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We revisit the construction of hydrodynamic actions describing the stress tensor dynamics of holographic systems dual to low dimensional gravity, namely, JT gravity and AdS$_3$ gravity. While there are existing techniques to construct such actions, for instance the Schwarzian for JT gravity, these rely on special features of low dimensional gravity. Here we wish to formulate the construction of hydrodynamic actions in a way that can be generalised to higher dimensional settings. We construct hydrodynamic actions describing perturbations about thermal states by placing the linearised Einstein-Hilbert action partially on-shell -- that is by solving a subset of the Einstein equations. The result is an action for hydrodynamic modes, which are realised in the bulk as relative diffeomorphisms between the horizon and the asymptotic boundary. We examine the relationship between boundary conditions on metric perturbations at the black hole horizon and symmetries of the hydrodynamic action. In particular, we identify novel boundary conditions, consistent with our variational principle, for which the hydrodynamic actions we construct manifestly realise certain horizon symmetries recently proposed in by Knysh et. al. |