Nambu brackets with constraint functionals

TL;DR

This paper explores how Nambu brackets can be expressed using constraint functionals in Hamiltonian systems with multiple constants of motion, especially when these constants form a polynomial algebra, providing detailed examples.

Contribution

It demonstrates that Nambu brackets can be formulated in terms of constraint functionals for systems with polynomial algebra of constants of motion, extending the understanding of their structure.

Findings

Nambu brackets can be expressed via constraint functionals in certain Hamiltonian systems.

The paper provides multiple detailed examples illustrating this formulation.

It clarifies the relationship between constants of motion and Nambu brackets in polynomial algebras.

Abstract

If a Hamiltonian dynamical system with $n$ degrees of freedom admits $m$ constants of motion more than $2n-1$, then there exist some functional relations between the constants of motion. Among these relations the number of functionally independent ones are $s=m-(2n-1)$. It is shown that for such a system in which the constants of motion constitute a polynomial algebra closing in Poisson bracket, the Nambu brackets can be written in terms of these $s$ constraint functionals. The exemplification is very rich and several of them are analyzed in the text.