Calculating geodesic curves on the graph of the gaussian function

Question

I am trying to calculate the equation of the shortest path between two points on the graph of the function $z=e^{-a(x^2+y^2)}$. I think the paramtrization that gives the most comfortable metric $g_{ij}$ is this: $$\varphi(r,\theta)=(r\cos\theta,r\sin\theta,e^{-ar^2})$$ the metric is $$(g_{ij})=\begin{pmatrix} 1+4a^2r^2e^{-2ar^2} & 0\\ 0&r^2 \end{pmatrix}$$ The geodesic equations and euler-lagrange equations are both turn out really 'ugly'. Is there any other way to find geodesic on this surface? I'll appreciate guidance.