Beginning at 31:03 Dr. Susskind presents an intuitively very satisfying derivation of the Euler-La Grange equation(s). But, I'm not convinced it is rigorous. It seems his choice of variation is not the only possible choice for the neighborhood he selected. The reason this matters to me is because, in his derivation velocity and position seem to be implicitly coupled. My understanding of the Euler-La Grange equations is that position and velocity are independent variables. Lemons Section 2.2 gives a more abstract and symbolic derivation. I understand that to say that: given any variation in position, there are an infinite number of variations in velocity possible, and vis-versa. Can these two derivations be shown to be equivalent?