On the ubiquity of classical harmonic oscillators and a universal equation for the natural frequency of a perturbed system

A new perspective on the ubiquity of classical harmonic oscillators is presented based on the two-variable Taylor expansion of a perturbed system's total energy E(q,q), where q(t) is the system displacement as a function of time t and q(t)=dq/dt. This generalised approach permits derivation of the lossless oscillator equation from energy arguments only, yielding a universal equation for the oscillation frequency ω=(∂2E/∂q2)/(∂2E/∂q2) which may be applied to arbitrary systems without the need to form system-specific linearised models. As illustrated by a range of examples, this perspective gives a unifying explanation for the prevalence of harmonic oscillators in classical physics, can be extended to include damping effects and driving forces, and is a powerful tool for simplifying the analyses of perturbed systems.