On p. 5 in $\S$3 pf the book of Mechanics by Landau & Lifshitz, it is claimed that
[...] for a free particle, the homogeneity of space and time implies that Lagrangian cannot depend on position or time, explicitly.[...]
However, in my understanding, Lagrangian of a system is the function that determines the equation of motion, i.e given a the initial conditions and the Lagrangian of the system, we can determine the future configuration of the system, as in the case Newton's second law.
However, we also do know that adding a constant to our Lagrangian, or a time derivative of a function of position and time, the equation of motion does not change, hence, we get an "equivalent" Lagrangian in the sense that both functions lead us to the same conclusion about the dynamics of the system at hand.
Given this, I cannot understand why the Lagrangian of a free particle cannot depend on the position or time, explicitly.
I mean it is clear that if that is the case, we have a simple Lagrangian that satisfy all the properties that you would expect it to have; however, why this is the only case that a free particle can have as a Lagrangian.
Note: I have read this question, but I still cannot understand why does the origin would have a privileged status in that case.