2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74945This paper shows that the time $t$ map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed boundary conditions is smooth, a result already known for Dirichlet boundary conditions.35 pagesDifferential GeometryAnalysis of PDEs35Q35; 35Q53; 53D17; 53D22; 53D25; 58B20; 58B25; 58D05text