I'm talking about this: http://www.cs.cornell.edu/courses/cs6650/2008fa/images/thumb_EL.jpg In the derivation when you minimize action you assume that all the variations in coordinates are independent and thus conclude that each term has to be zero. When this isn't the case anymore one doesn't reach this conclusion. Question 1: Is the above correct, If yes, here follows the real reason for this post: Question 2: One can also derive the E-L equations from d'Alemberts principle of virtual work. One arrives at the same equations. However it seems that no explicit assumption of the variables being independent was ever made. Where does this assumption hide in this derivation?