Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems

We will present a consistent description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-\underline{in}dependent Hamiltonian system on the conventional symplectic phase space. The extended Hamiltonian \(H_{1}\) and the pertaining extended symplectic structure that establish the proper canonical extension of a conventional Hamiltonian \(H\) will be derived from a generalized formulation of Hamilton's variational principle. The extended canonical transformation theory then naturally permits transformations that also map the time scales of original and destination system, while preserving the extended Hamiltonian \(H_{1}\), and hence the form of the canonical equations derived from \(H_{1}\). The Lorentz transformation, as well as time scaling transformations in celestial mechanics, will be shown to represent particular canonical transformations in the symplectic extended phase space. Furthermore, the generalized canonical transformation approach allows to directly map explicitly time-dependent Hamiltonians into time-independent ones. An ``extended'' generating function that defines transformations of this kind will be presented for the time-dependent damped harmonic oscillator and for a general class of explicitly time-dependent potentials. In the appendix, we will reestablish the proper form of the extended Hamiltonian \(H_{1}\) by means of a Legendre transformation of the extended Lagrangian \(L_{1}\).