Equations: ${\ddot{x}}_{1,2}=-{\omega }_0^2{x}_{1,2}-\kappa (x_{1,2}-x_{2,1})$. Normal modes: symmetric $x_1+x_2$ at ${\omega }_0$, antisymmetric $x_1-x_2$ at $\sqrt{{\omega }_0^2+2\kappa }$. Identical pendulums mean $\Delta =0$; the spring is $V$; the modes are the fully mixed $45^{\circ }$ combinations; starting one pendulum and watching the motion transfer completely to the other and back is the beat, at the mode-frequency difference — the exact analogue of the ammonia nitrogen oscillating L→R at the splitting frequency. Detune the pendulums ($\Delta \ne 0$) and the transfer becomes incomplete by the same $V^2/({\Delta }^2+V^2)$ factor as the Rabi formula.

The mathematics is the same because both are two coupled linear oscillators. The physics parts ways at interpretation: classically $x_1(t)$ is the observable, continuously and completely knowable; quantum mechanically the amplitudes are NOT observables — a measurement asks “which side?”, returns one bit, and collapses the state. Beats in amplitude become oscillating probabilities, and only an ensemble reveals the sinusoid. Everything on this page up to measurement is honest classical wave mechanics; the Born rule is the quantum content.