As far as I understand, Hamilton's principle (a.k.a. "least action" or "stationary action") requires that you know both the initial and final location of a particle. Then, based on the requirement that the action must be stationary along any "possible" path, it will tell you what path(s) the particle possibly could have taken between those two points. What I'm wondering is, is there some sort of "initial value" form of this principle? That is, if I know the initial position and the initial velocity of a particle, can I determine the path it's going to take by minimizing (stationary-izing) something? Is that something, in fact, the Lagrangian?