Poisson-Dirac submanifolds as a paradigm for imposing constraints in non-dissipative plasma models

Formulating reduced models for plasma dynamics often involves imposing algebraic constraints on dynamical fields, such as flow incompressibility or Ohm's Law. When the reduced model aims to describe non-dissipative processes, these constraints should be imposed in a manner that preserves Hamiltonian structure. Today, the two main approaches for structure-preserving constraints are Lagrange multipliers, on the variational side, and Dirac brackets, on the Hamiltonian side. I will show that these techniques are special cases of a more general method based on the notion of Poisson-Dirac submanifolds. I will review the mathematical theory showing that Poisson-Dirac submanifolds encode the most general class of constraints that lead to the reduced model inheriting a Poisson bracket structure from its parent model. Then I will illustrate the general theory with several examples.