Poisson Bracket - Definition

Definition

Let M be symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form ω which is both closed ( dω = 0) and non-degenerate, in the following sense: when viewed as a map, ω is invertible to obtain . d is the exterior derivative operation intrinsic to the manifold structure of M, and is the interior product or contraction operation, which is equivalent to θ(ξ) on 1-forms θ.

Using the axioms of the exterior calculus, one can derive:

Here denotes the Lie bracket on smooth vector fields, whose properties essentially define the manifold structure of M.

If v is such that, we may call it ω-coclosed (or just coclosed). Similarly, if for some function f, we may call v ω-coexact (or just coexact). Given that dω = 0, the expression above implies that the Lie bracket of two coclosed vector fields is always a coexact vector field, because when v and w are both coclosed, the only nonzero term in the expression is . And because the exterior derivative obeys, all coexact vector fields are coclosed; so the Lie bracket is closed both on the space of coclosed vector fields and on the subspace within it consisting of the coexact vector fields. In the language of abstract algebra, the coclosed vector fields form a subalgebra of the Lie algebra of smooth vector fields on M, and the coexact vector fields form an algebraic ideal of this subalgebra.

Given the existence of the inverse map, every smooth real-valued function f on M may be associated with a coexact vector field . (Two functions are associated with the same vector field if and only if their difference is in the kernel of d, i. e., constant on each connected component of M.) We therefore define the Poisson bracket on (M, ω), a bilinear operation on differentiable functions, under which the (smooth) functions form an algebra. It is given by:

The skew-symmetry of the Poisson bracket is ensured by the axioms of the exterior calculus and the condition dω = 0. Because the map is pointwise linear and skew-symmetric in this sense, some authors associate it with a bivector, which is not an object often encountered in the exterior calculus. In this form it is called the Poisson bivector or the Poisson structure on the symplectic manifold, and the Poisson bracket written simply .

The Poisson bracket on smooth functions corresponds to the Lie bracket on coexact vector fields and inherits its properties. It therefore satisfies the Jacobi identity:

The Poisson bracket with respect to a particular scalar field f corresponds to the Lie derivative with respect to . Consequently, it is a derivation; that is, it satisfies Leibniz' law:

also known as the "Poisson property". It is a fundamental property of manifolds that the commutator of the Lie derivative operations with respect to two vector fields is equivalent to the Lie derivative with respect to some vector field, namely, their Lie bracket. The parallel role of the Poisson bracket is apparent from a rearrangement of the Jacobi identity:

If the Poisson bracket of f and g vanishes ({f,g} = 0), then f and g are said to be in mutual involution, and the operations of taking the Poisson bracket with respect to f and with respect to g commute.