According to this post, it seems like we should use the $$\mathbb{L}~=~L \mathrm{d}t~=~ L \dot{t}\mathrm{d}\lambda \tag1$$ rather than $$L(v'^2) = L(v^2)+ \frac{df}{dt}\tag2$$ when working with special relativity and relating different Lagrangians. However, the latter can be understood as this: if we think the Lagrangian in different frames really describes the same motion, and if $$L' = L+ \frac{df}{dt}\tag3$$ describes the same motion with $L$ in the coordinate system we choose, when we convert the physical quantities in another frame into the frame we are working with, i.e using Lorentz transformation to transform quantities like velocities, we should expect (2) to hold. Why it's that not the case?
Edit: One of my premises is that all the Lagrangians that describe the same motion in one coordinate system must only differ by a total derivative of a function, which seems to appear in Landau's argument for free particle's Lagrangian for Galilean invariance. But is it true?
