Coordinate free description of Noether current

by Ivo Terek   Last Updated July 10, 2019 05:20 AM - source

If $Q$ is a differentiable manifold and $L:TQ\to \Bbb R$ is a Lagrangian invariant under a $1$-parameter group $(\varphi_s)_{s\in \Bbb R}$ of diffeomorphisms of $Q$, then the Noether charge $\mathscr{J}:TQ\to \Bbb R$ defined by $$\mathscr{J}(x,v) = \mathbb{F}L(x,v)\left(\frac{\rm d}{{\rm d}s}\bigg|_{s=0}\varphi_s(x)\right),$$where $\mathbb{F}L:TQ\to T^*Q$ is the fiber derivative of $L$, is constant along curves $x:[a,b]\to Q$ which are critical points of the action functional of $L$. This is Noether's Theorem stated in the cleanest way I know.

I am interested in the version of the theorem for Lagrangians whose domain is $TQ^{\oplus m}$ for some $m\geq 1$, i.e., with more than one tangent vector as input. Coordinates $(q^1,\ldots, q^n)$ in $Q$ induce coordinates in each copy of $TQ$ inside $TQ^{\oplus m}$, and so we get coordinates $$(q^1,\ldots , q^n, v^1_{(1)},\ldots ,v^n_{(1)},\ldots, v^1_{(m)},\ldots, v^n_{(m)}).$$Fix $\Omega\subseteq \Bbb R^m$ a compact subset with non-empty interior and regular boundary (and coordinates $(u^1,\ldots u^m)$), so that the domain of the action functional of $L:TQ^{\oplus m}\to \Bbb R$ consists of "$m$-surfaces" $x:\Omega\to Q$. For a Lagrangian like this invariant over $(\varphi_s)_{s \in \Bbb R}$, I got that that $$\sum_{\ell=1}^m\frac{\partial}{\partial u^\ell}\left( \sum_{k=1}^n \frac{\partial L}{\partial v^k_{(\ell)}}(x(u),\nabla x(u))\frac{\partial q^k}{\partial s}(0,u)\right) = 0.$$ This is clearly the divergence of something. I don't know how to describe in an intrinsic way what is this something, in terms of (partial?) fiber derivatives or whatever. I would like to possibly describe this as some map $TQ^{\oplus m}\to ?$ that is constant along critical $m$-surfaces.

Physics texts are completely unintelligible for me, and the few mathematics texts that could possibly say something useful about this discuss a level of generality that goes way over my head. Help?

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