Coordinate Transformations — First Order Systems of ODEs

I just learned a new trick to transform systems of ordinary differential equations in Cartesian coordinates into polar coordinates. It is kind of basic, but leads to interesting physical interpretations, so I will record it here.

As everyone knows, $𝑥 = 𝑟 cos (𝜃)$ and $𝑦 = 𝑟 sin (𝜃)$ . We will use these formula below.

By definition $𝑟^{2} = 𝑥^{2} + 𝑦^{2}$ . Taking the derivative with respect to time $𝑡$ ,

2 𝑟 \dot{𝑟} = 2 𝑥 \dot{𝑥} + 2 𝑦 \dot{𝑦} \Rightarrow 𝑟 \dot{𝑟} = 𝑥 \dot{𝑥} + 𝑦 \dot{𝑦} .

A formula for $\dot{𝑟}$ . We can derive a similar formula for $\dot{𝜃}$ . Differentiate the definitions of $𝑥$ and $𝑦$ with respect to $𝑡$ :

\begin{aligned} \dot{𝑥} & = \dot{𝑟} cos (𝜃) - 𝑟 \dot{𝜃} sin (𝜃) \\ \dot{𝑦} & = \dot{𝑟} sin (𝜃) + 𝑟 \dot{𝜃} cos (𝜃), \end{aligned}

which follows from application of the product rule and chain rule. To isolate $\dot{𝜃}$ , we use a technique. We multiply the equation for $\dot{𝑥}$ by a term proportional to $sin (𝜃)$ , and the equation for $\dot{𝑦}$ with a term proportional to $cos (𝜃)$ . We do this so we can apply the well known identity ${cos}^{2} (𝜃) + {sin}^{2} (𝜃) = 1$ .

\begin{aligned} \dot{𝑥} 𝑦 & = \dot{𝑟} cos (𝜃) 𝑟 sin (𝜃) - 𝑟^{2} \dot{𝜃} {sin}^{2} (𝜃) \\ \dot{𝑦} 𝑥 & = \dot{𝑟} sin (𝜃) 𝑟 cos (𝜃) + 𝑟^{2} \dot{𝜃} {cos}^{2} (𝜃), \end{aligned}

Now, subtracting the two expressions allows us to isolate $\dot{𝜃}$ with the identity:

\begin{aligned} \dot{𝑦} 𝑥 - \dot{𝑥} 𝑦 & = \dot{𝑟} sin (𝜃) 𝑟 cos (𝜃) + 𝑟^{2} \dot{𝜃} {cos}^{2} (𝜃) - (\dot{𝑟} cos (𝜃) 𝑟 sin (𝜃) - 𝑟^{2} \dot{𝜃} {sin}^{2} (𝜃)) \\ = 𝑟^{2} \dot{𝜃} {cos}^{2} (𝜃) + 𝑟^{2} \dot{𝜃} {sin}^{2} (𝜃) \\ = 𝑟^{2} \dot{𝜃} ({cos}^{2} (𝜃) + {sin}^{2} (𝜃)) \\ = 𝑟^{2} \dot{𝜃} . \end{aligned}

Relatively simple combinations of the original expressions in terms of $𝑥$ , $𝑦$ , $\dot{𝑥}$ , and $\dot{𝑦}$ yield simple expressions in polar coordinates. The term $𝑟^{2} \dot{𝜃}$ bears similarity to the $𝑧$ -component of a cross product $\vec{𝑟} \times \vec{𝑣}$ , where $\vec{𝑟}$ is the position vector and $\vec{𝑣}$ is the velocity vector. The term $𝑟 \dot{𝑟}$ bears similarity to a dot product $\vec{𝑟} \cdot \vec{𝑣}$ . These are fascinating similarities to me.