As we know, the variational principle can be used as the fundamental principle of mechanics. Without knowing Newton's laws, the Lagrangian could be derived from symmetric consideration. As the simplest case, the Lagrangian of a free particle could be derived from Galileian invariance, or the homogeneity of space and time. But I never saw a rigrious proof. I think we just "guess" the form of the Lagrangian for some reason rather than prove it. In Landau's famous "Mechanics", he said that the Lagragian of a free particle must be a function of v^2 only. This may be a cool argument of Landau's style, full of physics intuition, but not a rigrious proof. Is it possible to prove that "if space and time are homogeneous, the Lagrangian must be of the form L(v)+df(x,t)/dt"?