattens and the walls eventually curve over into the next period of the cosine. The departures from the parabolic approximation are what make the large-angle pendulum nonlinear.

Every stable equilibrium in physics sits at the bottom of a potential well. The quadratic approximation near the minimum always gives simple harmonic motion. The interesting physics — anharmonicity, amplitude-dependent frequency, chaos — comes from the higher-order terms in the shape of the well.