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If a Hamiltonian dynamical system with $n$ degrees of freedom admits $m$constants of motion more than $2n-1$, then there exist some functionalrelations between the constants of motion. Among these relations the number offunctionally independent ones are $s=m-(2n-1)$. It is shown that for such asystem in which the constants of motion constitute a polynomial algebra closingin Poisson bracket, the Nambu brackets can be written in terms of these $s$constraint functionals. The exemplification is very rich and several of themare analyzed in the text.Comment: 15 page

NAMBU BRACKETS WITH CONSTRAINT FUNCTIONALS