Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

TIME BOUNDARY TERMS AND DIRAC CONSTRAINTS

Time boundary terms usually added to action principles are systematically handled in the framework of Dirac's canonical analysis. The procedure begins with the introduction of the boundary term into the integral Hamiltonian action and then the resulting action is interpreted as a Lagrangian one to which Dirac's method is applied. Once the general theory is developed, the current procedure is implemented and illustrated in various examples which are originally endowed with different types of constraints.

QUANTIZATION AND THE ISSUE OF TIME FOR VARIOUS TWO-DIMENSIONAL MODELS OF GRAVITY

It is shown that the models of 2D Liouville Gravity, 2D Black Hole- and R2-Gravity are embedded in the Katanaev-Volovich model of 2D NonEinsteinian Gravity. Different approaches to the formulation of a quantum theory for the above systems are then presented: The Dirac constraints can be solved exactly in the momentum representation, the path integral can be integrated out, and the constraint algebra can be explicitely canonically abelianized, thus allowing also for a (superficial) reduced phase space quantization. Non-trivial dynamics are obtained by means of time dependent gauges. All of these approaches lead to the same finite dimensional quantum mechanical system.