The Lagrangian of Einstein's special relativity with universal parameter c (Script SRc) is invariant under Poincaré transformation, which preserves Lorentz metric ημν. The Script SRc has been extended to be one which is invariant under de Sitter transformation that preserves so-called Beltrami metric Bμν. There are two universal parameters, c and R, in this Special Relativity (denoted as Script SRcR). The Lagrangian-Hamiltonian formulism of Script SRcR 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 Script SRcR-free particle is performed. The physics related to it is discussed.