Coupled Springs and Normal Modes

Two equal masses on a frictionless track, joined to each other and to two walls by three identical springs. Almost any way you start them looks like a complicated dance, but it is really just two simple motions added together: a slow in-phase mode where both masses move the same way at $\omega_+ = \sqrt{k/m}$, and a fast out-of-phase mode where they move oppositely at $\omega_- = \sqrt{3k/m}$. Press + mode or - mode to see each one alone, pure and single-frequency. Start them generically and the energy sloshes back and forth between the masses, a beat. The diagnostic tracks both displacements over time; the readout shows how much of each mode is present, $A_\pm = (x_1 \pm x_2)/2$.

Figure 1. Coupled springs and normal modes. Top: two masses and three springs between fixed walls. Bottom: the two displacements versus time, a single frequency in a pure mode and a beat in a generic mix. Method: velocity-Verlet, with the analytic eigenmode decomposition.

x1 start0.60

x2 start0.00

WHAT TO TRY

Press + mode: both masses move together as one, a single slow frequency, and the two traces sit right on top of each other.
Press - mode: the masses move oppositely at a faster single frequency, and the traces become mirror images.
Press generic: both modes are on, the energy sloshes from one mass to the other and back, a beat. The readout shows A+ and A- both nonzero.
Set the start displacements by hand: equal values give the pure slow mode, equal and opposite give the pure fast mode, anything else is a mix.