In classical mechanics, there is a symmetry definition for a lagrangian as invariance under $$L\rightarrow L+\dfrac{dF(x)}{dt}$$ or even $$L\rightarrow L+\dfrac{dF(x,\dot{x})}{dt}$$ But, what is the Noether current and procedure when we extended the definition to the whole jet coordinates, beginning with $$L\rightarrow L+\dfrac{dF(x,\dot{x},\ddot{x})}{dt}$$ and more generally $$L\rightarrow L+\dfrac{dF(x,\dot{x},\ddot{x},\ldots)}{dt}$$ where the dots signal the full jet coordinates of the X variable? What about a jet bundle approach to the two Noether's theorem, even with the local variation trick $\varepsilon (t)$? That is, what are the Noether currents and the local $\varepsilon (t)$ trick extension to jet variables?
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