# Lagrangian mechanics and initial conditions vs boundary conditions

It bothers me that many basic books on the classical mechanics don't discuss the following difference between "Newton's laws" and the "Principle of stationary action". Newton's laws can predict the behavior of a system if one sets initial positions and velocities. On the contrary, if we want to construct an action functional we have to set an initial and a final position. From the mathematical point of view, this is the difference between initial and boundary value problem, which have different qualitative behavior (BVPs may have many solutions or no solution).

My question: Which approach is more reasonable - the initial value problem or the boundary value problem?

• More reasonable for what? They are different types of problem and each has its uses. – Nathaniel Jan 26 '15 at 12:50
• Possible duplicates: physics.stackexchange.com/q/38348/2451 and links therein. – Qmechanic Jan 26 '15 at 12:53
• Thanks, I don't understand why I missed the other answer! – user127911 Jan 26 '15 at 13:02

• I've always had this intuitive view: Suppose we have a path $q$ of a particle that runs from point $A$ to point $B$. The action principle tells us that the action is stationary along $q$. We therefore conclude that $q$ satisfies the equations of motion. But the converse is not generally true, right? If $q$ satisfies the equations of motion (the EL equations), then there is no need for it to hit $B$ assuming only that it starts at $A$. Is this right? – Ryan Unger Jan 26 '15 at 13:44