a symplectomorphism
If a symplectic manifold is the geometric stage of mechanics, a symplectomorphism is a change of stage that the physics cannot tell apart from the original — a diffeomorphism that preserves the area-measuring 2-form exactly. Just as an isometry preserves a metric and a holomorphic map preserves complex structure, a symplectomorphism is the structure-preserving map of symplectic geometry. In classical mechanics these are exactly the canonical transformations: changes of position-momentum coordinates that keep Hamilton's equations in their standard form.
Precisely, given symplectic manifolds (M, omega_M) and (N, omega_N), a symplectomorphism is a diffeomorphism f: M -> N with f^* omega_N = omega_M, where f^* is the pullback of forms. When M = N this is a symmetry of the symplectic manifold, and the symplectomorphisms form an infinite-dimensional group Symp(M, omega) under composition. Because omega^n is preserved, every symplectomorphism automatically preserves the canonical volume — so it is in particular volume-preserving (this is the abstract form of Liouville's theorem). A central source of examples: the time-t flow of any Hamiltonian vector field is a symplectomorphism, so the dynamics of any Hamiltonian system is a one-parameter family of symplectomorphisms.
Why it matters: the central drama of symplectic topology is that Symp is much smaller and more rigid than the group of volume-preserving diffeomorphisms, even though both preserve volume. Gromov's nonsqueezing theorem is the sharpest statement of this: a volume-preserving map could squeeze a ball into a thin tube, but a symplectomorphism cannot. A common slip is to assume 'preserves volume' equals 'symplectomorphism' in dimension 2n > 2; it does not. Volume preservation is genuinely weaker, and the gap between them is exactly what symplectic rigidity measures.
On (R^2, dp ^ dq) the linear map (q, p) -> (q + p, p) (a shear) satisfies f^* (dp ^ dq) = dp ^ d(q + p) = dp ^ dq, so it is a symplectomorphism — it preserves area, as a shear should. By contrast (q, p) -> (2q, p) doubles area, so it is not symplectic. In higher dimensions linear symplectomorphisms of (R^{2n}, omega_0) are exactly the matrices in the symplectic group Sp(2n, R).
A shear preserves dp ^ dq and is symplectic; a stretch that changes area is not.
In dimension 2, area-preserving and symplectic coincide (omega IS the area form). Only from dimension 4 upward do symplectomorphisms become strictly rarer than volume-preserving maps — that strictness is the whole content of symplectic rigidity.