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Would a relativistic system be properly described by 4N coordinates [tex]x_i,y_i,z_i,t_i[/tex], with a time variable for each particle? If so, how can we ever speak of the Lagrangian of the system as a whole or the Hamiltonian of the system as a whole? Or the action of the system as a whole?

In general, can we find a single quantity to parametrize the state of a system of N particles?