Rotating Frames and Fictitious Forces

#physics
We consider equations of motion in non-inertial frames, which are accelerating, and try to find the fictitious forces that appear in those frames.
They are mainly two, Centrifugal and Coriolis.

Define two frames,

• $S$ which is fixed, inertial frame. $\hat{i},\hat{j},\hat{k}$ are the basis vectors.
• $S^{\prime }$ which is rotating with the frame we are examining, ${\hat{e}}_1,{\hat{e}}_2,{\hat{e}}_3$ are the basis vectors, and this frame is spinning relative to $S$ with angular velocity $\omega$.

By using Transport Theorem, we already know the relation between a vector in frame $S$ and in frame $S^{\prime }$.

Now we differentiate $$\mathbf{v}{S} = \mathbf{v} + \omega \times \mathbf{r}$$and again to obtain the entire equation

$$\mathbf{a}{S} = \mathbf{a} - \underbrace{ \dot{\omega} \times \mathbf{r} }{ \pu{ Euler } } - \underbrace{ 2\omega \times \mathbf{v} }{ \pu{ Coriolis } } - \underbrace{ \omega \times (\omega \times \mathbf{r}) }{ \pu{ Centrifugal } } $$

Euler force is also called Azimuthal Force

Harmonic Oscillator