Condition for minimal Harmonic Oscillator Action

TL;DR

This paper proves that the harmonic oscillator's action is minimized only when the elapsed time is less than π divided by its natural frequency, providing a clear condition for minimal action.

Contribution

It offers an elementary proof establishing the precise condition under which the harmonic oscillator's action is minimized.

Findings

Action is minimized if and only if τ < π/ω

Provides a clear criterion for minimal harmonic oscillator action

Clarifies the relationship between elapsed time and action minimization

Abstract

We provide an elementary proof that the action for the physical trajectory of the one-dimensional harmonic oscillator is guaranteed to be a minimum if and only if $\tau < \pi/\omega$, where $\tau$ is the elapsed time and $\omega$ is the oscillator's natural frequency.