Let's say I have a Riemannian manifold with known metric $g$. I want to compute the geodesics of this manifold, specifically with respect to the Levi-Civita connection.
In practice, (i.e. with a computer), how would I go about this? I'm wondering what kind of algorithms exist for computing/estimating geodesics. Looking for the 'typical method' and/or references.
This may not be helpful, but I know the definition of a geodesic is a curve $\gamma: [0,t] \rightarrow M$ that minimizes the energy $$E(\gamma) = \int_0^t ||\gamma'(s)||^2 ds$$
where $|| \cdot ||$ is with respect to the metric $g$.
This seems difficult to optimize in practice, and in addition, how do I find this for the Levi-Civita connection in particular?
