From Landau & Lifshitz The Classical Theory Of Fields it is said:
To determine the action integral for a free material particle (a particle not under the influence of any external force), we note that this integral must not depend on our choice of reference system, that is, it must be invariant under Lorentz transformations.
This seems understandable. But in comments on this answer Ján Lalinský says that "there is no good physical reason why action should be invariant". Further he suggests another Lagrangian than that given in L&L, namely, if we denote L&L Lagrangian as $L_0$, then the example could be $L_0+Cv_x$, which is clearly anisotropic. Clearly, the equations of motion must not change with this Lagrangian, because $Cv_x$ is a total time derivative (of $Cx$).
On the other hand, in this answer Luboš Motl says that "the invariance of the action follows from special relativity – and special relativity is right (not only) because it is experimentally verified. [snip] If $S$ depended on the inertial system, so would the terms in the equations $\delta S=0$, and these laws of motion couldn't be Lorentz-covariant".
How could I connect L&L and Luboš's arguments with Ján's example? Can both sides be simultaneously right? They seem to contradict each other.
Best Answer
Jan's example is in some sense trivial, since he has added a term which is essentially adding a constant to the action, and a constant is Lorentz invariant. It's just the constant. So what you really want is a Lagrangian whose resulting action is Lorentz invariant.
The Lubos answer is correct -- the action must be a Lorentz invariant. The Lagrangian need not be, so long as it can be written as $L = L_{\textrm{lorentz invariant}} + \dot{f}$ or some other such formulation that yields a Lorentz invariant action. This distinction between the action being Lorentz invariant and the Lagrangian being Lorentz invariant is important, since it is frequently a source of the anomalies (http://en.wikipedia.org/wiki/Anomaly_(physics)) that appear in QFT.