FTP, what principle says that the correct path is the one on which this value is a minimum?
Feynman used this principle as a basis for constructing his path integral formulation of quantum mechanics.  While the Newtonian approach constructs trajectories point by point, this principle asks why a particular trajectory happens, and is thus more global.  In theory, all one needs to do is calculate the time integral of the Lagrangian for every trajectory.  FTP, what principle says that the correct path is the one on which this value is a minimum?
Feynman used this principle as a basis for constructing his path integral formulation of quantum mechanics.  
principle of least action