Symplectomorphism

Also called symplectic map.
It is a bijective $f$ map from a symplectic manifold $(M,\omega )$ to another one $(N,{\omega }^{\prime })$ that preserves the symplectic forms, i.e.

For example, if the symplectic manifold is the phase space of a classical mechanical system, then it is called a canonical transformation.

In the context of symplectic geometry, symplectomorphisms from $M$ to itself are the natural idea for a symmetry of a symplectic manifold $(M,\omega )$. They constitute symplectic group actions.

A special case are Hamiltonian symmetrys.