While the idea is obviously correct, the paper itself suffers from extremely sloppy writing. They discretize the integral with a discrete sum, but then forget

While the idea is obviously correct, the paper itself suffers from extremely sloppy writing. They discretize the integral with a discrete sum, but then forget to discretize the variables by substituting x with x(t_i) or at least x_i, same for dot x. They put the objective function x hat = argmin S(X) last, when it is the most important aspect. In the equation where x hat must fulfill the Euler lagrange equation for all t, they butchered the application of the derivative with respect to a constant point. It should look more like this: https://wikimedia.org/api/rest_v1/media/math/render/svg/6efe74342a2cc42ad4fc5e120bf5c46d76777f4e https://wikimedia.org/api/rest_v1/media/math/render/svg/6efe... You need to explicitly pass in the x(t), dot x (t) and t as arguments into the derivative. Their notation implies that you have to take the derivative with respect to a constant (not at a point) which always returns zero (a blatantly banal property) or that the function (=the laws of physics) behind x(t) varies over time (shudder). Overall this was extremely unpleasant to read even though the approach is neat.