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.
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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
\[\begin{equation*}\mathcal{L} = \frac{1}{2} m \left(\dot{\theta}^2(x_r^2+y_r^2) +2\dot{\theta}(x_r\dot{y}_r-y_r\dot{x}_r)+\dot{x}_r^2+\dot{y}_r^2\right).\end{equation*}\]
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.
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