  Classical Hamiltonian mechanics is realized by the action of a Poisson
bracket on a Hamiltonian function. The Hamiltonian function is a constant of
motion (the energy) of the system. The properties of the Poisson bracket are
encapsulated in the symplectic 2-form, a closed second order differential form.
Due to closure, the symplectic 2-form is preserved by the Hamiltonian flow, and
it assigns an invariant (Liouville) measure on the phase space through the
Lie-Darboux theorem. In this paper we propose a generalization of classical
Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson
bracket is replaced with a generalized Poisson bracket acting on a pair of
Hamiltonian functions, while the symplectic 2-form is replaced by a symplectic
3-form. We show that, using the closure of the symplectic 3-form, a result
analogous to the classical Lie-Darboux theorem holds: locally, there exist
smooth coordinates such that the components of the symplectic 3-form are
constants, and the phase space is endowed with a preserved volume element.
Furthermore, as in the classical theory, the Jacobi identity for the
generalized Poisson bracket mathematically expresses the closure of the
associated symplectic form. As a consequence, constant skew-symmetric third
order contravariant tensors always define generalized Poisson brackets. This is
in contrast with generalizations of Hamiltonian mechanics postulating the
fundamental identity as replacement for the Jacobi identity. In particular, we
find that the fundamental identity represents a stronger requirement than the
closure of the symplectic 3-form.
