A Method for Preserving Geometric Meaning in the Calculus of Variations

We show that the Calculus of Moving Surfaces offers distinct advantages over the Euler-Lagrange equation for optimization problems originating in Geometry. An extension of Tensor Calculus, it provides tools for analyzing invariant representations rather than specific coordinate representations of relevant quantities, which allows us to avoid the myriad of hardships associated with the use of coordinates from untenable complexity of analytical expressions to virtual impossibility of recovering the geometric interpretation of the final result. As an illustration, we solve the brachistochrone problem and give a geometric characterization for the desired curve in terms of its curvature.