# Noether (quasi)symmetry with jet coordinates?

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?

• Sorry, I mistyped what I wanted to be answered (I know the answer, included the complicated new formulae, but I just wanted to know if people do it with free coordinate tools or with the full coordinated approach in jet bundles). – riemannium Oct 23 '18 at 19:07
• Yes, Noether's theorems work for (quasi)symmetry. Related: physics.stackexchange.com/q/387320/2451 , physics.stackexchange.com/q/123098/2451 and links therein. – Qmechanic Oct 23 '18 at 19:25