Newtons laws of motion are Galilean invariant. But the Newtonian Lagrangian and Newtonian action for a particle are not Galilean invariant. Similarly we want the Euler-Lagrange (EL) equations in quantum field theory to be Lorentz invariant. But how can we say that the Lagrangian (I mean Lagrangian density) should always be Lorentz invariant? Is there any deep reason for that?
One reason I thought of is since we quantize the classical relativistic Lagrangians which are already invariant we get invariant Lagrangians. But then I am not sure why even in classical field theory the Lagrangians are invariant.
I found in some places that Lagrangian must be invariant for the EL equations to be invariant. But that is not obvious. In some places it is said that action must be invariant and since the volume element $d^4x$ is invariant so must Lagrangian densities. But then why should action be invariant?
Edit: This answer explained it using group theory which I haven't studied yet. If there is an answer without using Group theory you can write it otherwise close this question.