Editing Rotating reference frame (section)

=== Relation between accelerations in the two frames ===
Acceleration is the second time derivative of position, or the first time derivative of velocity

:<math> 
\mathbf{a}_{\mathrm{i}} \ \stackrel{\mathrm{def}}{=}\   
\left( \frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}}\right)_{\mathrm{i}} = 
\left( \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} \right)_{\mathrm{i}} = 
\left[  \left( \frac{\mathrm{d}}{\mathrm{d}t} \right)_{\mathrm{r}} + \boldsymbol\Omega \times \right]
\left[\left( \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \right)_{\mathrm{r}} + \boldsymbol\Omega \times \mathbf{r} \right] \ ,
</math>
where subscript <math>\mathrm{i}</math> means the inertial frame of reference, <math>\mathrm{r}</math> the rotating frame of reference, and where the expression, again, <math>\boldsymbol\Omega \times</math> in the bracketed expression on the left is to be interpreted as an [[Operator (mathematics)|operator]] working onto the bracketed expression on the right.

As <math>\boldsymbol\Omega\times\boldsymbol\Omega=\boldsymbol 0</math>, the first time derivatives of <math>\boldsymbol\Omega</math> inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide.
Carrying out the [[Derivative|differentiation]]s and re-arranging some terms yields the acceleration ''relative to the rotating'' reference frame, <math>\mathbf{a}_{\mathrm{r}}</math>

:<math> 
\mathbf{a}_{\mathrm{r}} = 
\mathbf{a}_{\mathrm{i}} -
2 \boldsymbol\Omega \times \mathbf{v}_{\mathrm{r}} -
\boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{r}) -
\frac{\mathrm{d}\boldsymbol\Omega}{\mathrm{d}t} \times \mathbf{r}
</math>

where <math>\mathbf{a}_{\mathrm{r}} \ \stackrel{\mathrm{def}}{=}\   \left( \tfrac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}} \right)_{\mathrm{r}}</math> is the apparent acceleration in the rotating reference frame, the term <math>-\boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{r})</math> represents [[centrifugal acceleration]], and the term <math>-2 \boldsymbol\Omega \times \mathbf{v}_{\mathrm{r}}</math> is the [[Coriolis acceleration]]. The last term, <math>-\tfrac{\mathrm{d}\boldsymbol\Omega}{\mathrm{d}t} \times \mathbf{r}</math>, is the [[Euler acceleration]] and is zero in uniformly rotating frames.