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

Abstract

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,q̇), where q(t) is the system displacement as a function of time t and q̇(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/∂q̇2) 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.

Statistical analysis of coupled oscillations with a single fixed endpoint and its application to carbon nanomaterials

“…Thus, the coupled harmonic oscillators considered in this study are insufficient for accurately describe the graphene sheet. However, if the displacement is small and restricted to a one-dimensional direction, then the graphene sheet can exhibit statistical properties similar to those of coupled harmonic oscillators [26]. We calculate the average of X n 2 for the different initial velocities via MD simulations (micro-canonical ensemble), in which the time step for numerical integration is set to 0.5 fs and T = 500 ps.…”