I've been doing some calculations of geodesics in different Riemannian Manifolds. More precisely I'm trying to compute, given two points on a Riemannian Manifold, the smallest distance between those points.
I started by trying to compute the distance of two points on a 2-sphere. That was easy. Then I thought to consider not a 2-sphere but a 2-sphere with some kind of a wormhole in it, something like a doughnut...
I thought of using the following metric: $$ ds^2=(1-b(r)/r)^{-1}dr^2+r^2 d\theta^2 + r^2\sin^2\theta d\varphi^2, $$ where $b(r)$ is some function that should describe the throat of the wormhole - as it can be seen here. First question: Is this correct for what I'm looking for?
Then, given this metric, the distance is given by: $$ l=\int{ds}=\int_{\theta_1}^{\theta_2}[(1-b(r)/r)^{-1}\dot{r}^2+r^2+ r^2\sin^2\theta \dot{\varphi}^2]^{1/2}d\theta, $$ right? And then, I should make use of the Euler-Lagrange equations...Am I correct?
Mark L. Irons posted a very nice and detailed discussion of geodesics on round tori. More detail in the PDF version.
Jim Belk gives simple tips for finding geodesics.
If you have access to jstor and enjoy reading old papers with lots of explicit computations, take a look at The Geodesic Lines on the Helicoid by S. E. Rasor. The author derives equations for all geodesics on both catenoid and helicoid. Most of these equations involve elliptic integrals.