Coupled springs: normal modes and beats

What you are seeing: two equal masses on a frictionless track, coupled by three identical springs between two fixed walls. The classical small-oscillation problem has two normal modes: a symmetric in-phase mode at $\omega_+ = \sqrt{k/m}$, and an antisymmetric out-of-phase mode at $\omega_- = \sqrt{3k/m}$. Generic initial conditions excite both modes; the visible motion is the superposition.

Press the + mode or - mode button to start in a pure eigenmode. With the generic initial condition, watch energy slosh between the two masses: the upper trace (left mass position) and lower trace (right mass position) show the beat pattern at envelope frequency $|\omega_- - \omega_+|/2 \approx 0.366$. The phase portrait $(x_1, x_2)$ on the right traces out a Lissajous figure whose closure depends on whether $\omega_-/\omega_+ = \sqrt{3}$ is rational (it is not), so the trajectory is dense in its bounding parallelogram.