Symplektische

Symplektische, in German the feminine form of the adjective symplektisch, denotes the mathematical notion of symplectic. In geometry and mathematical physics, a symplectic structure on a smooth manifold M is given by a closed, nondegenerate differential 2-form ω. The pair (M, ω) is called a symplectic manifold, and the dimension of M must be even (2n).

Such a form ω is skew-symmetric and nondegenerate: for each p ∈ M the map v ↦ ω_p(v, ·) is a linear isomorphism T_pM → T_p^M. The condition dω = 0 makes ω a symplectic form, providing a geometric framework that does not rely on a metric.

Darboux's theorem states that locally there exist coordinates (x1,...,xn, y1,...,yn) in which ω = ∑_i dx_i ∧ dy_i, so all symplectic manifolds are locally equivalent. Globally, however, ω can have nontrivial topology or cohomology, leading to a variety of global phenomena.

Canonical examples include R^{2n} with ω0 = ∑_i dx_i ∧ dy_i and the cotangent bundle TQ with its canonical symplectic form. The group preserving ω is the symplectic group Sp(2n, R). In Hamiltonian mechanics, Hamiltonian vector fields X_H defined by i_{X_H} ω = dH generate flows that conserve ω, and the Poisson bracket {F, G} is defined from ω.

Extensions and related subjects include holomorphic symplectic forms on complex manifolds, symplectic reduction (Marsden–Weinstein theory), and connections to contact geometry in odd dimensions, as well as quantization approaches in mathematical physics.