Euler-Poincar\'e formulation of hybrid plasma models

Three different hybrid Vlasov-fluid systems are derived by applying reduction by symmetry to Hamilton's variational principle. In particular, the discussion focuses on the Euler-Poincar\'e formulation of three major hybrid MHD models, which are compared in the same framework. These are the current-coupling scheme and two different variants of the pressure-coupling scheme. The Kelvin-Noether theorem is presented explicitly for each scheme, together with the Poincar\'e invariants for its hot particle trajectories. Extensions of Ertel's relation for the potential vorticity and for its gradient are also found in each case, as well as new expressions of cross helicity invariants.