2D Polar Coordinate
The two-dimensional polar coordinate uses the two parameter vectors \(\boldsymbol{r}\) and angle vector \(\boldsymbol{\phi}\). It is simply defined by unit vectors,
\[
\boldsymbol{r} = r\boldsymbol{\hat r}
\]
\[ \boldsymbol{\phi} = \phi\boldsymbol{\hat \phi} \]
Cartesian to Polar
We make the transformation given \(x, y\) by,
\[\begin{split}
\begin{align}
\begin{pmatrix}
x = r \cos \phi\\
y = r \sin \phi
\end{pmatrix}
\longleftrightarrow
\begin{pmatrix}
r = \sqrt{x^2 + y^2}\\
\phi = \arctan (y/x)
\end{pmatrix}
\end{align}
\end{split}\]
Velocity
There is an unfamiliar property for the rate of change in polar coordinates compared to cartesian coordinates. Most prominent is velocity, \(\boldsymbol{\dot r}\) which is,
\[ \boldsymbol{\dot r} = \frac{dr}{dt} \boldsymbol{\hat r} + r\frac{d\boldsymbol{\hat r}}{dt} \]
Which may be simplified to,
\[\begin{split}\begin{align}
\boldsymbol{v} \equiv \boldsymbol{\dot r}\\
v_r \equiv \dot r \quad v_\phi &\equiv r \dot \phi
\end{align}\end{split}\]
\[\boxed{\boldsymbol{v} \equiv v_r \boldsymbol{\hat r} + v_\phi \boldsymbol{\hat \phi}} \]
Derivation
Naively one may assume \(d\boldsymbol{\hat r}/dt = 0\) however this is not the case because \(\boldsymbol{\hat r}\) is allowed to change directions over time. Let’s take for granted and assume \(d\boldsymbol{\hat r}/dt = \dot \phi \boldsymbol{\hat \phi}\)
\[\begin{split} \begin{align}
\boldsymbol{\dot r} &= \frac{dr}{dt} \boldsymbol{\hat r} + r\frac{d \phi}{dt}\boldsymbol{\hat \phi} \\
\boldsymbol{\dot r} &= \dot r \boldsymbol{\hat r} + r \dot \phi \boldsymbol{\hat \phi}
\end{align}
\end{split}\]
More familiar uses the identity,
\[\begin{split}\begin{align}
\boldsymbol{v} \equiv \boldsymbol{\dot r}\\
v_r \equiv \dot r \quad v_\phi &\equiv r \dot \phi
\end{align}\end{split}\]
\[\boxed{\boldsymbol{v} \equiv v_r \boldsymbol{\hat r} + v_\phi \boldsymbol{\hat \phi}} \]
Acceleration
The second time derivative or acceleration is defined as,
\[ \boldsymbol{a} \equiv \boldsymbol{\ddot r} = \boldsymbol{\dot v} \]
Knowing \(d \boldsymbol{\hat \phi}/dt\) gives us acceleration as,
\[ \frac{d\boldsymbol{\hat\phi}}{dt} = -\frac{d\phi}{dt}\boldsymbol{\hat r} \]
\[ \boldsymbol{a} = (\ddot r - r\phi^2)\boldsymbol{\hat r} + (r \ddot \phi + 2\dot r \dot \phi)\boldsymbol{\hat \phi} \]
Given that \(\dot r = 0\) thus \(\ddot r = 0\) gives us a somewhat more familiar form of acceleration,
\[ a = -r\dot \phi^2 \boldsymbol{\hat r} + r\ddot \phi \boldsymbol{\hat \phi} \]
\[ a_\text{c} \equiv r\dot \phi^2 = r\omega^2, \qquad a_\perp \equiv r\ddot \phi = r\alpha\]
\[ a = a_c \boldsymbol{\hat r} + a_\perp \boldsymbol{\hat \phi} \]
\(\alpha\) : angular acceleration
\(a_c\) : centripetal acceleration
\(a_\perp\) : tangential acceleration