[gpsx]

I'm not exactly sure from what perspective you ask this question. As far as I know people misspeak when they say to minimize the lagrangian rather than find the stationary points (min max or maybe saddle). Applied to classical physics in the absence of quantum mechanics I don't think there is a good answer. It is just a rule. But, when combined with qunatum mechanics, it explains how classical physics is an approximation to quantum mechanics.

In the path integral formulation of quantum machanics, the evolution of a wave function from state 1 to a later state 2 can be thought of as occuring as a superposition of all possible paths between the two states, with each path contributing a factor of exp(iS) where S is the lagrangian. This means all paths either obeying classical physics or not.

In situations where classical physics is valid these contributions from different paths changes very quickly. The contributions from neighboring paths cancel each other out except at stationary points in the lagrangian, where there is a zero change between neighboring paths. Hence, we see classical physics are the trajectories that are the stationary points of the lagrangian.