Earlier works adopting the same parametrization for the fluid degrees of freedom include, e.g., Refs. [12, 13].

To avoid confusion, let us stress that this condition does not imply that our variational problem is constrained. Locally, the condition of invertibility is simply the in-equality $\det (\partial \varphi /\partial x)\ne 0$ and does not lead to any constraints on the variations of $\varphi$.

Given a system with Lagrangian $\mathcal{L}(\dot{q},\dots )$, where the ellipsis denotes other fields, the fact that the conjugate momentum ($p$) to $q$ is conserved can be used to integrate out $q$. The dynamics is then described by the effective Lagrangian ${\mathcal{L}}_{\mathrm{eff}}=\mathcal{L}-p\dot{q}$, where all $\dot{q}$ dependence should be eliminated using $p(\dot{q},\dots )=\mathrm{\text{constant}}$ [17]. In our example, $\psi$ plays the role of $q$. Thus, $-p\dot{q}\to -{\Pi }_{\psi }{\partial }_{\tau }\psi =-{\Pi }_{\psi }(y-u·A)$.

Of course, the same would be true if the entropy current were a Noether current, provided the coupling to external sources preserves the corresponding symmetry. The difference is that the entropy current would depend on the sources in this case. A priori, there is nothing wrong with this, and this may lead to an alternative dictionary between field theory and hydrodynamics. This ambiguity may be related to the “integration constants” of anomalous hydrodynamics [10].

In the interest of full disclosure, we should mention that this happened to us for the examples discussed in Secs. IV A and IV B.

The two statements are of course related. For instance, if one couples the fluid to dynamical gravity, the rate of graviton emission at leading order in $G_N$ is determined by the on-shell fluid energy momentum.