Equations for geodesics (in polar coordinates)?

Question

How are these equations obtained

$$\frac{d}{dt}(Ep' + F\theta') = \frac{1}{2}\left( E_p (p')^2 + 2F_p (p'\theta') + G_p (\theta')^2\right)$$ and

$$\frac{d}{dt}(Fp' + G\theta') = \frac{1}{2}\left( E_{\theta} (p')^2 + 2F_{\theta} (p'\theta') + G_{\theta} (\theta')^2\right)$$

for geodesics in polar coordinates (E=1, F=0)?

Answer 1

I think I've got it.

given the metric \begin{equation} ds=\sqrt{E(\rho,\theta)u'^{2}+2 \rho' \theta' F(\rho,\theta)+G(\rho,\theta) \theta'^{2}} \end{equation} we compute the partial derivatives: \begin{equation} \frac{\partial ^2s}{\partial \rho}=\frac{1}{2 \partial s} \big(\rho'^{2} E_{\rho}+2 \rho' \theta' F_{\rho}+\theta'^2 G_{\rho} \big), \end{equation} \begin{equation} \frac{\partial ^2s}{\partial \rho'}=\frac{1}{\partial s} \big(\rho' E+\theta' F \big), \end{equation} \begin{equation} \frac{\partial ^2 s}{\partial \theta}=\frac{1}{2 \partial s} \big(\rho'^{2} E_{\theta}+2 \rho' \theta' F_{\theta}+\theta'^2 G_{\theta} \big), \end{equation} \begin{equation} \frac{\partial ^2 s}{\partial \theta'}=\frac{1}{\partial s} \big(\rho' F+ \theta' G \big). \end{equation} where $E_x$, $F_x$, and $G_x$ are the derivative of $E$, $F$, and $G$ with respect to the coordinate $x ={\rho,\theta}$, and $\rho'$ and $\theta'$ are the derivative with respect to the parameter $t$.

By combining the above, we get to the desired equations.