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author | Thomas Albers <thomas@thomaslabs.org> | 2023-03-08 23:43:00 +0100 |
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committer | Thomas Albers <thomas@thomaslabs.org> | 2023-03-08 23:43:00 +0100 |
commit | 61b5ce20f25c5785e41574998a12c6d06eb05a5e (patch) | |
tree | 20e2225b4f30b15d8dee30351041d1f33d42b34a /src/math/polar_coordinates.org | |
parent | 561bac75579391c14e47eaccfabdf9eda98855da (diff) |
Diffstat (limited to 'src/math/polar_coordinates.org')
-rw-r--r-- | src/math/polar_coordinates.org | 34 |
1 files changed, 34 insertions, 0 deletions
diff --git a/src/math/polar_coordinates.org b/src/math/polar_coordinates.org new file mode 100644 index 0000000..0d26138 --- /dev/null +++ b/src/math/polar_coordinates.org @@ -0,0 +1,34 @@ +#+title:Polar Coordinates +#+setupfile: ../../math_options.org + +* 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*} |