  In the conventional formalism of physics, with 1-time, systems with different
Hamiltonians or Lagrangians have different physical interpretations and are
considered to be independent systems unrelated to each other. However, in this
paper we construct explicitly canonical maps in 1T phase space (including
timelike components, specifically the Hamiltonian) to show that it is
appropriate to regard various 1T-physics systems, with different Lagrangians or
Hamiltonians, as being duals of each other. This concept is similar in spirit
to dualities discovered in more complicated examples in field theory or string
theory. Our approach makes it evident that such generalized dualities are
widespread. This suggests that, as a general phenomenon, there are hidden
relations and hidden symmetries that conventional 1T-physics does not capture,
implying the existence of a more unified formulation of physics that naturally
supplies the hidden information. In fact, we show that 2T-physics in
(d+2)-dimensions is the generator of these dualities in 1T-physics in
d-dimensions by providing a holographic perspective that unifies all the dual
1T systems into one. The unifying ingredient is a gauge symmetry in phase
space. Via such dualities it is then possible to gain new insights toward new
physical predictions not suspected before, and suggest new methods of
computation that yield results not obtained before. As an illustration, we will
provide concrete examples of 1T-systems in classical mechanics that are solved
analytically for the first time via our dualities. These dualities in classical
mechanics have counterparts in quantum mechanics and field theory, and in some
simpler cases they have already been constructed in field theory. We comment on
the impact of our approach on the meaning of spacetime and on the development
of new computational methods based on dualities.
