In lagrangian variation we are trying to minimize the action S = ∫t2t1 L dt. Consider a simple case of free particle. Imagine In a world that everyone one only knows how to solve ODE, Using euler lagrange equation, one has d2x/dt2 = 0 , give that we know the initial position of particle in the phase space, the people can solve for the motion. Now imagine in a world that everyone only know variation principle. (They have some ways to measure action in every possible path and thus find out the least action one). They need to vary the path while KEEPING BOTH INITIAL point and end point fixed in the phase space. Then they can vary the path and find out the true one. My question is, why in the first kind of world people only need to know about initial position in phase space but in the second kind of world people must know about the ending position in the phase space too ? This "inconsistency of information" bothers me a lot. I appreciate anyone's help in explain or pointing out my misconception. Thanks.