Nambu tensors and commuting vector fields

Abstract

Takhtajan has recently studied the consistency conditions for Nambu brackets, and suggested that they have to be skew-symmetric, and satisfy Leibnitz rule and the Fundamental Identity (FI, it is a generalization of the Jacobi identity). If the n'th order Nambu brackets in dimension N is written in terms of the Nambu tensor η, the FI implies two conditions on it, one algebraic and one differential. The algebraic part of FI implies decomposability of ηand in this letter we show that the Nambu bracket can then be written in terms of the usual totally antisymmetric n-dimensional tensor and n vector fields D. Our main result is that the differential part of the FI is satisfied iff the vector fields D commute. Examples are provided by integrable Hamiltonian systems. It turns out that then the Nambu bracket itself guarantees that the motions stays on the manifold defined by the constants of motion of the integrable system, while the n-1 Nambu Hamiltonians determine the (possibly non-integrable) motion on this manifold.