  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 $L_{1}$.
