Not really. You will get a different Euler-Lagrange equation.

Now, kudos to Jose for all that manual labor.

If we wished to make things a notch simpler, we could let the plane curve be expressed locally as a graph,

and proceed from there. Anyhow, you possibly won't get anything better than what

wikipedia and

wolfram have to offer under the lemma "Geodesic". That is term assigned to curves that satisfy the Euler-Lagrange equations concerning the variation of length on a surface (and, more generally, on a manifold).

Here's food for thought. Is every geodesic a length minimizing curve between any two of its points?