Casimir invariants and the Jacobi identity in Dirac's theory of constrained Hamiltonian systems

TL;DR

This paper demonstrates that in Dirac's constrained Hamiltonian systems, the Jacobi identity naturally follows from constraints being Casimir invariants, ensuring its validity throughout phase space.

Contribution

It establishes that the Jacobi identity results from constraints being Casimir invariants, valid everywhere in phase space, regardless of the invertibility of the Poisson matrix.

Findings

Jacobi identity derived from Casimir constraints

Proof valid throughout phase space, not just on constraint surface

Illustrated with finite-dimensional and Vlasov-Poisson examples

Abstract

We consider constrained Hamiltonian systems in the framework of Dirac's theory. We show that the Jacobi identity results from imposing that the constraints are Casimir invariants, regardless of the fact that the matrix of Poisson brackets between constraints is invertible or not. We point out that the proof we provide ensures the validity of the Jacobi identity everywhere in phase space, and not just on the surface defined by the constraints. Two examples are considered: A finite dimensional system with an odd number of constraints, and the Vlasov-Poisson reduction from Vlasov-Maxwell equations.