 Solution

The inertial Lagrangian is $L=m|\dot{\mathbf{r}}{|}^2/2-U(\mathbf{r},t)$. Put $\mathbf{r}=R(t){\mathbf{r}}^{\prime }$ and define $U^{\prime }({\mathbf{r}}^{\prime },t)=U(R(t){\mathbf{r}}^{\prime },t)$. The transport identity and invariance of the Euclidean norm under a rotation giveIts expansion contains the ordinary kinetic energy, $m{\dot{\mathbf{r}}}^{\prime }\cdot ({\mathbf{\Omega }}^{\prime }\times {\mathbf{r}}^{\prime })$, and $m|{\mathbf{\Omega }}^{\prime }\times {\mathbf{r}}^{\prime }{|}^2/2$. These velocity and position terms generate the apparent forces in the rotating frame.