Scientific paper ID 1642 : 2018/3 COMPUTING OF NON-INTEGRABILITY WITH COMPUTER ALGEBRA METHODSGeorgi Georgiev Abstract. In this paper it is shown that the Hamiltonian system with Dyson potential is analytical non-integrable and formal non integrable. The Dyson system, appeared in 1962 by Dyson statistical physics research of the energy levels of one-dimensional Coulomb’s gas. There is connection between the of system of point vortices - this is result by Calogero and Perelomov in 1978. Hamiltonian system has two integrals, that is why for integrability it is important the first nontrivial case . Borisov and Kozlov in 1998 had proved that the system in case is non-integrable in analytical first integrals using splitting separarises method. In this paper it is proved the same statement, but way is different. Proof based of Duistermaat’s idea when a system has a family of periodic solutions around equilibrium, and if the period function is infinitely branched, then if the system has additional analytical first integral - this integral is a constant. We call that the system with Hamiltonian H formally integrable if there exist formal power series in involution, where are functionally independent and Taylor expansion of H is a formal power series in . An asymptotic behavior near equilibrium is like an integrable system. The formal integrability gives asymptotic information about the flow. For the proof of formal non-integrability in the paper is used Theory of Ziglin - Morales-Ruiz-Ramis. Computer algebra method is the Kovacic algorithm using a software for education, engineering and research Maple. Потенциал на Дайсън Хамилтонови системи Неинтегруемост Компютърни алгебри.Dyson potential Hamiltonian systems Non-integrability Computer Algebra Georgi GeorgievBIBLIOGRAPHY[1] Dyson F.J., (1962), Statistical theory of the enrgy levels of complex systems, I, II, III, J. Math Phys., 3, 140-156, 157-165, 166-175. [2] Calogero F., Perelomov A. M., (1978), Properties of Certain Matrices Related to the Equilibrium Configuration of the One-Dimensional Many-Body Problems with the Pair Potentials, and , Commun. math. Phys. 59, no.12, 109-116. [3] Kozlov V.V., (1995), Symmetries, Topology, and Resonace in Hamiltonian Mechanics, Izhevsk, Udmurt Univ. Moskow (in Russian). [4] Borisov A. V., Kozlov V.V., (1998), Nonitegrability Of a System of Interacting Particles with the Dyson Potential, Doklady Akademii Nauk, Vol 306, no. 1, 30- -31. [5] Duistermaat J. J., (1984), Non-integrability of the 1:1:2- resonance, Ergod. Th. & Dynam. Sys., 4, 553- -568. [6] Christov O., Georgiev G. (2015), On the integrability of a system describing the stationary solutions in Bose-Fermi mixtures, Chaos Solitons and Fractals, 77, 8, 138- -148. [7] Morales-Ruiz J., (1999), Differential Galois Theory and Non-integrability of Hamiltonian Systems, Birkheauser. [8] Yoshida H., (1988), Non-Integrability of the Truncated Toda Lattice Hamiltonian at Any Order, Commun. Math. Phys., 116, 529- -538.

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