De-sitter Invariant Lagrangian In The Form Of The Free Particle In The Theory Of Relativity

De-Sitter Invariant relativity generalizes the Einstein's special relativity. It is the relativity without gravity in dS-spacetime, based upon the principle of relativity (PoR) and the postulate on universal constants. In addition to the light speed c, there exists another universal constant R as the curvature radius of dS-spacetime. With an identification of the universal length with respect to the cosmological constant A, R := ((3/∧)^1/2, it is a relativity with cosmological constant.In this relativity, there exists two kinds of simultaneity, which are the simultaneity defined by the Beltrami time coordinate and that defined by the proper time. In the first case, the Beltrami coordinate system is the inertial frame and the observers are the inertial observer. The motion of the light signal and the free particle satisfy the inertia law. Thus, we can define the observables which are invariant and satisfy the generalized Einstein's equation. In this paper, we have defined the Lagrangian function and action for the free particle. Through taking variation on the action, we can get the motion equation for the particle and re-check there exactly exist the inertial motion and inertial frame. The dS-spacetime is de-Sitter group (?)_∧-invariant, and the generators of the (?)_∧ are the killing vectors, we have taken the Lie derivative of the action with respect to the Killing, then we obtain the Neother's charges corresponding to the symmetries. In this paper, we have also obtained the symplectic 2-form and phase space for this relativity. Induced by the symplectic structure, we have obtained the Poisson bracket and the Lie algebra for the conservative quantities with respect to the Poisson bracket.