First part of Clausius heat theorem in terms of Noether's theorem

After Helmholtz, the mechanical foundation of thermodynamics included the First Law $dE=\delta Q+\delta W$, and the first part of the Clausius heat theorem $\delta Q^{\text{rev}}\mathrm{/}T=dS$. The resulting invariance of the entropy $S$ for quasistatic changes in thermally isolated systems invites a connection with Noether’s theorem (only established later). In this quest, we continue an idea, first brought up by Wald in black hole thermodynamics and by Sasa $\mathit{\text{et}}\mathit{\text{al.}}$ in various contexts. We follow both Lagrangian and Hamiltonian frameworks, and emphasize the role of Killing equations for deriving a First Law for thermodynamically consistent trajectories, to end up with an expression of ``heat over temperature’’ as an exact differential of a Noether charge.