Principle of relativity - a second, equivalent form, using invariants Most people state the principle of relativity like this:

"The rules of physics must take the same form in all inertial frames."

Question: is this an equivalent way of saying the same thing:

"The rules of physics must be expressible using only (Lorentz) invariants."

Note the word expressible. A given rule could be expressed in two different ways. A statement using action integrals with a Lagrangian would be an example of using only Lorentz invariants. A statement using covariant differential equation would not.
 A: As far as I know you might be able to do that, but I'm not entirely sure what counts.  For instance, consider the invariant interval $s^2$, it is invariant.  And the equation $s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta y^2$, has the same form in all inertial frames, so we have a covariant right hand side for how to compute the invariant left hand side.  So now when we see a covariant expression that has the above right hand side in it we can replace it that expression with the above left hand side invariant.  So maybe your question is about whether we can take all the known covariant expressions used in the laws of physics and find appropriate invariants to replace them with.  And hopefully avoid reducing to a tautology like A=A that means we buried all the real physics in our definitions of our invariants.  Its not entirely silly since deviation from the laws of physics can be expressed covariantly, so if you made an invariant for each deviation $D_i$, then the invariant laws of physics might be $0=D_i$, with all the important stuff going on in the definitions of the various $D_i$.
But let me further say how much work might be needed, since you give an example of Lagrangians.  In Landau and Lifshitz's The Clasical Theory of Fields page 48 eqaution 16.1 they write $S=\int mcds-\frac{e}{c}A_idx^i$ for the action which looks nice and invariant, but it's just an action, not a law of physics.  But then in equation 16.4 they give the Lagrangian $L=-mc^2\sqrt{1-\frac{v^2}{c^2}}+\frac{e}{c}A\cdot v-e\phi$, which is not invariant (they want $S=\int L dt$).  So your prototypical example of how to do your program actually itself would require work.
That said, it might already have been done, people like to work on geometrical versions of physics.  Or even taking the action, and just saying that you want a path that extremizes it might be enough.
