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

After Helmholtz, the mechanical foundation of thermodynamics included the First Law d E = δ Q + δ W, and the first part of the Clausius heat theorem δ Qʳev/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 et 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.