Variational Methods (Finite Constraints - Geodesics Example)

[@user-hx2wo8hf7k]

Finally! Thanks a lot, I've got it. When doing this myself I got quite confused with all those fractional powers...Still there is one thing I can't figure out... Namely, aren't we getting the same result if we simply remove the power 1/2 from equation (2), meaning that in that case we would be integrating the kinetic energy? This would save a lot of space and efforts... As far as I see a geodesic line occurs when a material point moves on a surface by inertia. Thus, the Lagrangian function (for this classical problem) is the kinetic energy T plus lambda(t)*G and there are no square roots here. Then writing out the classical E-L equations yields the desired equations for geodesics. Could that square root (on a legal basis, of course) be removed from (2) immediately at the very beginning?