On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets

TL;DR

This paper investigates how projectors associated with Poisson brackets influence constrained Hamiltonian systems, linking them to Dirac brackets, and illustrates their role through plasma physics examples.

Contribution

It clarifies the role of projectors in Poisson brackets for constrained systems and connects them with Dirac brackets using practical plasma physics examples.

Findings

Projectors modify the explicit dependence and functional derivatives in brackets.

Dirac brackets can be understood through the action of projectors.

Applications to plasma physics demonstrate the theoretical insights.

Abstract

The role of projectors associated with Poisson brackets of constrained Hamiltonian systems is analyzed. Projectors act in two instances in a bracket: in the explicit dependence on the variables and in the computation of the functional derivatives. The role of these projectors is investigated by using Dirac's theory of constrained Hamiltonian systems. Results are illustrated by three examples taken from plasma physics: magnetohydrodynamics, the Vlasov-Maxwell system, and the linear two-species Vlasov system with quasineutrality.