Show Lorentz invariance for Euler-Lagrange's equations- how?

[Bapelsin]

Hello,

I need help showing that the Euler-Lagrange equations are Lorentz invariant (if Einstein's extended energy concept is used). Is there an easy way to show this? Any help would be very much appreciated.

 

[Gigi]

This and a lot more of that you can find in a book by Greiner called relativistic quantum mechanics. Good luck!

 

[samalkhaiat]

I need help showing that the Euler-Lagrange equations are Lorentz invariant

Welcome to PF.
The E-L equations are statements derived covariantly from an invariant action integral. Therefore they are covariant with respect to Lorentz group.

(if Einstein's extended energy concept is used).

What is this, and what relation does it have with the Euler-Lagrange equations?


sam

 

[Bapelsin]

Gigi: Thanks for your tip, but I don't think I can obtain that book soon enough. But maybe later. :-)

samalkhaiat: Thank you! First of all I want to tell you that I am more or less a layman when it comes to this (I haven't studied natural sciences for almost three years, and I'm I only half-way to obtain a MSc degree in engineering physics) so I'm not sure I'm following you completely. So covariances and tensors and even less group theory etc. isn't my strong side. This was originally supposed to serve as a short background for a philosophical paper in science theory.

My physics professor tells me I'm right concerning this, but I don't know how to show this. As I don't know how to insert formulas here, I'm attaching the relevant ones as .GIF images. By extended energy concept I simply mean the famous formula E = mc².

 

[Bapelsin]

I forgot to mention that my main problem is that I don't know how a Lorentz' transformation in generalized coordinates look like. I've only dealt with such in Cartesian coordinates (Google etc. doesn't help me out much at all with this).