  Hamiltonian dynamics are characterized by a function, called the Hamiltonian,
and a Poisson bracket. The Hamiltonian is a conserved quantity due to the
anti-symmetry of the Poisson bracket. The Poisson bracket satisfies the Jacobi
identity which is usually more intricate and more complex to comprehend than
the conservation of the Hamiltonian. Here we investigate the importance of the
Jacobi identity in the dynamics by considering three different types of
conservative flows in R3 : Hamiltonian, almost-Poisson and metriplectic. The
comparison of their dynamics reveals the importance of the Jacobi identity in
structuring the resulting phase space.
