Lagrangian and Hamiltonian dynamics with imaginary time

Certain aspects of Lagrangian and Hamiltonian dynamics are investigated within the framework of an extended complex phase space approach, characterized by the transformation q = q(1) + i q(2), p = p(1) + i p(2) and by the presence of the imaginary time variable tau in the complex variable z = t +i tau in a local chart. It is an application of imaginary time which is essential in connecting quantum mechanics with statistical physics. We argue that the novel complexified approach enhances the system of dynamical equations obtained in the sense that the new derived equations appear as certain combinations of former equations. Furthermore, it was shown that the physics we experience in the real time is somewhat different from what is experienced in imaginary region. Further consequences are discussed in some details.