satisfying the Jacobi identity and the product rule is defined.
Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well.
This observation is important in studying the classical limit of quantum mechanics—the non-commutative algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.
In this case, add the extra requirement for all scalars s. For each g in a Poisson ring A, the operation
If a non-degenerate Poisson ring is isomorphic as a commutative ring to the algebra of smooth functions on a manifold M, then M must be a symplectic manifold and
is the Poisson bracket defined by the symplectic form.
This article incorporates material from Poisson Ring on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.