UDC 531.395:512.622 Algebraic Dynamics on a Single Worldline: Vieta Formulas and Conservation Laws V. V. Kassandrov∗, I. Sh. <...> Khasanov∗, N. V. Markova† ∗ Educational-Scientific Institute of Gravitation and Cosmology † Department of Higher Mathematics Peoples’ Friendship University of Russia 6, Mikluho-Macklaya str., Moscow, Russia, 117198 In development of the old conjecture of Stuckelberg, Wheeler and Feynman on the socalled “one electron Universe”, we elaborate a purely algebraic construction of an ensemble of identical pointlike particles occupying the same worldline and moving in concord with each other. <...> In the proposed construction one does not make use of any differential equations of motion, Lagrangians, etc. <...> Instead, we define a “unique” worldline implicitly, by a system of nonlinear polynomial equations containing a time-like parameter. <...> Then at each instant there is a whole set of solutions setting the coordinates of particles-copies localized on the unique worldline and moving along it. <...> There naturally arise two different kinds of such particles which correspond to real or complex conjugate roots of the initial system of polynomial equations, respectively. <...> At some particular time instants, one encounters the transitions between these two kinds of particles-roots that model the processes of annihilation or creation of a pair “particle-antiparticle”. <...> We restrict by consideration of nonrelativistic collective dynamics of the ensemble of such particles on a plane. <...> Making use of the techniques of resultants of polynomials, the generating system reduces to a pair of polynomial equations for one unknown, with coefficients depending on time. <...> Then the well-known Vieta formulas predetermine the existence of time-independent constraints on the positions of particlesroots and their time derivatives. <...> We demonstrate that for a very wide class of the initial polynomials (with polynomial dependence of the coefficients on time) these constraints always take place and can be naturally interpreted as the conservation laws for total momentum, angular momentum and (the analogue of) total mechanical energy of the “closed” system of particles. <...> Introduction elberg, Wheeler and Feynman on the possibility of unified dynamics of an ensemble of identical point-like particles located at different places of a unique worldline. <...> E.C.G. Stueckelberg [1, 2] was, perhaps, the first who considered a worldline containing segments with superliminar velocities and forbidden, therefore, in the canonical theory of relativity; this assumption explicitly results in the notion of a “multi-particle” wordline. <...> Later on J.A. Wheeler (in his famous telephone call to R. Feynman, see [3 <...>