Incomplete Dirac reduction of constrained Hamiltonian systems

TL;DR

This paper introduces a method to construct Dirac-Poisson brackets using pseudoinverses for constrained Hamiltonian systems, enabling incomplete reductions even with potential obstacles from first-class constraints.

Contribution

It presents a novel approach to Dirac reduction employing pseudoinverses, addressing limitations posed by first-class constraints in Hamiltonian systems.

Findings

Dirac-Poisson brackets can be constructed with pseudoinverses.

The method allows incomplete reduction of Hamiltonian systems.

Discussion on the uniqueness of Dirac brackets.

Abstract

First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac's theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac-Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed.