Newsgroups: sci.physics.research,sci.math,sci.physics
From: torre@cc.usu.edu (Charles Torre)
Subject: Symmetries of a Lagrangian (was Re: This Week's Finds...)
Date: Fri, 20 Nov 1998 03:54:58 GMT

In article <72v9op$sf7$2@pravda.ucr.edu>, nurban@crib.corepower.com 
(Nathan Urban) writes:
> 
> What's the recipe?  I've always wondered how you can determine all of
> the symmetries of a Lagrangian; it seems like it could be quite subtle.
> 


For symmetries that can be described, infinitesimally, as
transformations that at each spacetime point are
constructed from the fields and a finite number of field
derivatives at that point, one can use "jet space"
techniques.  

The Lagrangian can be viewed as a function on a large, but
finite dimensional, space: the jet space (the set of
spacetime points, the space of fields at a point, the space
of derivatives at a point, etc.). Write down the
infinitesimal symmetry condition (Lagrangian is invariant
up to a total divergence), and use the fact that, at any
one point, the jets of a field can be specified
arbitrarily. This leads to an overdetermined system of
linear PDE's for the infinitesimal symmetry transformation,
which are perfectly tractable if you have enough time to
wade through them all.   

One often first solves a related problem, namely, to find
all symmetries of the field equations.  The symmetries of
the Lagrangian are guaranteed to be a subset of the
symmetries of the field equations.  The jet space strategy
is again available for finding symmetries of the field
equations.  One can test all the equation symmetries
to see which ones are in fact Lagrangian symmetries.

This is, of course, a terribly brief attempt to summarize
the situation. Have a look at 

P. Olver, "Applications of Lie groups to differential
equations" (Springer, 1993)

 G. Bluman and S. Kumei,
"Symmetries and differential equations" (Springer, 1989).


 Charles Torre
 torre@cc.usu.edu