Lagrangian mechanics and geodesics in configuration space?

Question

In lagrangian mechanics Is the path that take a System in the configuration space between initial and final state is identical to the geodesic which connect the two points?

Answer 1

I think this will depend upon the details of the actual action. In many common cases it will be.

Moreover, we require the configuration manifold to be at least a pseudo-Riemannian manifold in order to even define the notion of a geodesic.