3. Transformation rules for polar coordinates#
The relation between Cartesian coordinates and polar coordinates are
(3.34)#\[x = r \cos (\theta), \quad y = r \sin (\theta)\]
Differentiating both sides with respect to time results in
(3.35)#\[\begin{split}\begin{aligned}
\dot{x} &= \dot{r} \cos (\theta) - r \sin (\theta) \dot{\theta} \\
\dot{y} &= \dot{r} \sin (\theta) + r \cos (\theta) \dot{\theta}
\end{aligned}\end{split}\]
Now note that
\[\begin{split}
\begin{aligned}
\sin(\theta) \dot{x} + \cos(\theta) \dot{y} &= \dot{r} \\
\cos(\theta) \dot{y} - \sin(\theta) \dot{x} &= r \dot{\theta}
\end{aligned}
\end{split}\]
Substituting \(\sin{\theta}=y/r\) and \(\cos(\theta) = x/r\) we obtain
(3.36)#\[\begin{split}\begin{aligned}
\dot{r} = \frac{x \dot{x} + y \dot{y}}{r} \\
\dot{\theta} = \frac{x \dot{y} - y \dot{x}}{r^2}
\end{aligned}\end{split}\]
Equations ((3.36)) are used to transform the ODE to polar coordinates. The inverse transform is defined by ((3.35)).