= Solution
Suppress primes on the angular <velocity> while retaining rotating components. The <canonical momentum> is $\partial L'/\partial\dot{\mathbf r}'=m(\dot{\mathbf r}'+\boldsymbol\Omega\times\mathbf r')$. Varying $L'$ with respect to position and applying the <Euler-Lagrange equations> gives
$$
\boxed{m\ddot{\mathbf r}'=-\nabla' U'-2m\boldsymbol\Omega\times\dot{\mathbf r}'-m\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r')-m\dot{\boldsymbol\Omega}\times\mathbf r'.}
$$
The terms are the physical <force>, the velocity-dependent <Coriolis force>, the outward <centrifugal force>, and the <Euler force> caused by changing angular <velocity>. The common origin is inertial; an accelerating origin would introduce another translational apparent <force>.
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