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.
Best Answer
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.