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I'm doing a small research project on applications of Group theory and chose to investigate Noether's theorem. Evidently, Noether's theorem at its highest level does contain lots of elements of Lie Groups. But my level of knowledge doesn't reach those high levels and therefore I've stuck to the easy proofs (I'll post a link to illustrate what I mean by this) related to:

translational symmetry and conservation of momentum

rotational symmetry and conservation of angular momentum

time symmetry and conservation of energy

I wanted to know however if what I'm about to say is justified:

I have the transformation $$\vec{r}_a \to \vec{r}_a +\epsilon \hat{n} \qquad \dot{\vec{r}}\to \dot{\vec{r}} \qquad t\to t$$

Can I say these transformations define the Lie group representation of euclidean translations if my Lagrangian function is invariant under them?

I was wondering if I could have other transformations and say also "this is the Lie group representation for a rotation", etc.

Does it make sense to say such a thing? I think it does since they do fulfill each of the requirements of a symmetry group.

Thanks.

Relevant link:

http://phys.columbia.edu/~nicolis/NewFiles/Noether_theorem.pdf

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    $\begingroup$ Just to get a feel of your knowledge level -- do you know what a group is? $\endgroup$ – Neal Sep 25 '15 at 0:33
  • $\begingroup$ Yes. I literally have a 5 day knowledge of abstract algebra and a little bit of group theory. As it was a project that was assigned to me. $\endgroup$ – DLV Sep 25 '15 at 0:33
  • $\begingroup$ Is my question way-off or something? $\endgroup$ – DLV Sep 25 '15 at 0:51
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    $\begingroup$ Not really. Just knowing that you have 5 days of abstract algebra under your belt, instead of (say) two years, is useful in tailoring an answer so it's useful. $\endgroup$ – Neal Sep 25 '15 at 0:53
  • $\begingroup$ Okay, thanks. In case you're not writing a response: could you tell me if I can continue presenting Noether's theorem in this fashion for a quick presentation? Thanks again. $\endgroup$ – DLV Sep 25 '15 at 0:59
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Let's state informally the general form of the Noether theorem: to every one-parameter group of diffeomorphism of the configuration manifold of a lagrangian system which preserves the lagrangian function, there correspond a first integral of the equations of motion.

Let's state the general form of the Noether theorem (cfr. [1]).

Theorem (Noether). Suppose $M$ is a smooth manifold, $L : TM \to \mathbb R$ is a smooth lagrangian and that the couple $(M,L)$ admits a one-parameter group of diffeomorphism $h^s : M \to M$, $s \in \mathbb R$, $h^0 = id$ (identity). Then the Euler-Lagrange equations corresponding to $L$ have a first integral $I : TM \to \mathbb R$, which in local coordinates $\{q\}$ on $M$ is given by

$$I(q,\dot q) = \frac{\partial L}{\partial \dot q} \frac{d h^s(q)}{dq}\Big\vert_{s=0} $$

Here "admits" means that $L(h_* v) = L(v)$. This notation means that if $v \in TM$ is a vector in $P \in M$ (i.e, an element of $T_p M$, the tagent space to $M$ in $P$), then $h_* v$ is an element of $T_{P'} M$, $P' = h(P)$, obviously again an element of $TM$.

This does not involve Lie groups. However, it generically happens that relevant one-parameter groups of diffeomorphism are actually Lie groups. Remember that a Lie group is a smooth manifold $G$ which is also a group, endowed with an operation of inversion and an operation of composition (both continuous with respect to the appropriate topology of $G$ or $G \times G$). A realization of $G$ is a map which associates to each $g \in G$ a transformation $T(G)$ of some space $M$, such that preserves group properties (see [2]). A realization is faithful if it is 1-1, and is a representation if $M$ is a vector space.

Example. Let $S^2$ be the unit 2-sphere in $\mathbb R^3$. A rotation by an angle $\theta$ about an axis, for example the $x$-axis, is associated with the element $\exp{\{\theta L_1\}}$ of $SO(3)$, where

$$L_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.$$

$SO(3)$ is known to be a Lie group. If we see $S^2$ as an abstract manifold, then this is a realization of the $SO(3)$. Explicitly, the coordinate transformation induced is

$$x' = x$$ $$y' = y \cos \theta - z \sin \theta$$ $$z' = y \sin \theta + z \cos \theta$$

It is possible to consider this as a coordinate transformation of $\mathbb R^3$. This way, we can speak of a representation of $SO(3)$ in terms of matrices which transform vectors of $\mathbb R^3$.

It is not hard to see that the first integral associated to the rotational invariance is the angular momentum. For other invariances the mechanism works the same way.


[1] Arnold, Mathematical methods of classical mechanics;

[2] Schutz, Geometrical methods of mathemathcal physics.

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