math/polar_coordinates.org


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

#+TITLE:Polar Coordinates
#+SETUPFILE: ../math_options.org
#+LATEX_HEADER: \usepackage{bm}
#+LATEX_HEADER: \usepackage{mathtools}
#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}#1}{\text{d}#2}}
#+LATEX_HEADER: \newcommand{\unitv}[1]{\bm{\hat{e}}_#1}

* Disclaimer
This site as of now just a technology demonstration and its claims
should not be taken as true (even though I myself am pretty confident
they are)
* Coordinate transformations
\begin{align*}
x &= r \cos\theta\\
y &= r \sin\theta
\end{align*}

* Local unit vectors
\begin{align*}
\unitv{r} &= \cos\theta \unitv{x} + \sin\theta \unitv{y}\\
\unitv{\theta} &= -\sin\theta \unitv{x} + \cos\theta \unitv{y}
\end{align*}

* Kinematic in polar coordinates
** Time derivatives of the local unit vectors
\begin{align*}
\deriv{\unitv{r}}{t} &= \dot{\theta}\unitv{\theta}\\
\deriv{\unitv{\theta}}{t} &= -\dot{\theta}\unitv{r}
\end{align*}
** Position vector and its time derivatives
\begin{align*}
\bm{r} &= r\unitv{r}\\
\bm{v} &= \dot{r}\unitv{r} + r\dot\theta\unitv{\theta}\\
\bm{a} &= \left(\ddot{r} - r\dot\theta^2\right)\unitv{r}
+ \left(2\dot{r}\dot\theta + r\ddot\theta\right)\unitv{\theta}
\end{align*}