normal modes

Take a system of coupled oscillators, masses joined by springs, or the atoms of a molecule, and at first it looks hopelessly tangled: push one part and everything jiggles at once. But there exist special collective patterns of motion in which every part oscillates at the same single frequency, all moving in fixed proportion. These are the normal modes, and any motion whatsoever is a superposition of them.

Near a stable equilibrium you expand the Lagrangian to quadratic order: L = (1/2) q_dot^T M q_dot - (1/2) q^T K q, with a mass matrix M and a stiffness matrix K. Seeking solutions of the form q(t) = a e^(i omega t) leads to the generalized eigenvalue problem (K - omega^2 M) a = 0. The eigenvalues omega^2 are the squared normal-mode frequencies and the eigenvectors a are the mode shapes. In terms of the normal coordinates the system decouples into independent simple harmonic oscillators.

You meet normal modes everywhere: molecular vibrations (read off infrared spectra), crystal lattice vibrations (which quantize into phonons), coupled pendulums, and engineering vibration analysis; they are also the classical prelude to quantum field theory, where a field is infinitely many normal modes each quantized. The one caveat is that the whole decomposition relies on the small-oscillation, harmonic approximation; at large amplitude, anharmonic terms couple the modes and they are no longer independent.