Center-of-mass variables in the relativistic Lagrangian dynamics of a system of particles

To separate the motion of a relativisticN-particle system as a whole from its internal motion, we propose center-of-mass variables in an arbitrary (geometrical) form of Lagrangian dynamics. In terms of these variables, we construct a representation of the Poincaré group $\mathcal P(1.3)$ by Lie–Bäcklund vector fields; we find expressions for transformation of the center-of-mass variables under the influence of finite transformations of this group. We obtain a class of Lagrangians that depend on derivatives of not higher than the second order. We construct ten conservation laws corresponding to the symmetry with respect to $\mathcal P(1.3)$P. We analyze the motion of the system as a whole. The transition to the Hamiltonian description is considered.