Necessity for action to be a Lorentz scalar

[HJ Farnsworth]

Hello,

On p.573 of Jackson 2nd Ed. (section 12.1), he says, "From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion are determined by the extremum condition, $\delta A=0$."

I agree that if the action is a Lorentz scalar, then that is sufficient to assure that the equations of motion are the same in all frames: Lorentz scalar, so Lorentz invariant, so action is minimized in all frames when it is minimized in one frame, since it is the same in all frames as it is in that one frame.

However, Jackson seems to imply not only that it is sufficient, but that it is necessary as well. I do not see why this is the case - it seems that the action could vary from frame to frame, but still be minimized in all frames, resulting in the same equation of motion in all frames.

Is inter-frame invariance of the action necessary, and if so why?

Thanks very much for any help that you can give.

-HJ Farnsworth

 

[vanhees71]

You are completely right! In order for the equations of motion to be invariant around the stationary point, not the action itself must be invariant but only its variation.

Example: Take the electromagnetic potentials in Coulomb gauge and write down the corresponding action by working in the constraint with a Lagrange multiplier. This action is not Lorentz invariant, but the equations of motion, Maxwell's equations, are!

 

[HJ Farnsworth]

Excellent, thank you very much!