I need to find the Lagrangian for charged particles in EM fields considering relativistic effects. Is action integral Lorentz invariant. $$A = \int_{t_1}^{t_2} L (q_i, \dot q_i, t) dt $$
According to my note
According to the first postulate of special relativity, the action integral $A$ must be invariant because the equation of motion is determined by extreme condition $\delta A = 0$.
I do not understand how does this make $A$ invariant.
The action is obviously not invariant because energy is different in different frames. What's invariant is the trajectory that makes the action stationary. Specifically, transforming frames will add a total derivative to the Lagrangian, thereby adding a constant to the action. See, for example, the first few pages of Mechanics by Landau and Lifshitz.
In general, invariance of the action under some transformations, such as Lorenz rotations, is an extra condition that we impose on a theory. This is not related to the extreme condition that just gives trajectories along which the classical evolution goes.
In your case, I guess, there is some confusion in formulation of the idea. According to the 1st postulate of special relativity, all physical laws are the same in any inertial frame (related by a Lorenz transformation). To realise this theoretically, we need to make our Lagrangian invariant under such transformations. Then it will give the same physics in all frames.
Hence, one should impose some extra condition to satisfy the 1st postulate.
