PoissonKlammer

PoissonKlammer, commonly written Poisson-Klammer in German-language sources and known in English as the Poisson bracket, is a bilinear, antisymmetric operation on the algebra of smooth functions on a Poisson manifold. It plays a central role in Hamiltonian mechanics and symplectic geometry by encoding the infinitesimal flow generated by a function and by governing the time evolution of observables.

On a symplectic manifold (M, ω), each smooth function f defines a Hamiltonian vector field X_f via i_{X_f} ω = df. The PoissonKlammer of two functions f and g is defined by {f,g} = ω(X_f, X_g) = df(X_g) = -dg(X_f). In canonical coordinates (q^i, p_i) this yields the standard relations {q^i, q^j} = 0, {p_i, p_j} = 0, and {q^i, p_j} = δ^i_j. The bracket satisfies bilinearity, antisymmetry, the Jacobi identity, and the Leibniz rule {fg,h} = f{g,h} + g{f,h}.

Beyond symplectic manifolds, the PoissonKlammer extends to Poisson manifolds through a Poisson bivector π, with {f,g} = π(df,dg). In this setting the bracket remains a derivation in each argument and induces the Hamiltonian dynamics df/dt = {f,H}.

Applications include formulation of classical mechanics, the study of integrable systems, and the bridge to quantum mechanics via deformation quantization, where commutators replicate the classical PoissonKlammer in the limit ħ → 0. The term reflects historical and linguistic variations, with Poisson-Klammer and PoissonKlammer used interchangeably in literature.