> If you take the geodesic form of the Lagrange equations, then you can take the starting positions and generalized velocities and keep parallel transporting the velocities along themselves. That will produce a solution that AFAICT is always physical and the process does not require solving an optimization problem.
> Daniel, Isn't that the general idea that Einstein used in his relativity formulation?
What is the general relationship between the Langrangian dynamics and the general theory of relativity? Since both have a geodesic formulation, their metrics must agree enough to give the same paths of motion in gravitational fields -- the way that nature actually behaves.