Symmetry and conservation.... which is first?

[Eiren]

I have a question.

According to Noether's theorem,
"For each symmetry of the Lagrangian, there is a conserved quantity."

But soon I thought that I can also prove
"For each conserved quantiry, there is a symmetry of the Lagrangian."

Actually I can prove the second statement if I start prove from back, although it seems unnatural...So... Symmetry and Conservation.
Which is first?
Which concept is more fundamental than another?
(the image is from http://www.people.fas.harvard.edu/~djmorin/chap6.pdf )

 

[Orodruin]

Neither is more fundamental, it depends on your point of view. They are equivalent statements. Such statements appear in many different contexts and if you have come to the level of Noether's theorem you should already be familiar with them.

 

[haushofer]

That's a matter of methodology, I guess. I think in the early days the conservation principles were considered to be more fundamental, but nowadays the symmetries are considered to be the fundamental thing. You start from some symmetry principles, which restricts the form of the Lagrangian you can write down, and from that the conservation laws are derived. It's not always a clear cut; e.g. in general relativity you have to use more principles than just this.

Hope this helps.

 

[vanhees71]

Yes, it's a one-to-one-relationship: Each one-parameter Lie symmetry of a Hamiltonian system leads to the conservation of the generator of this one-parameter Lie group and vice versa any conserved quantity is the generator of a one-parameter Lie symmetry. Noether has been even more general, including also gauge symmetries for systems with constraints. For a nice review, see

http://arxiv.org/abs/hep-th/0009058