canonical transformation | symplectomorphism

A map T:(M_1, ω_1)->(M_2, ω_2) between the symplectic manifolds (M_1, ω_1) and (M_2, ω_2) which is a diffeomorphism and T^*(ω_2) = ω_1, where T^* is the pullback map induced by T (i.e., the derivative of the diffeomorphism T acting on tangent vectors). A symplectic diffeomorphism is also known as a symplectomorphism or canonical transformation.

diffeomorphism | pullback map | symplectic manifold