# Rotating coordinate system

## Alexei Gilchrist

In a rotating coordinate system, a free particle moves in a way that appears to be affected by three forces: the centrifugal force, the Coriolis force and the Euler force. These forces arise only from the rotation of the coordinate system.

Consider a particle with some initial velocity but no forces acting upon it. In the “lab frame” the Lagrangian is just

$\begin{equation*}\mathcal{L} = \frac{1}{2} m (\dot{x}^2+\dot{y}^2).\end{equation*}$
which leads to motion with a constant velocity as per Newton’s first law.

However, if we consider what the motion of the particle looks like in a coordinate frame that is rotating with respect to the lab frame by some $$\theta(t)$$, we will find all sorts of “fictitious” forces show up. Consider the diagram below

The coordinates in which the particle is undergoing free motion ($$x$$ and $$y$$) and the coordinates in the rotating axes ($$x_r$$ and $$y_r$$) are related to each other by

\begin{align*}x &= x_r\cos\theta - y_r\sin\theta \\ y &= x_r\sin\theta + y_r\cos\theta\end{align*}
so that the velocities are
\begin{align*}\dot{x} &= \dot{x}_r\cos\theta -x_r\sin\theta\dot{\theta}- \dot{y}_r\sin\theta -y_r\cos\theta\dot{\theta} \\ \dot{y} &= \dot{x}_r\sin\theta +x_r\cos\theta\dot{\theta}+ \dot{y}_r\cos\theta - y_r\sin\theta\dot{\theta}\end{align*}

Substituting these values in for the Lagrangian we have

$\begin{equation*}\mathcal{L} = \frac{1}{2} m \left(\dot{\theta}^2(\dot{x}_r^2+\dot{y}_r^2) +2\dot{\theta}(x_r\dot{y}_r-y_r\dot{x}_r)+\dot{x}_r^2+\dot{y}_r^2\right).\end{equation*}$
The Euler-Lagrange equations then yield
\begin{align*}m\ddot{x}_r & = m\dot{\theta}^2x_r + 2m\dot{\theta}\dot{y}_r + m\ddot{\theta}y_r \\ m\ddot{y}_r & = m\dot{\theta}^2y_r -2m\dot{\theta}\dot{x}_r - m\ddot{\theta}x_r .\end{align*}
These terms are the centrifugal force, the Coriolis force, and the Euler force. That is, in a rotating coordinate system three forces suddenly appear that are due only to the rotation of our point of view:
$\begin{equation*}F_\mathrm{centrifugal} = m\dot{\theta}^2\begin{pmatrix} x_r \\ y_r\end{pmatrix} \quad F_\mathrm{Coriolis} = 2m\dot{\theta}\begin{pmatrix} \dot{y}_r \\ -\dot{x}_r\end{pmatrix} \quad F_\mathrm{Euler} = m\ddot{\theta}\begin{pmatrix} y_r \\ -x_r\end{pmatrix}\end{equation*}$
The centrifugal force is directed radially outwards and it’s the force that presses you against the side of a car as it turns a corner. The Coriolis force depends both on the angular velocity of the coordinate system and on the velocity within that coordinate system. It has a significant effect on weather patterns on Earth but not on the direction of a toilet’s flush! Finally, the Euler force depends on the acceleration of the coordinate system.