= Solution
<Hamilton's equations> are $\dot q_j=\partial H/\partial p_j$ and $\dot p_j=-\partial H/\partial q_j$. The <Arnold-Liouville theorem> states that $n$ functionally independent first integrals in involution on a $2n$-dimensional <symplectic manifold> give local integrability; a compact connected regular common level is an $n$-torus. Near such a torus there are <action-angle variables> $(I,\vartheta)$ with $H=H(I)$, $\dot I=0$ and $\dot\vartheta=\partial H/\partial I$.
For the displayed diagonal oscillator <Hamiltonian> set $E_k=(p_k^2+\omega_k^2q_k^2)/2$. Each is constant, since $\dot q_k=p_k$ and $\dot p_k=-\omega_k^2q_k$. Different $E_k$ depend on disjoint canonical pairs, so their <Poisson brackets> vanish. Their differentials are independent wherever every $dE_k$ is nonzero, an open dense set; this proves complete integrability. For nonzero frequencies put $\Omega_k=|\omega_k|$. A positive-energy level in each pair is an ellipse with semiaxes $\sqrt{2E_k}$ and $\sqrt{2E_k}/\Omega_k$. Its enclosed area is $2\pi E_k/\Omega_k$, so the <action variables> are
$$
\boxed{I_k=\frac1{2\pi}\oint p_k\,dq_k=\frac{E_k}{|\omega_k|},\qquad H=\sum_k|\omega_k|I_k.}
$$
For the usual positive oscillator frequencies this is $I_k=E_k/\omega_k$. If a printed “constant” frequency is allowed to be zero, that pair is a free particle: $p_k$ is a commuting integral, but its generic level is noncompact and has no closed-cycle oscillator action. Complete integrability persists, while the torus/action formula requires nonzero frequencies.
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