Polar coordinates

Polar coordinates are a two-dimensional set of coordinates where each point on the plane is determined by its distance from a reference point called the pole and its angle from a reference direction called the polar axis. The distance is called the radial coordinate, while the angle is called angular coordinate or sometimes azimuth.

Relation to Cartesian coordinates

Polar coordinates (r,θ)(r,\theta) can be converted to Cartesian coordinates (x,y)(x,y) by using the Coordinate transformation

φ:R2→R2,φ(r,θ)=(rcos⁡θ,rsin⁡θ)=(x,y)\varphi:\mathbb{R}^{2}\to \mathbb{R}^{2},\qquad \varphi(r,\theta)=(r\cos \theta,r\sin \theta)=(x,y)

Motion

The distance from the origin is represented as vector r\mathbf{r}, with a unit vector R=r/∣r∣\mathbf{R}=\mathbf{r}/|\mathbf{r}| representing its axis. As a unit vector, R\mathbf{R} can be written like R=(cos⁡θ,sin⁡θ)\mathbf{R}=(\cos\theta,\sin\theta). P=(−sin⁡θ,cos⁡θ)\mathbf{P}=(-\sin\theta,\cos\theta) is perpendicular to R\mathbf{R} and is just a Rotation by π/2\pi/2 of it. The two vectors constitute a Basis, and with the position vector they form the Moving frame

{r(t);R(t),P(t)}\{\mathbf{r(t)};\mathbf{R}(t),\mathbf{P}(t)\}

center

The derivatives of the frame are, in antisymmetric matrix form,

(R˙P˙)=(0θ˙−θ˙0)(RP)\begin{pmatrix}\dot{\mathbf{R}} \\ \dot{\mathbf{P}}\end{pmatrix}=\begin{pmatrix}0 & \dot{\theta} \\ -\dot{\theta} & 0\end{pmatrix}\begin{pmatrix}\mathbf{R} \\ \mathbf{P}\end{pmatrix}

The velocity is

v=r˙=ddt(rR)=r˙R+rR˙=r˙R+rθ˙P\mathbf{v}=\dot{\mathbf{r}}=\frac{d}{dt}(r\mathbf{R})=\dot{r}\mathbf{R}+r\dot{\mathbf{R}}=\dot{r}\mathbf{R}+r\dot{\theta}\mathbf{P}

and the acceleration is

a=v˙=r¨R+r˙R˙+ddt(rθ˙)P+rθ˙P˙=(r¨−rθ˙2)R+(rθ¨+2r˙θ˙)P\mathbf{a}=\dot{\mathbf{v}}=\ddot{r}\mathbf{R}+\dot{r}\dot{\mathbf{R}}+ \frac{d}{dt}(r\dot{\theta})\mathbf{P}+r\dot{\theta}\dot{\mathbf{P}}=(\ddot{r}-r\dot{\theta}^{2})\mathbf{R}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\mathbf{P}