Symplectic geometry is one of the most important research subjects.In particular,the investigation of the symplectic structures and the symplectic theory are closely related to the analytic mechanics.Hamiltonian mechanics is a subject established on the symplectic manifolds,the symplectic structure play an important role in connecting the Hamiltonian function and Hamiltonian vector field,see[1-3].This paper was prepared with a nature idea,that is,how can we construct a new Hamiltonian system or Lagrangian system by using the known Hamiltonian systems or Lagrangian system and what relations are there between the new constructed Hamiltonian system and Lagrangian system.In this paper,we first introduce a kind of special functions,special vector fields and special symplectic structure on a product manifold and a new Hamiltonian system on the symplectic product manifold is composed of two known Hamiltonian systems on the factor symplectic manifolds.