Dirac bracket and Nambu structures

TL;DR

This paper explores the relationship between Dirac and Nambu brackets, showing how Nambu structures can be derived from Dirac brackets in constrained Hamiltonian systems, and introduces a generalized Nambu structure for such systems.

Contribution

It establishes a connection between Dirac and Nambu brackets and introduces a generalized Nambu structure for constrained dynamics.

Findings

Nambu brackets can be related to Dirac brackets when expressed as generalized Poisson structures.

The associated Nambu structure reduces to the Dirac bracket when certain constraints are fixed.

A new Nambu structure, called 'Dirac-Nambu' bracket, can describe constrained dynamics in general phase spaces.

Abstract

A relation between the Dirac bracket (DB) and Nambu bracket (NB) is presented. The Nambu bracket can be related with Dirac bracket if we can write the DB as a generalized Poisson structure. The NB associated with DB have all the standard properties of the original DB. When the dimension of the phase space is $s+2$ where $s$ is the number of second class constraints, the associated Nambu structure has $s+2$ entries and reduces to the Dirac bracket when $s$ of its entries are fixed to be the second class constraints. In general, when the dimension of phase space is $d=r+s$ a new Nambu structure that describes correctly the constrained dynamics can also be constructed but in thsi case addicional conditidionts are requiered. In that case the associated NB corresponds to a ``Dirac-Nambu'' bracket with $r$ entries.