Oscillations, Symmetry and Diagonalization
From a single mass on a spring to a vibrating string, oscillatory systems are often taught in the typical classroom via a sequence of seemingly ad-hoc guesses: exponentials in time, normal modes in space, and Fourier series in the continuum limit. In this essay, we look at oscillations from a linear-operator lens, and use symmetries with which those operators commute to show how diagonalization naturally decouples the dynamics.