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.