2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221939In this study, firstly, the k-th order extension of complex product manifold is consid- ered. Then the higher order vertical, complete lifts of geometric structures on product manifold to its extended spaces are given. Also higher order lifts of tensor field of type (1,1) are presented. And then extended contact manifolds are defined. Finally higher order vertical and complete lifts of time dependent complex Hamiltonian equations on contact manifold to its extensions are introduced. In conclusion, geometric meaning of Hamiltonian mechanical systems is discussed.Dynamical SystemsMathematical Physics53C15, 70H05, PACS: 02.40text