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author | Thomas Albers <thomas@thomaslabs.org> | 2022-07-27 18:13:20 +0200 |
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committer | Thomas Albers <thomas@thomaslabs.org> | 2022-07-27 18:13:20 +0200 |
commit | 561bac75579391c14e47eaccfabdf9eda98855da (patch) | |
tree | 00a65bb3c590a072beb9552ccdea8bfa096b8502 /math/polar_coordinates.org |
Initial commit
Diffstat (limited to 'math/polar_coordinates.org')
-rw-r--r-- | math/polar_coordinates.org | 36 |
1 files changed, 36 insertions, 0 deletions
diff --git a/math/polar_coordinates.org b/math/polar_coordinates.org new file mode 100644 index 0000000..ff07be3 --- /dev/null +++ b/math/polar_coordinates.org @@ -0,0 +1,36 @@ +#+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*} |