A comparison of Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetofluid

The imposition of the incompressibility constraint in fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in a variational formulation [1]. An alternative approach is the imposition of this constraint in the Eulerian variable description by a generalization of Dirac's constraint method [2,3] using noncanonical
Poisson brackets [4]. It will be shown how to impose the compressibility constraint using Dirac's method in terms of canonical Poisson brackets, and a comparison emphasizing the pros and cons of the various methods will be discussed. Both finite- and infinite-dimensional examples will be given. In addition thedefinition and use of energy for stability will be described along with the extension to magnetofluid dynamics.

[1] J. L. Lagrange, Me'canique Analytique (Paris, 1788).
[2] P. J. Morrison et al., Ann. Phys. 324, 1747 (2009).
[3] C. Chandre et al., Phys.~Lett. A 376, 737 (2012).
[4] P. J. Morrison, Rev. Mod. Phys. 70, 467 (1998).