Lagrangian mechanics, as you know, is very useful in today's physics. But there is a point that I can't understand. In cases where we can write [itex] L=T-V [/itex], Lagrangian mechanics is very useful because for some problems, it gives us a easier way than Newtonian mechanics to derive the equations of motion. But in cases where we can't write [itex] L=T-V [/itex], it seems that people just look for a Lagrangian to give the right equations of motion. But that is useless! For example, we know that for Newtonian gravity we have [itex] \nabla^2 \Phi=4\pi G \rho [/itex] and look for a Lagrangian density to give that same equation and after finding it, they say we have the variational formulation of Newtonian gravity. My question is, why not just use the equation we know? Why should we use something we know to derive something that gives the same thing we knew? That seems non-sense! Of course its useful for something we're just trying to understand. I mean, when we have no law for some system, we can look for something in it that always gets extremum and then find a variational theory for it and then find the laws of motion.