src/math/polar_coordinates.org


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#+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*}