I'm in trouble with comprehending derivation of Lagrangian of a particle in non-inertial , translational and rotational frame of reference by Landau's Mechanics.
More precisely I don't understand why the following can happen.
$$ L^\prime=\frac{1}{2}m{v^\prime}^2-m\mathbf{W}(t)\cdot\mathbf{r}^\prime-U \tag{39.4} $$
The velocity $\mathbf{v}'$ of the particle relative to $K'$ is composed of its velocity $\mathbf{v}$ relative to $K$ and the velocity $\mathbf{\Omega}\times\mathbf{r}$ of its rotation with $K$: $\mathbf{v}^\prime=\mathbf{v}+\mathbf{\Omega}\times\mathbf{r}$ (since the radius vectors $\mathbf{r}$ and $\mathbf{r}'$ in the frames $K$ and $K'$ coincide). Substituting this in the Lagrangian (39.4), we obtain
$$ L=\frac{1}{2}mv^2+m\mathbf{v}\cdot\mathbf{\Omega}\times \mathbf{r}+\frac{1}{2}m(\mathbf{\Omega}\times \mathbf{r})^2-m\mathbf{W}\cdot \mathbf{r}-U \tag{39.6} $$
(in p.127, L.D. Landau and E.M. Lifshitz Mechanics )
The problem is the rectlinear term. Why the second term in (39.4) $m\mathbf{W}\cdot\mathbf{r}^\prime$ simply converts into the third term in (39.6) $m\mathbf{W}\cdot\mathbf{r}$. Radius vectors $\mathbf{r}$ and $\mathbf{r}'$ have rotational relation, hence I guess it can't be just replaced each other.