Poisson algebra of smooth functions - Imported Knowlpedia Content

Definition

Let $(M,\omega)$ be a symplectic manifold. Its Poisson algebra of smooth functions is the real unital commutative algebra $C^\infty(M,\mathbb R)$ , with pointwise multiplication, together with the Poisson bracket $\{-,-\}$ . The bracket is real-bilinear and antisymmetric, satisfies the Jacobi identity, and obeys the Leibniz rule

\{f,gh\}=\{f,g\}h+g\{f,h\}.

Thus it is simultaneously a Lie algebra and a commutative associative algebra, with the Lie bracket acting as a derivation in either argument.

Geometric meaning

The bracket records the action of Hamiltonian dynamics on observables:

\{f,g\}=X_g(f)

under the convention $\iota_{X_g}\omega=dg$ . Therefore $\{f,H\}$ is the rate of change of $f$ along the Hamiltonian flow of $H$ . On a connected symplectic manifold, the Poisson center consists exactly of the constant functions. These standard properties are developed in Marsden and Ratiu, §10.1.

Maps and examples

The canonical bracket on $\mathbb R^{2n}$ satisfies $\{q^i,p_j\}=\delta^i_j$ . A symplectomorphism induces an isomorphism of Poisson algebras by pullback. More generally, a Poisson map is precisely a smooth map whose pullback preserves brackets.

For a general Poisson manifold, the same axioms define a Poisson algebra on smooth functions, but its Poisson center may contain nonconstant Casimir functions Vaisman, Chapter 1.

Conventions and scope

The real-valued algebra is used here. One may instead complexify to $C^\infty(M,\mathbb C)$ and extend the bracket complex-bilinearly. The phrase “classical observable algebra” often refers to a chosen subalgebra of admissible observables rather than all smooth functions, especially in systems with constraints or analytic restrictions.

References

Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Springer, 1999. DOI record. Relevant: §10.1, Poisson brackets and Hamiltonian dynamics.
Izu Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994. DOI record. Relevant: Chapter 1, Poisson algebras and Poisson manifolds.