davidbrown3 : Rotating Frames of Reference

Consider two frames of reference, $A$ and $B$. Frame $A$ is stationary and can be said to be inertial. Frame $B$ is rotating relative to frame $A$.

A point in space, $P$ can be located using a vector observed in either frame of reference and expressed using a combination of their respective $i$, $j$ and $k$ components;

$v^A = [x_{i}^A;x_{j}^A;x_{k}^A]$

$v^B = [x_{i}^B;x_{j}^B;x_{k}^B]$

The velocity of point $P$ as observed from either frame of reference can also be expressed using the derivative of the same components;

$\dot{v^A} = [\dot{x_{i}^A};\dot{x_{j}^A};\dot{x_{k}^A}]$

$\dot{v^B} = [\dot{x_{i}^B};\dot{x_{j}^B};\dot{x_{k}^B}]$

If the rotation matrix that transforms reference frame $A$ to $B$ is known and expressed as $R_{A}^{B}$ then an additional expression can be written that expresses the position vector expressed in reference $A$ to the same vector expressed in reference $B$;

$v^B =R_{A}^{B}\cdot{v}^A$

By chain-rule differentiation it is also possible to write an expression relating the velocity of point $P$ expressed in reference $A$ to the same velocity expressed in reference $B$;

$\dot{v^B} = \dot{R_{A}^{B}}\cdot{v}^A+R_{A}^{B} \cdot\dot{v^A}$

Using the same method again, an expression relating the acceleration of point $P​$ expressed in reference $A​$ to the same acceleration expressed in reference $B​$ can be written;

$\ddot{v^B} = (\ddot{R_{A}^{B}}\cdot{v}^A+\dot{R_{A}^{B}}\cdot{\dot{v}^A})+(R_{A}^{B} \cdot\ddot{v^A}+\dot{R_{A}^{B}} \cdot\dot{v^A})$

The expressions of this can be grouped to form the equation;

$\ddot{v^B} = 2\cdot\dot{R_{A}^{B}} \cdot\dot{v^A}+\ddot{R_{A}^{B}}\cdot{v}^A+R_{A}^{B} \cdot\ddot{v^A}$

From Netwons first law we can relate acceleration in an inertial frame to the force applied;

$F^A= M\cdot\ddot{v^A}$

Inserting this into the previous equation;

$\ddot{v^B} = 2\cdot\dot{R_{A}^{B}} \cdot\dot{v^A}+\ddot{R_{A}^{B}}\cdot{v}^A+R_{A}^{B} \frac{F^A}{M}$

It is seen that the Coriolis and Centripetal terms are interpreted as if they themselves were forces, which is why they are often called 'Apparent Forces'. However, there is no physical origin behind them nor do they create any reaction force. Their existence is purely a result of matrix transformations.