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I'm doing a project in multivariable analysis regarding Kepler's laws of planetary motions and the following equation was a recommended equality to use, but none of the variables were actually defined:

$$\dot{\vec{r}}=\dot{r}\hat{r}+r\dot\theta\hat\theta$$

$r$ and its variations obviously refer to the radius and its derivative (I'm assuming in relation to time?), but what does $\dot\theta$ and $\hat\theta$ refer to in this case?

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The vector equation you posted is written is polar coordinates. The $\hat{\theta}$ vector is a unity vector orthogonal to the radius vector and it points tangent to the circle (or the elipse, your orbit etc...) in the direction where the angle theta changes (see image below). The theta dot is the variation rate of the angle theta. Is a derivative with respect to time. It measures the angular velocity of, for example, a planet in its orbit.

enter image description here

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  • $\begingroup$ The $\hat{\theta}$ vector is not necessarily tangent to the ellipse. The velocity vector is the one tangent to the trajectory. In the case of elliptical orbit, the velocity has non-zero components along both $\hat{\theta}$ and $\hat{r}$. With the exception of the two extreme points. $\endgroup$ – nasu Sep 18 '18 at 17:22
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I think that $\hat{r}$ is the unit vector along the radius direction and $\hat{\theta}$ the unit vector along the direct orthogonal direction to the radius.

For instance $\vec{r} = r\hat{r}$

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We can prove that for a central force motion the angular momentum vector of the planet remains constant (i.e. $l=mr^2\dot{\theta}$= constant and also direction of $l$, where $l$= angular momentum, $r$= distance from the center of force, $\dot{\theta}$= angular velocity of the planet). This implies the motion of the planet lies in a plane. Thus we use polar co-ordinates.

Now, for a central force motion, $$\vec{r}=r\hat{r}$$ \begin{align} \dot{\vec{r}}&=\frac{d\vec{r}}{dt}=\frac{dr}{dt}\hat{r}+r\frac{d\hat{r}}{dt} \\&=\dot{r}\hat{r}+r\frac{d\hat{r}}{d\theta}\cdot\frac{d\theta}{dt} \\&=\dot{r}\hat{r}+r\dot{\theta}{\hat{\theta}} \end{align}

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