Relation of Noether's theorem and group theory

[davidbenari]

I'm doing a small research project on group theory and its applications. The topic I wanted to investigate was Noether's theorem.

I've only seen the easy proofs regarding translational symmetry, time symmetry and rotational symmetry (I'll post a link to illustrate what I mean by "the easy proofs"). I know there are way more abstract proofs dealing with Lagrangian densities and whatnot and "conserved currents" and "divergent Lagrange relations".

My research doesn't have to be too extensive to be honest, so I was wondering if what I'm planning to do seems reasonable (and makes sense) to you.

What I plan to do is show the "easy proofs", interpret them, and then say that the transformations of the type (For $L(q(t),\dot{q}(t),t)$):

$\vec{r} \to \vec{r}+\epsilon\hat{n}$ ( and leave other coordinates the same)

or $\vec{r} \to \vec{r}+\epsilon \vec{r} \times \hat{n}$ (and leave other coordinates the same)

or $t \to t+\epsilon$ (and leave all other coordinates the same)

Define a Lie group, because they are a symmetry group of the Lagrangian (under certain conditions) and because they have continuous parameters and satisfy the relevant axioms for a group.

But this doesn't seem like a blatant application of group theory. I could've studied these transformations without even knowing group theory existed. So I was wondering what you thought?

Is Noether's theorem really an application of group theory? In what way? What should I investigate specifically?

Link for "easy proofs" : http://phys.columbia.edu/~nicolis/NewFiles/Noether_theorem.pdf

 

[Orodruin]

The symmetry transformations that you deal with in Noether's theorem are going to form a group (as any set of symmetries of a system will do). This is the connection to group theory. For example, the set of translations form an (Abelian) group and the corresponding conserved quantity is linear momentum and so on. In addition, momentum is the generator of the group of translations.

 

[davidbenari]

Okay, nice. Can I in addition say that they will form Lie groups since they depend on continuous parameters? Also, is a Lie group necessarily a symmetry group?

 

[Orodruin]

Also, is a Lie group necessarily a symmetry group?

No, this depends on your theory. For example, a system does not necessarily need to be invariant under translations.