my question is are the solutions of the Euler -Lagrange Equation always OPTIMAL , in the sense of the best solution
for example for the 'Brachistochrone' problem could exist a piecewise linear function (non differentiable) that gave an 'smaller' time to descend from point A to point B ??
and the same for Differential Geometry and minimal surfaces , could exist better solution to these problems ( in any spatial dimension) but these solutions do not follow Euler Lagrange Equation ?