 Rotating reference frame

A rotating reference frame uses spatial axes whose orientation changes relative to an inertial frame. Coordinates in those axes can simplify an apparatus rotating at a known angular velocity $\mathbf{\Omega }$, but the equations of motion acquire inertial terms because the basis vectors themselves change.

For a shared fixed origin and constant $\mathbf{\Omega }$, a particle at rotating-frame position $\mathbf{r}$ and velocity $\mathbf{u}$ satisfiesThe additional terms are the Coriolis force and centrifugal force. A time-dependent $\mathbf{\Omega }$ adds an Euler term $-m\dot{\mathbf{\Omega }}\times \mathbf{r}$; a translating origin can add another term. The rotating-axes theorem derives these contributions by differentiating the position twice, rather than treating them as new interactions with another body.^[ref]

A rotating reference frame is a frame of reference in which the observer is undergoing rotation around an axis. In such frames, the laws of physics remain valid, but additional apparent forces, known as fictitious or pseudo-forces, must be considered due to the rotation. Two primary examples of these fictitious forces are the Coriolis force and the centrifugal force.