  Lagrangian of the Einstein's special relativity with universal parameter $c$
($\mathcal{SR}_c$) is invariant under Poincar\'e transformation which preserves
Lorentz metric $\eta_{\mu\nu}$. The $\mathcal{SR}_c$ has been extended to be
one which is invariant under de Sitter transformation that preserves so called
Beltrami metric $B_{\mu\nu}$. There are two universal parameters $c$ and $R$ in
this Special Relativity (denote it as $\mathcal{SR}_{cR}$). The
Lagrangian-Hamiltonian formulism of $\mathcal{SR}_{cR}$ is formulated in this
paper. The canonic energy, canonic momenta, and 10 Noether charges
corresponding to the space-time's de Sitter symmetry are derived. The canonical
quantization of the mechanics for $\mathcal{SR}_{cR}$-free particle is
performed. The physics related to it is discussed.
