In Noether's theorem you want to be able to say that the functional

$$J[x,y,z] = \smallint_{t_1}^{t_2} \mathcal{L}(t,x,y,z,x',y',z')dt$$

is invariant with respect to a continuous one-parameter group of transformations of the form

$$T(t,x,y,z,\varepsilon) = (t^*(\varepsilon),x^*(\varepsilon),y^*(\varepsilon),z^*(\varepsilon)) = (t,x\cos(\varepsilon)+y\sin(\varepsilon),-x\cos(\varepsilon)+y\sin(\varepsilon),z)$$

(in the case of conservation of angular momentum).

How would I say this in mathematical notation rigorously? Is there a nice way to talk about one-parameter groups acting on functionals? For clarity I see $J$ as a function on a function space, something like

$$J : \mathcal{C}^1[a,b] \times \mathcal{C}^1[a,b] \times \mathcal{C}^1[a,b] \rightarrow \mathbb{R} \| \ (x,y,z) \mapsto J[x,y,z]$$

& the one-parameter group of transformations as a map of the form

$$T : \mathbb{R}^4 \times \mathbb{R} \rightarrow \mathbb{R}^3 | (t,x,y,z,\varepsilon) \mapsto T(t,x,y,z,\varepsilon) = (t^*,x^*,y^*,z^*)$$

Thus as it stands I'm kind of going on intuition when I form $J[x^*,y^*,z^*]$, the notation has gaps which intuition plugs but I'm sure there's a nicer & more consistent/cogent way to do all this!

I'm hoping there's something like $J[\vec{r}] = J[x,y,z]$ implying $J[T(\vec{r})] = J[\vec{r}]$, though obviously that is massively flawed, how would I say the next best thing rigorously? If something like group actions applies would you mind being a bit explicit as I see things like this being mentioned & no notation is offered to justify it & I get lost, thanks!


1 Answer 1


Firstly I think you should give yourself more credit; nothing you have said is "massively flawed" in my opinion. You remark that

If something like group actions applies would you mind being a bit explicit

This is quite on-point! Often the one-parameter groups of transformations considered in physics can, in fact, be considered a particular kind of group action, a so-called flow. A flow $\phi$ on a set $X$ is an action of the additive group of real numbers $\mathbb R$ acting on $X$. In other words, $\phi$ is a mapping $\mathbb R\times X\to X$ satisfying the following properties: \begin{align} \phi_0 = \mathrm{id}_X, \qquad \phi_t\circ\phi_s = \phi_{t+s} \end{align} where $\mathrm{id}_X$ is the identity on $X$. Here, and later in the response, I'll be using the notation that the $\mathbb R$ argument of the flow is written as the subscript of $\phi$, so for some $(t,x)\in \mathbb R\times X$, we would write $\phi_t(x)$.

In the context of Noether's theorem, we deform parameterized curves of the form $\mathbf \gamma :[t_a, t_b]\to Q$, where $Q$ is the configuration manifold of the system, and we investigate how this affects the action.

To put this in the language of flows, we suppose that $\mathscr C$ denotes the set of all such curves that we wish to consider (there are usually some tacit continuity and/or smoothness assumptions etc.), and we investigate flows $\phi$ on $\mathscr C$. The notation would therefore look like $\phi_\epsilon(\gamma)$; this would represent the curve $\gamma$ "flowed" under $\phi$ by an amount $\epsilon$. I am using $\epsilon$ here instead of $t$ as a generic flow parameter so as to not confuse it with time. The flow $\phi$ then induces a one-parameter transformation on any action functional $J:\mathscr C\to \mathbb R$ as follows: \begin{align} J_{\phi_\epsilon}[\gamma] = J[\phi_\epsilon(\gamma)]. \end{align} We then say that the action $J$ is invariant under the flow $\phi$ provided \begin{align} J_{\phi_\epsilon} = J \end{align} for all $\epsilon\in \mathbb R$. In practice, note that the flow may only be defined for some open interval on the reals, and in this case it is called a local flow, but he gist of everything remains the same.


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