In Landaus mechanics he claims the homogeneity of space/time allows us to drop the $q$/$t$ dependence of the lagrangian for free space. Even though I don’t see a way to “prove” this I guess we can accept it. However he then claims that a Galilean boost will let the Lagrangian differ up to a total time derivative.
What gives for this distinction? Why are these transformations regarded as so drastically different despite the fact that any two observers whose positions are related by some boost will not be able to differentiate space?
