A note on Nambu mechanics

Abstract

In Nambu mechanics an autonomous system of first-order ordinary differential equations du i /dt=F i(u) (i=1,…,n) is constructed with the help of (n−1) smooth functionsI i . These smooth functions are first integrals of this dynamical system. If the functionsI i are polynomials, then the system is algebraic completely integrable. We discuss the question whether the first integrals determine uniquely the autonomous system of first-order differential equations. Then we give a generalization of Nambu mechanics.