Polar Coordinates#

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,

\[\begin{split} \begin{align} \begin{pmatrix} x = r \cos \phi\\ y = r \sin \phi \end{pmatrix} \longleftrightarrow \begin{pmatrix} r = \sqrt{x^2 + y^2}\\ \phi = \arctan (y/x) \end{pmatrix} \end{align} \end{split}\]

  • This transformation is not mathematically general since we use \(\arctan\). More properly use \(\arctan2\)

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 time1. 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

1

When learning vectors in cartesian coordinates we take for granted that \(d\boldsymbol{\hat x}/dt = 0\) which is definitely true since \(\boldsymbol{\hat x}\) never changes directions.