Hamilton-Pontryagin principle for incompressible ideal fluids

We develop the Hamilton-Pontryagin principle for Lagrangians with advective parameters, which yields an implicit analogue of Euler-Poincaré equations with advective parameters. Then, we derive the reduced Hamilton-Pontryagin principle and illustrate it with the example of incompressible ideal fluids, where the configuration space is given by the group of (volume preserving) diffeomorphisms. Incorporating pressure and momentum densities as Lagrange multipliers into the Hamilton-Pontryagin principle, we finally show that the dynamics of incompressible ideal fluids can be effectively formulated in the context of implicit Euler-Poincaré equations.