How to obtain the geodesic equation from the action?

+1 vote

Hi!
I would be grateful if someone tells me how to obtain the geodesic equation from, for example, the cadabra notebook

I mean: is there a command such as "geodesic(_);" to readily obtain it?
Thanks a lot!
Mario

I don't understand your question. Do you want to obtain the equation from a variational principle or obtaining the differential equations from a set of connection components?

Hi doxdrum,
Thank you for paying attention to this possibly dumb question. I am new to cadabra, slowly figuring out how to use it. My question is if there is a simple instruction of the kind "geodesic(action,...)" to immediately give the well-known geodesic equation, or the equation of motion usually presented in the form

x"^a+\Gamma^a_{bc} x'^b x'^c=0

where \Gamma is the metric connection of g_{ab} could be directly obtained from the action.
Thank you very much!

+1 vote

So far there is no build-in command that spits the geodesic equations.

However, I'd like to clarify: the geodesic equation can be obtain by extremising (one half of) the Riemannian line element for a given metric, and it is given by the expression stated by @skyfold in the comments of the question.

It is, nonetheless, straightforward to obtain the differential equations for the geodesic---once one knows the metric---. The only nontrivial ingredient is to define the (Levi-Civita) connection in terms of the metric, and calculate the coefficients. This has been shown in the notebook about Schwarzschild spacetime.