I read a book by Professor Liu Chuan. When talking about action principle, the author says:
In classical mechamics, the true trajectory of a system is unique. Thus a trajectory obtained by action principle in one coordinate frame should be the trajectory after Lorentz transformation. To fullfill this aim, assuming the action is Lorenzt invariant is the simplist choice.
Here is my question:
Wouldn't it be allowed if $S1$ (before transformation) and $S2$ (after transformation) differ by a const, sinces actions differing by constants would give the same Euler-Lagragian equation?
How could "action is Lorentz invariant" guarantee "one trajectory in this coordinate frame is the trajectory in another frame"?
