  The Poincar\'e-Snyder relativity was introduced in an earlier paper of ours
as an extended form of Einstein relativity obtained by appropriate limiting
setting of the full Quantum Relativity. The latter, with fundamental constants
$\hbar$ and $G$ built into the symmetry, is supposed to be the relativity of
quantum space-time. Studying the mechanics of Poincar\'e-Snyder relativity is
an important means to get to confront the great challenge of constructing the
dynamics of Quantum Relativity. The mechanics will also be of interest on its
own, plausibly yielding prediction accessible to experiments. We write the
straightforward canonical formulation here, and show that it yields sensible
physics pictures. Besides the free particle case, we also give an explicit
analysis of two particle collision as dictated by the formulation, as well as
the case of a particle rebouncing from an insurmountable potential barrier in
the time direction. The very interesting solution of particle-antiparticle
creation and annihilation as interpreted in a the usual time evolution picture
can be obtained, in the simple classical mechanics setting. We consider that a
nontrivial success of the theory, giving confidence that the whole background
approach is sensible and plausibly on the right track. We also sketch the
quantum mechanics formulation, direct from the familiar canoincal quantization,
matching to a relativity group geometric quantization formulation in a previous
publication.
