= Solution
The inertial <Lagrangian> is $L=m|\dot{\mathbf r}|^2/2-U(\mathbf r,t)$. Put $\mathbf r=R(t)\mathbf r'$ and define $U'(\mathbf r',t)=U(R(t)\mathbf r',t)$. The transport identity and invariance of the <Euclidean norm> under a <rotation> give
$$
\boxed{L'=\frac m2|\dot{\mathbf r}'+\boldsymbol\Omega'\times\mathbf r'|^2-U'(\mathbf r',t).}
$$
Its expansion contains the ordinary <kinetic energy>, $m\dot{\mathbf r}'\cdot(\boldsymbol\Omega'\times\mathbf r')$, and $m|\boldsymbol\Omega'\times\mathbf r'|^2/2$. These velocity and position terms generate the apparent forces in the rotating frame.
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