Question about the concepts of Noether charge and Noether current

I read that a noether current occurs when the lagrangian assume vector values. Well, what are noether current and noether charge in comparison to elementary classical mechanics notions of Noether's theorem?

• "in comparison to elementary classical mechanics notions of noether theorem" - can you define "elementary classical mechanics notions of noether theorem", so that we know what to compare to? Nov 28, 2018 at 19:53
• Also, where did you read that "a noether current occurs when the lagrangian assume vector values"? Nov 28, 2018 at 20:06

If $$Q$$ is a configuration manifold and $$G$$ is a Lie group acting on $$Q$$ (usually $$G = \Bbb R$$ with a $$1$$-parameter group of diffeomorphisms of $$Q$$) leaving a Lagrangian $$L:TQ\to \Bbb R$$ invariant, for every $$X\in \mathfrak{g}$$ we have the Noether charge generated by $$X$$, the map $$\mathscr{J}^X\colon TQ\to \Bbb R$$ given by $$\mathscr{J}^X(x,v) =\mathbb{F}L(x,v)X^\#_x,$$where $$X^\#\in\mathfrak{X}(Q)$$ is the action field of $$X$$ and $$\mathbb{F}L$$ is the fiber derivative of $$L$$. The content of Noether's theorem is that $$\mathscr{J}^X$$ is constant along critical points of the action functional associated to $$L$$.
If $$m>1$$ we look at multivariable Lagrangians $$L:TQ^{\oplus m}\to \Bbb R$$, which have partial fiber derivatives. For $$X\in \mathfrak{g}$$, the Noether current generated by $$X$$ is the map $$\mathscr{J}^X:TQ^{\oplus m}\to \Bbb R^m$$ whose $$j$$-th component is the $$j$$-th partial fiber derivative of $$L$$ evaluated in the action field of $$X$$, as above. Here, Noether's theorem says that if $$L$$ is $$G$$-invariant and we have a critical point $$x:\Omega\subseteq \Bbb R^m\to Q$$ of the action functional associated to $$L$$, then the composition of $$\mathscr{J}^X$$ with the $$1$$-jet of $$x$$, which is a bona fide vector field on $$\Omega$$, has zero divergence.