Symplectomorphisms

A symplectomorphism is a diffeomorphism between symplectic manifolds that preserves the symplectic structure. If (M, ω) is a symplectic manifold, a diffeomorphism φ: M → M is a symplectomorphism when φ^*ω = ω. More generally, between (M, ω_M) and (N, ω_N) a map φ is a symplectomorphism if φ^*ω_N = ω_M. The set of all symplectomorphisms of (M, ω) forms a group under composition, denoted Symp(M, ω).

Symplectomorphisms preserve the nondegenerate closed 2-form ω, and hence preserve the associated volume form ω^n on a 2n-dimensional manifold. They also preserve Hamiltonian dynamics and Poisson brackets: for smooth functions f, g on M, the Poisson bracket is invariant under composition with a symplectomorphism.

Locally, symplectomorphisms are constrained by Darboux's theorem, which states that every symplectic manifold looks like standard Euclidean space with the form ω0 = Σ_i dx_i ∧ dy_i in a neighborhood of any point; thus symplectomorphisms act as canonical changes of coordinates locally.

Examples include canonical transformations in classical mechanics, which are symplectomorphisms of phase space. The set of linear symplectomorphisms of R^{2n} with the standard form constitutes the group Sp(2n, R). Infinitesimally, vector fields X with L_X ω = 0 are symplectic; their flows yield one-parameter subgroups of Symp(M, ω). Those generated by Hamiltonians, via i_{X_H} ω = dH, are called Hamiltonian diffeomorphisms and form a notable subgroup, Ham(M, ω).