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Two or more oscillators connected by springs form a coupled system where the equation of motion for each mass depends on the displacements of its neighbors. The system has multiple natural frequencies and exhibits energy exchange.
From your study of the spring-mass system you know that a single mass on a spring oscillates at its natural frequency ω₀ = √(k/m), and from your work on systems of linear ODEs and eigenvalues and eigenvectors you know how to find the solution structure of coupled linear equations. Coupled oscillators are where these two streams of knowledge meet. When you connect two masses with springs, each mass no longer oscillates independently — the displacement of one affects the force on the other, and the system as a whole develops its own characteristic behavior.
Consider the simplest case: two identical masses m connected by three springs (one on each outer wall, one coupling spring between them), each spring with constant k. The equation of motion for the left mass is m·ẍ₁ = -kx₁ + k_c(x₂ - x₁), and symmetrically for the right. Here x₁ and x₂ are displacements from equilibrium, and k_c is the coupling spring constant. The force on each mass now depends on *both* positions — the system is described by a 2×2 matrix equation, not two independent scalar equations. Writing it in matrix form, Mẍ = -Kx, where M is the mass matrix and K is the stiffness matrix, you immediately recognize the structure you studied in eigenvalue problems.
The key insight is that the coupled system has exactly two normal modes — special configurations where both masses oscillate at the same frequency and maintain a fixed ratio of displacements. To find them, you substitute the trial solution x = v·cos(ωt) and reduce the problem to the eigenvalue equation (K - ω²M)v = 0. The eigenvalues ω² give the two natural frequencies; the corresponding eigenvectors v give the mode shapes. For the symmetric two-mass system, the first mode (symmetric mode) has both masses moving in the same direction with frequency ω₁ = √(k/m) — the coupling spring carries no force because it never stretches. The second mode (antisymmetric mode) has the masses moving in opposite directions with higher frequency ω₂ = √((k + 2k_c)/m) — the coupling spring is alternately compressed and stretched, raising the restoring force and therefore the frequency.
The general motion is a superposition of normal modes. If you start both masses moving identically, you excite only the first mode and they oscillate in unison forever. If you displace one mass while holding the other fixed, you excite both modes simultaneously. What you observe then is a phenomenon called beats: the energy initially concentrated in one mass gradually transfers to the other, then back again, at a beat frequency equal to the difference of the two normal mode frequencies. This energy exchange is the hallmark of coupled oscillators and has direct analogs in quantum mechanics (energy transfer between coupled quantum states) and electromagnetism (coupled resonant circuits).
For larger systems with N masses, the approach scales directly: the eigenvalue equation produces N normal mode frequencies and N mode shapes. The general solution is always a superposition with 2N free constants (N amplitudes and N phases) set by initial conditions. The lesson is architectural: even an arbitrarily complex network of coupled oscillators has a clean mode decomposition, and once you find it, all the complicated correlated motion becomes independent harmonic oscillators in disguise. Your upcoming study of normal modes will work this out systematically for continuous media, where the "masses" become infinitely many and the normal modes become standing waves.