  We re-examine the time-dependent harmonic oscillator $\ddot q=-\omega^2 q$
under various regularity assumptions. Where $\omega(t)$ is continuously
differentiable we reduce its integration to that of a single first order
equation, i.e. the Hamilton equation for the angle variable $\psi$ alone (the
action variable ${\cal I}$ does not appear). Reformulating the generic Cauchy
problem for $\psi(t)$ as a Volterra-type integral equation and applying the
fixed point theorem we obtain a sequence $\{\psi^{(h)}\}_{h\in\mathbb{N}_0}$
converging uniformly to $\psi$ in every compact time interval; if $\omega$
varies slowly or little, already $\psi^{(0)}$ approximates $\psi$ well for
rather long time lapses. The discontinuities of $\omega$, if any, determine
those of $\psi,{\cal I}$. The zeroes of $q,\dot q$ are investigated with the
help of two Riccati equations. As a demo of the potential of our approach, we
briefly argue that it yields good approximations of the solutions (as well as
exact bounds on them) and thereby may simplify the study of: the stability of
the trivial one; the occurrence of parametric resonance when $\omega(t)$ is
periodic; the adiabatic invariance of ${\cal I}$; the asymptotic expansions in
a slow time parameter $\varepsilon$; etc.
