The evolution of a cluster of particles and the invariants of a time-dependent oscillator

A simple physical realization is found for the time-dependent harmonic oscillator equation $\text{q}\text{?}\text{+ φ(t)q = 0}$, for its auxiliary equation $\text{g}\text{?}\text{9 + φ(t)? = h}{}^2\text{?}{}^{\mathrm{?3}}$, and for the Ermakov invariant. Through generalization of the model, a set of invariants is obtained for a system of equations of the form .