Does Noether's Theorem Apply in Galley's Extension to Lagrangian Mechanics?

Question

I've been trying to digest The Classical Mechanics of Non-conservative Systems by Chad Galley.  It describes an approach to applying Lagrangian mechanics to non-conservative systems via variable doubling.  The approach is basically to have one variable handling causal interactions with the environment, while the other handles anti-causal.  Then he takes the "physical limit," forcing the two to coincide.
I've been trying to work through whether I can apply Noether's Theorem in this system.  It looks like it would apply, because the larger system (with 2x the variables) is just Lagrangian mechanics, but one may not be able to follow conservative paths because of the physical limit, which introduces non-conservative elements.  However, I'm not certain if there are any assumptions made in Noether's Theorem which get violated with this construction.
Does Nother's Theorem apply in this variable-doubled system, or is there an assumption made by Noether which falls apart in this new construction?

Qmechanic · Accepted Answer

In principle Noether's theorem (NT) itself also works for Galley's extended action functional (5) and its non-conservative potential $K$, which are still local in time. Galley's peculiar boundary conditions are not a problem as they are not relevant for NT.

As usual it is your job to provide an off-shell quasisymmetry of the action in the first place. The issue is rather whether the resulting on-shell conservation law is useful/non-trivial, in particular when one goes to Galley's physical limit $q^j_-to 0$.

For a brief summary of Galley's construction, see e.g. section II.3 in my Phys.SE answer here.

References:

C.R. Galley, The classical mechanics of non-conservative systems, arXiv:1210.2745.

Does Noether's Theorem Apply in Galley's Extension to Lagrangian Mechanics?

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