Finding all symmetries of a given Lagrangian

[Rocky Raccoon]

Is there a systematically way of finding all space-time symmetries of a given Lagrangian? E.g. given a electromagnetic Lagrangian, can I somehow derive that the symmetries in question are conformal ones?

Thanks.

 

[dextercioby]

Well, hmmm, my take on this is: it was quite accidental and fortunate that SO(2,4) was found as a symmetry group of L=-1/4 F:F. I haven't read the original article, but L was actually built to accommodate the Lorentz (and more generally the Poincare) symmetry of electromagnetism.

Actually, some people tend to do it the other way around and find it natural: assume the symmetry (how ?) and from it derive the equations of motion and eventually derive the simplest Lagrangian (density) satisfying the symmetries and leading to the equations of motion.

So to give you an answer, if you have the Lagrangian, then you already know the symmetries. At least most of them.

 

[Rocky Raccoon]

I thought there may be some equation with solutions that would give possible transformations that leave the action integral invariant?

 

[samalkhaiat]

Each symmetry imposes its own invariance-condition on the Lagrangian. In the conformal symmetry case, you only need to examine you Lagrangian against scale and Poincare' invariance. see posts 12-16 in
www.physicsforums.com/showthread.php?t=172461

 

[Rocky Raccoon]

I am aware of that. But what if I have a Lagrangian and no clue as to which invariance conditions to check? Is there any way of making the Lagrangian tell me that it's invariant wrt (and only wrt) conformal transformations?

 

[samalkhaiat]

Did you look at the link I gave you? Any way, here is another way "to let your Lagrangian tell you" it is conformally-invariant: 1) translate your Lagrangian to curved space-time, then 2) see if the curved space Lagrangian is invariant under Weyl rescaling of the metric. If it is Weyl invariant, then the flat space (your original) Lagrangian is conformally invariant.