An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems

TL;DR

This paper extends Dirac and Gotay-Nester theories to handle integrable Dirac dynamical systems with foliated constraint submanifolds, introducing a new constraint algorithm and applying it to LC circuits.

Contribution

It develops a unified framework combining Dirac and Gotay-Nester theories for integrable systems, including a new constraint algorithm and adapted evolution equations.

Findings

Extended the constraint algorithm for foliated systems

Applied the theory to LC circuit examples

Showed duality between Dirac and Gotay-Nester approaches

Abstract

This paper extends the Gotay-Nester and the Dirac theories of constrained systems in order to deal with Dirac dynamical systems in the integrable case. Integrable Dirac dynamical systems are viewed as constrained systems where the constraint submanifolds are foliated, the case considered in Gotay-Nester theory being the particular case where the foliation has only one leaf. A Constraint Algorithm for Dirac dynamical systems (CAD), which extends the Gotay-Nester algorithm, is developed. Evolution equations are written using a Dirac bracket adapted to the foliations and an abridged total energy which coincides with the total Hamiltonian in the particular case considered by Dirac. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are dual and that using a combination of results from both theories may have advantages in dealing with a given example, rather than using systematically one or the other.