Acceleration in polar coordinates ($\mathbf{a}=(\ddot{r}-r{\dot{\theta }}^2)\hat{\mathbf{r}}+(2\dot{r}\dot{\theta }+r\ddot{\theta })\hat{\mathbf{\theta }}$)

Differentiating $\mathbf{r}=r\hat{\mathbf{r}}$ twice and using $\dot{\hat{\mathbf{r}}}=\dot{\theta }\hat{\mathbf{\theta }}$ and $\dot{\hat{\mathbf{\theta }}}=-\dot{\theta }\hat{\mathbf{r}}$ gives the displayed acceleration. For a central force, its angular component vanishes, so $r^2\dot{\theta }$ is constant and expresses conservation of angular momentum.