The notion of a Hamiltonian vector field leads to a skew-symmetric, bilinear operation on the differentiable functions on a symplectic manifold M, the Poisson bracket, defined by the formula
where denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds:
where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g. As a consequence, the Poisson bracket satisfies the Jacobi identity
which means that the vector space of differentiable functions on M, endowed with the Poisson bracket, has the structure of a Lie algebra over R, and the assignment f ↦ Xf is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if M is connected).
Read more about this topic: Hamiltonian Vector Field