Editing Noether's theorem (section)

== Informal statement of the theorem ==
All fine technical points aside, Noether's theorem can be stated informally as:

{{blockquote|If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.<ref>{{cite book |author=Thompson, W.J. |title=Angular Momentum: an illustrated guide to rotational symmetries for physical systems |publisher=Wiley |year=1994 |isbn=0-471-55264-X |volume=1 |page=5 |url=https://books.google.com/books?id=O25fXV4z0B0C&pg=PA5}}</ref>}}

A more sophisticated version of the theorem involving fields states that:

{{blockquote|To every continuous [[Symmetry in physics|symmetry]] generated by local actions there corresponds a [[conserved current]] and vice versa.}}

The word "symmetry" in the above statement refers more precisely to the [[general covariance|covariance]] of the form that a physical law takes with respect to a one-dimensional [[Lie group]] of transformations satisfying certain technical criteria. The [[conservation law]] of a [[physical quantity]] is usually expressed as a [[continuity equation]].

The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern terminology, the  conserved quantity is called the ''Noether charge'', while the flow carrying that charge is called the ''Noether current''. The Noether current is defined [[up to]] a [[solenoidal]] (divergenceless) vector field.

In the context of gravitation, [[Felix Klein]]'s statement of Noether's theorem for action ''I'' stipulates for the invariants:<ref>Nina Byers (1998) [http://cwp.library.ucla.edu/articles/noether.asg/noether.html "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws"]. In Proceedings of a Symposium on the Heritage of Emmy Noether, held on 2–4 December 1996, at the Bar-Ilan University, Israel, Appendix B.</ref> 
{{blockquote|If an integral I is invariant under a continuous group ''G''<sub>''ρ''</sub> with ''ρ'' parameters, then ''ρ'' linearly independent combinations of the Lagrangian expressions are divergences.}}