= Inertia-tensor acceleration of a steadily rotating stellar pattern
{title2=$\tfrac12\ddot I=\Omega^2\operatorname{diag}(I_{22}-I_{11},I_{11}-I_{22},0)$}
Use the position second-moment tensor $I_{ij}=\int\rho x_ix_j\,d^3x$, not the engineering moment-of-inertia convention. If a density pattern rotates steadily about its third principal axis, $I(t)=R(\Omega t)I(0)R(\Omega t)^T$. Differentiating twice at a principal-axis instant gives the displayed tensor. The <tensor virial theorem> then determines the pattern speed from the difference of its first two diagonal components. This density-pattern speed is distinct from stellar mean streaming; in the axisymmetric limit $I_{11}=I_{22}$, this difference cannot determine it.
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