Inverting a normal harmonic oscillator: Physical interpretation and applications

Understand

A harmonic oscillator with time-dependent mass $m(t)$ and a time-dependent (squared) frequency $\omega^2(t)$ occurs in the modelling of several physical systems.

• It is generally believed that systems, with $m(t)>0$ and $\omega^2(t)>0$ (normal oscillator) are stable while systems with $m(t)>0$ and $\omega^2(t)<0$ (inverted oscillator) are unstable.
• We show that it is possible to represent the \textit{same} physical system either as a normal oscillator or as an inverted oscillator by redefinition of dynamical variables.
• While we expect the physics to be invariant under such redefinitions, it is not obvious how this invariance actually comes about.