Canonical and canonoid transformations for Hamiltonian systems on (co)symplectic and (co)contact manifolds

In this paper, we present canonical and canonoid transformations considered as global geometrical objects for Hamiltonian systems. Under the mathematical formalisms of symplectic, cosymplectic, contact, and cocontact geometries, the canonoid transformations are defined for (co)symplectic and (co)contact Hamiltonian systems. The local characterizations of these transformations are derived explicitly, and it is demonstrated that for a given canonoid transformation, there exist constants of motion associated with it.