symplectically

Symplectically describes properties, transformations, or statements that respect the structure of symplectic geometry. In mathematics, a symplectic form ω is a closed, nondegenerate differential 2-form on a smooth manifold. A map f between symplectic manifolds (M, ω_M) and (N, ω_N) is said to be symplectic if it preserves the form: fω_N = ω_M. In linear algebra, a linear transformation A on a 2n-dimensional vector space with a standard skew-symmetric form J is symplectic when A^T J A = J; such transformations form the symplectic group Sp(2n, R). In coordinates, the canonical symplectic form on R^{2n} is ω = sum_i dx_i ∧ dy_i, and a map is symplectic if it preserves this expression.

The adverb is commonly used in the context of dynamics and physics. A Hamiltonian flow is symplectic, meaning its time-t maps are symplectomorphisms and therefore preserve the symplectic structure; this preservation under evolution is closely tied to conservation laws and phase-space volume in Hamiltonian mechanics. A key theoretical result is Darboux’s theorem, which asserts that locally every symplectic form looks standard, so many statements can be made in a coordinate-free, symplectic-invariant way.

In geometric terms, two objects can be described as symplectically equivalent if there exists a symplectomorphism between them. The term helps distinguish properties that depend on the symplectic structure from those that depend on other geometric structures, such as Riemannian metrics. Overall, symplectically indicates adherence to or preservation of a symplectic framework.