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?