  Nambu dynamics is a generalized Hamiltonian dynamics of more than two
variables, whose time evolutions are given by the Nambu bracket, a
generalization of the canonical Poisson bracket. Nambu dynamics can always be
represented in the form of noncanonical Hamiltonian dynamics by defining the
noncanonical Poisson bracket by means of the Nambu bracket. For the time
evolution to be consistent, the Nambu bracket must satisfy the fundamental
identity, while the noncanonical Poisson bracket must satisfy the Jacobi
identity. However, in many degrees of freedom systems, it is well known that
the fundamental identity does not hold. In the present paper we show that, even
if the fundamental identity is violated, the Jacobi identity for the
corresponding noncanonical Hamiltonian dynamics could hold. As an example, we
evaluate these identities for a semiclassical system of two coupled
oscillators.
