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The incompressibility constraint for fluid flow was imposed by Lagrange inthe so-called Lagrangian variable description using his method of multipliersin the Lagrangian (variational) formulation. An alternative is the impositionof incompressibility in the Eulerian variable description by a generalizationof Dirac's constraint method using noncanonical Poisson brackets. Here it isshown how to impose the incompressibility constraint using Dirac's method interms of both the canonical Poisson brackets in the Lagrangian variabledescription and the noncanonical Poisson brackets in the Eulerian description,allowing for the advection of density. Both cases give dynamics ofinfinite-dimensional geodesic flow on the group of volume preservingdiffeomorphisms and explicit expressions for this dynamics in terms of theconstraints and original variables is given. Because Lagrangian and Eulerianconservation laws are not identical, comparison of the various methods is made.

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics